# Patent

## Title
Estimation of fast shear azimuth, methods and apparatus

## Abstract

Various embodiments include apparatus and methods to operate to record a plurality of acoustic waveforms, to generate an objective function based on the plurality of acoustic waveforms, and to estimate a global minimum of the objective function. The plurality of acoustic waveforms can correspond to a plurality of acoustic dipole receivers azimuthally disposed around a tool to which the receivers are attached. Additional apparatus, systems, and methods are disclosed.

## Background
RELATED APPLICATION
This application is a continuation-in-part of U.S. application Ser. No. 13/878,727, filed 10 Apr. 2013, which is a U.S. National Stage Filing under 35 U.S.C. 371 from International Application No. PCT/US2012/052679, filed on 28 Aug. 2012, and published as WO 2013/039690 A1 on 21 Mar. 2013, which application claims priority benefit to U.S. Provisional Patent Application Ser. No. 61/533,620, filed 12 Sep. 2011, and to U.S. Provisional Patent Application Ser. No. 61/533,420, filed 12 Sep. 2011, which applications and publication are incorporated herein by reference in their entirety.


BACKGROUND
Conventional algorithms use only numerical searching for anisotropy processing of cross-dipole acoustic waveforms, to minimize an objective function whose parameters are the azimuth angle of the X-dipole transmitter relative to the fast principal flexural wave axis, and a set of auxiliary parameters. The auxiliary parameters are used to characterize the received waveforms as functions of slowness and borehole radius, among others. Sometimes the processing is used to determine the location and amount of stress-induced anisotropy in a geological formation.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is an acoustic tool operating according the various embodiments of the invention.
FIG. 2 is a flow chart illustrating anisotropy processing according to various embodiments of the invention.
FIG. 3 is a more detailed flow chart illustrating analytic anisotropy processing according to various embodiments of the invention.
FIG. 4 is a more detailed flow chart illustrating eigenvalue alignment according to various embodiments of the invention.
FIGS. 5-6 illustrate system embodiments of the invention.
FIG. 7 is a flow chart illustrating several methods according to various embodiments of the invention.
FIG. 8 is a block diagram of an article according to various embodiments of the invention.
FIG. 9 is a graph of the angle of the fast principal wave vs. frequency according to various embodiments of the invention.
FIG. 10 is a graph of the fast and slow principal wave dispersion curves according to various embodiments of the invention.
FIG. 11 is a block diagram of an apparatus according to various embodiments of the invention.
FIG. 12 is a representation of transverse isotropy, with the axis of symmetry being the z-axis according to various embodiments of the invention.
FIG. 13 is a representation of horizontal transverse isotropy anisotropy, where the axis of symmetry is along the slow principal axis and perpendicular to the borehole according to various embodiments of the invention.
FIG. 14 is a schematic representation of a new generic tool with many dipole receivers spaced in azimuth around the tool according to various embodiments of the invention.
FIG. 15 is a top view of the tool of FIG. 14 according to various embodiments of the invention.
FIG. 16 is a representation of ambiguities with respect to azimuthal angle according to various embodiments of the invention.
FIG. 17 is a flow diagram of a processor-implemented method of estimating an azimuth angle according to various embodiments of the invention.
FIG. 18 is a representation of ambiguities with respect to azimuthal angle according to various embodiments of the invention.



DETAILED DESCRIPTION
The following detailed description refers to the accompanying drawings that show, by way of illustration and not limitation, various embodiments in which the invention may be practiced. These embodiments are described in sufficient detail to enable those skilled in the art to practice these and other embodiments. Other embodiments may be utilized, and structural, logical, and electrical changes may be made to these embodiments. The various embodiments are not necessarily mutually exclusive, as some embodiments can be combined with one or more other embodiments to form new embodiments. The following detailed description is, therefore, not to be taken in a limiting sense.
Algorithms for anisotropy processing of cross-dipole acoustic waveforms can operate to minimize an objective function whose parameters are the azimuth angle, θ, of the X-dipole transmitter relative to the fast principal flexural wave axis and a set of auxiliary parameters. The auxiliary parameters are used to characterize the received waveforms as functions of slowness, borehole radius, etc. Minimizing the objective function with respect to all the parameters can provide a desired anisotropy angle. The dependence of the objective function on angle is based on the assumption that the cross-dipole acoustic waveforms obey the Alford rotation equations, i.e. the 2×2 cross-dipole waveform matrix, Xkl, can be diagonalized by a similarity transformation, TT(θ)XT(θ), using a real 2×2 orthonormal matrix, T(θ). The objective function measures how well this is accomplished, for example, by minimizing the off-diagonal elements of the similarity transformation with respect to angle.
Embodiments of the invention use the fact that the waveform matrix can be exactly diagonalized by a similarity transformation using a unitary matrix, U. The eigenvalues of the diagonalized matrix are direct estimates of the fast and slow principal waves. Objective functions can then be formed; these functions are dependent upon the eigenvalues, as well as T(θ) and Xkl.
When processing cross-dipole acoustic waveform data for anisotropy using the Alford rotation relationship between in-line and cross-line data, the objective function can be minimized with respect to angle and the auxiliary parameters using a numerical search method or brute force. However, using the embodiments described herein it is also possible, using the eigenvalues, to solve for the angles as a function of frequency analytically—without recourse to numerical search methods, which are not guaranteed to find the global minimum, and can be computationally expensive. For example, a 90° jump in angle as a function of frequency can be used as an indicator of stress induced anisotropy.
In the detailed description that follows, it is noted that some of the apparatus and systems discussed herein are well known to those of ordinary skill in the art, and thus, the details of their operation are not disclosed in detail, in the interest of economy and clarity. Those that wish to learn more about these apparatus and systems are encouraged to refer to U.S. Pat. Nos. 4,912,979; 5,712,829; 5,808,963; 6,718,266; 6,791,899; 7,310,285; and 7,623,412; and U.S. Patent Publication Nos. 2009/0185446. Any and all portions of the apparatus, systems, and methods described in each of these documents may be used to realize the various embodiments described herein.
Data Acquisition and Processing
FIG. 1 shows a cross-dipole acoustic tool 100 in a borehole aligned along the z-axis that can be used in conjunction with a method described herein. The tool 100 includes two azimuthally orthogonal dipole-transmitter arrays 102 and two azimuthally orthogonal dipole-receiver arrays 106. Each array has at least one element aligned along the tool axis. Specified in the traditional way, the recorded acoustic waveforms are denoted by in-line components, (XX, YY), and cross-line components, (XY, YX). The first letter refers to the transmitter, and the second letter to the receiver. Thus, for example, the XY recorded waveform is formed by firing the X-directed dipole transmitter and receiving at the Y-directed dipole receiver. The X and Y arrays are aligned along the z-axis such that the m'th element of array X is at the same z-position as the m'th element of array Y and is denoted as ring m. The fast principal wave axis of the formation is directed at angle θ relative to the x-axis. The Alford rotation equation assumes the waveforms transform as a second order tensor. The relationship can be expressed as shown in equation (1) as follows:

TTXjT=Dj,   (1)

where




  
    
      
        T
        ⁡
        
          (
          θ
          )
        
      
      =
      
        
          [
          
            
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  θ
                
              
              
                
                  
                    -
                    sin
                  
                  ⁢
                  
                    
                  
                  ⁢
                  θ
                
              
            
            
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  θ
                
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  θ
                
              
            
          
          ]
        
        ≡
        
          [
          
            
              
                
                  c
                  θ
                
              
              
                
                  -
                  
                    s
                    θ
                  
                
              
            
            
              
                
                  s
                  θ
                
              
              
                
                  c
                  θ
                
              
            
          
          ]
        
      
    
    ,
    
      

    
    ⁢
    
      
        X
        j
      
      =
      
        [
        
          
            
              
                XX
                j
              
            
            
              
                XY
                j
              
            
          
          
            
              
                YX
                j
              
            
            
              
                YY
                j
              
            
          
        
        ]
      
    
    ,
    
      

    
    ⁢
    and
  




  
    
      D
      j
    
    =
    
      
        [
        
          
            
              
                FP
                j
              
            
            
              0
            
          
          
            
              0
            
            
              
                SP
                j
              
            
          
        
        ]
      
      .
    
  



The real orthonormal matrix, T(θ), describes the Alford rotation, XXj,YYj,XYj, and YXj are the in-line and cross-line waveforms from transmitter ring m to receiver ring n, and j represents a data point j={m,n,t or ω}. FPj and SPj are the fast and slow principal waves. The fast principal axis is at angle θ relative to the x-axis. Written explicitly, the elements of equation (1) are:

(1,1)cθ2XXj+sθ2YYj+sθcθ(XYj+YXj)=FPj,

(2,2)cθ2YYj+sθ2XXj−sθcθ(XYj+YXj)=SPj,

(1,2)cθ2XYj−sθ2YXj−sθcθ(XXj−YYj)=0,

(2,1)cθ2YXj−sθ2XYj−sθcθ(XXj−YYj)=0.   (2)

Note these equations are approximate; in general there does not exist a matrix, T(θ), that exactly diagonalizes the cross-dipole data. However, in a similar vein, one can always find a unitary matrix, U, that exactly diagonalizes the cross-dipole data. The corresponding eigenvalues are direct estimates of the fast and slow principal flexural waves,

U′XjU=Dj,   (3)

where





  
    
      
        
          D
          j
        
        =
        
          
            [
            
              
                
                  
                    D
                    11
                    j
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    D
                    22
                    j
                  
                
              
            
            ]
          
          ≈
          
            
              [
              
                
                  
                    
                      FP
                      j
                    
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    
                      SP
                      j
                    
                  
                
              
              ]
            
            .
          
        
      
    
    
      
        (
        4
        )
      
    
  




Processing Methods

FIG. 2 is a flow chart illustrating anisotropy processing according to various embodiments of the invention. The anisotropy algorithm 211 can be used to solve for the angle θ by defining an objective function, O([TTXjT]kl,s), and minimizing the objective function with respect to (θ, s), where s is an auxiliary parameter space that defines the stacking of the points, j, used in the objective function and/or models the waveforms, Xj. Generally, s includes at least a slowness parameter, but may include other parameters such as borehole radius and formation density. The minimization is done using analytical methods, as described in more detail below. Often there are two minima in angle separated by 90°, corresponding to the fast and slow principal axis, which are to be resolved.
(a) A More Detailed Algorithm
FIG. 3 is a more detailed flow chart illustrating analytic anisotropy processing according to various embodiments of the invention. Instead of starting with an objective function formulated from the elements of equation (2), the more detailed algorithm in FIG. 3 uses an objective function formulated from the elements of equation (2) and the eigenvalues from the unitary transformation of equation (3). The algorithm 311 thus starts with implementing an optional noise rejection algorithm, after which the eigenvalues and unitary matrices are computed for each of the points in the data space j={m, n,t or ω}. Noise rejection processing of acquired acoustic waveform data can be accomplished in many different ways, well-known to those of ordinary skill in the art.
The waveforms can be represented in either the frequency or time domains. Note that the unitary transformation is not unique. The columns of U and the corresponding eigenvalues can be swapped so that the (1,1) element of D corresponds to the slow principal wave instead of the fast principal wave. Thus, the eigenvalues from different data points should be aligned relative to one another before they can be used in the objective function. After alignment, the objective function is globally minimized with respect to the angle and auxiliary parameters to determine the angle of the fast anisotropy axis and slowness of the principal waves.
A more detailed embodiment of blocks 321 in FIG. 3 is shown in FIG. 4, to illustrate eigenvalue alignment according to various embodiments of the invention.
(b) Alignment of Eigenvalues Across Receivers (Block 421)
Note there is a 180° symmetry in equation (2), but replacing θ→θ+90° exchanges the eigenvalues and eigenvectors. Since U is almost real and orthonormal, we can align the eigenvalues across receivers as shown below, assuming one transmitter ring and N receiver rings.
The following definition may be employed:

un=U11(jn)+iU21(jn),   (5)

where n is the receiver ring index, and a waveform data point is specified by jn={n,t or ω}. In addition, the following definition may also be employed:

θn=angle(u1*un), n=2, . . . , N.   (6)

The eigenvalues can be aligned relative to one another across receiver ring by exchanging the eigenvalues of receiver ring n>1 if

(45°<θn<135°)|(−135°<θn<−45°).   (7)

(c) Estimate of Eigenvalue Propagators (Block 425)

After aligning eigenvalues with respect to receiver rings, it is possible to estimate propagators from receiver ring to receiver ring at a given frequency. Assuming the acoustic tool is substantially centered and only the fast and slow principal flexural waves are present, one can represent the eigenvalues as:

Dkkjn≈Ak(ω)zk−n(ω), zk=eiωskd,   (8)

As of yet it has not been determined which slowness value, s1 or s2, corresponds to the fast principal wave. According to equation (8), a frequency semblance algorithm can be used to determine the propagator, Zest.
One embodiment makes use of the equation (9), as follows:




  
    
      
        
          
            z
            k
          
          =
          
            
              
                ∑
                
                  n
                  =
                  1
                
                
                  N
                  -
                  1
                
              
              ⁢
              
                
                  
                    (
                    
                      D
                      kk
                      
                        j
                        
                          n
                          +
                          1
                        
                      
                    
                    )
                  
                  *
                
                ⁢
                
                  D
                  kk
                  
                    j
                    n
                  
                
              
            
            
              
                ∑
                
                  n
                  =
                  1
                
                
                  N
                  -
                  1
                
              
              ⁢
              
                
                  
                    (
                    
                      D
                      kk
                      
                        j
                        
                          n
                          +
                          1
                        
                      
                    
                    )
                  
                  *
                
                ⁢
                
                  D
                  kk
                  
                    j
                    
                      n
                      +
                      1
                    
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        9
        )
      
    
  




where





  
    
      
        
          
            Z
            est
          
          ⁡
          
            (
            ω
            )
          
        
        =
        
          
            [
            
              
                
                  
                    
                      z
                      1
                    
                    ⁡
                    
                      (
                      ω
                      )
                    
                  
                
                
                  
                    
                  
                
              
              
                
                  
                    
                  
                
                
                  
                    
                      z
                      2
                    
                    ⁡
                    
                      (
                      ω
                      )
                    
                  
                
              
            
            ]
          
          .
        
      
    
    
      
        (
        10
        )
      
    
  




Note that using the eigenvalues directly makes it unnecessary to do a numerical search in slowness or angle to minimize the objective function if the minimization in the angle is done analytically. Also note that the propagator measures the slowness across the receiver array. The average slowness from transmitter to first receiver is absorbed in the amplitudes, Ak(ω). Thus this method intrinsically provides depth resolution equal to the receiver array length.
(d) Alignment of Eigenvalues Across Frequency (Block 429)
After computing the propagator, alignment across frequency can be accomplished to determine whether the first eigenvalue corresponds to the fast principal wave, or the slow principal wave. Assume the objective function is defined over a bandwidth Δω starting at frequency ωST. Typically the bandwidth will be narrow, because it is useful to detect in-situ stress by solving for the angle as a function of ωST by computing the relative phase of the propagators as follows:

Ω(ω)=angle(z1(ω)*z2(ω)), ωST≦ω≦ωST+Δω. (11)

The eigenvalues are aligned across frequency by comparing the phase. If sign(Ω(ω))≠sign(Ω(ωST)), the diagonal elements of D and Zest at ω are exchanged. If sign(Ω(ωST))>0, then the first eigenvalue corresponds to the fast principal wave; otherwise it corresponds to the slow principal wave. By tracking sign(Ω(ωST)) as a function of ωST the eigenvalues can be aligned across the frequency spectrum as well. Then a 90° jump in angle can be used to indicate stress induced anisotropy.
(e) The Objective Function (Block 433)
Having aligned the waveforms, the objective function can be defined. The objective function can be chosen to simplify the analytic estimation of angle. In this case, the objective function is defined as




  
    
      
        
          O
          ⁡
          
            (
            θ
            )
          
        
        ≡
        
          
            ∑
            
              
                
                  
                    j
                    n
                  
                  =
                  
                    (
                    
                      ω
                      ,
                      n
                    
                    )
                  
                
                
                  
                    j
                    m
                  
                  =
                  
                    (
                    
                      ω
                      ,
                      m
                    
                    )
                  
                
              
              
                k
                ,
                
                  l
                  =
                  1
                
                ,
                2
              
            
            
              B
              ⁡
              
                (
                
                  
                    ω
                    ST
                  
                  ,
                  Δω
                
                )
              
            
          
          ⁢
          
            
              
                
                
                  
                    
                      T
                      T
                    
                    ⁢
                    
                      X
                      
                        j
                        n
                      
                    
                    ⁢
                    T
                  
                  -
                  
                    
                      D
                      
                        j
                        m
                      
                    
                    ⁢
                    
                      Z
                      est
                      
                        m
                        -
                        n
                      
                    
                  
                
                
              
              kl
              2
            
            .
          
        
      
    
    
      
        (
        12
        )
      
    
  




When minimized over angle, this objective function will operate to determine the best real orthonormal T(θ) that fits the waveforms to the eigenvalues. As the process continues, it is mathematically expedient to change the notation as shown in equation (13):




  
    
      
        
          
            O
            ⁡
            
              (
              θ
              )
            
          
          ≡
          
            
              ∑
              
                
                  j
                  =
                  
                    {
                    
                      
                        j
                        n
                      
                      ,
                      
                        j
                        m
                      
                    
                    }
                  
                
                
                  k
                  ,
                  
                    l
                    =
                    1
                  
                  ,
                  2
                
              
              
                B
                ⁡
                
                  (
                  
                    
                      ω
                      ST
                    
                    ,
                    
                      Δ
                      ⁢
                      
                        
                      
                      ⁢
                      ω
                    
                  
                  )
                
              
            
            ⁢
            
              
                
                
                  
                    
                      T
                      T
                    
                    ⁢
                    
                      X
                      j
                    
                    ⁢
                    T
                  
                  -
                  
                    Δ
                    j
                  
                
                
              
              kl
              2
            
          
        
        ,
      
    
    
      
        (
        13
        )
      
    
  




where

Δj≡DjmZestm−n,

Xj=Xjn∀jm.   (14)

(f) Analytic Minimization of the Angle (Block 437)

Due to the choice of objective function, minimization with respect to angle is greatly simplified. The result is shown in equation (14):




  
    
      
        
          
            tan
            ⁡
            
              (
              
                2
                ⁢
                θ
              
              )
            
          
          =
          
            
              Re
              [
              
                
                  
                    a
                    ⇀
                  
                  *
                
                ·
                
                  d
                  ⇀
                
              
              ]
            
            
              Re
              [
              
                
                  
                    b
                    ⇀
                  
                  *
                
                ·
                
                  d
                  ⇀
                
              
              ]
            
          
        
        ,
      
    
    
      
        (
        14
        )
      
    
  




where the vectors are defined as:

aj=X12j+X21j,

bj=X22j−X11j,

dj=Δ22j−Δ11j.   (15)

Solving equation (14) gives two possible global minimization angles separated by 90°. They are substituted into equation (13) to determine the true global minimum, θg(ωST).
FIG. 9 shows the angle of the fast principal wave as a function frequency for a synthetic crossed dipole. Here, the X-dipole is oriented at 21.8° relative to the fast principal wave axis. The data in this case was generated using a spatially constant elasticity matrix, so there is no 90° jump.
(g) Refinement of Slowness Estimates (Block 441)
Returning now to FIG. 4, at block 441, it is noted that slowness estimates can be refined by rotating the waveforms using θg(ω) into the fast and slow principal waves using equation (2). A frequency semblance algorithm can be used to detect slowness.
FIG. 10 shows the frequency dispersion curves for the same synthetic waveforms computed using a matrix pencil algorithm: a computationally efficient method that can be used for estimating signal parameters, which is well-known to those of ordinary skill in the art. In this case, it is seen that the fast wave has a slowness of about 155 μs/ft., and the slow wave has a slowness of about 175 μs/ft.
Additional Features
Previous solutions to the problem of estimating anisotropy minimize the objective function using a numerical search. The proposed approach uses the eigenvalues derived from a unitary similarity transformation as estimates of the fast and slow principal waves. These estimates can then be used in the objective function as well as the components of equation (2). This makes the objective function more robust, so that it is possible to minimize the objective function analytically.
The ability to detect stress induced anisotropy is a useful aspect of formation evaluation and can be a governing factor in the optimization of the development of a reservoir. Conventional software measures anisotropy using a numerical search or brute force minimization of an objective function in the time or frequency domain. When operating in the time domain the objective function uses the early arrivals of the cross-dipole waveforms over a narrow time interval. After solving for the angle of anisotropy, the fast and slow principal waves are generated using equation (2). Individual depths are analyzed for the presence of stress induced anisotropy by plotting an overlay of the fast and slow dispersion curves. Visual inspection of the dispersion curves for a cross-over in slowness is indicative of stress induced anisotropy. If there is no cross-over, then the anisotropy may be due to some other cause, such as intrinsic anisotropy.
Visual inspection by the operator of two very similar and sometimes noisy dispersion curves can be impractical for processing an entire well bore log, so that individual depths are only spot-checked.
Frequency domain embodiments using the aligned eigenvalues have the advantage of directly measuring the angle as a function of frequency. Stress induced anisotropy is detected by a jump of approximately 90° in the angle as a function of frequency, providing a more reliable indicator that is more amenable to automation than conventional methods. Some embodiments shown herein also provide a depth resolution approximately equal to the receiver array length.
FIG. 11 is a block diagram of an apparatus 1100, according to various embodiments of the invention. The apparatus 1100 may comprise a number of components, including those described in the following paragraphs.
For example, an apparatus 1100 may comprise a set of receiver and transmitter arrays 1104, recording logic 1140, and one or more signal processors 1130. The arrays 1104 may be configured as azimuthally orthogonal dipole receiver arrays and azimuthally orthogonal transmitter arrays, attached to a down hole tool, similar to or identical to the tool 100 shown in FIG. 1. The logic 1140 may be configured to record a plurality of acoustic waveforms corresponding to acoustic waves received at the azimuthally orthogonal dipole receiver arrays, the waves being generated by the azimuthally orthogonal transmitter arrays.
The signal processor 1130 may be configured to estimate a global minimum of an objective function with respect to an azimuth angle and a set (possibly null) of auxiliary parameters, wherein the azimuth angle corresponds to an orientation of the transmitter arrays and the receiver arrays relative to fast and slow principal flexural wave axes. The signal processor 1130 may be further configured to estimate fast and slow principal flexural waves propagating along the axes and characterized by the waveforms as ordered complex eigenvalues of a waveform matrix diagonalized by a unitary similarity transformation at each data point of the waveforms, and the objective function depending on the waveforms (which may comprise cross-dipole waveforms), the ordered eigenvalues, the azimuth angle, and (optionally) the set of auxiliary parameters. The signal processor may also be configured to remove existing ambiguities associated with the fast and slow principal flexural wave axes.
The apparatus 1100 may further comprise a memory 1150 to receive and store values 1134 associated with the global minimum of the objective function, such as values corresponding to the azimuth angle and the auxiliary parameters at the global minimum. In various embodiments, the apparatus may comprise a telemetry transmitter 1124 to communicate values associated with the global minimum of the objective function, to include at least the azimuth angle, to a logging facility 1156 at the surface 1166. The logging facility 1156 may include the signal processor 1130.
In the apparatus 1100, the signal processor 1130 may be located at the surface, or be at least partially housed by the downhole tool 100. The surface facility may comprise a display (see element 496 of FIGS. 5, 6) to display various elements of the acquired waveform data, the azimuth angle, the global minimum of the objective function, formation properties, and formation anisotropy information.
FIGS. 5-6 illustrate system embodiments of the invention. For example, FIG. 5 illustrates a drilling rig system 564 embodiment of the invention, and FIG. 6 illustrates a wireline system 464 embodiment of the invention. Thus, systems 464, 564 may comprise portions of a tool body 470 as part of a wireline logging operation, or of a downhole tool 524 as part of a downhole drilling operation.
Drilling of oil and gas wells is commonly carried out using a string of drill pipes connected together so as to form a drilling string that is lowered through a rotary table 410 into a wellbore or borehole 412. Turning now to FIG. 5, it can be seen how a system 564 may form a portion of a drilling rig 502 located at the surface 504 of a well 506. The drilling rig 502 may provide support for a drill string 508. The drill string 508 may operate to penetrate a rotary table 410 for drilling a borehole 412 through subsurface formations 414. The drill string 508 may include a Kelly 516, drill pipe 518, and a bottom hole assembly 520, which can be located at the lower portion of the drill pipe 518. In various embodiments, apparatus 1100 may be carried as part of the drill string 508 or the downhole tool 524.
The bottom hole assembly 520 may include drill collars 522, a downhole tool 524, and a drill bit 526. The drill bit 526 may operate to create a borehole 412 by penetrating the surface 504 and subsurface formations 414. The downhole tool 524 may comprise any of a number of different types of tools including MWD (measurement while drilling) tools, LWD tools, and others.
During drilling operations, the drill string 508, which can include the Kelly 516, the drill pipe 518, and the bottom hole assembly 520, may be rotated by the rotary table 410. In addition to, or alternatively, the bottom hole assembly 520 may also be rotated by a motor (e.g., a mud motor) that is located downhole. The drill collars 522 may be used to add weight to the drill bit 526. The drill collars 522 may also operate to stiffen the bottom hole assembly 520, allowing the bottom hole assembly 520 to transfer the added weight to the drill bit 526, and in turn, to assist the drill bit 526 in penetrating the surface 504 and subsurface formations 414.
During drilling operations, a mud pump 532 may pump drilling fluid (sometimes known by those of skill in the art as “drilling mud”) from a mud pit 534 through a hose 536 into the drill pipe 518 and down to the drill bit 526. The drilling fluid can flow out from the drill bit 526 and be returned to the surface 504 through an annular area 540 between the drill pipe 518 and the sides of the borehole 412. The drilling fluid may then be returned to the mud pit 534, where such fluid is filtered. In various embodiments, the drilling fluid can be used to cool the drill bit 526, as well as to provide lubrication for the drill bit 526 during drilling operations. Additionally, the drilling fluid may be used to remove subsurface formation 414 cuttings created by operating the drill bit 526.
FIG. 6 shows a well during wireline logging operations. A drilling platform 486 is equipped with a derrick 480 that supports a hoist 490. Here it is assumed that the drilling string has been temporarily removed from the borehole 412 to allow a wireline logging tool body 470, such as a probe or sonde that carries a sonic tool, to be lowered by wireline or logging cable 474 into the borehole 412. Typically, the tool body 470 is lowered to the bottom of the region of interest and subsequently pulled upward at a substantially constant speed.
During the upward trip, apparatus 1100 included in the tool body 470 may be used to perform measurements in the borehole 412 as they pass by. The measurement data can be communicated to a surface logging facility 492 for storage, processing, and analysis. The logging facility 492 may be provided with electronic equipment for various types of signal processing, which may be implemented by any one or more of the components of the apparatus 1100 shown in FIG. 11. The log data is similar to that which may be gathered and analyzed during drilling operations (e.g., during logging while drilling (LWD) operations).
The boreholes 110; 412; apparatus 300; logging facility 492; display 396; rotary table 410; formation 414; systems 464, 564; tool body 470; drilling platform 486; derrick 480; hoist 490; logging cable 474; drilling rig 502; well 506; drill string 508; Kelly 516; drill pipe 518; bottom hole assembly 520; drill collars 522; downhole tool 524; drill bit 526; mud pump 532; mud pit 534; and hose 536 may all be characterized as “modules” herein. Such modules may include hardware circuitry, and/or a processor and/or memory circuits, software program modules and objects, and/or firmware, and combinations thereof, as desired by the architect of the apparatus 1100 and systems 464, 564, and as appropriate for particular implementations of various embodiments. For example, in various embodiments, such modules may be included in an apparatus and/or system operation simulation package, such as a software electrical signal simulation package, a power usage and distribution simulation package, a power/heat dissipation simulation package, and/or a combination of software and hardware used to simulate the operation of various potential embodiments.
It should also be understood that the apparatus and systems of various embodiments can be used in applications other than for drilling operations, and thus, various embodiments are not to be so limited. The illustrations of apparatus 1100 and systems 464, 564 are intended to provide a general understanding of the structure of various embodiments, and they are not intended to serve as a complete description of all the elements and features of apparatus and systems that might make use of the structures described herein.
Applications that may incorporate the novel apparatus and systems of various embodiments include a variety of electronic systems, such as televisions, cellular telephones, personal computers, workstations, radios, video players, vehicles, and location technology (e.g., GPS (Global Positioning System) location technology), signal processing for geothermal tools and smart transducer interface node telemetry systems, among others. Various embodiments include a number of methods.
Thus, a system 464, 564 may comprise a down hole tool and one or more apparatus 1100, as described previously. The downhole tool may comprise a wireline tool or a measurement while drilling tool, among others. In various embodiments, the azimuthally orthogonal dipole receiver arrays and azimuthally orthogonal transmitter arrays, as part of the apparatus, are each attached to the down hole tool. The recording logic, as part of the apparatus, may be attached to the tool, or form part of a surface computer. Similarly, the signal processor, as part of the apparatus, may also be attached to the tool, or form part of a surface computer. Thus, the signal processor can be attached to the tool and the results of estimating the global minimum of the objective function can be sent to the surface, or the signal processor can operate entirely on the surface. In this way, signal processing that is not completed down hole may be completed at the surface. Telemetry data obtained from the recording logic may be used in the processing at the surface. Various embodiments include a number of methods.
FIG. 7 is a flow chart illustrating several methods according to various embodiments of the invention. For example, a processor-implemented method 711 to execute on one or more processors that perform methods to estimate an azimuth angle and to determine geological formation properties may comprise: recording a plurality of acoustic waveforms comprising cross-dipole waveforms and corresponding to acoustic waves received at azimuthally orthogonal dipole receiver arrays surrounded by a geological formation, the waves being generated by azimuthally orthogonal dipole transmitter arrays, wherein fast and slow principal flexural waves characterized by the waveforms are estimated as the ordered complex eigenvalues of a waveform matrix diagonalized by a unitary similarity transformation at each data point of the waveforms; defining an objective function dependent on the cross-dipole waveforms, the eigenvalues, an azimuth angle corresponding to an orientation of the transmitter and receiver arrays relative to fast and slow principal wave axes, and a set (possibly null) of auxiliary parameters; minimizing the objective function with respect to said angle and the set of auxiliary parameters; and determining at least one property of the geological formation based on the global minimum.
In various embodiments, a processor-implemented method 711 of estimating an azimuth angle, to execute on one or more processors that perform the method, comprises: at block 721, recording a plurality of acoustic waveforms corresponding to acoustic waves received at azimuthally orthogonal dipole receiver arrays surrounded by a geological formation, the waves being generated by azimuthally orthogonal transmitter arrays, the azimuth angle corresponding to an orientation of the receiver arrays and the transmitter arrays relative to fast and slow principal flexural wave axes, wherein fast and slow principal flexural waves propagating along the axes and characterized by the waveforms being estimated as the ordered (optionally complex) eigenvalues of a waveform matrix diagonalized by a unitary similarity transformation; at block 723, defining an objective function dependent on the waveforms (which may comprise cross-dipole waveforms), the ordered eigenvalues, the azimuth angle, and (optionally) a set of auxiliary parameters; at block 727, minimizing the objective function with respect to the azimuth angle (and optionally, the set of auxiliary parameters); and at block 737, determining at least one anisotropy angle of the geological formation based on the global minimum.
The matrix represents a set of data points in a space defined by receiver ring index position and time. The matrix may comprise a cross-dipole waveform matrix. The matrix may also represent a set of data points in a space defined by receiver ring index position and frequency.
The method 711 may further comprise, at block 725, diagonalizing the waveform matrix by a unitary similarity transformation at each data point of the waveforms, by computing a unitary matrix having columns comprising complex eigenvectors of the waveform matrix at each of the data points of the waveforms; and computing the eigenvalues of the eigenvectors.
Analytical or numerical methods (e.g., the Matlab® software “eig” function) may be used to diagonalize the matrix. Thus, diagonalizing the waveform matrix is accomplished using an analytical method or a numerical method.
In various embodiments, the method 711 may comprise, at block 725: ordering the eigenvalues by ordering eigenvectors and corresponding ones of the eigenvalues across data points of the waveforms such that a difference in angle between the eigenvectors across the data points is minimized, and (optionally) ordering the eigenvectors and the corresponding ones of the eigenvalues across the data points using a parameter correlated in a known way with the difference in the angle.
In various embodiments, the method 711 may comprise, at block 725: ordering eigenvectors and corresponding ones of the eigenvalues across receiver rings at each frequency in a space defined by data points in the waveforms; estimating eigenvalue propagators across the receiver rings from the ordered (across the receiver rings) eigenvalues at each frequency in the space; and ordering the eigenvectors, the corresponding ones of the eigenvalues, and the propagators over frequency by comparing relative phases associated with the estimates of the eigenvalue propagators at one frequency to relative phases associated with the estimates of the eigenvalue propagators at another frequency.
Ordering eigenvectors and eigenvalues as described above is relevant to waveforms in the frequency domain. When the waveforms are in the time domain, eigenvalue alignment is possible but more difficult due to the oscillatory nature of the waveforms. Various time domain embodiments described herein only use the cross-dipole waveforms in the objective function and not the eigenvalues found from a unitary transformation of the cross-dipole waveforms. Use of the eigenvalues in the objective function is optional in either the time or frequency domain and is not meant to limit embodiments taught herein.
In various embodiments, the objective function is defined by utilizing all combinations of receiver rings by estimating eigenvalues at receiver ring index m from application of estimated propagators to eigenvalues at receiver ring index n. When the waveforms are in the time domain an embodiment utilizing all combinations of receiver rings comprises shifting the waveforms according to their receiver ring index by using the auxiliary parameter slowness values.
Minimizing the objective function may be accomplished analytically or by using a numerical search algorithm. Minimizing the objective function may also be accomplished by estimating eigenvalue propagators from the eigenvalues, without minimizing the objective function over a set of auxiliary parameters.
The method of not minimizing the objective function over a set of auxiliary parameters is used in an embodiment that takes advantage of a frequency domain approach, with propagator estimates. In other embodiments that operate in either the time or frequency domain, the objective function is often minimized over a set of auxiliary parameters.
In various embodiments, determining at least one anisotropy angle of the geological formation comprises: implementing the activities described previously for a data point space defined by multiple frequency bands having different starting frequencies or by multiple different time periods having different starting times.
In various embodiments, depending on the form of the objective function and constraints on auxiliary slowness parameters, adding 90° or 45° (modulo 180°) to the anisotropy angle estimated for the one of the bands or time periods to resolve a min/max ambiguity or a fast/slow ambiguity may be necessary. Multiple examples are shown later to illustrate this.
In various embodiments, ordering the eigenvectors and corresponding ones of the eigenvalues comprises switching the corresponding ones of the eigenvalues, and the propagators.
In various embodiments, the method 711 comprises, at block 733 (if objective function min/max or fast/slow axes ambiguities are determined to exist at block 729), removing fast/slow ambiguities or min/max ambiguities as exemplified in the various embodiments.
Various embodiments of the method 711 comprise, at block 741, displaying the global minimum in conjunction with at least one anisotropy angle.
Various embodiments of the method 711 comprise, at block 741, displaying an indication of stress induced anisotropy when an estimated version of the azimuth angle exhibits approximately a 90° jump as a function of start frequency, and dispersion curves for fast and slow principal flexural waves propagating along the axes, wherein the dispersion curves are created from the estimated version of the azimuth angle and the waveforms at each one of the start frequencies.
Various embodiments of the method 711 comprise, at block 745, detecting and characterizing formation fractures and/or a formation ambient stress field, based on the at least one anisotropy angle and an absolute orientation of a down hole tool carrying the transmitter arrays and the receiver arrays.
Additional activities forming a part of the methods 711 are listed in the activities for methods claimed below, and described above with respect to FIGS. 1-6 and 8-11. Thus, it should be noted that the methods described herein do not have to be executed in the order described, or in any particular order. Moreover, various activities described with respect to the methods identified herein can be executed in iterative, serial, or parallel fashion. Some activities may be added, and some of the included activities may be left out. Information, including parameters, commands, operands, and other data, can be sent and received in the form of one or more carrier waves.
FIG. 8 is a block diagram of an article 800 of manufacture, including a specific machine 802, according to various embodiments of the invention. Upon reading and comprehending the content of this disclosure, one of ordinary skill in the art will understand the manner in which a software program can be launched from a computer-readable medium in a computer-based system to execute the functions defined in the software program.
One of ordinary skill in the art will further understand the various programming languages that may be employed to create one or more software programs designed to implement and perform the methods disclosed herein. The programs may be structured in an object-orientated format using an object-oriented language such as Java or C++. In various embodiments, the programs can be structured in a procedure-orientated format using a procedural language, such as assembly or C. The software components may communicate using any of a number of mechanisms well known to those of ordinary skill in the art, such as application program interfaces or interprocess communication techniques, including remote procedure calls. The teachings of various embodiments are not limited to any particular programming language or environment. Thus, other embodiments may be realized.
For example, an article 800 of manufacture, such as a computer, a memory system, a magnetic or optical disk, some other storage device, and/or any type of electronic device or system may include one or more processors 804 coupled to a machine-readable medium 808 such as a memory (e.g., removable storage media, as well as any memory including an electrical, optical, or electromagnetic conductor comprising non-transitory, tangible media) having instructions 812 stored thereon (e.g., computer program instructions), which when executed by the one or more processors 804 result in the machine 802 performing any of the actions described with respect to the methods above.
The machine 802 may take the form of a specific computer system having a processor 804 coupled to a number of components directly, and/or using a bus 816. Thus, the machine 802 may be similar to or identical to the workstation 454 shown in FIGS. 5 and 6, or the processor in the apparatus 1100 of FIG. 11.
Turning now to FIG. 8, it can be seen that the components of the machine 802 may include main memory 820, static or non-volatile memory 824, and mass storage 806. Other components coupled to the processor 804 may include an input device 832, such as a keyboard, or a cursor control device 836, such as a mouse. An output device 828, such as a video display, may be located apart from the machine 802 (as shown), or made as an integral part of the machine 802.
A network interface device 840 to couple the processor 804 and other components to a network 844 may also be coupled to the bus 816. The instructions 812 may be transmitted or received over the network 844 via the network interface device 840 utilizing any one of a number of well-known transfer protocols (e.g., HyperText Transfer Protocol). Any of these elements coupled to the bus 816 may be absent, present singly, or present in plural numbers, depending on the specific embodiment to be realized.
The processor 804, the memories 820, 824, and the storage device 806 may each include instructions 812 which, when executed, cause the machine 802 to perform any one or more of the methods described herein. In various embodiments, the machine 802 operates as a standalone device or may be connected (e.g., networked) to other machines. In a networked environment, the machine 802 may operate in the capacity of a server or a client machine in server-client network environment, or as a peer machine in a peer-to-peer (or distributed) network environment.
The machine 802 may comprise a personal computer (PC), a workstation, a tablet PC, a set-top box (STB), a PDA, a cellular telephone, a web appliance, a network router, switch or bridge, server, client, or any specific machine capable of executing a set of instructions (sequential or otherwise) that direct actions to be taken by that machine to implement the methods and functions described herein. Further, while only a single machine 802 is illustrated, the term “machine” shall also be taken to include any collection of machines that individually or jointly execute a set (or multiple sets) of instructions to perform any one or more of the methodologies discussed herein.
While the machine-readable medium 808 is shown as a single medium, the term “machine-readable medium” should be taken to include a single medium or multiple media (e.g., a centralized or distributed database, and/or associated caches and servers, and or a variety of storage media, such as the registers of the processor 804, memories 820, 824, and the storage device 806 that store the one or more sets of instructions 812. The term “machine-readable medium” shall also be taken to include any medium that is capable of storing, encoding or carrying a set of instructions for execution by the machine and that cause the machine 802 to perform any one or more of the methodologies of the present invention, or that is capable of storing, encoding or carrying data structures utilized by or associated with such a set of instructions. The terms “machine-readable medium” or “computer-readable medium” shall accordingly be taken to include tangible media, such as solid-state memories and optical and magnetic media.
Various embodiments may be implemented as a stand-alone application (e.g., without any network capabilities), a client-server application or a peer-to-peer (or distributed) application. Embodiments may also, for example, be deployed by Software-as-a-Service (SaaS), an Application Service Provider (ASP), or utility computing providers, in addition to being sold or licensed via traditional channels.
Many rocks in earth formations exhibit anisotropic characteristics. A common form of anisotropy in sedimentary rocks is transverse isotropy (TI). This form of anisotropy has an axis of rotational symmetry. The material properties of the rock are the same for any direction in the plane perpendicular to the symmetry axis, but different from the material properties along the symmetry axis. An example of transverse isotropy in a sedimentary rock is shown in FIG. 12. FIG. 12 is a representation of transverse isotropy, with the axis of symmetry being the z-axis.
When the symmetry axis is orthogonal to the acoustic tool in the borehole, the anisotropy is called horizontal transverse isotropy (HTI). FIG. 13 is a representation of horizontal transverse isotropy anisotropy, where the axis of symmetry is along the slow principal axis (SP) and perpendicular to the borehole. In this example, the anisotropy is due to aligned fractures. The velocity of acoustic shear waves propagating along the z-axis (out of FIG. 13) and polarized in the fracture plane along the fast principal axis (FP) are faster than shear waves propagating along the z-axis and polarized along the SP.
Traditional acoustic tools use cross-dipole technology to detect this type of anisotropy. Cross-dipole acoustic tools have two orthogonal dipole transmitters aligned along the x and y axes in FIGS. 12 and 13. Two orthogonal dipole receivers (also aligned along the x and y axes in the figure) record the waveforms. Thus, four different waveforms are recorded by individually firing each of the transmitters while recording on both receivers. The waveforms are processed in the time, t, or frequency, ω, domain to determine φ, the angle of the fast principal axis relative to the tool axis. The angle φ and the tool orientation are used to determine the angle of the fast principal axis relative to north. Cross-dipole measurements are routinely made in common well logging practice. One important application for cross-dipole measurements is fracture detection and analysis in open and cased boreholes. The cased-hole application is useful for evaluating hydraulic fracture stimulation. Another important application is identifying stress-induced anisotropy for use in estimating formation stress orientation and magnitude. This information is useful for optimizing the hydraulic fracture process.
As technology advances, more sophisticated acoustic tools are being developed. Electronic and mechanical components are decreasing in size, which can allow many dipole receivers to be disposed azimuthally spaced around the tool instead of having a single pair of orthogonal dipole receivers. Thus, many waveforms at different angles around the tool at the same distance along the length of the tool may be recorded. The following embodiments can provide mechanisms and methods to process the additional information from the increased number of receivers to be disposed azimuthally spaced around the tool to improve angle estimation accuracy and stability.
With the transmitters of FIG. 1 limited to one X-dipole transmitter and one Y-dipole transmitter, FIG. 1 can represent a traditional cross-dipole acoustic tool in a borehole aligned along the z-axis. The tool consists primarily of two azimuthally orthogonal dipole-transmitters and two azimuthally orthogonal dipole-receiver arrays. Each receiver array consists of at least one element aligned along the tool axis. Specified in the traditional way, the recorded acoustic waveforms are denoted by in-line components, (XX, YY), and cross-line components, (XY, YX). The first letter refers to the transmitter, and the second letter to the receiver. Thus, for example, the XY recorded waveform is formed by firing the X-directed dipole transmitter and receiving at the Y-directed dipole receiver. The X and Y receiver arrays are aligned along the z-axis such that the m'th element of receiver array X is at the same z-position as the m'th element of receiver array Y and is denoted as ring m.
FIG. 14 is a schematic representation of a new generic tool 1410 with many dipole receivers spaced in azimuth around the tool 1410. The tool 1410 can include receiver rings 1415-1, 1415-2, . . . 1415-M. Each of the receiver rings can include acoustic dipole receivers 1405-1 . . . 1405-N azimuthally disposed around the tool 1410. An acoustic dipole receiver is an arrangement of two acoustic receivers on opposite sides of a tool to which it is attached such that a signal received by the acoustic dipole receiver is the difference of signals received at the two acoustic receivers.
FIG. 15 is a top view of the tool 1410 of FIG. 14. The X-dipole transmitter is directed at angle φ relative to the fast principal wave axis of the formation and the dipole receivers are directed at angles 0≦θn<π relative to the X-dipole. Thus, each dipole receiver is at angle φn=φ+θn relative to the fast axis. In practice the receivers may be equally spaced but this is not a limitation of the method. The equations described herein are valid for unequally spaced receivers. Note that the orientation of the angle φ in FIG. 15 has changed relative to FIG. 13. The angle from the X-Dipole to the fast axis is −φ. In the classic cross-dipole configuration the X and Y transmitters are aligned with the X and Y receivers and the 2×2 set of waveforms (XX, XY, YX, YY) satisfy the Alford rotation. The Alford rotation equation assumes the waveforms transform as a second order tensor. Thus,

TTXjT=Dj,   (16)

where





  
    
      
        
          
            T
            =
            
              
                [
                
                  
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                      
                    
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                      
                    
                  
                  
                    
                      
                        
                          -
                          sin
                        
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                      
                    
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                      
                    
                  
                
                ]
              
              ≡
              
                [
                
                  
                    
                      
                        c
                        φ
                      
                    
                    
                      
                        s
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                        c
                        φ
                      
                    
                  
                
                ]
              
            
          
          ,
          
            

          
          ⁢
          
            
              X
              j
            
            =
            
              [
              
                
                  
                    
                      XX
                      j
                    
                  
                  
                    
                      XY
                      j
                    
                  
                
                
                  
                    
                      YX
                      j
                    
                  
                  
                    
                      YY
                      j
                    
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          and
        
        ⁢
        
          

        
        ⁢
        
          
            D
            j
          
          =
          
            
              [
              
                
                  
                    
                      D
                      11
                      j
                    
                  
                  
                    0
                  
                
                
                  
                    0
                  
                  
                    
                      D
                      22
                      j
                    
                  
                
              
              ]
            
            .
          
        
      
    
    
      
        (
        17
        )
      
    
  




XXj, YYj, XYj, and YXj are the in-line and cross-line waveforms from the transmitter to receiver ring mR. The index j represents a data point j={mR,t or ω}. D11j and D22j are the fast and slow principal waves. Matrix variables have reserved superscripts for data point indices and subscripts for the 2×2 matrix element indices.

When dealing with many receivers, the notation changes. Define Xn,j as the waveform recorded at the dipole receiver with azimuth index n when firing the X-dipole transmitter, and Yn,j as the waveform recorded at the dipole receiver with azimuth index n when firing the Y-dipole transmitter. Using this notation, a more general 2×2 waveform can be defined as




  
    
      
        
          
            
              X
              nmj
            
            ≡
            
              [
              
                
                  
                    
                      X
                      11
                      nmj
                    
                  
                  
                    
                      X
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                      nmj
                    
                  
                
                
                  
                    
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                      nmj
                    
                  
                  
                    
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                      22
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              ]
            
          
          =
          
            
              [
              
                
                  
                    
                      X
                      
                        n
                        ,
                        j
                      
                    
                  
                  
                    
                      X
                      
                        m
                        ,
                        j
                      
                    
                  
                
                
                  
                    
                      Y
                      
                        n
                        ,
                        j
                      
                    
                  
                  
                    
                      Y
                      
                        m
                        ,
                        j
                      
                    
                  
                
              
              ]
            
            ⁢
            
              {
              
                I
                -
                
                  
                    c
                    
                      (
                      
                        
                          θ
                          n
                        
                        -
                        
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                          m
                        
                      
                      )
                    
                  
                  ⁢
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              }
            
            ⁢
            
              T
              nm
            
          
        
        ,
      
    
    
      
        (
        18
        )
      
    
  




where n≠m are any two azimuth indices,





  
    
      
        
          
            T
            nm
          
          =
          
            
              [
              
                
                  
                    
                      c
                      
                        θ
                        n
                      
                    
                  
                  
                    
                      s
                      
                        θ
                        n
                      
                    
                  
                
                
                  
                    
                      c
                      
                        θ
                        m
                      
                    
                  
                  
                    
                      s
                      
                        θ
                        m
                      
                    
                  
                
              
              ]
            
            
              s
              
                (
                
                  
                    θ
                    n
                  
                  -
                  
                    θ
                    m
                  
                
                )
              
              2
            
          
        
        ,
        
          I
          =
          
            [
            
              
                
                  1
                
                
                  0
                
              
              
                
                  0
                
                
                  1
                
              
            
            ]
          
        
        ,
        
          
            and
            ⁢
            
              
            
            ⁢
            P
          
          =
          
            
              [
              
                
                  
                    0
                  
                  
                    1
                  
                
                
                  
                    1
                  
                  
                    0
                  
                
              
              ]
            
            .
          
        
      
    
    
      
        (
        19
        )
      
    
  




It can be shown that the 2×2 waveform in equation (18) also satisfies the Alford rotation. For any n≠m, the following is obtained

TTXnmjT=Dj,   (20)

where Dj is independent of the azimuth indices. Note that equation (18) is symmetric with respect to permutation of m and n. Thus, for N dipole receivers, there are N(N−1)/2 independent 2×2 waveforms. For example, if 8 receivers are distributed uniformly around the tool, taking the difference of opposite receiver waveforms gives 4 dipole receiver waveforms and 6 independent 2×2 waveforms. The notation can be further simplified by defining a single index, nA, into the ordered set of 2×2 waveforms. Explicitly for the example N=4, the following is specified

nA=1→(m.n)=(0,1),

nA=2→(m.n)=(0,2),

nA=3→(m.n)=(0,3),

nA=4→(m.n)=(1,2),

nA=5→(m.n)=(1,3),

nA=6→(m.n)=(2,3),   (21)

and the generalized Alford rotation is expressed as

TTXnA,jT=Dj.   (22)

Traditional cross-dipole algorithms minimize an objective function dependent upon the Alford rotation of equations (16) and (17) to determine the angle, φ. Since the Alford rotation is preserved for the general 2×2 waveforms, any such objective function may be modified to use all the generalized waveforms. It is clear that increasing the number of waveforms processed may result in improved accuracy and stability of the algorithm.
Processing Methods
This section discusses two embodiments showing the use of the generalized waveforms in different objective functions. The objective functions can be minimized analytically for the angle of anisotropy. See, for example, U.S. Patent Publication 2014/0222346 (A1). A first embodiment uses the following objective function which is applicable in either the time or frequency domain,




  
    
      
        
          O
          ⁡
          
            (
            
              θ
              ,
              
                
                  ω
                  ST
                
                ⁢
                
                  
                
                ⁢
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                ⁢
                
                  
                
                ⁢
                
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                  ST
                
              
            
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              ∑
              
                
                  
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                ⁢
                
                  
                
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              ∑
              
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                ≠
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              {
              
                
                  
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                        )
                      
                      kl
                    
                    
                  
                  2
                
                +
                
                  
                    
                    
                      
                        (
                        
                          
                            T
                            θ
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                          ⁢
                          
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              }
            
          
        
      
    
    
      
        (
        23
        )
      
    
  




If the objective function is in the time domain, the summation is over a time window, T, starting at tST. If the objective function is in the frequency domain the summation is over a frequency band, B, starting at ωST. θ is the test angle and




  
    
      
        
          T
          θ
        
        =
        
          
            [
            
              
                
                  
                    c
                    θ
                  
                
                
                  
                    s
                    θ
                  
                
              
              
                
                  
                    -
                    
                      s
                      θ
                    
                  
                
                
                  
                    c
                    θ
                  
                
              
            
            ]
          
          .
        
      
    
    
      
        (
        24
        )
      
    
  




Each of the moduli in the objective function should vanish and the objective function will be minimized for θ=φ. The term j can be defined to absorb the azimuth indices such that j={mA,nA,mR,t or ω}≡{mA,nA,jR} and define the inner product,




  
    
      
        
          
            a
            ⇀
          
          ·
          
            b
            ⇀
          
        
        =
        
          
            ∑
            
              
                j
                R
              
              
                
                  m
                  A
                
                ≠
                
                  n
                  A
                
              
            
            
              B
              ⁢
              
                
              
              ⁢
              or
              ⁢
              
                
              
              ⁢
              T
            
          
          ⁢
          
            
              a
              j
            
            ⁢
            
              
                b
                j
              
              .
            
          
        
      
    
    
      
        (
        25
        )
      
    
  




See, for example, U.S. Patent Publication 2014/0222346 (A1). The solution for the minimizing angle is





  
    
      
        
          tan
          ⁡
          
            (
            
              4
              ⁢
              φ
            
            )
          
        
        =
        
          -
          
            
              
                Re
                [
                
                  
                    
                      
                        a
                        ⇀
                      
                      *
                    
                    ·
                    
                      d
                      ⇀
                    
                  
                  +
                  
                    
                      
                        b
                        ⇀
                      
                      *
                    
                    ·
                    
                      c
                      ⇀
                    
                  
                
                ]
              
              
                Re
                [
                
                  
                    
                      
                        a
                        ⇀
                      
                      *
                    
                    ·
                    
                      b
                      ⇀
                    
                  
                  -
                  
                    
                      
                        c
                        ⇀
                      
                      *
                    
                    ·
                    
                      d
                      ⇀
                    
                  
                
                ]
              
            
            .
          
        
      
    
    
      
        (
        26
        )
      
    
  




where

aj=X11mA,jR−X22mAjR,

bj=X11nA,jR−X22nAjR,

cj=X12mA,jR+X21mAjR,

dj=X12nA,jR+X21nAjR,   (27)

The asterisk is complex conjugation, which is only applicable in the frequency domain. If the objective function is defined in the time domain, the vectors are real. If φ is defined as the angle of the fast axis relative to the X-transmitter then φ→−φ in equation (10). φ is ambiguous by a multiple of π/4, (φ→φ+nπ/4) as shown below in FIG. 16. FIG. 16 is a representation of ambiguities with respect to azimuthal angle.
There is no concern about the it ambiguity, since it has no physical significance. The π/4 ambiguity corresponds to a minimum and a maximum of the objective function, where the minimum is wanted. This can be determined by substituting φ and φ+π/4 back into the objective function of equation (23) and selecting the angle that minimizes the objective function. This leaves the φ and φ+π/2 ambiguity corresponding to the fast and slow formation axes. This can be resolved by rotating the cross-dipole waveforms by φ into the fast and slow waveforms using equation (22) and comparing their slowness using either a time semblance algorithm or frequency semblance algorithm as appropriate. Semblance algorithms are known procedures.
A second embodiment is an extension of a cross-dipole algorithm described herein with respect to a single X,Y transmitter and two receiver rings indexed by m and n, where each receiver ring has two orthogonal acoustic receivers. First, a frequency domain objective function is given above as




  
    
      
        
          O
          ⁡
          
            (
            
              θ
              ,
              
                ω
                ST
              
            
            )
          
        
        =
        
          
            ∑
            
              
                
                  j
                  n
                
                ,
                
                  j
                  m
                
              
              
                
                  k
                  ,
                  l
                
                ⁢
                
                  
                
              
            
            
              B
              ⁡
              
                (
                
                  
                    ω
                    ST
                  
                  ,
                  W
                
                )
              
            
          
          ⁢
          
            
              
                
                
                  
                    (
                    
                      
                        
                          T
                          θ
                          T
                        
                        ⁢
                        
                          X
                          
                            j
                            n
                          
                        
                        ⁢
                        
                          T
                          θ
                        
                      
                      -
                      
                        
                          D
                          
                            j
                            m
                          
                        
                        ⁢
                        
                          Z
                          est
                          
                            m
                            -
                            n
                          
                        
                      
                    
                    )
                  
                  kl
                
                
              
              2
            
            .
          
        
      
    
    
      
        (
        28
        )
      
    
  




In equation (28), jn=(n, ω) and jm=(m,ω) where m and n are receiver indices, and the frequency summation is over a frequency band starting at ωST with width W. The Fourier transformed cross-dipole waveform is





  
    
      
        
          
            X
            
              j
              n
            
          
          =
          
            [
            
              
                
                  
                    
                      XX
                      n
                    
                    ⁡
                    
                      (
                      ω
                      )
                    
                  
                
                
                  
                    
                      XY
                      n
                    
                    ⁡
                    
                      (
                      ω
                      )
                    
                  
                
              
              
                
                  
                    
                      YX
                      n
                    
                    ⁡
                    
                      (
                      ω
                      )
                    
                  
                
                
                  
                    
                      YY
                      n
                    
                    ⁡
                    
                      (
                      ω
                      )
                    
                  
                
              
            
            ]
          
        
        ,
      
    
    
      
        (
        29
        )
      
    
  




where n is receiver index. The diagonal matrix Djn can be computed by diagonalizing the cross-dipole waveform of equation (29) with a unitary transformation. It is an estimate of the fast and slow waveforms in the frequency domain at receiver n,





  
    
      
        
          
            U
            ′
          
          ⁢
          
            X
            
              j
              n
            
          
          ⁢
          U
        
        =
        
          
            [
            
              
                
                  
                    D
                    11
                    
                      j
                      n
                    
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    D
                    22
                    
                      j
                      n
                    
                  
                
              
            
            ]
          
          .
        
      
    
    
      
        (
        30
        )
      
    
  




In equation (30), U′ is the complex conjugate transpose of U. Using methods described herein it can ensured that the D11 element is the fast wave and the D22 element is the slow wave. The diagonal matrix Zest is given by





  
    
      
        
          Z
          est
        
        =
        
          
            [
            
              
                
                  
                    z
                    1
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    z
                    2
                  
                
              
            
            ]
          
          .
        
      
    
    
      
        (
        31
        )
      
    
  




A method for calculating Zest has been described earlier herein. The diagonal matrix Zestm−n (Zest raised to the m−n power) propagates the matrix Djm at receiver m to the matrix Djn at receiver n. The notation can be simplified by defining j=(jm, jn), then




  
    
      
        
          
            O
            ⁡
            
              (
              
                θ
                ,
                
                  ω
                  ST
                
              
              )
            
          
          =
          
            
              ∑
              
                j
                ,
                k
                ,
                l
              
            
            ⁢
            
              
                
                
                  
                    (
                    
                      
                        
                          T
                          θ
                          T
                        
                        ⁢
                        
                          X
                          j
                        
                        ⁢
                        
                          T
                          θ
                        
                      
                      -
                      
                        Δ
                        j
                      
                    
                    )
                  
                  kl
                
                
              
              2
            
          
        
        ,
      
    
    
      
        (
        32
        )
      
    
  




where

Xj=Xjn,

Δj=DjmZestm−n.   (33)

The solution for the angle of anisotropy may be found as




  
    
      
        
          tan
          ⁡
          
            (
            
              2
              ⁢
              
                
              
              ⁢
              φ
            
            )
          
        
        =
        
          
            
              Re
              ⁡
              
                [
                
                  
                    
                      a
                      ⇀
                    
                    *
                  
                  ·
                  d
                
                ]
              
            
            
              Re
              ⁡
              
                [
                
                  
                    
                      b
                      ⇀
                    
                    *
                  
                  ·
                  
                    d
                    ⇀
                  
                
                ]
              
            
          
          .
        
      
    
    
      
        (
        34
        )
      
    
  




where vectors have been defined by

dj=Δ22j−Δ11j,aj=XYj+YXj,bj=YYj−XXj.   (35)

See, for example, U.S. Patent Publication 2014/0222346 (A1).

If φ is defined as the angle of the fast axis relative to the X-transmitter then φ→−φ in equation (34). φ is ambiguous by a multiple of π/2. The π/2 ambiguity corresponds to a minimum and a maximum of the objective function, where the minimum is wanted. This can be determined by substituting φ and φ+π/2 back into the objective function equation (32) and selecting the angle that minimizes the objective function. There is no fast-slow ambiguity in this example. The extension of the objective function to generalized waveforms is




  
    
      
        
          
            
              
                O
                ⁡
                
                  (
                  
                    θ
                    ,
                    
                      ω
                      ST
                    
                  
                  )
                
              
              =
              
              ⁢
              
                
                  ∑
                  
                    
                      
                        j
                        n
                      
                      ,
                      
                        j
                        m
                      
                    
                    
                      
                        
                          m
                          A
                        
                        ,
                        
                          n
                          A
                        
                      
                      
                        k
                        ,
                        l
                      
                    
                  
                  
                    B
                    ⁡
                    
                      (
                      
                        
                          ω
                          ST
                        
                        ,
                        W
                      
                      )
                    
                  
                
                ⁢
                
                  
                    
                    
                      
                        (
                        
                          
                            
                              T
                              θ
                              T
                            
                            ⁢
                            
                              X
                              
                                
                                  m
                                  
                                    A
                                    ,
                                  
                                
                                ⁢
                                
                                  j
                                  n
                                
                              
                            
                            ⁢
                            
                              T
                              θ
                            
                          
                          -
                          
                            
                              D
                              
                                
                                  n
                                  A
                                
                                ,
                                
                                  j
                                  m
                                
                              
                            
                            ⁢
                            
                              Z
                              
                                est
                                ,
                                
                                  n
                                  A
                                
                              
                              
                                m
                                -
                                n
                              
                            
                          
                        
                        )
                      
                      kl
                    
                    
                  
                  2
                
              
            
          
        
        
          
            
              
                =
                
                ⁢
                
                  
                    ∑
                    
                      j
                      ,
                      k
                      ,
                      l
                    
                  
                  ⁢
                  
                    
                      
                      
                        
                          (
                          
                            
                              
                                T
                                θ
                                T
                              
                              ⁢
                              
                                X
                                j
                              
                              ⁢
                              
                                T
                                θ
                              
                            
                            -
                            
                              Δ
                              j
                            
                          
                          )
                        
                        kl
                      
                      
                    
                    2
                  
                
              
              ,
            
          
        
      
    
    
      
        (
        36
        )
      
    
  




where





  
    
      
        
          
            X
            j
          
          =
          
            X
            
              
                m
                A
              
              ,
              
                j
                n
              
            
          
        
        ,
        
          

        
        ⁢
        
          
            Δ
            j
          
          =
          
            
              D
              
                
                  n
                  A
                
                ,
                
                  j
                  m
                
              
            
            ⁢
            
              
                Z
                
                  est
                  ,
                  
                    n
                    A
                  
                
                
                  m
                  -
                  n
                
              
              .
            
          
        
      
    
    
      
        (
        37
        )
      
    
  




The solution for the angle of anisotropy can be determined as





  
    
      
        
          
            tan
            ⁡
            
              (
              
                2
                ⁢
                
                  
                
                ⁢
                φ
              
              )
            
          
          =
          
            
              Re
              ⁡
              
                [
                
                  
                    
                      a
                      ⇀
                    
                    *
                  
                  ·
                  d
                
                ]
              
            
            
              Re
              ⁡
              
                [
                
                  
                    
                      b
                      ⇀
                    
                    *
                  
                  ·
                  
                    d
                    ⇀
                  
                
                ]
              
            
          
        
        ,
        
          

        
        ⁢
        where
      
    
    
      
        (
        23
        )
      
    
  
  
    
      
        
          
            d
            j
          
          =
          
            
              Δ
              22
              
                
                  n
                  A
                
                ,
                
                  j
                  m
                
              
            
            -
            
              Δ
              11
              
                
                  n
                  A
                
                ,
                
                  j
                  m
                
              
            
          
        
        ,
        
          
            a
            j
          
          =
          
            
              X
              12
              
                
                  m
                  A
                
                ,
                
                  j
                  n
                
              
            
            +
            
              X
              21
              
                
                  m
                  A
                
                ,
                
                  j
                  n
                
              
            
          
        
        ,
        
          

        
        ⁢
        
          
            b
            j
          
          =
          
            
              X
              22
              
                
                  m
                  A
                
                ,
                
                  j
                  n
                
              
            
            -
            
              
                X
                11
                
                  
                    m
                    A
                  
                  ,
                  
                    j
                    n
                  
                
              
              .
            
          
        
      
    
    
      
        (
        24
        )
      
    
  




Again, φ is ambiguous by a multiple of π/2, which can be handled in the same way as noted previously.

Embodiments of algorithmic approaches can be designed to take advantage of additional azimuthal receivers available in more advanced acoustic tools. The algorithm forms a complete set of 2×2 waveforms from the receiver data that all satisfy the Alford rotation. Using the complete set of waveforms improves the accuracy and stability of the algorithm. The new algorithm also can solve for the angle of anisotropy analytically.
Cross-dipole measurements are routinely made in common well logging practice. One important application for cross-dipole measurements is fracture detection and analysis in open and cased boreholes. The cased-hole application is useful for evaluating hydraulic fracture stimulation. Another important application is identifying stress-induced anisotropy for use in estimating formation stress orientation and magnitude. This information is useful for optimizing the hydraulic fracture process. Better accuracy and stability in detecting anisotropy as taught herein may provide operational advantages.
Current HTI anisotropy detection algorithms based on the traditional crossed-dipole acoustic tool use a single crossed-dipole waveform to determine the angle of anisotropy. Earlier attempts to solve the problem suggest using numerical search methods such as very fast simulated annealing, gradient search, or brute force to minimize the objective function. Embodiments as taught herein may use a complete set of waveforms recorded at multiple azimuth positions and may solve for the angle of anisotropy analytically.
FIG. 17 is a flow diagram of a processor-implemented method 1700 of estimating an azimuth angle. The processor-implemented method can be executed by one or more processors. At 1710, a plurality of acoustic waveforms corresponding to acoustic waves that are received at a number of receiver rings are recorded. Each receiver ring can be disposed along a length of a longitudinal axis of a tool at a different distance from the other receiver rings along the length of the longitudinal axis, where the tool can be operatively surrounded by a geological formation. Each receiver ring can have a plurality of acoustic dipole receivers azimuthally disposed around the tool and can operatively receive acoustic waves, the acoustic waves being generated by orthogonal dipole transmitters in a transmitter ring attached to the tool.
At 1720, an objective function is generated with respect to an azimuth angle based on ordered sets of the plurality of acoustic waveforms received at each receiver ring. The azimuth angle may correspond to an orientation of the orthogonal dipole transmitters and the acoustic dipole receivers relative to fast and slow principal flexural wave axes. The processor-implemented method 1700 or a similar method can include forming the ordered sets of the plurality of acoustic waveforms from a receiver ring of the number of receiver rings having N dipole receivers providing N(N−1)/2 independent 2×2 waveforms, N being an integer greater than two.
The processor-implemented method 1700 or a similar method can include generating the objective function to include extending a first objective function to include generalized waveforms correlated to the plurality of acoustic dipole receivers azimuthally disposed around the tool in each ring, the first objective function generated with respect to in-line and cross-line waveforms from transmitter rings to receiver rings, each transmitter ring having only two dipole transmitters orthogonal to each other and each receiver ring having only two dipole receivers orthogonal to each other. The processor-implemented method 1700 or a similar method can include diagonalizing independent 2×2 waveforms, each independent 2×2 waveform being in-line and cross-line components according to two azimuth indices, the two azimuth indices not equal to each other, each azimuth index correlated to a different one of the acoustic dipole receivers of one of the receiver rings.
At 1730, a global minimum of the objective function with respect to the azimuth angle is estimated. The processor-implemented method 1700 or a similar method can include estimating the global minimum of the objective function to be accomplished analytically or by using a numerical search algorithm.
At 1740, an anisotropy angle of the geological formation is determined based on the global minimum. The processor-implemented method 1700 or a similar method can include determining the anisotropy angle of the geological formation to include implementing the generating, the estimating, and the determining for a data point space defined over multiple frequency bands having different starting frequencies or by multiple different time periods having different starting times.
The processor-implemented method 1700 or a similar method can include removing existing ambiguities associated with the fast and slow principal flexural wave axes or ambiguities associated with minima and maxima of the objective function. Removing existing objective function minimum/maximum ambiguity includes substituting the azimuth angle into the objective function and substituting an ambiguous angle (typically azimuth angle+π/2 or azimuth angle+π/4) into the objective function, and selecting the angle that minimizes the objective function. Removing existing fast slow principal axis ambiguity can include either constraining auxiliary parameters a priori to eliminate the ambiguity before minimizing the objective function; or eigenvalue alignment a priori to eliminate the ambiguity before minimizing the objective function; or rotating cross-dipole waveforms of the ordered sets of the plurality of acoustic waveforms by min/max ambiguity resolved azimuth angle into the fast and slow waveforms and comparing slowness of the cross-dipole waveforms using either a time semblance algorithm or frequency semblance algorithm.
In various embodiments, a non-transitory machine-readable storage device can comprise instructions stored thereon, which, when performed by a machine, cause the machine to perform operations, the operations comprising one or more features similar to or identical to features of methods and techniques described herein. The physical structures of such instructions may be operated on by one or more processors. Executing these physical structures can cause the machine to perform operations comprising recording a plurality of acoustic waveforms corresponding to acoustic waves received at a number of receiver rings, each receiver ring disposed along a length of a longitudinal axis of a tool at a different distance from the other receiver rings along the length of the longitudinal axis, the tool surrounded by a geological formation, each receiver ring having a plurality of acoustic dipole receivers azimuthally disposed around the tool and receiving acoustic waves, the acoustic waves being generated by orthogonal dipole transmitters in a transmitter ring attached to the tool; generating an objective function, based on ordered sets of the plurality of acoustic waveforms received at each receiver ring, with respect to an azimuth angle, wherein the azimuth angle corresponds to an orientation of the orthogonal dipole transmitters and the acoustic dipole receivers relative to fast and slow principal flexural wave axes; estimating a global minimum of the objective function with respect to the azimuth angle; and determining an anisotropy angle of the geological formation based on the global minimum. The instructions can include instructions to operate a tool or tools having sensors disposed downhole in a borehole to provide data to a processing unit in accordance with the teachings herein.
Instructions generating the objective function can include generating the objective function with respect to the azimuth angle and a set of auxiliary parameters. The auxiliary parameters may include fast and slow slowness values. Including fast and slow slowness values as auxiliary parameters may depend on the form of the objective function. Instructions to generate the objective function can include extending a first objective function to include generalized waveforms correlated to the plurality of acoustic dipole receivers azimuthally disposed around the tool in each ring, the first objective function generated with respect to in-line and cross-line waveforms from transmitter rings to receiver rings, each transmitter ring having only two dipole transmitters orthogonal to each other and each receiver ring having only two dipole receivers orthogonal to each other.
The instructions can include instructions to form the ordered sets of the plurality of acoustic waveforms from a receiver ring of the number of receiver rings having N dipole receivers providing N(N−1)/2 independent 2×2 waveforms, N being an integer greater than two. The instructions can include instructions to diagonalize independent 2×2 waveforms, each independent 2×2 waveform being in-line and cross-line components according to two azimuth indices, the two azimuth indices not equal to each other, each azimuth index correlated to a different one of the acoustic dipole receivers of one of the receiver rings.
Further, a machine-readable storage device, herein, is a physical device that stores data represented by physical structure within the device. Such a physical device is a non-transitory device. Examples of machine-readable storage devices can include, but are not limited to, read only memory (ROM), random access memory (RAM), a magnetic disk storage device, an optical storage device, a flash memory, and other electronic, magnetic, and/or optical memory devices.
In various embodiments, an apparatus can comprise: a tool having a longitudinal axis; a transmitter ring attached to the tool, the transmitter ring having orthogonal dipole transmitters; a number of receiver rings, each receiver ring disposed along a length of the longitudinal axis at a different distance from the other receiver rings along the length, each receiver ring having a plurality of acoustic dipole receivers azimuthally disposed around the tool; recording logic to record a plurality of acoustic waveforms corresponding to acoustic waves received at the acoustic dipole receivers when the tool is operated in a geological formation, the acoustic waves being generated by the orthogonal dipole transmitters; and a signal processor to generate an objective function, based on ordered sets of the plurality of acoustic waveforms received at each receiver ring, with respect to an azimuth angle, wherein the azimuth angle corresponds to an orientation of the orthogonal dipole transmitters and the acoustic dipole receivers relative to fast and slow principal flexural wave axes, with the signal processor structured to estimate a global minimum of the objective function with respect to the azimuth angle and to determine an anisotropy angle of the geological formation based on the global minimum. The tool, transmitter ring, orthogonal dipole transmitters, receiver rings, and plurality of acoustic dipole receivers may be realized by a structure similar to or identical to tool 1410 and its attached components as shown in FIG. 14. Such a tool can be used in a system architecture such as represented in FIG. 11 replacing tool 100 with corresponding changes in the other components such as the recording logic and signal processor to operate with a structure similar to or identical to tool 1410 as taught herein. The tool can comprise one of a wireline tool or a measurement while drilling tool. The tool may be realized as a wireline tool operable in a wireline system similar to or identical to the wireline system of FIG. 6. The tool may be realized as a measurement while drilling tool operable in a measurement while drilling system similar to or identical to the measurement while drilling system of FIG. 5.
The signal processor to generate an objective function, based on ordered sets of the plurality of acoustic waveforms received at the plurality of acoustic dipole receivers of each receiver ring, can be structured to generate the objective function with respect to the azimuth angle and a set of auxiliary parameters. The auxiliary parameters can include fast and slow slowness values. Including fast and slow slowness values as auxiliary parameters may depend on the form of the objective function.
In various embodiments, a receiver ring of the number of receiver rings can have N dipole receivers and can be operable to provide N(N−1)/2 independent 2×2 waveforms, N being an integer greater than two. The signal processor can be structured to diagonalize independent 2×2 waveforms, each independent 2×2 waveform being in-line and cross-line components according to two azimuth indices, the two azimuth indices not equal to each other, each azimuth index correlated to a different one of the acoustic dipole receivers of one of the receiver rings. The signal processor can be structured to remove existing ambiguities associated with the fast and slow principal flexural wave axes and/or objective function minima and maxima. The signal processor and other components may be structured to execute various functions as taught herein. Features of any appropriate permutations of apparatus/systems, processes, or other combinations of features, as taught herein, may be combined into a apparatus/system or process according to the teachings herein.
HTI Derivations
The following is a description of HTI derivations. The description is presented in a lecture format. FIG. 15 is a schematic representation of crossed dipole transmitters oriented at angle φ relative to the fast axis of an anisotropic formation with arbitrarily oriented dipole receivers. If we fire the X-dipole the waveform S can be decomposed into two components directed along the formation axes.

Xf=STfcφ≡Sfcφ

Xs=STssφ≡Sssφ,   (H1)

where Tf and Ts are the fast and slow formation transfer functions. Similarly firing the Y-dipole gives

Yf=−STfsφ≡−Sfsφ

Ys=STscφ≡Sscφ.   (H2)

When the transmitted pulse is received in dipole receiver n the fast and slow waveforms are combined as

Xn=Xfcφn+Xssφn=Sfcφcφn+Sssφsφn 

Yn=Yfcφn+Yssφn=−Sfsφcφn+Sscφsφn.   (H3)

where φn=φ+θn, and θn is the angle of the receiver relative to the X transmitter. We defined φ as the angle of the X transmitter relative to the fast axis. If we define φ as the angle of the fast axis relative to the X transmitter then φ→−φ in equations H1-H3. We wish to derive convenient Alford rotation equations for arbitrary receiver pairs (n1, n2≠n1). Rewrite equation (H3) as





  
    
      
        
          
            
              
                X
                
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
              
              =
              
              ⁢
              
                
                  
                    S
                    f
                  
                  ⁢
                  
                    c
                    φ
                  
                  ⁢
                  
                    c
                    
                      
                        φ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                      +
                      
                        (
                        
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                          -
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                        )
                      
                    
                  
                
                +
                
                  
                    S
                    s
                  
                  ⁢
                  
                    s
                    φ
                  
                  ⁢
                  
                    s
                    
                      
                        φ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                      +
                      
                        (
                        
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                          -
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                        )
                      
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    S
                    f
                  
                  ⁢
                  
                    c
                    φ
                  
                  ⁢
                  
                    c
                    
                      φ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                  ⁢
                  
                    c
                    
                      (
                      
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                        -
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                      )
                    
                  
                
                -
                
                  
                    S
                    f
                  
                  ⁢
                  
                    c
                    φ
                  
                  ⁢
                  
                    s
                    
                      φ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                  ⁢
                  
                    s
                    
                      
                        (
                        
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                          -
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                        )
                      
                      ⁢
                      
                        
                      
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    S
                    s
                  
                  ⁢
                  
                    s
                    φ
                  
                  ⁢
                  
                    s
                    
                      φ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                  ⁢
                  
                    c
                    
                      (
                      
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                        -
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                      )
                    
                  
                
                +
                
                  
                    S
                    s
                  
                  ⁢
                  
                    s
                    φ
                  
                  ⁢
                  
                    c
                    
                      φ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                  ⁢
                  
                    s
                    
                      (
                      
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                        -
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                      )
                    
                  
                
              
            
          
        
        
          
            
              
                =
                
                ⁢
                
                  
                    
                      X
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                    ⁢
                    
                      c
                      
                        (
                        
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                          -
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                        )
                      
                    
                  
                  +
                  
                    
                      (
                      
                        
                          
                            S
                            s
                          
                          ⁢
                          
                            s
                            φ
                          
                          ⁢
                          
                            c
                            
                              φ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              2
                            
                          
                        
                        -
                        
                          
                            S
                            f
                          
                          ⁢
                          
                            c
                            φ
                          
                          ⁢
                          
                            s
                            
                              φ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              2
                            
                          
                        
                      
                      )
                    
                    ⁢
                    
                      s
                      
                        (
                        
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                          -
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                        )
                      
                    
                  
                
              
              ,
              and
            
          
        
      
    
    
      
        (
        H4
        )
      
    
  
  
    
      
        
          
            
              
                Y
                
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
              
              =
              
              ⁢
              
                
                  
                    -
                    
                      S
                      f
                    
                  
                  ⁢
                  
                    s
                    φ
                  
                  ⁢
                  
                    c
                    
                      
                        φ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                      +
                      
                        (
                        
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                          -
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                        )
                      
                    
                  
                
                +
                
                  
                    S
                    s
                  
                  ⁢
                  
                    c
                    φ
                  
                  ⁢
                  
                    s
                    
                      
                        φ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                      +
                      
                        (
                        
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                          -
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                        )
                      
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    -
                    
                      S
                      f
                    
                  
                  ⁢
                  
                    s
                    φ
                  
                  ⁢
                  
                    c
                    
                      φ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                  ⁢
                  
                    c
                    
                      (
                      
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                        -
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                      )
                    
                  
                
                +
                
                  
                    S
                    f
                  
                  ⁢
                  
                    s
                    φ
                  
                  ⁢
                  
                    s
                    
                      φ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                  ⁢
                  
                    s
                    
                      (
                      
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                        -
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  
                    S
                    s
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    c
                    φ
                  
                  ⁢
                  
                    s
                    
                      φ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                  ⁢
                  
                    c
                    
                      (
                      
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                        -
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                      )
                    
                  
                
                +
                
                  
                    S
                    s
                  
                  ⁢
                  
                    c
                    φ
                  
                  ⁢
                  
                    c
                    
                      φ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                  ⁢
                  
                    s
                    
                      (
                      
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                        -
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                      )
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    Y
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                  ⁢
                  
                    c
                    
                      (
                      
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                        -
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                      )
                    
                  
                
                +
                
                  
                    (
                    
                      
                        
                          S
                          s
                        
                        ⁢
                        
                          c
                          φ
                        
                        ⁢
                        
                          c
                          
                            φ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                      
                      +
                      
                        
                          S
                          f
                        
                        ⁢
                        
                          s
                          φ
                        
                        ⁢
                        
                          s
                          
                            φ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    
                      s
                      
                        (
                        
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                          -
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                        )
                      
                    
                    .
                  
                
              
            
          
        
      
    
    
      
        (
        H5
        )
      
    
  




Combine equations (H4) and (H5) to get

sφXn1+cφYn1=(sφXn2+cφYn2)c(θn1−θn2)+Sscφn2s(θn1−θn2),   (H6)

and

cφXn1−sφYn1=(cφXn2−sφYn2)c(θn1−θn2)−Sfsφn2s(θn1−θn2).   (H7)

Equations (H6) and (H7) can be combined into a matrix equation,





  
    
      
        
          
            [
            
              
                
                  
                    c
                    φ
                  
                
                
                  
                    -
                    
                      s
                      φ
                    
                  
                
              
              
                
                  
                    s
                    φ
                  
                
                
                  
                    c
                    φ
                  
                
              
            
            ]
          
          ⁡
          
            [
            
              
                
                  
                    X
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                
              
              
                
                  
                    Y
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                
              
            
            ]
          
        
        =
        
          
            
              
                c
                
                  (
                  
                    
                      θ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                    -
                    
                      θ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                  )
                
              
              ⁡
              
                [
                
                  
                    
                      
                        c
                        φ
                      
                    
                    
                      
                        -
                        
                          s
                          φ
                        
                      
                    
                  
                  
                    
                      
                        s
                        φ
                      
                    
                    
                      
                        c
                        φ
                      
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      X
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                
                
                  
                    
                      Y
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                
              
              ]
            
          
          +
          
            
              
                
                  s
                  
                    (
                    
                      
                        θ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                      -
                      
                        θ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                    )
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          -
                          
                            s
                            
                              φ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              2
                            
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          c
                          
                            φ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        S
                        f
                      
                    
                  
                  
                    
                      
                        S
                        s
                      
                    
                  
                
                ]
              
            
            .
          
        
      
    
    
      
        (
        H8
        )
      
    
  




Permute the indices (n1, n2) in equation (H8) to get a second matrix equation,




  
    
      
        
          
            [
            
              
                
                  
                    c
                    φ
                  
                
                
                  
                    -
                    
                      s
                      φ
                    
                  
                
              
              
                
                  
                    s
                    φ
                  
                
                
                  
                    c
                    φ
                  
                
              
            
            ]
          
          ⁡
          
            [
            
              
                
                  
                    X
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                
              
              
                
                  
                    Y
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                
              
            
            ]
          
        
        =
        
          
            
              
                c
                
                  (
                  
                    
                      θ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                    -
                    
                      θ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                  )
                
              
              ⁡
              
                [
                
                  
                    
                      
                        c
                        φ
                      
                    
                    
                      
                        -
                        
                          s
                          φ
                        
                      
                    
                  
                  
                    
                      
                        s
                        φ
                      
                    
                    
                      
                        c
                        φ
                      
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      X
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
                
                  
                    
                      Y
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
              
              ]
            
          
          -
          
            
              
                
                  s
                  
                    (
                    
                      
                        θ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                      -
                      
                        θ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                    )
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          -
                          
                            s
                            
                              φ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              1
                            
                          
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          c
                          
                            φ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        S
                        f
                      
                    
                  
                  
                    
                      
                        S
                        s
                      
                    
                  
                
                ]
              
            
            .
          
        
      
    
    
      
        (
        H9
        )
      
    
  




Combine equations (H8) and (H9) to get





  
    
      
        
          
            [
            
              
                
                  
                    c
                    φ
                  
                
                
                  
                    -
                    
                      s
                      φ
                    
                  
                
              
              
                
                  
                    s
                    φ
                  
                
                
                  
                    c
                    φ
                  
                
              
            
            ]
          
          ⁡
          
            [
            
              
                
                  
                    X
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                
                
                  
                    X
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                
              
              
                
                  
                    Y
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                
                
                  
                    Y
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                
              
            
            ]
          
        
        =
        
          
            
              
                c
                
                  (
                  
                    
                      θ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                    -
                    
                      θ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                  )
                
              
              ⁡
              
                [
                
                  
                    
                      
                        c
                        φ
                      
                    
                    
                      
                        -
                        
                          s
                          φ
                        
                      
                    
                  
                  
                    
                      
                        s
                        φ
                      
                    
                    
                      
                        c
                        φ
                      
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      X
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                  
                    
                      X
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
                
                  
                    
                      Y
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                  
                    
                      Y
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
              
              ]
            
          
          +
          
            
              s
              
                (
                
                  
                    θ
                    ⁢
                    
                      
                    
                    ⁢
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    1
                  
                  -
                  
                    θ
                    ⁢
                    
                      
                    
                    ⁢
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    2
                  
                
                )
              
            
            ⁡
            
              [
              
                
                  
                    
                      
                        -
                        
                          s
                          
                            φ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                      
                      ⁢
                      
                        S
                        f
                      
                    
                  
                  
                    
                      
                        s
                        
                          φ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                      ⁢
                      
                        S
                        f
                      
                    
                  
                
                
                  
                    
                      
                        c
                        
                          φ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                      ⁢
                      
                        S
                        s
                      
                    
                  
                  
                    
                      
                        -
                        
                          c
                          
                            φ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                      
                      ⁢
                      
                        S
                        s
                      
                    
                  
                
              
              ]
            
          
        
      
    
    
      
        (
        H10
        )
      
    
  




Substitute the identity




  
    
      
        
          
            [
            
              
                
                  
                    S
                    f
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    S
                    s
                  
                
              
            
            ]
          
          ⁡
          
            [
            
              
                
                  
                    -
                    
                      s
                      
                        φ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                
                
                  
                    s
                    
                      φ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                
              
              
                
                  
                    c
                    
                      φ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                
                
                  
                    -
                    
                      c
                      
                        φ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
              
            
            ]
          
        
        =
        
          [
          
            
              
                
                  
                    -
                    
                      s
                      
                        φ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                  ⁢
                  
                    S
                    f
                  
                
              
              
                
                  
                    s
                    
                      φ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                  ⁢
                  
                    S
                    f
                  
                
              
            
            
              
                
                  
                    c
                    
                      φ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                  ⁢
                  
                    S
                    s
                  
                
              
              
                
                  
                    -
                    
                      c
                      
                        φ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                  ⁢
                  
                    S
                    s
                  
                
              
            
          
          ]
        
      
    
    
      
        (
        H11
        )
      
    
  




into equation (H10),





  
    
      
        
          
            [
            
              
                
                  
                    c
                    φ
                  
                
                
                  
                    -
                    
                      s
                      φ
                    
                  
                
              
              
                
                  
                    s
                    φ
                  
                
                
                  
                    c
                    φ
                  
                
              
            
            ]
          
          ⁡
          
            [
            
              
                
                  
                    X
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                
                
                  
                    X
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                
              
              
                
                  
                    Y
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                
                
                  
                    Y
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                
              
            
            ]
          
        
        =
        
          
            
              
                c
                
                  (
                  
                    
                      θ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                    -
                    
                      θ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                  )
                
              
              ⁡
              
                [
                
                  
                    
                      
                        c
                        φ
                      
                    
                    
                      
                        -
                        
                          s
                          φ
                        
                      
                    
                  
                  
                    
                      
                        s
                        φ
                      
                    
                    
                      
                        c
                        φ
                      
                    
                  
                
                ]
              
            
            ⁡
            
              [
              
                
                  
                    
                      X
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                  
                    
                      X
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
                
                  
                    
                      Y
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                  
                    
                      Y
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
              
              ]
            
          
          +
          
            
              
                
                  s
                  
                    (
                    
                      
                        θ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                      -
                      
                        θ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                    )
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          S
                          f
                        
                      
                      
                        0
                      
                    
                    
                      
                        0
                      
                      
                        
                          S
                          s
                        
                      
                    
                  
                  ]
                
              
              ⁡
              
                [
                
                  
                    
                      
                        -
                        
                          s
                          
                            φ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                      
                    
                    
                      
                        s
                        
                          φ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                    
                  
                  
                    
                      
                        c
                        
                          φ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                    
                    
                      
                        -
                        
                          c
                          
                            φ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                      
                    
                  
                
                ]
              
            
            .
          
        
      
    
    
      
        (
        H12
        )
      
    
  




Define the orthonormal transformation




  
    
      
        
          
            T
            φ
          
          =
          
            [
            
              
                
                  
                    c
                    φ
                  
                
                
                  
                    s
                    φ
                  
                
              
              
                
                  
                    -
                    
                      s
                      φ
                    
                  
                
                
                  
                    c
                    φ
                  
                
              
            
            ]
          
        
        ,
      
    
    
      
        (
        H13
        )
      
    
  




where TφT=Tφ−1 and solve for the fast and slow waves,





  
    
      
        
          
            
              T
              φ
              T
            
            ⁢
            
              {
              
                
                  [
                  
                    
                      
                        
                          X
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                      
                      
                        
                          X
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                      
                    
                    
                      
                        
                          Y
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                      
                      
                        
                          Y
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                      
                    
                  
                  ]
                
                -
                
                  
                    c
                    
                      (
                      
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                        -
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                      )
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            X
                            
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              2
                            
                          
                        
                        
                          
                            X
                            
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              1
                            
                          
                        
                      
                      
                        
                          
                            Y
                            
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              2
                            
                          
                        
                        
                          
                            Y
                            
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              1
                            
                          
                        
                      
                    
                    ]
                  
                
              
              }
            
            ⁢
            
              
                R
                
                  -
                  1
                
              
              
                s
                
                  (
                  
                    
                      θ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                    -
                    
                      θ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                  )
                
              
            
          
          =
          
            [
            
              
                
                  
                    S
                    f
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    S
                    s
                  
                
              
            
            ]
          
        
        ,
      
    
    
      
        (
        H14
        )
      
    
  
  
    
      
        
          
        
        ⁢
        where
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          R
          ≡
          
            
              [
              
                
                  
                    
                      -
                      
                        s
                        
                          φ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                    
                  
                  
                    
                      s
                      
                        φ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
                
                  
                    
                      c
                      
                        φ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                  
                    
                      -
                      
                        c
                        
                          φ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                    
                  
                
              
              ]
            
            .
          
        
      
    
    
      
        (
        H15
        )
      
    
  




Equation (H14) can be further manipulated to look like an Alford rotation. The inverse of R is




  
    
      
        
          
            
              
                R
                
                  -
                  1
                
              
              ≡
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          -
                          
                            c
                            
                              φ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              1
                            
                          
                        
                      
                      
                        
                          -
                          
                            s
                            
                              φ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              1
                            
                          
                        
                      
                    
                    
                      
                        
                          -
                          
                            c
                            
                              φ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              2
                            
                          
                        
                      
                      
                        
                          -
                          
                            s
                            
                              φ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              2
                            
                          
                        
                      
                    
                  
                  ]
                
                
                  (
                  
                    
                      
                        s
                        
                          φ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                      ⁢
                      
                        c
                        
                          φ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                    
                    -
                    
                      
                        s
                        
                          φ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                      ⁢
                      
                        c
                        
                          φ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                    
                  
                  )
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                -
                
                  
                    [
                    
                      
                        
                          
                            c
                            
                              φ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              1
                            
                          
                        
                        
                          
                            s
                            
                              φ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              1
                            
                          
                        
                      
                      
                        
                          
                            c
                            
                              φ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              2
                            
                          
                        
                        
                          
                            s
                            
                              φ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              2
                            
                          
                        
                      
                    
                    ]
                  
                  
                    s
                    
                      (
                      
                        
                          φ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                        -
                        
                          φ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                      )
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    [
                    
                      
                        
                          
                            c
                            
                              φ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              1
                            
                          
                        
                        
                          
                            s
                            
                              φ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              1
                            
                          
                        
                      
                      
                        
                          
                            c
                            
                              φ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              2
                            
                          
                        
                        
                          
                            s
                            
                              φ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              2
                            
                          
                        
                      
                    
                    ]
                  
                  
                    s
                    
                      (
                      
                        
                          φ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                        -
                        
                          φ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                      )
                    
                  
                
                .
              
            
          
        
      
    
    
      
        (
        H16
        )
      
    
  




The matrix in the numerator on the right hand side of equation (H16) can be factored as





  
    
      
        
          
            
              
                [
                
                  
                    
                      
                        c
                        
                          φ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                    
                    
                      
                        s
                        
                          φ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                    
                  
                  
                    
                      
                        c
                        
                          φ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                    
                    
                      
                        s
                        
                          φ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                    
                  
                
                ]
              
              =
              
              ⁢
              
                [
                
                  
                    
                      
                        (
                        
                          
                            
                              c
                              φ
                            
                            ⁢
                            
                              c
                              
                                θ
                                ⁢
                                
                                  
                                
                                ⁢
                                n
                                ⁢
                                
                                  
                                
                                ⁢
                                1
                              
                            
                          
                          -
                          
                            
                              s
                              φ
                            
                            ⁢
                            
                              s
                              
                                θ
                                ⁢
                                
                                  
                                
                                ⁢
                                n
                                ⁢
                                
                                  
                                
                                ⁢
                                1
                              
                            
                          
                        
                        )
                      
                    
                    
                      
                        (
                        
                          
                            
                              s
                              φ
                            
                            ⁢
                            
                              c
                              
                                θ
                                ⁢
                                
                                  
                                
                                ⁢
                                n
                                ⁢
                                
                                  
                                
                                ⁢
                                1
                              
                            
                          
                          +
                          
                            
                              c
                              φ
                            
                            ⁢
                            
                              s
                              
                                θ
                                ⁢
                                
                                  
                                
                                ⁢
                                n
                                ⁢
                                
                                  
                                
                                ⁢
                                1
                              
                            
                          
                        
                        )
                      
                    
                  
                  
                    
                      
                        (
                        
                          
                            
                              c
                              φ
                            
                            ⁢
                            
                              c
                              
                                θ
                                ⁢
                                
                                  
                                
                                ⁢
                                n
                                ⁢
                                
                                  
                                
                                ⁢
                                2
                              
                            
                          
                          -
                          
                            
                              s
                              φ
                            
                            ⁢
                            
                              s
                              
                                θ
                                ⁢
                                
                                  
                                
                                ⁢
                                n
                                ⁢
                                
                                  
                                
                                ⁢
                                2
                              
                            
                          
                        
                        )
                      
                    
                    
                      
                        (
                        
                          
                            
                              s
                              φ
                            
                            ⁢
                            
                              c
                              
                                θ
                                ⁢
                                
                                  
                                
                                ⁢
                                n
                                ⁢
                                
                                  
                                
                                ⁢
                                2
                              
                            
                          
                          +
                          
                            
                              c
                              φ
                            
                            ⁢
                            
                              s
                              
                                θ
                                ⁢
                                
                                  
                                
                                ⁢
                                n
                                ⁢
                                
                                  
                                
                                ⁢
                                2
                              
                            
                          
                        
                        )
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    [
                    
                      
                        
                          
                            c
                            
                              θ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              1
                            
                          
                        
                        
                          
                            s
                            
                              θ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              1
                            
                          
                        
                      
                      
                        
                          
                            c
                            
                              θ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              2
                            
                          
                        
                        
                          
                            s
                            
                              θ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              2
                            
                          
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            c
                            
                              φ
                              ⁢
                              
                                
                              
                            
                          
                        
                        
                          
                            s
                            
                              φ
                              ⁢
                              
                                
                              
                            
                          
                        
                      
                      
                        
                          
                            -
                            
                              s
                              
                                φ
                                ⁢
                                
                                  
                                
                              
                            
                          
                        
                        
                          
                            c
                            
                              φ
                              ⁢
                              
                                
                              
                            
                          
                        
                      
                    
                    ]
                  
                
                .
              
            
          
        
      
    
    
      
        (
        H17
        )
      
    
  




Substituting equation (H16) and (H17) into (H14) gives





  
    
      
        
          
            
              T
              φ
              T
            
            ⁢
            
              {
              
                
                  [
                  
                    
                      
                        
                          X
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                      
                      
                        
                          X
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                      
                    
                    
                      
                        
                          Y
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            1
                          
                        
                      
                      
                        
                          Y
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            2
                          
                        
                      
                    
                  
                  ]
                
                -
                
                  
                    c
                    
                      (
                      
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                        -
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                      )
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            X
                            
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              2
                            
                          
                        
                        
                          
                            X
                            
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              1
                            
                          
                        
                      
                      
                        
                          
                            Y
                            
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              2
                            
                          
                        
                        
                          
                            Y
                            
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              1
                            
                          
                        
                      
                    
                    ]
                  
                
              
              }
            
            ⁢
            
              T
              12
            
            ⁢
            
              T
              φ
            
          
          =
          
            [
            
              
                
                  
                    S
                    f
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    S
                    s
                  
                
              
            
            ]
          
        
        ,
      
    
    
      
        (
        H18
        )
      
    
  
  
    
      
        
          where
          ⁢
          
            
          
          ⁢
          
            T
            12
          
        
        =
        
          
            
              [
              
                
                  
                    
                      c
                      
                        θ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                  
                    
                      s
                      
                        θ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        1
                      
                    
                  
                
                
                  
                    
                      c
                      
                        θ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                  
                    
                      s
                      
                        θ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        2
                      
                    
                  
                
              
              ]
            
            
              s
              
                (
                
                  
                    θ
                    ⁢
                    
                      
                    
                    ⁢
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    1
                  
                  -
                  
                    θ
                    ⁢
                    
                      
                    
                    ⁢
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    2
                  
                
                )
              
              2
            
          
          .
        
      
    
    
      
        (
        H19
        )
      
    
  




Finally substitute




  
    
      
        
          
            [
            
              
                
                  
                    X
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                
                
                  
                    X
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                
              
              
                
                  
                    Y
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      2
                    
                  
                
                
                  
                    Y
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      1
                    
                  
                
              
            
            ]
          
          =
          
            
              
                [
                
                  
                    
                      
                        X
                        
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                    
                    
                      
                        X
                        
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                    
                  
                  
                    
                      
                        Y
                        
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                    
                    
                      
                        Y
                        
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                    
                  
                
                ]
              
              ⁡
              
                [
                
                  
                    
                      0
                    
                    
                      1
                    
                  
                  
                    
                      1
                    
                    
                      0
                    
                  
                
                ]
              
            
            ≡
            
              
                [
                
                  
                    
                      
                        X
                        
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                    
                    
                      
                        X
                        
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          2
                        
                      
                    
                  
                  
                    
                      
                        Y
                        
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          1
                        
                      
                    
                    
                      
                        Y
                        n2
                      
                    
                  
                
                ]
              
              ⁢
              P
            
          
        
        ,
      
    
    
      
        (
        H20
        )
      
    
  




into equation (H18) to get the final form for the Alford rotation for any pair of receivers (n, m≠n),





  
    
      
        
          
            
              T
              φ
              T
            
            ⁢
            
              X
              
                n
                ⁢
                
                  
                
                ⁢
                m
              
            
            ⁢
            
              T
              φ
            
          
          =
          
            [
            
              
                
                  
                    S
                    f
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    S
                    s
                  
                
              
            
            ]
          
        
        ,
        where
      
    
    
      
        (
        H21
        )
      
    
  
  
    
      
        
          
            X
            
              n
              ⁢
              
                
              
              ⁢
              m
            
          
          ≡
          
            
              [
              
                
                  
                    
                      X
                      n
                    
                  
                  
                    
                      X
                      m
                    
                  
                
                
                  
                    
                      Y
                      n
                    
                  
                  
                    
                      Y
                      m
                    
                  
                
              
              ]
            
            ⁢
            
              {
              
                I
                -
                
                  
                    c
                    
                      (
                      
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          n
                        
                        -
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          m
                        
                      
                      )
                    
                  
                  ⁢
                  P
                
              
              }
            
            ⁢
            
              T
              
                n
                ⁢
                
                  
                
                ⁢
                m
              
            
          
        
        ,
        
          

        
        ⁢
        
          
            T
            
              n
              ⁢
              
                
              
              ⁢
              m
            
          
          =
          
            
              
                [
                
                  
                    
                      
                        c
                        
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                          
                          ⁢
                          
                            
                          
                        
                      
                    
                    
                      
                        s
                        
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                          
                          ⁢
                          
                            
                          
                        
                      
                    
                  
                  
                    
                      
                        c
                        
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                          
                          ⁢
                          
                            
                          
                        
                      
                    
                    
                      
                        s
                        
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                          
                          ⁢
                          
                            
                          
                        
                      
                    
                  
                
                ]
              
              
                s
                
                  (
                  
                    
                      θ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                    
                    ⁢
                    
                      
                    
                    -
                    
                      θ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                    
                  
                  ⁢
                  
                    
                  
                  )
                
                2
              
            
            .
          
        
      
    
    
      
        (
        H22
        )
      
    
  




We have changed the notation to superscripts for the matrix variables to avoid confusion with the matrix element indices. Note that equation (H22) is symmetric with respect to permutation of m and n. This is easily shown using the identity P2=I and the fact that multiplying by P on the left switches rows instead of columns,




  
    
      
        
          
            
              
                X
                
                  m
                  ⁢
                  
                    
                  
                  ⁢
                  n
                
              
              ≡
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          X
                          m
                        
                      
                      
                        
                          X
                          n
                        
                      
                    
                    
                      
                        
                          Y
                          m
                        
                      
                      
                        
                          Y
                          n
                        
                      
                    
                  
                  ]
                
                ⁢
                
                  {
                  
                    I
                    -
                    
                      
                        c
                        
                          (
                          
                            
                              θ
                              ⁢
                              
                                
                              
                              ⁢
                              m
                            
                            -
                            
                              θ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                            
                          
                          )
                        
                      
                      ⁢
                      P
                    
                  
                  }
                
                ⁢
                
                  T
                  
                    m
                    ⁢
                    
                      
                    
                    ⁢
                    n
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          X
                          n
                        
                      
                      
                        
                          X
                          m
                        
                      
                    
                    
                      
                        
                          Y
                          n
                        
                      
                      
                        
                          Y
                          m
                        
                      
                    
                  
                  ]
                
                ⁢
                P
                ⁢
                
                  {
                  
                    I
                    -
                    
                      
                        c
                        
                          (
                          
                            
                              θ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                            
                            -
                            
                              θ
                              ⁢
                              
                                
                              
                              ⁢
                              m
                            
                          
                          )
                        
                      
                      ⁢
                      P
                    
                  
                  }
                
                ⁢
                
                  PT
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    m
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          X
                          n
                        
                      
                      
                        
                          X
                          m
                        
                      
                    
                    
                      
                        
                          Y
                          n
                        
                      
                      
                        
                          Y
                          m
                        
                      
                    
                  
                  ]
                
                ⁢
                
                  {
                  
                    I
                    -
                    
                      
                        c
                        
                          (
                          
                            
                              θ
                              ⁢
                              
                                
                              
                              ⁢
                              n
                            
                            -
                            
                              θ
                              ⁢
                              
                                
                              
                              ⁢
                              m
                            
                          
                          )
                        
                      
                      ⁢
                      P
                    
                  
                  }
                
                ⁢
                
                  P
                  2
                
                ⁢
                
                  T
                  
                    
                      
                    
                    ⁢
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      m
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                X
                
                  
                    
                  
                  ⁢
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    m
                  
                
              
            
          
        
      
    
    
      
        (
        
          H
          ⁢
          
            
          
          ⁢
          23
        
        )
      
    
  




Therefore the number of independent 2×2 waveforms satisfying the Alford rotation is N(N−1)/2 where N is the number of dipole receivers. As an example if we have 8 azimuthal receivers, we can form four dipole receivers and 6 independent 2×2 waveforms indexed by {(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)}. As another example consider the classic cross dipole receiver configuration. Then we have a single cross-dipole waveform, X01, with θ0=0 and θ1=π/2. In this case equation (H22) reduces to




  
    
      
        
          
            X
            01
          
          =
          
            
              [
              
                
                  
                    
                      X
                      0
                    
                  
                  
                    
                      X
                      1
                    
                  
                
                
                  
                    
                      Y
                      0
                    
                  
                  
                    
                      Y
                      1
                    
                  
                
              
              ]
            
            ≡
            
              [
              
                
                  
                    XX
                  
                  
                    XY
                  
                
                
                  
                    YX
                  
                  
                    YY
                  
                
              
              ]
            
          
        
        ,
      
    
    
      
        (
        
          H
          ⁢
          
            
          
          ⁢
          24
        
        )
      
    
  




and equation (H21) becomes





  
    
      
        
          
            
              
                [
                
                  
                    
                      
                        S
                        f
                      
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        S
                        s
                      
                    
                  
                
                ]
              
              =
              
              ⁢
              
                
                  
                    [
                    
                      
                        
                          
                            c
                            φ
                          
                        
                        
                          
                            -
                            
                              s
                              φ
                            
                          
                        
                      
                      
                        
                          
                            s
                            φ
                          
                        
                        
                          
                            c
                            φ
                          
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          XX
                        
                        
                          XY
                        
                      
                      
                        
                          YX
                        
                        
                          YY
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          c
                          φ
                        
                      
                      
                        
                          s
                          φ
                        
                      
                    
                    
                      
                        
                          -
                          
                            s
                            φ
                          
                        
                      
                      
                        
                          c
                          φ
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          c
                          φ
                        
                      
                      
                        
                          -
                          
                            s
                            φ
                          
                        
                      
                    
                    
                      
                        
                          s
                          φ
                        
                      
                      
                        
                          c
                          φ
                        
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        
                          (
                          
                            
                              XXc
                              φ
                            
                            -
                            
                              XYs
                              φ
                            
                          
                          )
                        
                      
                      
                        
                          (
                          
                            
                              XXs
                              φ
                            
                            +
                            
                              XYc
                              φ
                            
                          
                          )
                        
                      
                    
                    
                      
                        
                          (
                          
                            
                              YXc
                              φ
                            
                            -
                            
                              YYs
                              φ
                            
                          
                          )
                        
                      
                      
                        
                          (
                          
                            
                              YXs
                              φ
                            
                            -
                            
                              YYc
                              φ
                            
                          
                          )
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                [
                
                  
                    
                      
                        (
                        
                          
                            
                              
                                
                                  
                                    c
                                    φ
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        XXc
                                        φ
                                      
                                      -
                                      
                                        XYs
                                        φ
                                      
                                    
                                    )
                                  
                                
                                -
                                
                                  
                                    s
                                    φ
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        YXc
                                        φ
                                      
                                      -
                                      
                                        YYs
                                        φ
                                      
                                    
                                    )
                                  
                                
                              
                            
                          
                          
                            
                              
                                
                                  
                                    s
                                    φ
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        XXc
                                        φ
                                      
                                      -
                                      
                                        XYs
                                        φ
                                      
                                    
                                    )
                                  
                                
                                +
                                
                                  
                                    c
                                    φ
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        XYc
                                        φ
                                      
                                      -
                                      
                                        YYs
                                        φ
                                      
                                    
                                    )
                                  
                                
                              
                            
                          
                        
                        )
                      
                    
                    
                      
                        (
                        
                          
                            
                              
                                
                                  
                                    c
                                    φ
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        XXs
                                        φ
                                      
                                      +
                                      
                                        XYc
                                        φ
                                      
                                    
                                    )
                                  
                                
                                -
                                
                                  
                                    s
                                    φ
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        YXs
                                        φ
                                      
                                      +
                                      
                                        YYc
                                        φ
                                      
                                    
                                    )
                                  
                                
                              
                            
                          
                          
                            
                              
                                
                                  
                                    s
                                    φ
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        XXs
                                        φ
                                      
                                      +
                                      
                                        XYc
                                        φ
                                      
                                    
                                    )
                                  
                                
                                +
                                
                                  
                                    c
                                    φ
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        YXs
                                        φ
                                      
                                      +
                                      
                                        YYc
                                        φ
                                      
                                    
                                    )
                                  
                                
                              
                            
                          
                        
                        )
                      
                    
                  
                
                ]
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          (
                          
                            
                              
                                
                                  
                                    
                                      c
                                      φ
                                      2
                                    
                                    ⁢
                                    XX
                                  
                                  +
                                  
                                    
                                      s
                                      φ
                                      2
                                    
                                    ⁢
                                    YY
                                  
                                  -
                                  
                                    
                                      s
                                      φ
                                    
                                    ⁢
                                    
                                      
                                        c
                                        φ
                                      
                                      ⁡
                                      
                                        (
                                        
                                          XY
                                          +
                                          YX
                                        
                                        )
                                      
                                    
                                  
                                
                              
                            
                            
                              
                                
                                  
                                    
                                      c
                                      φ
                                      2
                                    
                                    ⁢
                                    YX
                                  
                                  +
                                  
                                    
                                      s
                                      φ
                                      2
                                    
                                    ⁢
                                    XY
                                  
                                  +
                                  
                                    
                                      s
                                      φ
                                    
                                    ⁢
                                    
                                      
                                        c
                                        φ
                                      
                                      ⁡
                                      
                                        (
                                        
                                          XX
                                          +
                                          YY
                                        
                                        )
                                      
                                    
                                  
                                
                              
                            
                          
                          )
                        
                      
                      
                        
                          (
                          
                            
                              
                                
                                  
                                    
                                      c
                                      φ
                                      2
                                    
                                    ⁢
                                    XY
                                  
                                  +
                                  
                                    
                                      s
                                      φ
                                      2
                                    
                                    ⁢
                                    YX
                                  
                                  -
                                  
                                    
                                      s
                                      φ
                                    
                                    ⁢
                                    
                                      
                                        c
                                        φ
                                      
                                      ⁡
                                      
                                        (
                                        
                                          XX
                                          -
                                          YY
                                        
                                        )
                                      
                                    
                                  
                                
                              
                            
                            
                              
                                
                                  
                                    
                                      s
                                      φ
                                      2
                                    
                                    ⁢
                                    XX
                                  
                                  +
                                  
                                    
                                      c
                                      φ
                                      2
                                    
                                    ⁢
                                    YY
                                  
                                  +
                                  
                                    
                                      s
                                      φ
                                    
                                    ⁢
                                    
                                      
                                        c
                                        φ
                                      
                                      ⁡
                                      
                                        (
                                        
                                          XY
                                          +
                                          YX
                                        
                                        )
                                      
                                    
                                  
                                
                              
                            
                          
                          )
                        
                      
                    
                  
                  ]
                
                .
              
            
          
        
      
    
    
      
        (
        H25
        )
      
    
  




The classic cross-dipole waveform can also be expressed in terms of the fast and slow waves as





  
    
      
        
          
            
              
                [
                
                  
                    
                      XX
                    
                    
                      XY
                    
                  
                  
                    
                      YX
                    
                    
                      YY
                    
                  
                
                ]
              
              =
              
              ⁢
              
                
                  
                    [
                    
                      
                        
                          
                            c
                            φ
                          
                        
                        
                          
                            s
                            φ
                          
                        
                      
                      
                        
                          
                            -
                            
                              s
                              φ
                            
                          
                        
                        
                          
                            c
                            φ
                          
                        
                      
                    
                    ]
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            S
                            f
                          
                        
                        
                          0
                        
                      
                      
                        
                          0
                        
                        
                          
                            S
                            s
                          
                        
                      
                    
                    ]
                  
                
                ⁡
                
                  [
                  
                    
                      
                        
                          c
                          φ
                        
                      
                      
                        
                          -
                          
                            s
                            φ
                          
                        
                      
                    
                    
                      
                        
                          s
                          φ
                        
                      
                      
                        
                          c
                          φ
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          c
                          φ
                        
                      
                      
                        
                          s
                          φ
                        
                      
                    
                    
                      
                        
                          -
                          
                            s
                            φ
                          
                        
                      
                      
                        
                          c
                          φ
                        
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        
                          
                            S
                            f
                          
                          ⁢
                          
                            c
                            φ
                          
                        
                      
                      
                        
                          
                            -
                            
                              S
                              f
                            
                          
                          ⁢
                          
                            s
                            φ
                          
                        
                      
                    
                    
                      
                        
                          
                            S
                            s
                          
                          ⁢
                          
                            s
                            φ
                          
                        
                      
                      
                        
                          
                            S
                            s
                          
                          ⁢
                          
                            c
                            φ
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          
                            
                              S
                              f
                            
                            ⁢
                            
                              c
                              φ
                              2
                            
                          
                          +
                          
                            
                              S
                              s
                            
                            ⁢
                            
                              s
                              φ
                              2
                            
                          
                        
                      
                      
                        
                          
                            (
                            
                              
                                S
                                s
                              
                              -
                              
                                S
                                f
                              
                            
                            )
                          
                          ⁢
                          
                            s
                            φ
                          
                          ⁢
                          
                            c
                            φ
                          
                        
                      
                    
                    
                      
                        
                          
                            (
                            
                              
                                S
                                s
                              
                              -
                              
                                S
                                f
                              
                            
                            )
                          
                          ⁢
                          
                            s
                            φ
                          
                          ⁢
                          
                            c
                            φ
                          
                        
                      
                      
                        
                          
                            
                              S
                              f
                            
                            ⁢
                            
                              s
                              φ
                              2
                            
                          
                          +
                          
                            
                              S
                              s
                            
                            ⁢
                            
                              c
                              φ
                              2
                            
                          
                        
                      
                    
                  
                  ]
                
                .
              
            
          
        
      
    
    
      
        (
        H26
        )
      
    
  




Next we will derive the equations for analytically computing the angle for the time and frequency domain algorithms. These algorithms use the classic cross-dipole receiver configuration. Then we will derive the corresponding equations for the general Alford rotation. First notice that equation (H26) says XY=YX if the data satisfies the Alford rotation. Then we should use the sum of XY and YX in equation (H25) to cancel noise,




  
    
      
        
          [
          
            
              
                
                  S
                  f
                
              
              
                0
              
            
            
              
                0
              
              
                
                  S
                  s
                
              
            
          
          ]
        
        =
        
          
            [
            
              
                
                  
                    
                      
                        
                          (
                          
                            
                              
                                c
                                φ
                                2
                              
                              ⁢
                              XX
                            
                            +
                            
                              
                                s
                                φ
                                2
                              
                              ⁢
                              YY
                            
                            -
                            
                              
                                s
                                φ
                              
                              ⁢
                              
                                
                                  c
                                  φ
                                
                                ⁡
                                
                                  (
                                  
                                    XY
                                    +
                                    YX
                                  
                                  )
                                
                              
                            
                          
                          )
                        
                      
                    
                    
                      
                        
                          0.5
                          ⁢
                          
                            (
                            
                              
                                
                                  c
                                  
                                    2
                                    ⁢
                                    φ
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    XY
                                    +
                                    YX
                                  
                                  )
                                
                              
                              +
                              
                                
                                  s
                                  
                                    2
                                    ⁢
                                    φ
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    XX
                                    -
                                    YY
                                  
                                  )
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                
                
                  
                    
                      
                        
                          0.5
                          ⁢
                          
                            (
                            
                              
                                
                                  c
                                  
                                    2
                                    ⁢
                                    φ
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    XY
                                    +
                                    YX
                                  
                                  )
                                
                              
                              +
                              
                                
                                  s
                                  
                                    2
                                    ⁢
                                    φ
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    XX
                                    -
                                    YY
                                  
                                  )
                                
                              
                            
                            )
                          
                        
                      
                    
                    
                      
                        
                          (
                          
                            
                              
                                s
                                φ
                                2
                              
                              ⁢
                              XX
                            
                            +
                            
                              
                                c
                                φ
                                2
                              
                              ⁢
                              YY
                            
                            +
                            
                              
                                s
                                φ
                              
                              ⁢
                              
                                
                                  c
                                  φ
                                
                                ⁡
                                
                                  (
                                  
                                    XY
                                    +
                                    YX
                                  
                                  )
                                
                              
                            
                          
                          )
                        
                      
                    
                  
                
              
            
            ]
          
          .
        
      
    
    
      
        (
        H27
        )
      
    
  




The time domain algorithm assumes the fast and slow principal waves (Sf, Ss) are essentially the same except for a time shift. Thus a fast wave at receiver m can be converted into a slow wave at receiver n by the following time shift,

Ss,n(t+T1+Tsh)=Sf,m(t+T1),   (H28)

where

T1=Ts+sf(zm−zmid), and Tsh=δs·zm+ss(zn−zm).   (H29)

In equation (H29), the slowness of the fast principal wave is denoted by sf and the slowness of the slow principal wave is denoted by ss and δs=ss−sf. Ts is a start time that positions the start of the processing window at the beginning of the fast flexural wave at mid-array. In practice we don't know this start time a priori, so the XX flexural wave is used to estimate Ts. zm is the distance from the transmitter to the m'th receiver. Note the receiver indices correspond to receiver rings at different depths on the tool, not azimuth positions. T1 can be quantized to an integer sample index, but Tsh has a fractional component that should be accounted for to avoid quantization error. The slow waveform should be interpolated to t+T1+Tsh. Now we change the notation and define waveforms for any test rotation angle, θ, fast slowness, s1, and differential slowness, δs=s2−s1,




  
    
      
        
          
            
              
                FP
                
                  j
                  F
                
              
              ≡
              
              ⁢
              
                
                  S
                  
                    f
                    ,
                    m
                  
                
                ⁡
                
                  (
                  
                    
                      t
                      k
                    
                    +
                    
                      T
                      1
                    
                  
                  )
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    c
                    θ
                    2
                  
                  ⁢
                  
                    
                      XX
                      m
                    
                    ⁡
                    
                      (
                      
                        
                          t
                          k
                        
                        +
                        
                          T
                          1
                        
                      
                      )
                    
                  
                
                +
                
                  
                    s
                    θ
                    2
                  
                  ⁢
                  
                    
                      YY
                      m
                    
                    ⁡
                    
                      (
                      
                        
                          t
                          k
                        
                        +
                        
                          T
                          1
                        
                      
                      )
                    
                  
                
                -
              
            
          
        
        
          
            
              
              ⁢
              
                
                  s
                  θ
                
                ⁢
                
                  
                    c
                    θ
                  
                  ⁡
                  
                    (
                    
                      
                        
                          XY
                          m
                        
                        ⁡
                        
                          (
                          
                            
                              t
                              k
                            
                            +
                            
                              T
                              1
                            
                          
                          )
                        
                      
                      +
                      
                        
                          YX
                          m
                        
                        ⁡
                        
                          (
                          
                            
                              t
                              k
                            
                            +
                            
                              T
                              1
                            
                          
                          )
                        
                      
                    
                    )
                  
                
              
            
          
        
        
          
            
              
                ≡
                
                ⁢
                
                  
                    
                      c
                      θ
                      2
                    
                    ⁢
                    
                      XX
                      
                        j
                        F
                      
                    
                  
                  +
                  
                    
                      s
                      θ
                      2
                    
                    ⁢
                    
                      YY
                      
                        j
                        F
                      
                    
                  
                  -
                  
                    
                      s
                      θ
                    
                    ⁢
                    
                      
                        c
                        θ
                      
                      ⁡
                      
                        (
                        
                          
                            XY
                            
                              j
                              F
                            
                          
                          +
                          
                            YX
                            
                              j
                              F
                            
                          
                        
                        )
                      
                    
                  
                
              
              ,
            
          
        
      
    
    
      
        (
        H30
        )
      
    
  
  
    
      
        
          
            
              
                SP
                
                  j
                  S
                
              
              ≡
              
              ⁢
              
                
                  S
                  
                    s
                    ,
                    n
                  
                
                ⁡
                
                  (
                  
                    
                      t
                      k
                    
                    +
                    
                      T
                      1
                    
                    +
                    
                      T
                      sh
                    
                  
                  )
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    s
                    θ
                    2
                  
                  ⁢
                  
                    
                      XX
                      n
                    
                    ⁡
                    
                      (
                      
                        
                          t
                          k
                        
                        +
                        
                          T
                          1
                        
                        +
                        
                          T
                          sh
                        
                      
                      )
                    
                  
                
                +
                
                  
                    c
                    θ
                    2
                  
                  ⁢
                  
                    
                      YY
                      n
                    
                    ⁡
                    
                      (
                      
                        
                          t
                          k
                        
                        +
                        
                          T
                          1
                        
                        +
                        
                          T
                          sh
                        
                      
                      )
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  s
                  θ
                
                ⁢
                
                  
                    c
                    θ
                  
                  ⁡
                  
                    (
                    
                      
                        
                          XY
                          n
                        
                        ⁡
                        
                          (
                          
                            
                              t
                              k
                            
                            +
                            
                              T
                              1
                            
                            +
                            
                              T
                              sh
                            
                          
                          )
                        
                      
                      +
                      
                        
                          YX
                          n
                        
                        ⁡
                        
                          (
                          
                            
                              t
                              k
                            
                            +
                            
                              T
                              1
                            
                            +
                            
                              T
                              sh
                            
                          
                          )
                        
                      
                    
                    )
                  
                
              
            
          
        
        
          
            
              
                ≡
                
                ⁢
                
                  
                    
                      s
                      θ
                      2
                    
                    ⁢
                    
                      XX
                      
                        j
                        S
                      
                    
                  
                  +
                  
                    
                      c
                      θ
                      2
                    
                    ⁢
                    
                      YY
                      
                        j
                        S
                      
                    
                  
                  +
                  
                    
                      s
                      θ
                    
                    ⁢
                    
                      
                        c
                        θ
                      
                      ⁡
                      
                        (
                        
                          
                            XY
                            
                              j
                              S
                            
                          
                          +
                          
                            YX
                            
                              j
                              S
                            
                          
                        
                        )
                      
                    
                  
                
              
              ,
            
          
        
      
    
    
      
        (
        H31
        )
      
    
  
  
    
      
        
          
            
              
                FP
                θ
                
                  j
                  F
                
              
              ≡
              
              ⁢
              
                
                  
                    c
                    
                      2
                      ⁢
                      θ
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          XY
                          m
                        
                        ⁡
                        
                          (
                          
                            
                              t
                              k
                            
                            +
                            
                              T
                              1
                            
                          
                          )
                        
                      
                      +
                      
                        
                          YX
                          m
                        
                        ⁡
                        
                          (
                          
                            
                              t
                              k
                            
                            +
                            
                              T
                              1
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  s
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
                ⁡
                
                  (
                  
                    
                      
                        XX
                        m
                      
                      ⁡
                      
                        (
                        
                          
                            t
                            k
                          
                          +
                          
                            T
                            1
                          
                        
                        )
                      
                    
                    -
                    
                      
                        YY
                        m
                      
                      ⁡
                      
                        (
                        
                          
                            t
                            k
                          
                          +
                          
                            T
                            1
                          
                        
                        )
                      
                    
                  
                  )
                
              
            
          
        
        
          
            
              
                ≡
                
                ⁢
                
                  
                    
                      c
                      
                        2
                        ⁢
                        θ
                      
                    
                    ⁡
                    
                      (
                      
                        
                          XY
                          
                            j
                            F
                          
                        
                        +
                        
                          YX
                          
                            j
                            F
                          
                        
                      
                      )
                    
                  
                  +
                  
                    
                      s
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                    
                    ⁡
                    
                      (
                      
                        
                          XX
                          
                            j
                            F
                          
                        
                        +
                        
                          YY
                          
                            j
                            F
                          
                        
                      
                      )
                    
                  
                
              
              ,
            
          
        
      
    
    
      
        (
        H32
        )
      
    
  
  
    
      
        
          
            
              
                FP
                θ
                
                  j
                  S
                
              
              ≡
              
              ⁢
              
                
                  
                    c
                    
                      2
                      ⁢
                      θ
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          XY
                          n
                        
                        ⁡
                        
                          (
                          
                            
                              t
                              k
                            
                            +
                            
                              T
                              1
                            
                            +
                            
                              T
                              sh
                            
                          
                          )
                        
                      
                      +
                      
                        
                          YX
                          n
                        
                        ⁡
                        
                          (
                          
                            
                              t
                              k
                            
                            +
                            
                              T
                              1
                            
                            +
                            
                              T
                              sh
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  s
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
                ⁡
                
                  (
                  
                    
                      
                        XX
                        n
                      
                      ⁡
                      
                        (
                        
                          
                            t
                            k
                          
                          +
                          
                            T
                            1
                          
                          +
                          
                            T
                            sh
                          
                        
                        )
                      
                    
                    -
                    
                      
                        YY
                        n
                      
                      ⁡
                      
                        (
                        
                          
                            t
                            k
                          
                          +
                          
                            T
                            1
                          
                          +
                          
                            T
                            sh
                          
                        
                        )
                      
                    
                  
                  )
                
              
            
          
        
        
          
            
              
                ≡
                
                ⁢
                
                  
                    
                      c
                      
                        2
                        ⁢
                        θ
                      
                    
                    ⁡
                    
                      (
                      
                        
                          XY
                          
                            j
                            S
                          
                        
                        +
                        
                          YX
                          
                            j
                            S
                          
                        
                      
                      )
                    
                  
                  +
                  
                    
                      s
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                    
                    ⁡
                    
                      (
                      
                        
                          XX
                          
                            j
                            S
                          
                        
                        +
                        
                          YY
                          
                            j
                            S
                          
                        
                      
                      )
                    
                  
                
              
              ,
            
          
        
      
    
    
      
        (
        
          H
          ⁢
          
            
          
          ⁢
          33
        
        )
      
    
  
  
    
      
        
          SP
          θ
        
        =
        
          -
          
            
              FP
              θ
            
            .
          
        
      
    
    
      
        (
        
          H
          ⁢
          
            
          
          ⁢
          34
        
        )
      
    
  




Thus jF and jS are shorthand notation for the indices in (k,m,n) space used to stack the fast and slow waveforms respectively. Conceptually both jF and jS can be functions of all three indices, but in this example jF is independent of n. In the literature FPθ and SPθ are found by taking the derivative of FP and SP with respect to θ and are called auxiliary principal waves. In reality they are simply proportional to the off-diagonal element of equation (H27). Next we define an objective function whose value is minimized when




  
    
      
        
          
            (
            
              θ
              ,
              
                s
                1
              
              ,
              
                δ
                ⁢
                
                  
                
                ⁢
                s
              
            
            )
          
          =
          
            
              
                (
                
                  φ
                  ,
                  
                    s
                    f
                  
                  ,
                  
                    
                      s
                      s
                    
                    -
                    
                      s
                      f
                    
                  
                
                )
              
              .
              
                

              
              ⁢
              
                O
                ⁡
                
                  (
                  
                    θ
                    ,
                    
                      s
                      1
                    
                    ,
                    
                      δ
                      ⁢
                      
                        
                      
                      ⁢
                      s
                    
                  
                  )
                
              
            
            =
            
              
                
                  ∑
                  
                    k
                    ,
                    m
                    ,
                    n
                  
                
                ⁢
                
                  
                
                ⁢
                
                  4
                  ⁢
                  
                    
                      (
                      
                        
                          SP
                          
                            j
                            s
                          
                        
                        -
                        
                          FP
                          
                            j
                            F
                          
                        
                      
                      )
                    
                    2
                  
                
              
              +
              
                
                  (
                  
                    
                      SP
                      θ
                      
                        j
                        S
                      
                    
                    -
                    
                      FP
                      θ
                      
                        j
                        F
                      
                    
                  
                  )
                
                2
              
            
          
        
        ,
      
    
    
      
        (
        H35
        )
      
    
  




where m and n are summed over the receiver indices and k is summed over a time window 0≦kΔt≦Tw. Substituting from equations (H30-H34) we have





  
    
      
        
          O
          ⁢
          
            (
            
              θ
              ,
              
                s
                1
              
              ,
              
                δ
                ⁢
                
                  
                
                ⁢
                s
              
            
            )
          
        
        =
        
          
            ∑
            
              k
              ,
              m
              ,
              n
            
          
          ⁢
          
            
              {
              
                
                  
                    
                      
                        
                          4
                          ⁡
                          
                            [
                            
                              
                                
                                  
                                    
                                      
                                        c
                                        θ
                                        2
                                      
                                      ⁡
                                      
                                        (
                                        
                                          YY
                                          
                                            
                                              j
                                              S
                                            
                                            -
                                            
                                              XX
                                              
                                                j
                                                F
                                              
                                            
                                          
                                        
                                        )
                                      
                                    
                                    +
                                    
                                      
                                        s
                                        θ
                                        2
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            XX
                                            
                                              j
                                              S
                                            
                                          
                                          -
                                          
                                            YY
                                            
                                              j
                                              F
                                            
                                          
                                        
                                        )
                                      
                                    
                                    +
                                  
                                
                              
                              
                                
                                  
                                    
                                      s
                                      θ
                                    
                                    ⁢
                                    
                                      
                                        c
                                        θ
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            XY
                                            
                                              j
                                              S
                                            
                                          
                                          +
                                          
                                            YX
                                            
                                              j
                                              S
                                            
                                          
                                          +
                                          
                                            XY
                                            
                                              j
                                              F
                                            
                                          
                                          +
                                          
                                            YX
                                            
                                              j
                                              F
                                            
                                          
                                        
                                        )
                                      
                                    
                                  
                                
                              
                            
                            ]
                          
                        
                        2
                      
                      +
                    
                  
                
                
                  
                    
                      
                        [
                        
                          
                            
                              
                                
                                  
                                    s
                                    
                                      2
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      θ
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        XX
                                        
                                          j
                                          S
                                        
                                      
                                      -
                                      
                                        YY
                                        
                                          j
                                          S
                                        
                                      
                                      +
                                      
                                        XX
                                        
                                          j
                                          F
                                        
                                      
                                      -
                                      
                                        YY
                                        
                                          j
                                          F
                                        
                                      
                                    
                                    )
                                  
                                
                                +
                              
                            
                          
                          
                            
                              
                                
                                  c
                                  
                                    2
                                    ⁢
                                    θ
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      XY
                                      
                                        j
                                        S
                                      
                                    
                                    +
                                    
                                      YX
                                      
                                        j
                                        S
                                      
                                    
                                    +
                                    
                                      XY
                                      
                                        j
                                        F
                                      
                                    
                                    +
                                    
                                      YX
                                      
                                        j
                                        F
                                      
                                    
                                  
                                  )
                                
                              
                            
                          
                        
                        ]
                      
                      2
                    
                  
                
              
              }
            
            .
          
        
      
    
    
      
        H36
        )
      
    
  




Substitute the identities cθ2=0.5(1+c2θ) and sθ2=0.5(1−c2θ) into equation (36) gives





  
    
      
        
          O
          ⁡
          
            (
            
              θ
              ,
              
                s
                1
              
              ,
              
                δ
                ⁢
                
                  
                
                ⁢
                s
              
            
            )
          
        
        =
        
          
            ∑
            
              k
              ,
              m
              ,
              n
            
          
          ⁢
          
            
          
          ⁢
          
            {
            
              
                
                  
                    
                      [
                      
                        
                          
                            
                              
                                (
                                
                                  
                                    
                                      
                                        
                                          YY
                                          
                                            j
                                            s
                                          
                                        
                                        -
                                        
                                          XX
                                          
                                            j
                                            F
                                          
                                        
                                        +
                                      
                                    
                                  
                                  
                                    
                                      
                                        
                                          XX
                                          
                                            j
                                            s
                                          
                                        
                                        -
                                        
                                          YY
                                          
                                            j
                                            F
                                          
                                        
                                      
                                    
                                  
                                
                                )
                              
                              +
                              
                                
                                  c
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                          
                                            YY
                                            
                                              j
                                              s
                                            
                                          
                                          -
                                          
                                            XX
                                            
                                              j
                                              F
                                            
                                          
                                          -
                                        
                                      
                                    
                                    
                                      
                                        
                                          
                                            XX
                                            
                                              j
                                              s
                                            
                                          
                                          +
                                          
                                            YY
                                            
                                              j
                                              F
                                            
                                          
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                
                                  s
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      XY
                                      
                                        j
                                        S
                                      
                                    
                                    +
                                    
                                      YX
                                      
                                        j
                                        S
                                      
                                    
                                    +
                                    
                                      XY
                                      
                                        j
                                        F
                                      
                                    
                                    +
                                    
                                      YX
                                      
                                        j
                                        F
                                      
                                    
                                  
                                  )
                                
                              
                              +
                            
                          
                        
                      
                      ]
                    
                    2
                  
                
              
              
                
                  
                    
                      [
                      
                        
                          
                            s
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  
                                    
                                      XX
                                      
                                        j
                                        S
                                      
                                    
                                    -
                                    
                                      YY
                                      
                                        j
                                        S
                                      
                                    
                                    +
                                  
                                
                              
                              
                                
                                  
                                    
                                      XX
                                      
                                        j
                                        F
                                      
                                    
                                    -
                                    
                                      YY
                                      
                                        j
                                        F
                                      
                                    
                                  
                                
                              
                            
                            )
                          
                        
                        +
                        
                          
                            c
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  
                                    
                                      XY
                                      
                                        j
                                        S
                                      
                                    
                                    +
                                    
                                      YX
                                      
                                        j
                                        S
                                      
                                    
                                    +
                                  
                                
                              
                              
                                
                                  
                                    
                                      XY
                                      
                                        j
                                        F
                                      
                                    
                                    +
                                    
                                      YX
                                      
                                        j
                                        F
                                      
                                    
                                  
                                
                              
                            
                            )
                          
                        
                      
                      ]
                    
                    2
                  
                
              
            
            }
          
        
      
    
    
      
        (
        H37
        )
      
    
  




We can further define j(k,m,n)=(jF(k,m,n), jS(k,m,n)). Using this compact notation we have




  
    
      
        
          
            
              
                O
                ⁢
                
                  (
                  
                    θ
                    ,
                    
                      s
                      1
                    
                    ,
                    
                      δ
                      ⁢
                      
                        
                      
                      ⁢
                      s
                    
                  
                  )
                
              
              =
              
              ⁢
              
                
                  ∑
                  j
                
                ⁢
                
                  
                
                ⁢
                
                  {
                  
                    
                      
                        [
                        
                          
                            a
                            j
                          
                          +
                          
                            
                              c
                              
                                2
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                            
                            ⁢
                            
                              b
                              j
                            
                          
                          +
                          
                            
                              s
                              
                                2
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                            
                            ⁢
                            
                              c
                              j
                            
                          
                        
                        ]
                      
                      2
                    
                    +
                    
                      
                        [
                        
                          
                            
                              -
                              
                                s
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                              
                            
                            ⁢
                            
                              b
                              j
                            
                          
                          +
                          
                            
                              c
                              
                                2
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                            
                            ⁢
                            
                              c
                              j
                            
                          
                        
                        ]
                      
                      2
                    
                  
                  }
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  a
                  2
                
                +
                
                  
                    b
                    2
                  
                  ⁢
                  
                    c
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                    2
                  
                
                +
                
                  
                    c
                    2
                  
                  ⁢
                  
                    s
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                    2
                  
                
                +
                
                  2
                  ⁢
                  
                    
                      a
                      ⇀
                    
                    ·
                    
                      b
                      ⇀
                    
                  
                  ⁢
                  
                    c
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                  
                
                +
                
                  2
                  ⁢
                  
                    
                      a
                      ⇀
                    
                    ·
                    
                      c
                      ⇀
                    
                  
                  ⁢
                  
                    s
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  2
                  ⁢
                  
                    
                      b
                      ⇀
                    
                    ·
                    
                      c
                      ⇀
                    
                  
                  ⁢
                  
                    s
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                  
                  ⁢
                  
                    c
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                  
                
                +
                
                  
                    b
                    2
                  
                  ⁢
                  
                    s
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                    2
                  
                
                +
                
                  
                    c
                    2
                  
                  ⁢
                  
                    c
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                    2
                  
                
                -
                
                  2
                  ⁢
                  
                    
                      b
                      ⇀
                    
                    ·
                    
                      c
                      ⇀
                    
                  
                  ⁢
                  
                    s
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                  
                  ⁢
                  
                    c
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  a
                  2
                
                +
                
                  b
                  2
                
                +
                
                  c
                  2
                
                +
                
                  2
                  ⁢
                  
                    
                      a
                      ⇀
                    
                    ·
                    
                      b
                      ⇀
                    
                  
                  ⁢
                  
                    c
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                  
                
                +
                
                  2
                  ⁢
                  
                    
                      a
                      ⇀
                    
                    ·
                    
                      c
                      ⇀
                    
                  
                  ⁢
                  
                    
                      s
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                    
                    .
                  
                
              
            
          
        
      
    
    
      
        (
        H38
        )
      
    
  
  
    
      
        
          
            where
            ⁢
            
              
            
            ⁢
            
              a
              j
            
          
          =
          
            
              YY
              
                j
                S
              
            
            -
            
              XX
              
                j
                F
              
            
            +
            
              XX
              
                j
                S
              
            
            -
            
              YY
              
                j
                F
              
            
          
        
        ,
        
          

        
        ⁢
        
          
            b
            j
          
          =
          
            
              YY
              
                j
                S
              
            
            -
            
              XX
              
                j
                F
              
            
            -
            
              XX
              
                j
                S
              
            
            +
            
              YY
              
                j
                F
              
            
          
        
        ,
        
          

        
        ⁢
        
          
            c
            j
          
          =
          
            
              XY
              
                j
                S
              
            
            +
            
              YX
              
                j
                S
              
            
            +
            
              XY
              
                j
                F
              
            
            +
            
              
                YX
                
                  j
                  F
                
              
              .
            
          
        
      
    
    
      
        (
        H39
        )
      
    
  




Minimizing with respect to θ gives





  
    
      
        
          
            
              
                a
                ⇀
              
              ·
              
                b
                ⇀
              
            
            ⁢
            
              s
              
                2
                ⁢
                φ
              
            
          
          -
          
            
              
                a
                ⇀
              
              ·
              
                c
                ⇀
              
            
            ⁢
            
              c
              
                2
                ⁢
                φ
              
            
          
        
        =
        
          
            0
            ⇒
            
              tan
              ⁡
              
                (
                
                  2
                  ⁢
                  φ
                
                )
              
            
          
          =
          
            
              
                
                  a
                  ⇀
                
                ·
                
                  c
                  ⇀
                
              
              
                
                  a
                  ⇀
                
                ·
                
                  b
                  ⇀
                
              
            
            .
          
        
      
    
    
      
        (
        H40
        )
      
    
  




If we define φ as the angle of the fast axis relative to the X transmitter then φ→−φ in equation (40). φ is ambiguous by a multiple of π/2, (φ→φ+nπ/2) as shown in FIG. 18. The it ambiguity is not of concern. It has no physical significance. The π/2 ambiguity corresponds to a minimum and a maximum of the objective function. We want the minimum. This is determined by substituting φ and φ+π/2 back into the objective function equation (H38) and selecting the angle that minimizes the objective function. This gives an angle for every fast slowness, s1, and differential slowness, δs=s2−s1>0. The global minimization is done by numerically minimizing over s1 and δs. The fast/slow ambiguity is automatically resolved by restricting δs to positive values. As a QC check we can minimize the cross-line energy only without slowness stacking. Then the objective function is (see equation (H27))




  
    
      
        
          
            
              
                O
                ⁡
                
                  (
                  θ
                  )
                
              
              =
              
              ⁢
              
                
                  ∑
                  
                    k
                    ,
                    m
                  
                
                ⁢
                
                  
                
                ⁢
                
                  [
                  
                    
                      
                        c
                        
                          2
                          ⁢
                          θ
                        
                      
                      ⁡
                      
                        (
                        
                          
                            
                              XY
                              m
                            
                            ⁡
                            
                              (
                              
                                
                                  t
                                  k
                                
                                +
                                
                                  T
                                  s
                                
                              
                              )
                            
                          
                          +
                          
                            
                              YX
                              m
                            
                            ⁡
                            
                              (
                              
                                
                                  t
                                  k
                                
                                +
                                
                                  T
                                  s
                                
                              
                              )
                            
                          
                        
                        )
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                ⁢
                
                  
                    s
                    
                      2
                      ⁢
                      θ
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          XX
                          m
                        
                        ⁡
                        
                          (
                          
                            
                              t
                              k
                            
                            +
                            
                              T
                              s
                            
                          
                          )
                        
                      
                      -
                      
                        
                          YY
                          m
                        
                        ⁡
                        
                          (
                          
                            
                              t
                              k
                            
                            +
                            
                              T
                              s
                            
                          
                          )
                        
                      
                    
                    )
                  
                
                ]
              
              2
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    
                      (
                      
                        k
                        ,
                        m
                      
                      )
                    
                  
                
                ⁢
                
                  
                
                ⁢
                
                  
                    (
                    
                      
                        
                          c
                          
                            2
                            ⁢
                            θ
                          
                        
                        ⁡
                        
                          (
                          
                            
                              XY
                              j
                            
                            +
                            
                              YX
                              j
                            
                          
                          )
                        
                      
                      +
                      
                        
                          s
                          
                            2
                            ⁢
                            θ
                          
                        
                        ⁡
                        
                          (
                          
                            
                              XX
                              j
                            
                            -
                            
                              YY
                              j
                            
                          
                          )
                        
                      
                    
                    )
                  
                  2
                
              
            
          
        
        
          
            
              
                ≡
                
                ⁢
                
                  
                    
                      c
                      
                        2
                        ⁢
                        θ
                      
                      2
                    
                    ⁢
                    
                      a
                      2
                    
                  
                  +
                  
                    
                      s
                      
                        2
                        ⁢
                        θ
                      
                      2
                    
                    ⁢
                    
                      b
                      2
                    
                  
                  +
                  
                    2
                    ⁢
                    
                      s
                      
                        2
                        ⁢
                        θ
                      
                    
                    ⁢
                    
                      c
                      
                        2
                        ⁢
                        θ
                      
                    
                    ⁢
                    
                      
                        a
                        ⇀
                      
                      ·
                      
                        b
                        ⇀
                      
                    
                  
                
              
              ,
            
          
        
      
    
    
      
        (
        H41
        )
      
    
  




where

aj=XYj+XYj,

bj=XXj−YYj.   (H42)

Minimizing with respect to θ gives





  
    
      
        
          
            
              s
              
                4
                ⁢
                φ
              
            
            ⁡
            
              (
              
                
                  b
                  2
                
                -
                
                  a
                  2
                
              
              )
            
          
          +
          
            2
            ⁢
            
              c
              
                4
                ⁢
                φ
              
            
            ⁢
            
              
                a
                ⇀
              
              ·
              
                b
                ⇀
              
            
          
        
        =
        
          
            0
            ⇒
            
              tan
              ⁡
              
                (
                
                  4
                  ⁢
                  φ
                
                )
              
            
          
          =
          
            -
            
              
                
                  2
                  ⁢
                  
                    
                      a
                      ⇀
                    
                    ·
                    
                      b
                      ⇀
                    
                  
                
                
                  (
                  
                    
                      b
                      2
                    
                    -
                    
                      a
                      2
                    
                  
                  )
                
              
              .
            
          
        
      
    
    
      
        (
        H43
        )
      
    
  




If we define φ as the angle of the fast axis relative to the X transmitter then φ→−φ in equation (40). φ is ambiguous by a multiple of π/4, (φ→φ+nπ/4) as in FIG. 16. The π ambiguity is of no concern. It has no physical significance. The π/4 ambiguity corresponds to a minimum and a maximum of the objective function. We want the minimum. This is determined by substituting φ and φ+π/4 back into the objective function of equation (41) and selecting the angle that minimizes the objective function. This leaves the φ and φ+π/2 ambiguity corresponding to the fast and slow formation axes. This is resolved by rotating the cross-dipole waveforms by φ into the fast and slow waveforms using equation (H27) and comparing there slowness. One simple way to do this is to plot them and compare their delay. If the assumed fast waveform lags the assumed slow waveform then the correct angle is φ+π/2. The ratio of cross-line energy after rotation to cross-line energy before rotation is

E=(c2θ2a2+s2θ2b2+2s2θc2θ{right arrow over (a)}·{right arrow over (b)})/a2.   (H44)

As an aside, if we started with the sum of the squares of the off-diagonal elements of equation (H25) instead of equation (H27), we arrive at the same result.

One issue with the objective function given in equation (H35) is the possibility of the cross-line term vanishing not because SPθjF and SPθjS are both small, but rather because they are 180° out of phase, i.e., SPθjF≈−SPθjS ≠0. The algorithm has been tested on many data sets and the angle estimate is correct despite this potential problem. However, the slowness values appear to be off. This could be due to dispersion or the out of phase problem. It doesn't matter because the angles are accurate and the slownesses are recomputed from the rotated waveforms. An alternative objective function that avoids the out of phase problem is




  
    
      
        
          O
          ⁢
          
            (
            
              θ
              ,
              
                s
                1
              
              ,
              
                δ
                ⁢
                
                  
                
                ⁢
                s
              
            
            )
          
        
        =
        
          
            
              ∑
              
                k
                ,
                m
                ,
                n
              
            
            ⁢
            
              
            
            ⁢
            
              4
              ⁢
              
                
                  (
                  
                    
                      SP
                      
                        j
                        S
                      
                    
                    -
                    
                      FP
                      
                        j
                        F
                      
                    
                  
                  )
                
                2
              
            
          
          +
          
            
              (
              
                SP
                θ
                
                  j
                  S
                
              
              )
            
            2
          
          +
          
            
              
                (
                
                  FP
                  θ
                  
                    j
                    F
                  
                
                )
              
              2
            
            .
          
        
      
    
    
      
        (
        H45
        )
      
    
  




This objective function can be minimized analytically, but it requires solving a fourth order polynomial in c2θ. The equations get quite messy and are given in the HTI Appendix.
Now we will derive the frequency domain algorithm equations. The objective function is




  
    
      
        
          O
          ⁡
          
            (
            
              θ
              ,
              
                ω
                ST
              
            
            )
          
        
        =
        
          
            ∑
            
              
                
                  
                    
                      j
                      n
                    
                    ,
                    
                      j
                      m
                    
                  
                
              
              
                
                  
                    k
                    ,
                    l
                  
                
              
            
            
              B
              ⁡
              
                (
                
                  
                    ω
                    ST
                  
                  ,
                  W
                
                )
              
            
          
          ⁢
          
            
          
          ⁢
          
            
              
                
                
                  
                    
                      T
                      θ
                      T
                    
                    ⁢
                    
                      X
                      
                        j
                        n
                      
                    
                    ⁢
                    
                      T
                      θ
                    
                  
                  -
                  
                    
                      D
                      
                        j
                        m
                      
                    
                    ⁢
                    
                      Z
                      est
                      
                        m
                        -
                        n
                      
                    
                  
                
                
              
              kl
              2
            
            .
          
        
      
    
    
      
        (
        H46
        )
      
    
  




In equation (H46), jn=(n,ω) and jm=(m,ω) where m and n are receiver indices, and the frequency summation is over a frequency band starting at ωST with width W. Tθ is given by equation (H13),





  
    T
    θ
  
  =
  
    
      [
      
        
          
            
              c
              θ
            
          
          
            
              s
              θ
            
          
        
        
          
            
              -
              
                s
                θ
              
            
          
          
            
              c
              θ
            
          
        
      
      ]
    
    .
  




The Fourier transformed cross-dipole waveform is





  
    
      
        
          
            X
            
              j
              n
            
          
          =
          
            [
            
              
                
                  
                    
                      XX
                      n
                    
                    ⁡
                    
                      (
                      ω
                      )
                    
                  
                
                
                  
                    
                      XY
                      n
                    
                    ⁡
                    
                      (
                      ω
                      )
                    
                  
                
              
              
                
                  
                    
                      YX
                      n
                    
                    ⁡
                    
                      (
                      ω
                      )
                    
                  
                
                
                  
                    
                      YY
                      n
                    
                    ⁡
                    
                      (
                      ω
                      )
                    
                  
                
              
            
            ]
          
        
        ,
      
    
    
      
        (
        H47
        )
      
    
  




where n is receiver index. The diagonal matrix Djn is computed by diagonalizing the cross-dipole waveform of equation (H47) with a unitary transformation. It is an estimate of the fast and slow waveforms in the frequency domain at receiver n,





  
    
      
        
          
            U
            ′
          
          ⁢
          
            X
            
              j
              n
            
          
          ⁢
          U
        
        =
        
          
            [
            
              
                
                  
                    D
                    11
                    
                      j
                      n
                    
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    D
                    22
                    
                      j
                      n
                    
                  
                
              
            
            ]
          
          .
        
      
    
    
      
        (
        H48
        )
      
    
  




Using methods described herein we can ensure that the D11 element is the fast wave and the D22 element is the slow wave. The diagonal matrix Zest is given by




  
    
      
        
          Z
          est
        
        =
        
          
            [
            
              
                
                  
                    z
                    1
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  
                    z
                    2
                  
                
              
            
            ]
          
          .
        
      
    
    
      
        (
        H49
        )
      
    
  




A method for calculating Zest is described in earlier sections. The diagonal matrix Zestm−n (Zest raised to the m−n power) propagates the matrix Djm at receiver m to the matrix Djn at receiver n. We simplify the notation by defining j=(jm, jn), then




  
    
      
        
          
            O
            ⁡
            
              (
              
                θ
                ,
                
                  ω
                  ST
                
              
              )
            
          
          =
          
            
              ∑
              
                j
                ,
                k
                ,
                l
              
            
            ⁢
            
              
            
            ⁢
            
              
                
                
                  
                    (
                    
                      
                        
                          T
                          θ
                          T
                        
                        ⁢
                        
                          X
                          j
                        
                        ⁢
                        
                          T
                          θ
                        
                      
                      -
                      
                        Δ
                        j
                      
                    
                    )
                  
                  kl
                
                
              
              2
            
          
        
        ,
        
          

        
        ⁢
        where
      
    
    
      
        (
        H50
        )
      
    
  
  
    
      
        
          
            X
            j
          
          =
          
            X
            
              j
              n
            
          
        
        ,
        
          

        
        ⁢
        
          
            Δ
            j
          
          =
          
            
              D
              
                j
                m
              
            
            ⁢
            
              
                Z
                est
                
                  m
                  -
                  n
                
              
              .
            
          
        
      
    
    
      
        (
        H51
        )
      
    
  




The rotated waveforms are given by equation (H27)




  
    
      
        
          
            T
            θ
            T
          
          ⁢
          
            X
            j
          
          ⁢
          
            T
            θ
          
        
        =
        
          [
          
            
              
                
                  (
                  
                    
                      
                        
                          
                            
                              c
                              θ
                              2
                            
                            ⁢
                            
                              XX
                              j
                            
                          
                          +
                          
                            
                              s
                              θ
                              2
                            
                            ⁢
                            
                              YY
                              j
                            
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            s
                            θ
                          
                          ⁢
                          
                            
                              c
                              θ
                            
                            ⁡
                            
                              (
                              
                                
                                  XY
                                  j
                                
                                +
                                
                                  YX
                                  j
                                
                              
                              )
                            
                          
                        
                      
                    
                  
                  )
                
              
              
                
                  0.5
                  ⁢
                  
                    (
                    
                      
                        
                          
                            
                              
                                c
                                
                                  2
                                  ⁢
                                  θ
                                
                              
                              ⁡
                              
                                (
                                
                                  
                                    XY
                                    j
                                  
                                  +
                                  
                                    YX
                                    j
                                  
                                
                                )
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              s
                              
                                2
                                ⁢
                                θ
                              
                            
                            ⁡
                            
                              (
                              
                                
                                  XX
                                  j
                                
                                -
                                
                                  YY
                                  j
                                
                              
                              )
                            
                          
                        
                      
                    
                    )
                  
                
              
            
            
              
                
                  0.5
                  ⁢
                  
                    (
                    
                      
                        
                          
                            
                              
                                c
                                
                                  2
                                  ⁢
                                  θ
                                
                              
                              ⁡
                              
                                (
                                
                                  
                                    XY
                                    j
                                  
                                  +
                                  
                                    YX
                                    j
                                  
                                
                                )
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              s
                              
                                2
                                ⁢
                                θ
                              
                            
                            ⁡
                            
                              (
                              
                                
                                  XX
                                  j
                                
                                -
                                
                                  YY
                                  j
                                
                              
                              )
                            
                          
                        
                      
                    
                    )
                  
                
              
              
                
                  (
                  
                    
                      
                        
                          
                            
                              s
                              θ
                              2
                            
                            ⁢
                            
                              XX
                              j
                            
                          
                          +
                          
                            
                              c
                              θ
                              2
                            
                            ⁢
                            
                              YY
                              j
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            s
                            θ
                          
                          ⁢
                          
                            
                              c
                              θ
                            
                            ⁡
                            
                              (
                              
                                
                                  XY
                                  j
                                
                                +
                                
                                  YX
                                  j
                                
                              
                              )
                            
                          
                        
                      
                    
                  
                  )
                
              
            
          
          ]
        
      
    
    
      
        (
        H52
        )
      
    
  




Working through the algebra we have




  
    
      
        
          
            
              
                O
                ⁡
                
                  (
                  
                    θ
                    ,
                    
                      ω
                      ST
                    
                  
                  )
                
              
              =
              
              ⁢
              
                
                  ∑
                  j
                
                ⁢
                
                  
                
                ⁢
                
                  {
                  
                    
                      
                        
                          
                            
                              
                              
                                
                                  
                                    c
                                    θ
                                    2
                                  
                                  ⁢
                                  
                                    XX
                                    j
                                  
                                
                                +
                                
                                  
                                    s
                                    θ
                                    2
                                  
                                  ⁢
                                  
                                    YY
                                    j
                                  
                                
                                -
                                
                                  
                                    s
                                    θ
                                  
                                  ⁢
                                  
                                    
                                      c
                                      θ
                                    
                                    ⁡
                                    
                                      (
                                      
                                        
                                          XY
                                          j
                                        
                                        +
                                        
                                          YX
                                          j
                                        
                                      
                                      )
                                    
                                  
                                
                                -
                                
                                  Δ
                                  11
                                  j
                                
                              
                              
                            
                            2
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              
                              
                                
                                  
                                    s
                                    θ
                                    2
                                  
                                  ⁢
                                  
                                    XX
                                    j
                                  
                                
                                +
                                
                                  
                                    c
                                    θ
                                    2
                                  
                                  ⁢
                                  
                                    YY
                                    j
                                  
                                
                                +
                                
                                  
                                    s
                                    θ
                                  
                                  ⁢
                                  
                                    
                                      c
                                      θ
                                    
                                    ⁡
                                    
                                      (
                                      
                                        
                                          XY
                                          j
                                        
                                        +
                                        
                                          YX
                                          j
                                        
                                      
                                      )
                                    
                                  
                                
                                -
                                
                                  Δ
                                  22
                                  j
                                
                              
                              
                            
                            2
                          
                          -
                        
                      
                    
                    
                      
                        
                          0.5
                          ·
                          
                            
                              
                              
                                
                                  
                                    c
                                    
                                      2
                                      ⁢
                                      θ
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        XY
                                        j
                                      
                                      +
                                      
                                        YX
                                        j
                                      
                                    
                                    )
                                  
                                
                                +
                                
                                  
                                    s
                                    
                                      2
                                      ⁢
                                      θ
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        XX
                                        j
                                      
                                      -
                                      
                                        YY
                                        j
                                      
                                    
                                    )
                                  
                                
                              
                              
                            
                            2
                          
                        
                      
                    
                  
                  }
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  ∑
                  j
                
                ⁢
                
                  
                
                ⁢
                
                  {
                  
                    
                      
                        
                          
                            
                              
                              
                                
                                  
                                    c
                                    θ
                                    2
                                  
                                  ⁢
                                  
                                    XX
                                    j
                                  
                                
                                +
                                
                                  
                                    s
                                    θ
                                    2
                                  
                                  ⁢
                                  
                                    YY
                                    j
                                  
                                
                                -
                                
                                  
                                    s
                                    θ
                                  
                                  ⁢
                                  
                                    
                                      c
                                      θ
                                    
                                    ⁡
                                    
                                      (
                                      
                                        
                                          XY
                                          j
                                        
                                        +
                                        
                                          YX
                                          j
                                        
                                      
                                      )
                                    
                                  
                                
                              
                              
                            
                            2
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            
                              
                              
                                Δ
                                11
                                j
                              
                              
                            
                            2
                          
                          -
                          
                            2
                            ⁢
                            
                              Re
                              [
                              
                                
                                  Δ
                                  11
                                  j
                                
                                (
                                
                                  
                                    
                                      c
                                      θ
                                      2
                                    
                                    ⁢
                                    
                                      XX
                                      j
                                    
                                  
                                  +
                                  
                                    
                                      s
                                      θ
                                      2
                                    
                                    ⁢
                                    
                                      YY
                                      j
                                    
                                  
                                  -
                                  
                                    
                                      s
                                      θ
                                    
                                    ⁢
                                    
                                      
                                        c
                                        θ
                                      
                                      (
                                      
                                        
                                          XY
                                          j
                                        
                                        +
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  
                                    YX
                                    j
                                  
                                  )
                                
                                )
                              
                              *
                            
                            ]
                          
                          +
                          
                            
                            
                              
                                
                                  s
                                  θ
                                  2
                                
                                ⁢
                                
                                  XX
                                  j
                                
                              
                              +
                              
                                
                                  c
                                  θ
                                  2
                                
                                ⁢
                                
                                  YY
                                  j
                                
                              
                              +
                              
                                
                                  s
                                  θ
                                
                                ⁢
                                
                                  
                                    c
                                    θ
                                  
                                  (
                                  
                                    
                                      XY
                                      j
                                    
                                    +
                                  
                                
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  YX
                                  j
                                
                                )
                              
                              
                            
                            2
                          
                          +
                          
                            
                              
                              
                                Δ
                                22
                                j
                              
                              
                            
                            2
                          
                          -
                          
                            2
                            ⁢
                            
                              Re
                              [
                              
                                
                                  Δ
                                  22
                                  j
                                
                                (
                                
                                  
                                    
                                      s
                                      θ
                                      2
                                    
                                    ⁢
                                    
                                      XX
                                      j
                                    
                                  
                                  +
                                  
                                    
                                      c
                                      θ
                                      2
                                    
                                    ⁢
                                    
                                      YY
                                      j
                                    
                                  
                                  +
                                
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  
                                    s
                                    θ
                                  
                                  ⁢
                                  
                                    
                                      c
                                      θ
                                    
                                    ⁡
                                    
                                      (
                                      
                                        
                                          XY
                                          j
                                        
                                        +
                                        
                                          YX
                                          j
                                        
                                      
                                      )
                                    
                                  
                                
                                )
                              
                              *
                            
                            ]
                          
                          +
                          
                            0.5
                            ·
                          
                        
                      
                    
                    
                      
                        
                          (
                          
                            
                              
                                c
                                
                                  2
                                  ⁢
                                  θ
                                
                                2
                              
                              ⁢
                              
                                
                                  
                                  
                                    
                                      XY
                                      j
                                    
                                    +
                                    
                                      YX
                                      j
                                    
                                  
                                  
                                
                                2
                              
                            
                            +
                            
                              
                                s
                                
                                  2
                                  ⁢
                                  θ
                                
                                2
                              
                              ⁢
                              
                                
                                  
                                  
                                    
                                      XX
                                      j
                                    
                                    -
                                    
                                      YY
                                      j
                                    
                                  
                                  
                                
                                2
                              
                            
                            +
                          
                        
                      
                    
                    
                      
                        
                          
                            2
                            ⁢
                            
                              s
                              
                                2
                                ⁢
                                θ
                              
                            
                            ⁢
                            
                              c
                              
                                2
                                ⁢
                                θ
                              
                            
                            ⁢
                            
                              Re
                              ⁡
                              
                                [
                                
                                  
                                    (
                                    
                                      
                                        XY
                                        j
                                      
                                      +
                                      
                                        YX
                                        j
                                      
                                    
                                    )
                                  
                                  ⁢
                                  
                                    
                                      (
                                      
                                        
                                          XX
                                          j
                                        
                                        -
                                        
                                          YY
                                          j
                                        
                                      
                                      )
                                    
                                    *
                                  
                                
                                ]
                              
                            
                          
                          )
                        
                      
                    
                  
                  }
                
              
            
          
        
      
    
    
      
        (
        H53
        )
      
    
  




Substitute the identities cθ2=0.5(1+c2θ) and sθ2=0.5(1−c2θ) into equation (H53) gives





  
    
      
        
          
            
              
                O
                ⁡
                
                  (
                  
                    θ
                    ,
                    
                      ω
                      ST
                    
                  
                  )
                
              
              =
              
              ⁢
              
                
                  ∑
                  j
                
                ⁢
                
                  
                
                ⁢
                
                  {
                  
                    
                      
                        
                          
                          
                            
                              0.5
                              ⁢
                              
                                (
                                
                                  
                                    XX
                                    j
                                  
                                  +
                                  
                                    YY
                                    j
                                  
                                
                                )
                              
                            
                            +
                            
                              0.5
                              ⁢
                              
                                
                                  c
                                  
                                    2
                                    ⁢
                                    θ
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      XX
                                      j
                                    
                                    -
                                    
                                      YY
                                      j
                                    
                                  
                                  )
                                
                              
                            
                            -
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                0.5
                                ⁢
                                
                                  
                                    s
                                    
                                      2
                                      ⁢
                                      θ
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        XY
                                        j
                                      
                                      +
                                      
                                        YX
                                        j
                                      
                                    
                                    )
                                  
                                
                              
                              
                            
                            2
                          
                          +
                          
                            
                              
                              
                                Δ
                                11
                                j
                              
                              
                            
                            2
                          
                          -
                        
                      
                    
                    
                      
                        
                          2
                          ⁢
                          
                            Re
                            [
                            
                              
                                Δ
                                11
                                j
                              
                              (
                              
                                
                                  
                                    c
                                    θ
                                    2
                                  
                                  ⁢
                                  
                                    XX
                                    j
                                  
                                
                                +
                                
                                  
                                    s
                                    θ
                                    2
                                  
                                  ⁢
                                  
                                    YY
                                    j
                                  
                                
                                -
                                
                                  
                                    s
                                    θ
                                  
                                  ⁢
                                  
                                    
                                      c
                                      θ
                                    
                                    (
                                    
                                      
                                        XY
                                        j
                                      
                                      +
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  
                                    YX
                                    j
                                  
                                  )
                                
                                )
                              
                              *
                            
                            ]
                          
                          +
                          
                            
                            
                              
                                0.5
                                ⁢
                                
                                  (
                                  
                                    
                                      XX
                                      j
                                    
                                    +
                                    
                                      YY
                                      j
                                    
                                  
                                  )
                                
                              
                              -
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  0.5
                                  ⁢
                                  
                                    
                                      c
                                      
                                        2
                                        ⁢
                                        θ
                                      
                                    
                                    ⁡
                                    
                                      (
                                      
                                        
                                          XX
                                          j
                                        
                                        -
                                        
                                          YY
                                          j
                                        
                                      
                                      )
                                    
                                  
                                
                                +
                                
                                  0.5
                                  ⁢
                                  
                                    
                                      s
                                      
                                        2
                                        ⁢
                                        θ
                                      
                                    
                                    ⁡
                                    
                                      (
                                      
                                        
                                          XY
                                          j
                                        
                                        +
                                        
                                          YX
                                          j
                                        
                                      
                                      )
                                    
                                  
                                
                              
                              
                            
                            2
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            
                              
                              
                                Δ
                                22
                                j
                              
                              
                            
                            2
                          
                          -
                          
                            2
                            ⁢
                            
                              Re
                              [
                              
                                
                                  Δ
                                  22
                                  j
                                
                                (
                                
                                  
                                    
                                      s
                                      θ
                                      2
                                    
                                    ⁢
                                    
                                      XX
                                      j
                                    
                                  
                                  +
                                  
                                    
                                      c
                                      θ
                                      2
                                    
                                    ⁢
                                    
                                      YY
                                      j
                                    
                                  
                                  +
                                
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  
                                    s
                                    θ
                                  
                                  ⁢
                                  
                                    
                                      c
                                      θ
                                    
                                    ⁡
                                    
                                      (
                                      
                                        
                                          XY
                                          j
                                        
                                        +
                                        
                                          YX
                                          j
                                        
                                      
                                      )
                                    
                                  
                                
                                )
                              
                              *
                            
                            ]
                          
                          +
                          
                            0.5
                            ·
                            
                              (
                              
                                
                                  
                                    c
                                    
                                      2
                                      ⁢
                                      θ
                                    
                                    2
                                  
                                  ⁢
                                  
                                    
                                      
                                      
                                        
                                          XY
                                          j
                                        
                                        +
                                        
                                          YX
                                          j
                                        
                                      
                                      
                                    
                                    2
                                  
                                
                                +
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              s
                              
                                2
                                ⁢
                                θ
                              
                              2
                            
                            ⁢
                            
                              
                                
                                
                                  
                                    XX
                                    j
                                  
                                  -
                                  
                                    YY
                                    j
                                  
                                
                                
                              
                              2
                            
                          
                          +
                          
                            2
                            ⁢
                            
                              s
                              
                                2
                                ⁢
                                θ
                              
                            
                            ⁢
                            
                              c
                              
                                2
                                ⁢
                                θ
                              
                            
                            ⁢
                            
                              Re
                              [
                              
                                (
                                
                                  
                                    XY
                                    j
                                  
                                  +
                                  
                                    YX
                                    j
                                  
                                
                                )
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                (
                                
                                  
                                    XX
                                    j
                                  
                                  -
                                  
                                    YY
                                    j
                                  
                                
                                )
                              
                              *
                            
                            ]
                          
                          )
                        
                      
                    
                  
                  }
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  ∑
                  j
                
                ⁢
                
                  
                
                ⁢
                
                  {
                  
                    
                      
                        
                          0.5
                          ⁢
                          
                            (
                            
                              
                                
                                  
                                  
                                    
                                      XX
                                      j
                                    
                                    +
                                    
                                      YY
                                      j
                                    
                                  
                                  
                                
                                2
                              
                              +
                              
                                
                                  c
                                  
                                    2
                                    ⁢
                                    θ
                                  
                                  2
                                
                                ⁢
                                
                                  
                                    
                                    
                                      
                                        XX
                                        j
                                      
                                      -
                                      
                                        YY
                                        j
                                      
                                    
                                    
                                  
                                  2
                                
                              
                              +
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              s
                              
                                2
                                ⁢
                                θ
                              
                              2
                            
                            ⁢
                            
                              
                                
                                
                                  
                                    XY
                                    j
                                  
                                  +
                                  
                                    YX
                                    j
                                  
                                
                                
                              
                              2
                            
                          
                          -
                          
                            2
                            ⁢
                            
                              s
                              
                                2
                                ⁢
                                θ
                              
                            
                            ⁢
                            
                              c
                              
                                2
                                ⁢
                                θ
                              
                            
                            ⁢
                            
                              Re
                              [
                              
                                (
                                
                                  
                                    XX
                                    j
                                  
                                  -
                                  
                                    YY
                                    j
                                  
                                
                                )
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      XY
                                      j
                                    
                                    +
                                    
                                      YX
                                      j
                                    
                                  
                                  )
                                
                                *
                              
                              ]
                            
                            )
                          
                          +
                          
                            
                              
                              
                                Δ
                                11
                                j
                              
                              
                            
                            2
                          
                          -
                          
                            2
                            ⁢
                            
                              Re
                              [
                              
                                
                                  Δ
                                  11
                                  j
                                
                                (
                                
                                  
                                    
                                      c
                                      θ
                                      2
                                    
                                    ⁢
                                    
                                      XX
                                      j
                                    
                                  
                                  +
                                
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  
                                    
                                      s
                                      θ
                                      2
                                    
                                    ⁢
                                    
                                      YY
                                      j
                                    
                                  
                                  -
                                  
                                    
                                      s
                                      θ
                                    
                                    ⁢
                                    
                                      
                                        c
                                        θ
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            XY
                                            j
                                          
                                          +
                                          
                                            YX
                                            j
                                          
                                        
                                        )
                                      
                                    
                                  
                                
                                )
                              
                              *
                            
                            ]
                          
                          +
                          
                            
                              
                              
                                Δ
                                22
                                j
                              
                              
                            
                            2
                          
                          -
                        
                      
                    
                    
                      
                        
                          2
                          ⁢
                          
                            Re
                            [
                            
                              
                                Δ
                                22
                                j
                              
                              (
                              
                                
                                  
                                    s
                                    θ
                                    2
                                  
                                  ⁢
                                  
                                    XX
                                    j
                                  
                                
                                +
                                
                                  
                                    c
                                    θ
                                    2
                                  
                                  ⁢
                                  
                                    YY
                                    j
                                  
                                
                                +
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  
                                    s
                                    θ
                                  
                                  ⁢
                                  
                                    
                                      c
                                      θ
                                    
                                    ⁡
                                    
                                      (
                                      
                                        
                                          XY
                                          j
                                        
                                        +
                                        
                                          YX
                                          j
                                        
                                      
                                      )
                                    
                                  
                                
                                )
                              
                              *
                            
                            ]
                          
                          +
                        
                      
                    
                    
                      
                        
                          0.5
                          ·
                          
                            (
                            
                              
                                
                                  c
                                  
                                    2
                                    ⁢
                                    θ
                                  
                                  2
                                
                                ⁢
                                
                                  
                                    
                                    
                                      
                                        XY
                                        j
                                      
                                      +
                                      
                                        YX
                                        j
                                      
                                    
                                    
                                  
                                  2
                                
                              
                              +
                              
                                
                                  s
                                  
                                    2
                                    ⁢
                                    θ
                                  
                                  2
                                
                                ⁢
                                
                                  
                                    
                                    
                                      
                                        XX
                                        j
                                      
                                      -
                                      
                                        YY
                                        j
                                      
                                    
                                    
                                  
                                  2
                                
                              
                              +
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            2
                            ⁢
                            
                              s
                              
                                2
                                ⁢
                                θ
                              
                            
                            ⁢
                            
                              c
                              
                                2
                                ⁢
                                θ
                              
                            
                            ⁢
                            
                              Re
                              ⁡
                              
                                [
                                
                                  
                                    (
                                    
                                      
                                        XY
                                        j
                                      
                                      +
                                      
                                        YX
                                        j
                                      
                                    
                                    )
                                  
                                  ⁢
                                  
                                    
                                      (
                                      
                                        
                                          XX
                                          j
                                        
                                        -
                                        
                                          YY
                                          j
                                        
                                      
                                      )
                                    
                                    *
                                  
                                
                                ]
                              
                            
                          
                          )
                        
                      
                    
                  
                  }
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  ∑
                  j
                
                ⁢
                
                  
                
                ⁢
                
                  
                    {
                    
                      
                        
                          
                            0.5
                            ⁢
                            
                              (
                              
                                
                                  
                                    
                                    
                                      
                                        XX
                                        j
                                      
                                      +
                                      
                                        YY
                                        j
                                      
                                    
                                    
                                  
                                  2
                                
                                +
                                
                                  
                                    
                                    
                                      
                                        XX
                                        j
                                      
                                      -
                                      
                                        YY
                                        j
                                      
                                    
                                    
                                  
                                  2
                                
                                +
                              
                            
                          
                        
                      
                      
                        
                          
                            
                              
                                
                                  
                                  
                                    
                                      XY
                                      j
                                    
                                    +
                                    
                                      YX
                                      j
                                    
                                  
                                  
                                
                                2
                              
                              )
                            
                            +
                            
                              
                                
                                
                                  Δ
                                  11
                                  j
                                
                                
                              
                              2
                            
                            -
                            
                              2
                              ⁢
                              
                                Re
                                [
                                
                                  
                                    Δ
                                    11
                                    j
                                  
                                  (
                                  
                                    
                                      
                                        c
                                        θ
                                        2
                                      
                                      ⁢
                                      
                                        XX
                                        j
                                      
                                    
                                    +
                                  
                                
                              
                            
                          
                        
                      
                      
                        
                          
                            
                              
                                
                                  
                                    
                                      
                                        s
                                        θ
                                        2
                                      
                                      ⁢
                                      
                                        YY
                                        j
                                      
                                    
                                    -
                                    
                                      
                                        s
                                        θ
                                      
                                      ⁢
                                      
                                        
                                          c
                                          θ
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              XY
                                              j
                                            
                                            +
                                            
                                              YX
                                              j
                                            
                                          
                                          )
                                        
                                      
                                    
                                  
                                  )
                                
                                *
                              
                              ]
                            
                            +
                            
                              
                                
                                
                                  Δ
                                  22
                                  j
                                
                                
                              
                              2
                            
                            -
                          
                        
                      
                      
                        
                          
                            2
                            ⁢
                            
                              Re
                              ⁡
                              
                                [
                                
                                  
                                    
                                      Δ
                                      22
                                      j
                                    
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            s
                                            θ
                                            2
                                          
                                          ⁢
                                          
                                            XX
                                            j
                                          
                                        
                                        +
                                        
                                          
                                            c
                                            θ
                                            2
                                          
                                          ⁢
                                          
                                            YY
                                            j
                                          
                                        
                                        +
                                        
                                          
                                            s
                                            θ
                                          
                                          ⁢
                                          
                                            
                                              c
                                              θ
                                            
                                            ⁡
                                            
                                              (
                                              
                                                
                                                  XY
                                                  j
                                                
                                                +
                                                
                                                  YX
                                                  j
                                                
                                              
                                              )
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                  *
                                
                                ]
                              
                            
                          
                        
                      
                    
                    }
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        H54
        )
      
    
  




Differentiating with respect to θ yields





  
    
      
        
          
            
              ∑
              j
            
            ⁢
            
              Re
              ⁡
              
                [
                
                  
                    s
                    
                      2
                      ⁢
                      θ
                    
                  
                  ⁢
                  
                    
                      
                        Δ
                        11
                        j
                      
                      ⁡
                      
                        (
                        
                          
                            (
                            
                              
                                YY
                                j
                              
                              -
                              
                                XX
                                j
                              
                            
                            )
                          
                          -
                          
                            
                              c
                              
                                2
                                ⁢
                                θ
                              
                            
                            ⁡
                            
                              (
                              
                                
                                  XY
                                  j
                                
                                +
                                
                                  YX
                                  j
                                
                              
                              )
                            
                          
                        
                        )
                      
                    
                    *
                  
                
                ]
              
            
          
          -
          
            Re
            ⁡
            
              [
              
                
                  s
                  
                    2
                    ⁢
                    θ
                  
                
                ⁢
                
                  
                    
                      Δ
                      22
                      j
                    
                    ⁡
                    
                      (
                      
                        
                          (
                          
                            
                              YY
                              j
                            
                            -
                            
                              XX
                              j
                            
                          
                          )
                        
                        -
                        
                          
                            c
                            
                              2
                              ⁢
                              θ
                            
                          
                          ⁡
                          
                            (
                            
                              
                                XY
                                j
                              
                              +
                              
                                YX
                                j
                              
                            
                            )
                          
                        
                      
                      )
                    
                  
                  *
                
              
              ]
            
          
        
        =
        0.
      
    
    
      
        (
        H55
        )
      
    
  




Rearranging gives

s2θRe[(Δ22j−Δ11j)(YYj−XXj)*]−c2θRe[(Δ22j−Δ11j)(XYj+YXj)*]=0.   (H56)

Define

dj=Δ22j−Δ11j,aj=XYj+YXj,bj=YYj−XXj   (H57)

to get





  
    
      
        
          tan
          ⁡
          
            (
            
              2
              ⁢
              φ
            
            )
          
        
        =
        
          
            
              Re
              [
              
                
                  
                    a
                    ⇀
                  
                  *
                
                ·
                d
              
              ]
            
            
              Re
              [
              
                
                  
                    b
                    ⇀
                  
                  *
                
                ·
                
                  d
                  ⇀
                
              
              ]
            
          
          .
        
      
    
    
      
        (
        H58
        )
      
    
  




If we define φ as the angle of the fast axis relative to the X transmitter then φ→−φ in equation (H58). φ is ambiguous by a multiple of π/2. This is handled the same way as noted previously.

Now we will move on to the general Alford rotation version of the algorithms. There are many ways to do this. First we need to take care of some notation since we are running out of letters. The generalized Alford rotation equations denoted different 2×2 waveforms by the indices (m.n)∈{(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)}. From now on this ordered set of indices will be denoted by a letter with subscript A. Letters without subscript A refer to receiver ring positions. For the 4 dipole receiver case we define

nA=1→(m.n)=(0,1),

nA=2→(m.n)=(0,2),

nA=3→(m.n)=(0,3),

nA=4→(m.n)=(1,2),

nA=5→(m.n)=(1,3),

nA=6→(m.n)=(2,3).   (H59)

Then the generalized Alford rotation equations look like this,




  
    
      
        
          
        
        ⁢
        
          
            
              
                
                  T
                  φ
                  T
                
                ⁢
                
                  X
                  
                    n
                    A
                  
                
                ⁢
                
                  T
                  φ
                
              
              =
              
                [
                
                  
                    
                      
                        S
                        f
                      
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        S
                        s
                      
                    
                  
                
                ]
              
            
            ,
            
              

            
            ⁢
            
              
            
            ⁢
            where
          
          ⁢
          
            

          
          ⁢
          
            
              
                X
                
                  n
                  A
                
              
              =
              
                
                  [
                  
                    
                      
                        
                          X
                          11
                          
                            n
                            A
                          
                        
                      
                      
                        
                          X
                          12
                          
                            n
                            A
                          
                        
                      
                    
                    
                      
                        
                          X
                          21
                          
                            n
                            A
                          
                        
                      
                      
                        
                          X
                          22
                          
                            n
                            A
                          
                        
                      
                    
                  
                  ]
                
                =
                
                  
                    [
                    
                      
                    
                    ⁢
                    
                      
                        
                          
                            X
                            
                              n
                              ⁡
                              
                                (
                                
                                  n
                                  A
                                
                                )
                              
                            
                          
                        
                        
                          
                            X
                            
                              m
                              ⁡
                              
                                (
                                
                                  n
                                  A
                                
                                )
                              
                            
                          
                        
                      
                      
                        
                          
                            Y
                            
                              n
                              ⁡
                              
                                (
                                
                                  n
                                  A
                                
                                )
                              
                            
                          
                        
                        
                          
                            Y
                            
                              m
                              ⁡
                              
                                (
                                
                                  n
                                  A
                                
                                )
                              
                            
                          
                        
                      
                    
                    ]
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    {
                    
                      I
                      -
                      
                        
                          c
                          
                            (
                            
                              
                                θ
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  n
                                  ⁡
                                  
                                    (
                                    
                                      n
                                      A
                                    
                                    )
                                  
                                
                              
                              -
                              
                                θ
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  m
                                  ⁡
                                  
                                    (
                                    
                                      n
                                      A
                                    
                                    )
                                  
                                
                              
                            
                            )
                          
                        
                        ⁢
                        P
                      
                    
                    }
                  
                  ⁢
                  
                    T
                    
                      
                        n
                        ⁡
                        
                          (
                          
                            n
                            A
                          
                          )
                        
                      
                      ⁢
                      
                        m
                        ⁡
                        
                          (
                          
                            n
                            A
                          
                          )
                        
                      
                    
                  
                
              
            
            ,
          
        
      
    
    
      
        (
        H60
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            T
            
              
                n
                ⁡
                
                  (
                  
                    n
                    A
                  
                  )
                
              
              ⁢
              
                m
                ⁡
                
                  (
                  
                    n
                    A
                  
                  )
                
              
            
          
          =
          
            
              
                [
                
                  
                
                ⁢
                
                  
                    
                      
                        c
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            n
                            ⁡
                            
                              (
                              
                                n
                                A
                              
                              )
                            
                          
                        
                      
                    
                    
                      
                        s
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            n
                            ⁡
                            
                              (
                              
                                n
                                A
                              
                              )
                            
                          
                        
                      
                    
                  
                  
                    
                      
                        c
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            m
                            ⁡
                            
                              (
                              
                                n
                                A
                              
                              )
                            
                          
                        
                      
                    
                    
                      
                        s
                        
                          θ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            m
                            ⁡
                            
                              (
                              
                                n
                                A
                              
                              )
                            
                          
                        
                      
                    
                  
                
                ]
              
              
                s
                
                  (
                  
                    
                      θ
                      ⁢
                      
                        
                      
                      ⁢
                      
                        n
                        ⁡
                        
                          (
                          
                            n
                            A
                          
                          )
                        
                      
                    
                    -
                    
                      θ
                      ⁢
                      
                        
                      
                      ⁢
                      
                        m
                        ⁡
                        
                          (
                          
                            n
                            A
                          
                          )
                        
                      
                    
                  
                  )
                
                2
              
            
            .
          
        
      
    
    
      
        (
        H61
        )
      
    
  




The general cross-dipole waveforms obey the same rules as the classic cross-dipole waveforms,




  
    
      
        
          
            
              
                
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          X
                          11
                          
                            n
                            A
                          
                        
                      
                      
                        
                          X
                          12
                          
                            n
                            A
                          
                        
                      
                    
                    
                      
                        
                          X
                          21
                          
                            n
                            A
                          
                        
                      
                      
                        
                          X
                          22
                          
                            n
                            A
                          
                        
                      
                    
                  
                  ]
                
                =
                
                ⁢
                
                  
                    
                      [
                      
                        
                          
                            
                              c
                              φ
                            
                          
                          
                            
                              s
                              φ
                            
                          
                        
                        
                          
                            
                              -
                              
                                s
                                φ
                              
                            
                          
                          
                            
                              c
                              φ
                            
                          
                        
                      
                      ]
                    
                    ⁡
                    
                      [
                      
                        
                          
                            
                              S
                              f
                            
                          
                          
                            0
                          
                        
                        
                          
                            0
                          
                          
                            
                              S
                              s
                            
                          
                        
                      
                      ]
                    
                  
                  ⁡
                  
                    [
                    
                      
                        
                          
                            c
                            φ
                          
                        
                        
                          
                            -
                            
                              s
                              φ
                            
                          
                        
                      
                      
                        
                          
                            s
                            φ
                          
                        
                        
                          
                            c
                            φ
                          
                        
                      
                    
                    ]
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  [
                  
                    
                      
                        
                          c
                          φ
                        
                      
                      
                        
                          s
                          φ
                        
                      
                    
                    
                      
                        
                          -
                          
                            s
                            φ
                          
                        
                      
                      
                        
                          c
                          φ
                        
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        
                          
                            S
                            f
                          
                          ⁢
                          
                            c
                            φ
                          
                        
                      
                      
                        
                          
                            -
                            
                              S
                              f
                            
                          
                          ⁢
                          
                            s
                            φ
                          
                        
                      
                    
                    
                      
                        
                          
                            S
                            s
                          
                          ⁢
                          
                            s
                            φ
                          
                        
                      
                      
                        
                          
                            S
                            s
                          
                          ⁢
                          
                            c
                            φ
                          
                        
                      
                    
                  
                  ]
                
              
            
          
        
        
          
            
              
                =
                
                ⁢
                
                  [
                  
                    
                      
                        
                          
                            
                              S
                              f
                            
                            ⁢
                            
                              c
                              φ
                              2
                            
                          
                          +
                          
                            
                              S
                              s
                            
                            ⁢
                            
                              s
                              φ
                              2
                            
                          
                        
                      
                      
                        
                          
                            (
                            
                              
                                S
                                s
                              
                              -
                              
                                S
                                f
                              
                            
                            )
                          
                          ⁢
                          
                            s
                            φ
                          
                          ⁢
                          
                            c
                            φ
                          
                        
                      
                    
                    
                      
                        
                          
                            (
                            
                              
                                S
                                s
                              
                              -
                              
                                S
                                f
                              
                            
                            )
                          
                          ⁢
                          
                            s
                            φ
                          
                          ⁢
                          
                            c
                            φ
                          
                        
                      
                      
                        
                          
                            
                              S
                              f
                            
                            ⁢
                            
                              s
                              φ
                              2
                            
                          
                          +
                          
                            
                              S
                              s
                            
                            ⁢
                            
                              c
                              φ
                              2
                            
                          
                        
                      
                    
                  
                  ]
                
              
              ,
              and
            
          
        
      
    
    
      
        (
        H62
        )
      
    
  
  
    
      
        
          [
          
            
              
                
                  S
                  f
                
              
              
                0
              
            
            
              
                0
              
              
                
                  S
                  s
                
              
            
          
          ]
        
        =
        
          [
          
            
              
                
                  (
                  
                    
                      
                        c
                        φ
                        2
                      
                      ⁢
                      
                        X
                        11
                        
                          n
                          A
                        
                      
                    
                    +
                    
                      
                        s
                        φ
                        2
                      
                      ⁢
                      
                        X
                        22
                        
                          n
                          A
                        
                      
                    
                    -
                    
                      
                        s
                        φ
                      
                      ⁢
                      
                        
                          c
                          φ
                        
                        ⁡
                        
                          (
                          
                            
                              X
                              12
                              
                                n
                                A
                              
                            
                            +
                            
                              X
                              21
                              
                                n
                                A
                              
                            
                          
                          )
                        
                      
                    
                  
                  )
                
              
              
                
                  0.5
                  ⁢
                  
                    (
                    
                      
                        
                          c
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            φ
                          
                        
                        ⁡
                        
                          (
                          
                            
                              X
                              12
                              
                                n
                                A
                              
                            
                            +
                            
                              X
                              21
                              
                                n
                                A
                              
                            
                          
                          )
                        
                      
                      +
                      
                        
                          s
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            φ
                          
                        
                        ⁡
                        
                          (
                          
                            
                              X
                              11
                              
                                n
                                A
                              
                            
                            -
                            
                              X
                              22
                              
                                n
                                A
                              
                            
                          
                          )
                        
                      
                    
                    )
                  
                
              
            
            
              
                
                  0.5
                  ⁢
                  
                    (
                    
                      
                        
                          c
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            φ
                          
                        
                        ⁡
                        
                          (
                          
                            
                              X
                              12
                              
                                n
                                A
                              
                            
                            +
                            
                              X
                              21
                              
                                n
                                A
                              
                            
                          
                          )
                        
                      
                      +
                      
                        
                          s
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            φ
                          
                        
                        ⁡
                        
                          (
                          
                            
                              X
                              11
                              
                                n
                                A
                              
                            
                            -
                            
                              X
                              22
                              
                                n
                                A
                              
                            
                          
                          )
                        
                      
                    
                    )
                  
                
              
              
                
                  (
                  
                    
                      
                        s
                        φ
                        2
                      
                      ⁢
                      
                        X
                        11
                        
                          n
                          A
                        
                      
                    
                    +
                    
                      
                        c
                        φ
                        2
                      
                      ⁢
                      
                        X
                        22
                        
                          n
                          A
                        
                      
                    
                    +
                    
                      
                        s
                        φ
                      
                      ⁢
                      
                        
                          c
                          φ
                        
                        ⁡
                        
                          (
                          
                            
                              X
                              12
                              
                                n
                                A
                              
                            
                            +
                            
                              X
                              21
                              
                                n
                                A
                              
                            
                          
                          )
                        
                      
                    
                  
                  )
                
              
            
          
          ]
        
      
    
    
      
        (
        H63
        )
      
    
  




A simple way to modify the algorithms is to redefine the generalized stacking index, j, to include the index nA of the generalized cross-dipole waveform. Then the solution for the angle has the same form. Thus for the time domain algorithm




  
    
      
        
          tan
          ⁡
          
            (
            
              2
              ⁢
              
                
              
              ⁢
              φ
            
            )
          
        
        =
        
          
            
              
                a
                ⇀
              
              ⁣
              
                ·
                
                  c
                  ⇀
                
              
            
            
              
                a
                ⇀
              
              ·
              
                b
                ⇀
              
            
          
          .
        
      
    
    
      
        (
        H64
        )
      
    
  




The generalized objective function is




  
    
      
        
          O
          ⁡
          
            (
            
              θ
              ,
              
                s
                1
              
              ,
              
                δ
                ⁢
                
                  
                
                ⁢
                s
              
            
            )
          
        
        =
        
          
            
              ∑
              
                
                  k
                  ,
                  m
                  ,
                  n
                
                
                  
                    m
                    A
                  
                  ,
                  
                    n
                    A
                  
                
              
            
            ⁢
            
              4
              ⁢
              
                
                  (
                  
                    
                      SP
                      
                        
                          m
                          A
                        
                        ,
                        
                          j
                          S
                        
                      
                    
                    -
                    
                      FP
                      
                        
                          n
                          A
                        
                        ,
                        
                          J
                          F
                        
                      
                    
                  
                  )
                
                2
              
            
          
          +
          
            
              (
              
                
                  SP
                  θ
                  
                    
                      m
                      A
                    
                    ,
                    
                      j
                      S
                    
                  
                
                -
                
                  FP
                  θ
                  
                    
                      n
                      A
                    
                    ,
                    
                      j
                      F
                    
                  
                
              
              )
            
            2
          
        
      
    
    
      
        (
        H65
        )
      
    
  




We re-define j(k,m,n,mA,nA)=((jF(k,m,n),mA), (jS(k,m,n),nA). Using this compact notation we have (see equation (H38)),





  
    
      
        
          
            
              
                O
                ⁡
                
                  (
                  
                    θ
                    ,
                    
                      s
                      1
                    
                    ,
                    
                      δ
                      ⁢
                      
                        
                      
                      ⁢
                      s
                    
                  
                  )
                
              
              =
              
              ⁢
              
                
                  ∑
                  j
                
                ⁢
                
                  {
                  
                    
                      
                        [
                        
                          
                            a
                            j
                          
                          +
                          
                            
                              c
                              
                                2
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                            
                            ⁢
                            
                              b
                              j
                            
                          
                          +
                          
                            
                              s
                              
                                2
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                            
                            ⁢
                            
                              c
                              j
                            
                          
                        
                        ]
                      
                      2
                    
                    +
                    
                      
                        [
                        
                          
                            
                              -
                              
                                s
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                              
                            
                            ⁢
                            
                              b
                              j
                            
                          
                          +
                          
                            
                              c
                              
                                2
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                            
                            ⁢
                            
                              c
                              j
                            
                          
                        
                        ]
                      
                      2
                    
                  
                  }
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  a
                  2
                
                +
                
                  
                    b
                    2
                  
                  ⁢
                  
                    c
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                    2
                  
                
                +
                
                  
                    c
                    2
                  
                  ⁢
                  
                    s
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                    2
                  
                
                +
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      a
                      ⇀
                    
                    ·
                    
                      b
                      ⇀
                    
                  
                  ⁢
                  
                    c
                    
                      2
                      ⁢
                      θ
                    
                  
                
                +
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      a
                      ⇀
                    
                    ·
                    
                      c
                      ⇀
                    
                  
                  ⁢
                  
                    s
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                  
                
                +
              
            
          
        
        
          
            
              
              ⁢
              
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      b
                      ⇀
                    
                    ·
                    
                      c
                      ⇀
                    
                  
                  ⁢
                  
                    s
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                  
                  ⁢
                  
                    c
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                  
                
                +
                
                  
                    b
                    2
                  
                  ⁢
                  
                    s
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                    2
                  
                
                +
                
                  
                    c
                    2
                  
                  ⁢
                  
                    c
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                    2
                  
                
                -
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      b
                      ⇀
                    
                    ·
                    
                      c
                      ⇀
                    
                  
                  ⁢
                  
                    s
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                  
                  ⁢
                  
                    c
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  a
                  2
                
                +
                
                  b
                  2
                
                +
                
                  c
                  2
                
                +
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      a
                      ⇀
                    
                    ·
                    
                      b
                      ⇀
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    c
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                  
                
                +
                
                  2
                  ⁢
                  
                    
                      a
                      ⇀
                    
                    ·
                    
                      c
                      ⇀
                    
                  
                  ⁢
                  
                    
                      s
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                    
                    .
                  
                
              
            
          
        
      
    
    
      
        (
        H66
        )
      
    
  




where

aj=X22mA,jS−X11nA,jF+X11mA,jS−X22nA,jF,

bj=X22mA,jS−X11nA,jF−X11mA,jS+X22nA,jF,

cj=X12mA,jS+X21mA,jS+X12nA,jF+X21nA,jF.   (H67)

There are many ways to incorporate the generalized waveforms. A less computationally intensive version is





  
    
      
        
          
            O
            ⁡
            
              (
              
                θ
                ,
                
                  s
                  1
                
                ,
                
                  δ
                  ⁢
                  
                    
                  
                  ⁢
                  s
                
              
              )
            
          
          =
          
            
              
                ∑
                
                  
                    k
                    ,
                    m
                    ,
                    n
                  
                  
                    m
                    A
                  
                
              
              ⁢
              
                4
                ⁢
                
                  
                    (
                    
                      
                        SP
                        
                          
                            m
                            A
                          
                          ,
                          
                            j
                            S
                          
                        
                      
                      -
                      
                        FP
                        
                          
                            m
                            A
                          
                          ,
                          
                            J
                            F
                          
                        
                      
                    
                    )
                  
                  2
                
              
            
            +
            
              
                (
                
                  
                    SP
                    θ
                    
                      
                        m
                        A
                      
                      ,
                      
                        j
                        S
                      
                    
                  
                  -
                  
                    FP
                    θ
                    
                      
                        m
                        A
                      
                      ,
                      
                        j
                        F
                      
                    
                  
                
                )
              
              2
            
          
        
        ,
      
    
    
      
        (
        H68
        )
      
    
  




in which case

aj=X22mA,jS−X11mA,jF+X11mA,jS−X22mA,jF,

bj=X22mA,jS−X11mA,jF−X11mA,jS+X22mA,jF,

cj=X12mA,jS+X21mA,jF+X12mA,jS+X21mA,jF.   (H68)

An extension of the frequency domain algorithm is




  
    
      
        
          
            
              
                O
                ⁡
                
                  (
                  
                    θ
                    ,
                    
                      ω
                      ST
                    
                  
                  )
                
              
              =
              
              ⁢
              
                
                  ∑
                  
                    
                      
                        
                          j
                          n
                        
                        ,
                        
                          j
                          m
                        
                      
                      
                        
                          m
                          A
                        
                        ,
                        
                          n
                          A
                        
                      
                    
                    
                      k
                      ,
                      l
                    
                  
                  
                    B
                    ⁡
                    
                      (
                      
                        
                          ω
                          ST
                        
                        ,
                        W
                      
                      )
                    
                  
                
                ⁢
                
                  
                    
                    
                      
                        (
                        
                          
                            
                              T
                              θ
                              T
                            
                            ⁢
                            
                              X
                              
                                
                                  m
                                  A
                                
                                ,
                                
                                  j
                                  n
                                
                              
                            
                            ⁢
                            
                              T
                              θ
                            
                          
                          -
                          
                            
                              D
                              
                                
                                  n
                                  A
                                
                                ,
                                
                                  j
                                  m
                                
                              
                            
                            ⁢
                            
                              Z
                              
                                est
                                ,
                                
                                  n
                                  A
                                
                              
                              
                                m
                                -
                                n
                              
                            
                          
                        
                        )
                      
                      kl
                    
                    
                  
                  2
                
              
            
          
        
        
          
            
              
                =
                
                ⁢
                
                  
                    ∑
                    
                      j
                      ,
                      k
                      ,
                      l
                    
                  
                  ⁢
                  
                    
                      
                      
                        
                          (
                          
                            
                              
                                T
                                θ
                                T
                              
                              ⁢
                              
                                X
                                j
                              
                              ⁢
                              
                                T
                                θ
                              
                            
                            -
                            
                              Δ
                              j
                            
                          
                          )
                        
                        kl
                      
                      
                    
                    2
                  
                
              
              ,
            
          
        
      
    
    
      
        (
        H70
        )
      
    
  




where

Xj=XmA,jn,

Δj=DnA,jmZest,nAm−n.   (H71)

The solution is





  
    
      
        
          
            tan
            ⁡
            
              (
              
                2
                ⁢
                
                  
                
                ⁢
                φ
              
              )
            
          
          =
          
            
              Re
              ⁡
              
                [
                
                  
                    
                      a
                      ⇀
                    
                    *
                  
                  ·
                  d
                
                ]
              
            
            
              Re
              ⁡
              
                [
                
                  
                    
                      b
                      ⇀
                    
                    *
                  
                  ·
                  
                    d
                    ⇀
                  
                
                ]
              
            
          
        
        ,
      
    
    
      
        (
        H72
        )
      
    
  




where

dj=Δ22nA,jm−Δ11nA,jm,aj=X12mA,jn+X21mA,jn+X21mA,jn,bj=X22mA,jn.   (H73)

An alternative embodiment of a frequency domain algorithm is




  
    
      
        
          O
          ⁡
          
            (
            
              θ
              ,
              
                ω
                ST
              
            
            )
          
        
        =
        
        ⁢
        
          
            
              ∑
              
                
                  j
                  n
                
                
                  
                    m
                    A
                  
                  ≠
                  
                    n
                    A
                  
                
              
              
                B
                ⁡
                
                  (
                  
                    
                      ω
                      ST
                    
                    ,
                    W
                  
                  )
                
              
            
            ⁢
            
              
                ∑
                k
              
              ⁢
              
                
                  
                  
                    
                      (
                      
                        
                          
                            T
                            θ
                            T
                          
                          ⁡
                          
                            (
                            
                              
                                X
                                
                                  
                                    m
                                    A
                                  
                                  ,
                                  
                                    j
                                    n
                                  
                                
                              
                              -
                              
                                X
                                
                                  
                                    n
                                    A
                                  
                                  ,
                                  
                                    j
                                    n
                                  
                                
                              
                            
                            )
                          
                        
                        ⁢
                        
                          T
                          θ
                        
                      
                      )
                    
                    kk
                  
                  
                
                2
              
            
          
          +
          
            
              ∑
              
                k
                ≠
                l
              
            
            ⁢
            
              {
              
                
                
                  
                    
                      (
                      
                        
                          T
                          θ
                          T
                        
                        ⁢
                        
                          X
                          
                            
                              m
                              A
                            
                            ,
                            
                              j
                              n
                            
                          
                        
                        ⁢
                        
                          T
                          θ
                        
                      
                      )
                    
                    kl
                  
                  ⁢
                  
                    
                      
                      2
                    
                    ⁢
                    
                      +
                      
                        
                          
                          
                            
                              (
                              
                                
                                  T
                                  θ
                                  T
                                
                                ⁢
                                
                                  X
                                  
                                    
                                      n
                                      A
                                    
                                    ,
                                    
                                      j
                                      n
                                    
                                  
                                
                                ⁢
                                
                                  T
                                  θ
                                
                              
                              )
                            
                            kl
                          
                          
                        
                        2
                      
                    
                    }
                  
                
              
            
          
        
      
    
    
      
        (
        H74
        )
      
    
  




Dropping the jn subscript for convenience, and substituting from equation (H63) we have





  
    
      
        
          O
          ⁢
          
            (
            
              θ
              ,
              
                ω
                ST
              
            
            )
          
        
        =
        
          
            ∑
            
              
                j
                n
              
              
                
                  m
                  A
                
                ≠
                
                  n
                  A
                
              
            
            
              B
              ⁡
              
                (
                
                  
                    ω
                    ST
                  
                  ,
                  W
                
                )
              
            
          
          ⁢
          
            {
            
              
                
                  
                    
                      
                        
                          
                            
                              
                              
                                
                                  
                                    
                                      
                                        
                                          c
                                          θ
                                          2
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              X
                                              11
                                              
                                                m
                                                A
                                              
                                            
                                            -
                                            
                                              X
                                              11
                                              
                                                n
                                                A
                                              
                                            
                                          
                                          )
                                        
                                      
                                      +
                                      
                                        
                                          s
                                          θ
                                          2
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              X
                                              22
                                              
                                                m
                                                A
                                              
                                            
                                            -
                                            
                                              X
                                              22
                                              
                                                n
                                                A
                                              
                                            
                                          
                                          )
                                        
                                      
                                      -
                                    
                                  
                                
                                
                                  
                                    
                                      
                                        s
                                        θ
                                      
                                      ⁢
                                      
                                        
                                          c
                                          θ
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              X
                                              12
                                              
                                                m
                                                A
                                              
                                            
                                            +
                                            
                                              X
                                              21
                                              
                                                m
                                                A
                                              
                                            
                                            -
                                            
                                              X
                                              12
                                              
                                                n
                                                A
                                              
                                            
                                            -
                                            
                                              X
                                              21
                                              
                                                n
                                                A
                                              
                                            
                                          
                                          )
                                        
                                      
                                    
                                  
                                
                              
                              
                            
                            2
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            
                              
                              
                                
                                  
                                    
                                      
                                        
                                          s
                                          θ
                                          2
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              X
                                              11
                                              
                                                m
                                                A
                                              
                                            
                                            -
                                            
                                              X
                                              11
                                              
                                                n
                                                A
                                              
                                            
                                          
                                          )
                                        
                                      
                                      +
                                      
                                        
                                          c
                                          θ
                                          2
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              X
                                              22
                                              
                                                m
                                                A
                                              
                                            
                                            -
                                            
                                              X
                                              22
                                              
                                                n
                                                A
                                              
                                            
                                          
                                          )
                                        
                                      
                                      +
                                    
                                  
                                
                                
                                  
                                    
                                      
                                        s
                                        θ
                                      
                                      ⁢
                                      
                                        
                                          c
                                          θ
                                        
                                        ⁡
                                        
                                          (
                                          
                                            
                                              X
                                              12
                                              
                                                m
                                                A
                                              
                                            
                                            +
                                            
                                              X
                                              21
                                              
                                                m
                                                A
                                              
                                            
                                            -
                                            
                                              X
                                              12
                                              
                                                n
                                                A
                                              
                                            
                                            -
                                            
                                              X
                                              21
                                              
                                                n
                                                A
                                              
                                            
                                          
                                          )
                                        
                                      
                                    
                                  
                                
                              
                              
                            
                            2
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            0.5
                            ·
                            
                              
                                
                                
                                  
                                    
                                      c
                                      
                                        2
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        θ
                                      
                                    
                                    ⁡
                                    
                                      (
                                      
                                        
                                          X
                                          12
                                          
                                            m
                                            A
                                          
                                        
                                        +
                                        
                                          X
                                          21
                                          
                                            m
                                            A
                                          
                                        
                                      
                                      )
                                    
                                  
                                  +
                                  
                                    
                                      s
                                      
                                        2
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        θ
                                      
                                    
                                    ⁡
                                    
                                      (
                                      
                                        
                                          X
                                          11
                                          
                                            m
                                            A
                                          
                                        
                                        -
                                        
                                          X
                                          22
                                          
                                            m
                                            A
                                          
                                        
                                      
                                      )
                                    
                                  
                                
                                
                              
                              2
                            
                          
                          +
                        
                      
                    
                  
                
              
              
                
                  
                    0.5
                    ·
                    
                      
                        
                        
                          
                            
                              c
                              
                                2
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                            
                            ⁡
                            
                              (
                              
                                
                                  X
                                  12
                                  
                                    n
                                    A
                                  
                                
                                +
                                
                                  X
                                  21
                                  
                                    n
                                    A
                                  
                                
                              
                              )
                            
                          
                          +
                          
                            
                              s
                              
                                2
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                            
                            ⁡
                            
                              (
                              
                                
                                  X
                                  11
                                  
                                    n
                                    A
                                  
                                
                                -
                                
                                  X
                                  22
                                  
                                    n
                                    A
                                  
                                
                              
                              )
                            
                          
                        
                        
                      
                      2
                    
                  
                
              
            
            }
          
        
      
    
    
      
        (
        H75
        )
      
    
  




Substitute the identities cθ2=0.5(1+c2θ) and sθ2=0.5(1−c2θ) into equation (H75) gives




  
    
      
        
          O
          ⁡
          
            (
            
              θ
              ,
              
                ω
                ST
              
            
            )
          
        
        =
        
          
            0.25
            ·
            
              
                ∑
                
                  
                    j
                    n
                  
                  
                    
                      m
                      A
                    
                    ≠
                    
                      n
                      A
                    
                  
                
                
                  B
                  ⁡
                  
                    (
                    
                      
                        ω
                        ST
                      
                      ,
                      W
                    
                    )
                  
                
              
              ⁢
              
                {
                
                  
                    
                      
                        
                        
                          
                            
                              (
                              
                                1
                                +
                                
                                  c
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              (
                              
                                
                                  X
                                  11
                                  
                                    m
                                    A
                                  
                                
                                -
                                
                                  X
                                  11
                                  
                                    n
                                    A
                                  
                                
                              
                              )
                            
                          
                          +
                          
                            (
                            
                              1
                              -
                              
                                c
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          (
                          
                            
                              X
                              22
                              
                                m
                                A
                              
                            
                            -
                            
                              X
                              22
                              
                                n
                                A
                              
                            
                          
                          )
                        
                        -
                        
                          
                            s
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                          
                          (
                          
                            
                              X
                              12
                              
                                m
                                A
                              
                            
                            +
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          
                            
                              
                                
                                  X
                                  21
                                  
                                    m
                                    A
                                  
                                
                                -
                                
                                  X
                                  12
                                  
                                    n
                                    A
                                  
                                
                                -
                                
                                  X
                                  21
                                  
                                    n
                                    A
                                  
                                
                              
                              )
                            
                            
                          
                          2
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                        
                          
                            
                              (
                              
                                1
                                -
                                
                                  c
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                
                              
                              )
                            
                            ⁢
                            
                              (
                              
                                
                                  X
                                  11
                                  
                                    m
                                    A
                                  
                                
                                -
                                
                                  X
                                  11
                                  
                                    n
                                    A
                                  
                                
                              
                              )
                            
                          
                          +
                          
                            (
                            
                              1
                              +
                              
                                c
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                              
                            
                            )
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          (
                          
                            
                              X
                              22
                              
                                m
                                A
                              
                            
                            -
                            
                              X
                              22
                              
                                n
                                A
                              
                            
                          
                          )
                        
                        +
                        
                          
                            s
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                          
                          (
                          
                            
                              X
                              12
                              
                                m
                                A
                              
                            
                            +
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          
                            
                              
                                
                                  X
                                  21
                                  
                                    m
                                    A
                                  
                                
                                -
                                
                                  X
                                  12
                                  
                                    n
                                    A
                                  
                                
                                -
                                
                                  X
                                  21
                                  
                                    n
                                    A
                                  
                                
                              
                              )
                            
                            
                          
                          2
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          2
                          ·
                          
                            
                              
                              
                                
                                  
                                    c
                                    
                                      2
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      θ
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        X
                                        12
                                        
                                          m
                                          A
                                        
                                      
                                      +
                                      
                                        X
                                        21
                                        
                                          m
                                          A
                                        
                                      
                                    
                                    )
                                  
                                
                                +
                                
                                  
                                    s
                                    
                                      2
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      θ
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        X
                                        11
                                        
                                          m
                                          A
                                        
                                      
                                      -
                                      
                                        X
                                        22
                                        
                                          m
                                          A
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              
                            
                            2
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        2
                        ·
                        
                          
                            
                            
                              
                                
                                  c
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      X
                                      12
                                      
                                        n
                                        A
                                      
                                    
                                    +
                                    
                                      X
                                      21
                                      
                                        n
                                        A
                                      
                                    
                                  
                                  )
                                
                              
                              +
                              
                                
                                  s
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      X
                                      11
                                      
                                        n
                                        A
                                      
                                    
                                    -
                                    
                                      X
                                      22
                                      
                                        n
                                        A
                                      
                                    
                                  
                                  )
                                
                              
                            
                            
                          
                          2
                        
                      
                    
                  
                
                }
              
            
          
          =
          
            
              0.25
              ·
              
                
                  
                    ∑
                    
                      B
                      ⁡
                      
                        (
                        
                          
                            ω
                            ST
                          
                          ,
                          W
                        
                        )
                      
                    
                  
                  
                    
                      j
                      n
                    
                    
                      
                        m
                        A
                      
                      ≠
                      
                        n
                        A
                      
                    
                  
                
                ⁢
                
                  {
                  
                    
                      
                        
                          
                            
                              
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              (
                                              
                                                
                                                  X
                                                  11
                                                  
                                                    m
                                                    A
                                                  
                                                
                                                -
                                                
                                                  X
                                                  11
                                                  
                                                    n
                                                    A
                                                  
                                                
                                                +
                                                
                                                  X
                                                  22
                                                  
                                                    m
                                                    A
                                                  
                                                
                                                -
                                                
                                                  X
                                                  22
                                                  
                                                    n
                                                    A
                                                  
                                                
                                              
                                              )
                                            
                                            +
                                          
                                        
                                      
                                      
                                        
                                          
                                            
                                              
                                                c
                                                
                                                  2
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  θ
                                                
                                              
                                              ⁡
                                              
                                                (
                                                
                                                  
                                                    X
                                                    11
                                                    
                                                      m
                                                      A
                                                    
                                                  
                                                  -
                                                  
                                                    X
                                                    11
                                                    
                                                      n
                                                      A
                                                    
                                                  
                                                  +
                                                  
                                                    X
                                                    22
                                                    
                                                      m
                                                      A
                                                    
                                                  
                                                  -
                                                  
                                                    X
                                                    22
                                                    
                                                      n
                                                      A
                                                    
                                                  
                                                
                                                )
                                              
                                            
                                            -
                                          
                                        
                                      
                                    
                                  
                                
                                
                                  
                                    
                                      
                                        s
                                        
                                          2
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          θ
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            X
                                            12
                                            
                                              m
                                              A
                                            
                                          
                                          +
                                          
                                            X
                                            21
                                            
                                              m
                                              A
                                            
                                          
                                          -
                                          
                                            X
                                            12
                                            
                                              n
                                              A
                                            
                                          
                                          -
                                          
                                            X
                                            21
                                            
                                              n
                                              A
                                            
                                          
                                        
                                        )
                                      
                                    
                                  
                                
                              
                              
                            
                            2
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            
                              
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              (
                                              
                                                
                                                  X
                                                  11
                                                  
                                                    m
                                                    A
                                                  
                                                
                                                -
                                                
                                                  X
                                                  11
                                                  
                                                    n
                                                    A
                                                  
                                                
                                                +
                                                
                                                  X
                                                  22
                                                  
                                                    m
                                                    A
                                                  
                                                
                                                -
                                                
                                                  X
                                                  22
                                                  
                                                    n
                                                    A
                                                  
                                                
                                              
                                              )
                                            
                                            -
                                          
                                        
                                      
                                      
                                        
                                          
                                            
                                              
                                                c
                                                
                                                  2
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  θ
                                                
                                              
                                              ⁡
                                              
                                                (
                                                
                                                  
                                                    X
                                                    11
                                                    
                                                      m
                                                      A
                                                    
                                                  
                                                  -
                                                  
                                                    X
                                                    11
                                                    
                                                      n
                                                      A
                                                    
                                                  
                                                  +
                                                  
                                                    X
                                                    22
                                                    
                                                      m
                                                      A
                                                    
                                                  
                                                  -
                                                  
                                                    X
                                                    22
                                                    
                                                      n
                                                      A
                                                    
                                                  
                                                
                                                )
                                              
                                            
                                            +
                                          
                                        
                                      
                                    
                                  
                                
                                
                                  
                                    
                                      
                                        s
                                        
                                          2
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          θ
                                        
                                      
                                      ⁡
                                      
                                        (
                                        
                                          
                                            X
                                            12
                                            
                                              m
                                              A
                                            
                                          
                                          +
                                          
                                            X
                                            21
                                            
                                              m
                                              A
                                            
                                          
                                          -
                                          
                                            X
                                            12
                                            
                                              n
                                              A
                                            
                                          
                                          -
                                          
                                            X
                                            21
                                            
                                              n
                                              A
                                            
                                          
                                        
                                        )
                                      
                                    
                                  
                                
                              
                              
                            
                            2
                          
                          +
                        
                      
                    
                    
                      
                        
                          2
                          ·
                          
                            (
                            
                              
                                
                                  c
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                  2
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                        
                                          
                                            X
                                            12
                                            
                                              m
                                              A
                                            
                                          
                                          +
                                          
                                            X
                                            21
                                            
                                              m
                                              A
                                            
                                          
                                        
                                        
                                      
                                      2
                                    
                                    +
                                    
                                      
                                        
                                        
                                          
                                            X
                                            12
                                            
                                              n
                                              A
                                            
                                          
                                          +
                                          
                                            X
                                            21
                                            
                                              n
                                              A
                                            
                                          
                                        
                                        
                                      
                                      2
                                    
                                  
                                  )
                                
                              
                              +
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                s
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                                2
                              
                              ⁡
                              
                                (
                                
                                  
                                    
                                      
                                      
                                        
                                          X
                                          11
                                          
                                            m
                                            A
                                          
                                        
                                        -
                                        
                                          X
                                          22
                                          
                                            m
                                            A
                                          
                                        
                                      
                                      
                                    
                                    2
                                  
                                  +
                                  
                                    
                                      
                                      
                                        
                                          X
                                          11
                                          
                                            n
                                            A
                                          
                                        
                                        -
                                        
                                          X
                                          22
                                          
                                            n
                                            A
                                          
                                        
                                      
                                      
                                    
                                    2
                                  
                                
                                )
                              
                            
                            )
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            4
                            ·
                            
                              s
                              
                                2
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                            
                          
                          ⁢
                          
                            c
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                          
                          ⁢
                          
                            Re
                            [
                            
                              
                                
                                  
                                    (
                                    
                                      
                                        X
                                        12
                                        
                                          m
                                          A
                                        
                                      
                                      +
                                      
                                        X
                                        21
                                        
                                          m
                                          A
                                        
                                      
                                    
                                    )
                                  
                                  *
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      X
                                      11
                                      
                                        m
                                        A
                                      
                                    
                                    -
                                    
                                      X
                                      22
                                      
                                        m
                                        A
                                      
                                    
                                  
                                  )
                                
                              
                              +
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                (
                                
                                  
                                    X
                                    12
                                    
                                      n
                                      A
                                    
                                  
                                  +
                                  
                                    X
                                    21
                                    
                                      n
                                      A
                                    
                                  
                                
                                )
                              
                              *
                            
                            ⁢
                            
                              (
                              
                                
                                  X
                                  11
                                  
                                    n
                                    A
                                  
                                
                                -
                                
                                  X
                                  22
                                  
                                    n
                                    A
                                  
                                
                              
                              )
                            
                          
                          ]
                        
                      
                    
                  
                  }
                
              
            
            =
            
              
                0.25
                ·
                
                  
                    
                      ∑
                      
                        
                          j
                          n
                        
                        
                          
                            m
                            A
                          
                          ≠
                          
                            n
                            A
                          
                        
                      
                    
                    
                      B
                      ⁡
                      
                        (
                        
                          
                            ω
                            ST
                          
                          ,
                          W
                        
                        )
                      
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          
                            
                              2
                              ·
                              
                                
                                  
                                  
                                    
                                      X
                                      11
                                      
                                        m
                                        A
                                      
                                    
                                    -
                                    
                                      X
                                      11
                                      
                                        n
                                        A
                                      
                                    
                                    +
                                    
                                      X
                                      22
                                      
                                        m
                                        A
                                      
                                    
                                    -
                                    
                                      X
                                      22
                                      
                                        n
                                        A
                                      
                                    
                                  
                                  
                                
                                2
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              
                                2
                                ·
                                
                                  c
                                  
                                    2
                                    ⁢
                                    θ
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  
                                  
                                    
                                      X
                                      11
                                      
                                        m
                                        A
                                      
                                    
                                    -
                                    
                                      X
                                      11
                                      
                                        n
                                        A
                                      
                                    
                                    -
                                    
                                      X
                                      22
                                      
                                        m
                                        A
                                      
                                    
                                    +
                                    
                                      X
                                      22
                                      
                                        n
                                        A
                                      
                                    
                                  
                                  
                                
                                2
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              
                                2
                                ·
                                
                                  s
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                  2
                                
                              
                              ⁢
                              
                                
                                  
                                  
                                    
                                      X
                                      12
                                      
                                        m
                                        A
                                      
                                    
                                    +
                                    
                                      X
                                      21
                                      
                                        m
                                        A
                                      
                                    
                                    -
                                    
                                      X
                                      12
                                      
                                        n
                                        A
                                      
                                    
                                    -
                                    
                                      X
                                      21
                                      
                                        n
                                        A
                                      
                                    
                                  
                                  
                                
                                2
                              
                            
                            -
                          
                        
                      
                      
                        
                          
                            
                              4
                              ·
                              
                                s
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                              
                            
                            ⁢
                            
                              c
                              
                                2
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                            
                            ⁢
                            
                              Re
                              [
                              
                                (
                                
                                  
                                    X
                                    11
                                    
                                      m
                                      A
                                    
                                  
                                  -
                                  
                                    X
                                    11
                                    
                                      n
                                      A
                                    
                                  
                                  -
                                  
                                    X
                                    22
                                    
                                      m
                                      A
                                    
                                  
                                  +
                                
                              
                            
                          
                        
                      
                      
                        
                          
                            
                              
                                
                                  
                                    
                                      X
                                      22
                                      
                                        n
                                        A
                                      
                                    
                                    )
                                  
                                  *
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      X
                                      12
                                      
                                        m
                                        A
                                      
                                    
                                    +
                                    
                                      X
                                      21
                                      
                                        m
                                        A
                                      
                                    
                                    -
                                    
                                      X
                                      12
                                      
                                        n
                                        A
                                      
                                    
                                    -
                                    
                                      X
                                      21
                                      
                                        n
                                        A
                                      
                                    
                                  
                                  )
                                
                              
                              ]
                            
                            +
                            
                              2
                              ·
                            
                          
                        
                      
                      
                        
                          
                            (
                            
                              
                                
                                  c
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                  2
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                        
                                          
                                            X
                                            12
                                            
                                              m
                                              A
                                            
                                          
                                          +
                                          
                                            X
                                            21
                                            
                                              m
                                              A
                                            
                                          
                                        
                                        
                                      
                                      2
                                    
                                    +
                                    
                                      
                                        
                                        
                                          
                                            X
                                            12
                                            
                                              n
                                              A
                                            
                                          
                                          +
                                          
                                            X
                                            21
                                            
                                              n
                                              A
                                            
                                          
                                        
                                        
                                      
                                      2
                                    
                                  
                                  )
                                
                              
                              +
                            
                          
                        
                      
                      
                        
                          
                            
                              
                                
                                  s
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                  2
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                        
                                          
                                            X
                                            11
                                            
                                              m
                                              A
                                            
                                          
                                          -
                                          
                                            X
                                            22
                                            
                                              m
                                              A
                                            
                                          
                                        
                                        
                                      
                                      2
                                    
                                    +
                                    
                                      
                                        
                                        
                                          
                                            X
                                            11
                                            
                                              n
                                              A
                                            
                                          
                                          -
                                          
                                            X
                                            22
                                            
                                              n
                                              A
                                            
                                          
                                        
                                        
                                      
                                      2
                                    
                                  
                                  )
                                
                              
                              )
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              4
                              ·
                              
                                s
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                              
                            
                            ⁢
                            
                              c
                              
                                2
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                            
                            ⁢
                            
                              Re
                              [
                              
                                
                                  (
                                  
                                    
                                      X
                                      12
                                      
                                        m
                                        A
                                      
                                    
                                    +
                                    
                                      X
                                      21
                                      
                                        m
                                        A
                                      
                                    
                                  
                                  )
                                
                                *
                              
                            
                          
                        
                      
                      
                        
                          
                            
                              (
                              
                                
                                  X
                                  11
                                  
                                    m
                                    A
                                  
                                
                                -
                                
                                  X
                                  22
                                  
                                    m
                                    A
                                  
                                
                              
                              )
                            
                            +
                            
                              
                                (
                                
                                  
                                    X
                                    12
                                    
                                      n
                                      A
                                    
                                  
                                  +
                                  
                                    X
                                    21
                                    
                                      n
                                      A
                                    
                                  
                                
                                )
                              
                              *
                            
                          
                        
                      
                      
                        
                          
                            
                              (
                              
                                
                                  X
                                  11
                                  
                                    n
                                    A
                                  
                                
                                -
                                
                                  X
                                  22
                                  
                                    n
                                    A
                                  
                                
                              
                              )
                            
                            ]
                          
                        
                      
                    
                    }
                  
                
              
              =
              
                
                  
                    
                      ∑
                      
                        
                          j
                          n
                        
                        
                          
                            m
                            A
                          
                          ≠
                          
                            n
                            A
                          
                        
                      
                    
                    
                      B
                      ⁡
                      
                        (
                        
                          
                            ω
                            ST
                          
                          ,
                          W
                        
                        )
                      
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          
                            
                              0.5
                              ·
                              
                                
                                  
                                  
                                    
                                      X
                                      11
                                      
                                        m
                                        A
                                      
                                    
                                    -
                                    
                                      X
                                      11
                                      
                                        n
                                        A
                                      
                                    
                                    +
                                    
                                      X
                                      22
                                      
                                        m
                                        A
                                      
                                    
                                    -
                                    
                                      X
                                      22
                                      
                                        n
                                        A
                                      
                                    
                                  
                                  
                                
                                2
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            0.5
                            ·
                            
                              
                                c
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                                2
                              
                              (
                              
                                
                                  
                                    
                                    
                                      
                                        X
                                        11
                                        
                                          m
                                          A
                                        
                                      
                                      -
                                      
                                        X
                                        22
                                        
                                          m
                                          A
                                        
                                      
                                    
                                    
                                  
                                  2
                                
                                +
                                
                                  
                                    
                                    
                                      
                                        X
                                        11
                                        
                                          n
                                          A
                                        
                                      
                                      -
                                      
                                        X
                                        22
                                        
                                          n
                                          A
                                        
                                      
                                    
                                    
                                  
                                  2
                                
                                -
                              
                            
                          
                        
                      
                      
                        
                          
                            
                              
                                2
                                ·
                                
                                  Re
                                  ⁡
                                  
                                    [
                                    
                                      
                                        
                                          (
                                          
                                            
                                              X
                                              11
                                              
                                                m
                                                A
                                              
                                            
                                            -
                                            
                                              X
                                              22
                                              
                                                m
                                                A
                                              
                                            
                                          
                                          )
                                        
                                        *
                                      
                                      ⁢
                                      
                                        (
                                        
                                          
                                            X
                                            11
                                            
                                              n
                                              A
                                            
                                          
                                          -
                                          
                                            X
                                            22
                                            
                                              n
                                              A
                                            
                                          
                                        
                                        )
                                      
                                    
                                    ]
                                  
                                
                              
                              )
                            
                            +
                          
                        
                      
                      
                        
                          
                            0.5
                            ·
                            
                              
                                s
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                                2
                              
                              (
                              
                                
                                  
                                    
                                    
                                      
                                        X
                                        12
                                        
                                          m
                                          A
                                        
                                      
                                      +
                                      
                                        X
                                        21
                                        
                                          m
                                          A
                                        
                                      
                                    
                                    
                                  
                                  2
                                
                                +
                                
                                  
                                    
                                    
                                      
                                        X
                                        12
                                        
                                          n
                                          A
                                        
                                      
                                      -
                                      
                                        X
                                        21
                                        
                                          n
                                          A
                                        
                                      
                                    
                                    
                                  
                                  2
                                
                                -
                              
                            
                          
                        
                      
                      
                        
                          
                            
                              
                                2
                                ·
                                
                                  Re
                                  ⁡
                                  
                                    [
                                    
                                      
                                        
                                          (
                                          
                                            
                                              X
                                              12
                                              
                                                m
                                                A
                                              
                                            
                                            +
                                            
                                              X
                                              21
                                              
                                                m
                                                A
                                              
                                            
                                          
                                          )
                                        
                                        *
                                      
                                      ⁢
                                      
                                        (
                                        
                                          
                                            X
                                            12
                                            
                                              n
                                              A
                                            
                                          
                                          +
                                          
                                            X
                                            21
                                            
                                              n
                                              A
                                            
                                          
                                        
                                        )
                                      
                                    
                                    ]
                                  
                                
                              
                              )
                            
                            -
                          
                        
                      
                      
                        
                          
                            
                              s
                              
                                2
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                            
                            ⁢
                            
                              c
                              
                                2
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                            
                            ⁢
                            
                              Re
                              [
                              
                                
                                  (
                                  
                                    
                                      X
                                      11
                                      
                                        m
                                        A
                                      
                                    
                                    -
                                    
                                      X
                                      11
                                      
                                        n
                                        A
                                      
                                    
                                    -
                                    
                                      X
                                      22
                                      
                                        m
                                        A
                                      
                                    
                                    +
                                    
                                      X
                                      22
                                      
                                        n
                                        A
                                      
                                    
                                  
                                  )
                                
                                *
                              
                            
                          
                        
                      
                      
                        
                          
                            
                              
                                (
                                
                                  
                                    X
                                    12
                                    
                                      m
                                      A
                                    
                                  
                                  +
                                  
                                    X
                                    21
                                    
                                      m
                                      A
                                    
                                  
                                  -
                                  
                                    X
                                    12
                                    
                                      n
                                      A
                                    
                                  
                                  -
                                  
                                    X
                                    21
                                    
                                      n
                                      A
                                    
                                  
                                
                                )
                              
                              ]
                            
                            +
                          
                        
                      
                      
                        
                          
                            0.5
                            ·
                            
                              (
                              
                                
                                  
                                    c
                                    
                                      2
                                      ⁢
                                      θ
                                    
                                    2
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        
                                          
                                          
                                            
                                              X
                                              12
                                              
                                                m
                                                A
                                              
                                            
                                            +
                                            
                                              X
                                              21
                                              
                                                m
                                                A
                                              
                                            
                                          
                                          
                                        
                                        2
                                      
                                      +
                                      
                                        
                                          
                                          
                                            
                                              X
                                              12
                                              
                                                n
                                                A
                                              
                                            
                                            +
                                            
                                              X
                                              21
                                              
                                                n
                                                A
                                              
                                            
                                          
                                          
                                        
                                        2
                                      
                                    
                                    )
                                  
                                
                                +
                              
                            
                          
                        
                      
                      
                        
                          
                            
                              
                                
                                  s
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                  2
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        
                                        
                                          
                                            X
                                            11
                                            
                                              m
                                              A
                                            
                                          
                                          -
                                          
                                            X
                                            22
                                            
                                              m
                                              A
                                            
                                          
                                        
                                        
                                      
                                      2
                                    
                                    +
                                    
                                      
                                        
                                        
                                          
                                            X
                                            11
                                            
                                              n
                                              A
                                            
                                          
                                          -
                                          
                                            X
                                            22
                                            
                                              n
                                              A
                                            
                                          
                                        
                                        
                                      
                                      2
                                    
                                  
                                  )
                                
                              
                              )
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              s
                              
                                2
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                            
                            ⁢
                            
                              c
                              
                                2
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                            
                            ⁢
                            
                              Re
                              [
                              
                                
                                  
                                    
                                      (
                                      
                                        
                                          X
                                          12
                                          
                                            m
                                            A
                                          
                                        
                                        +
                                        
                                          X
                                          21
                                          
                                            m
                                            A
                                          
                                        
                                      
                                      )
                                    
                                    *
                                  
                                  ⁢
                                  
                                    (
                                    
                                      
                                        X
                                        11
                                        
                                          m
                                          A
                                        
                                      
                                      -
                                      
                                        X
                                        22
                                        
                                          m
                                          A
                                        
                                      
                                    
                                    )
                                  
                                
                                +
                              
                            
                          
                        
                      
                      
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      X
                                      12
                                      
                                        n
                                        A
                                      
                                    
                                    +
                                    
                                      X
                                      21
                                      
                                        n
                                        A
                                      
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    X
                                    11
                                    
                                      n
                                      A
                                    
                                  
                                  -
                                  
                                    X
                                    22
                                    
                                      n
                                      A
                                    
                                  
                                
                                )
                              
                            
                            ]
                          
                        
                      
                    
                    }
                  
                
                =
                
                  
                    
                      
                        ∑
                        
                          
                            j
                            n
                          
                          
                            
                              m
                              A
                            
                            ≠
                            
                              n
                              A
                            
                          
                        
                      
                      
                        B
                        ⁡
                        
                          (
                          
                            
                              ω
                              ST
                            
                            ,
                            W
                          
                          )
                        
                      
                    
                    ⁢
                    
                      {
                      
                        
                          
                            
                              
                                0.5
                                ·
                                
                                  (
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                              
                                                
                                                  X
                                                  11
                                                  
                                                    m
                                                    A
                                                  
                                                
                                                -
                                                
                                                  X
                                                  11
                                                  
                                                    n
                                                    A
                                                  
                                                
                                                +
                                                
                                                  X
                                                  22
                                                  
                                                    m
                                                    A
                                                  
                                                
                                                -
                                                
                                                  X
                                                  22
                                                  
                                                    n
                                                    A
                                                  
                                                
                                              
                                              
                                            
                                            2
                                          
                                          +
                                        
                                      
                                    
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                    
                                                      
                                                        X
                                                        11
                                                        
                                                          m
                                                          A
                                                        
                                                      
                                                      -
                                                      
                                                        X
                                                        22
                                                        
                                                          m
                                                          A
                                                        
                                                      
                                                    
                                                    
                                                  
                                                  2
                                                
                                                +
                                                
                                                  
                                                    
                                                    
                                                      
                                                        X
                                                        11
                                                        
                                                          n
                                                          A
                                                        
                                                      
                                                      -
                                                      
                                                        X
                                                        22
                                                        
                                                          n
                                                          A
                                                        
                                                      
                                                    
                                                    
                                                  
                                                  2
                                                
                                                +
                                              
                                            
                                          
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                    
                                                      
                                                        X
                                                        12
                                                        
                                                          m
                                                          A
                                                        
                                                      
                                                      +
                                                      
                                                        X
                                                        21
                                                        
                                                          m
                                                          A
                                                        
                                                      
                                                    
                                                    
                                                  
                                                  2
                                                
                                                +
                                                
                                                  
                                                    
                                                    
                                                      
                                                        X
                                                        12
                                                        
                                                          n
                                                          A
                                                        
                                                      
                                                      +
                                                      
                                                        X
                                                        21
                                                        
                                                          n
                                                          A
                                                        
                                                      
                                                    
                                                    
                                                  
                                                  2
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                  )
                                
                              
                              -
                            
                          
                        
                        
                          
                            
                              
                                
                                  c
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                  2
                                
                                ⁢
                                
                                  Re
                                  ⁡
                                  
                                    [
                                    
                                      
                                        
                                          (
                                          
                                            
                                              X
                                              11
                                              
                                                m
                                                A
                                              
                                            
                                            -
                                            
                                              X
                                              22
                                              
                                                m
                                                A
                                              
                                            
                                          
                                          )
                                        
                                        *
                                      
                                      ⁢
                                      
                                        (
                                        
                                          
                                            X
                                            11
                                            
                                              n
                                              A
                                            
                                          
                                          -
                                          
                                            X
                                            22
                                            
                                              n
                                              A
                                            
                                          
                                        
                                        )
                                      
                                    
                                    ]
                                  
                                
                              
                              -
                            
                          
                        
                        
                          
                            
                              
                                
                                  s
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                  2
                                
                                ⁢
                                
                                  Re
                                  ⁡
                                  
                                    [
                                    
                                      
                                        
                                          (
                                          
                                            
                                              X
                                              12
                                              
                                                m
                                                A
                                              
                                            
                                            +
                                            
                                              X
                                              21
                                              
                                                m
                                                A
                                              
                                            
                                          
                                          )
                                        
                                        *
                                      
                                      ⁢
                                      
                                        (
                                        
                                          
                                            X
                                            12
                                            
                                              n
                                              A
                                            
                                          
                                          +
                                          
                                            X
                                            21
                                            
                                              m
                                              A
                                            
                                          
                                        
                                        )
                                      
                                    
                                    ]
                                  
                                
                              
                              -
                            
                          
                        
                        
                          
                            
                              
                                s
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                              
                              ⁢
                              
                                c
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                              
                              ⁢
                              
                                Re
                                [
                                
                                  
                                    (
                                    
                                      
                                        X
                                        11
                                        
                                          m
                                          A
                                        
                                      
                                      -
                                      
                                        X
                                        11
                                        
                                          n
                                          A
                                        
                                      
                                      -
                                      
                                        X
                                        22
                                        
                                          m
                                          A
                                        
                                      
                                      +
                                      
                                        X
                                        22
                                        
                                          n
                                          A
                                        
                                      
                                    
                                    )
                                  
                                  *
                                
                              
                            
                          
                        
                        
                          
                            
                              
                                
                                  (
                                  
                                    
                                      X
                                      12
                                      
                                        m
                                        A
                                      
                                    
                                    +
                                    
                                      X
                                      21
                                      
                                        m
                                        A
                                      
                                    
                                    -
                                    
                                      X
                                      12
                                      
                                        n
                                        A
                                      
                                    
                                    -
                                    
                                      X
                                      21
                                      
                                        n
                                        A
                                      
                                    
                                  
                                  )
                                
                                ]
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                s
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                              
                              ⁢
                              
                                c
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                              
                              ⁢
                              
                                Re
                                [
                                
                                  
                                    
                                      
                                        (
                                        
                                          
                                            X
                                            12
                                            
                                              m
                                              A
                                            
                                          
                                          -
                                          
                                            X
                                            21
                                            
                                              n
                                              A
                                            
                                          
                                        
                                        )
                                      
                                      *
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          X
                                          11
                                          
                                            m
                                            A
                                          
                                        
                                        -
                                        
                                          X
                                          22
                                          
                                            m
                                            A
                                          
                                        
                                      
                                      )
                                    
                                  
                                  +
                                
                              
                            
                          
                        
                        
                          
                            
                              
                                
                                  
                                    (
                                    
                                      
                                        X
                                        12
                                        
                                          n
                                          A
                                        
                                      
                                      +
                                      
                                        X
                                        21
                                        
                                          n
                                          A
                                        
                                      
                                    
                                    )
                                  
                                  *
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      X
                                      11
                                      
                                        n
                                        A
                                      
                                    
                                    -
                                    
                                      X
                                      22
                                      
                                        n
                                        A
                                      
                                    
                                  
                                  )
                                
                              
                              ]
                            
                          
                        
                      
                      }
                    
                  
                  =
                  
                    
                      
                        
                          ∑
                          
                            
                              j
                              n
                            
                            
                              
                                m
                                A
                              
                              ≠
                              
                                n
                                A
                              
                            
                          
                        
                        
                          B
                          ⁡
                          
                            (
                            
                              
                                ω
                                ST
                              
                              ,
                              W
                            
                            )
                          
                        
                      
                      ⁢
                      
                        {
                        
                          
                            
                              
                                
                                  0.5
                                  ·
                                  
                                    (
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                
                                                  
                                                    X
                                                    11
                                                    
                                                      m
                                                      A
                                                    
                                                  
                                                  -
                                                  
                                                    X
                                                    11
                                                    
                                                      n
                                                      A
                                                    
                                                  
                                                  +
                                                  
                                                    X
                                                    22
                                                    
                                                      m
                                                      A
                                                    
                                                  
                                                  -
                                                  
                                                    X
                                                    22
                                                    
                                                      n
                                                      A
                                                    
                                                  
                                                
                                                
                                              
                                              2
                                            
                                            +
                                          
                                        
                                      
                                      
                                        
                                          
                                            
                                              
                                                
                                                
                                                  
                                                    X
                                                    11
                                                    
                                                      m
                                                      A
                                                    
                                                  
                                                  -
                                                  
                                                    X
                                                    22
                                                    
                                                      m
                                                      A
                                                    
                                                  
                                                
                                                
                                              
                                              2
                                            
                                            +
                                            
                                              
                                                
                                                
                                                  
                                                    X
                                                    11
                                                    
                                                      n
                                                      A
                                                    
                                                  
                                                  -
                                                  
                                                    X
                                                    22
                                                    
                                                      n
                                                      A
                                                    
                                                  
                                                
                                                
                                              
                                              2
                                            
                                            +
                                          
                                        
                                      
                                      
                                        
                                          
                                            
                                              
                                                
                                                
                                                  
                                                    X
                                                    12
                                                    
                                                      m
                                                      A
                                                    
                                                  
                                                  +
                                                  
                                                    X
                                                    21
                                                    
                                                      m
                                                      A
                                                    
                                                  
                                                
                                                
                                              
                                              2
                                            
                                            +
                                            
                                              
                                                
                                                
                                                  
                                                    X
                                                    12
                                                    
                                                      n
                                                      A
                                                    
                                                  
                                                  +
                                                  
                                                    X
                                                    21
                                                    
                                                      n
                                                      A
                                                    
                                                  
                                                
                                                
                                              
                                              2
                                            
                                          
                                        
                                      
                                    
                                    )
                                  
                                
                                -
                              
                            
                          
                          
                            
                              
                                
                                  
                                    c
                                    
                                      2
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      θ
                                    
                                    2
                                  
                                  ⁢
                                  
                                    Re
                                    ⁡
                                    
                                      [
                                      
                                        
                                          
                                            (
                                            
                                              
                                                X
                                                11
                                                
                                                  m
                                                  A
                                                
                                              
                                              -
                                              
                                                X
                                                22
                                                
                                                  m
                                                  A
                                                
                                              
                                            
                                            )
                                          
                                          *
                                        
                                        ⁢
                                        
                                          (
                                          
                                            
                                              X
                                              11
                                              
                                                n
                                                A
                                              
                                            
                                            -
                                            
                                              X
                                              22
                                              
                                                m
                                                A
                                              
                                            
                                          
                                          )
                                        
                                      
                                      ]
                                    
                                  
                                
                                -
                              
                            
                          
                          
                            
                              
                                
                                  
                                    s
                                    
                                      2
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      θ
                                    
                                    2
                                  
                                  ⁢
                                  
                                    Re
                                    ⁡
                                    
                                      [
                                      
                                        
                                          
                                            (
                                            
                                              
                                                X
                                                12
                                                
                                                  m
                                                  A
                                                
                                              
                                              +
                                              
                                                X
                                                21
                                                
                                                  m
                                                  A
                                                
                                              
                                            
                                            )
                                          
                                          *
                                        
                                        ⁢
                                        
                                          (
                                          
                                            
                                              X
                                              12
                                              
                                                n
                                                A
                                              
                                            
                                            +
                                            
                                              X
                                              21
                                              
                                                n
                                                A
                                              
                                            
                                          
                                          )
                                        
                                      
                                      ]
                                    
                                  
                                
                                -
                              
                            
                          
                          
                            
                              
                                
                                  s
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                
                                ⁢
                                
                                  c
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                
                                ⁢
                                
                                  Re
                                  [
                                  
                                    
                                      (
                                      
                                        
                                          (
                                          
                                            
                                              X
                                              11
                                              
                                                m
                                                A
                                              
                                            
                                            -
                                            
                                              X
                                              22
                                              
                                                m
                                                A
                                              
                                            
                                          
                                          )
                                        
                                        -
                                        
                                          (
                                          
                                            
                                              X
                                              11
                                              
                                                n
                                                A
                                              
                                            
                                            -
                                            
                                              X
                                              22
                                              
                                                n
                                                A
                                              
                                            
                                          
                                          )
                                        
                                      
                                      )
                                    
                                    *
                                  
                                
                              
                            
                          
                          
                            
                              
                                
                                  
                                    (
                                    
                                      
                                        (
                                        
                                          
                                            X
                                            12
                                            
                                              m
                                              A
                                            
                                          
                                          +
                                          
                                            X
                                            21
                                            
                                              m
                                              A
                                            
                                          
                                        
                                        )
                                      
                                      -
                                      
                                        (
                                        
                                          
                                            X
                                            12
                                            
                                              n
                                              A
                                            
                                          
                                          -
                                          
                                            X
                                            21
                                            
                                              n
                                              A
                                            
                                          
                                        
                                        )
                                      
                                    
                                    )
                                  
                                  ]
                                
                                +
                              
                            
                          
                          
                            
                              
                                
                                  s
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                
                                ⁢
                                
                                  c
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                
                                ⁢
                                
                                  Re
                                  [
                                  
                                    
                                      
                                        
                                          (
                                          
                                            
                                              X
                                              12
                                              
                                                m
                                                A
                                              
                                            
                                            +
                                            
                                              X
                                              21
                                              
                                                m
                                                A
                                              
                                            
                                          
                                          )
                                        
                                        *
                                      
                                      ⁢
                                      
                                        (
                                        
                                          
                                            X
                                            11
                                            
                                              m
                                              A
                                            
                                          
                                          -
                                          
                                            X
                                            22
                                            
                                              m
                                              A
                                            
                                          
                                        
                                        )
                                      
                                    
                                    +
                                  
                                
                              
                            
                          
                          
                            
                              
                                
                                  
                                    
                                      (
                                      
                                        
                                          X
                                          12
                                          
                                            n
                                            A
                                          
                                        
                                        +
                                        
                                          X
                                          21
                                          
                                            n
                                            A
                                          
                                        
                                      
                                      )
                                    
                                    *
                                  
                                  ⁢
                                  
                                    (
                                    
                                      
                                        X
                                        11
                                        
                                          n
                                          A
                                        
                                      
                                      -
                                      
                                        X
                                        22
                                        
                                          n
                                          A
                                        
                                      
                                    
                                    )
                                  
                                
                                ]
                              
                            
                          
                        
                        }
                      
                    
                    =
                    
                      
                        
                          ∑
                          
                            
                              j
                              n
                            
                            
                              
                                m
                                A
                              
                              ≠
                              
                                n
                                A
                              
                            
                          
                        
                        
                          B
                          ⁡
                          
                            (
                            
                              
                                ω
                                ST
                              
                              ,
                              W
                            
                            )
                          
                        
                      
                      ⁢
                      
                        
                          {
                          
                            
                              
                                
                                  
                                    0.5
                                    ·
                                    
                                      (
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                  
                                                    
                                                      X
                                                      11
                                                      
                                                        m
                                                        A
                                                      
                                                    
                                                    -
                                                    
                                                      X
                                                      11
                                                      
                                                        n
                                                        A
                                                      
                                                    
                                                    +
                                                    
                                                      X
                                                      22
                                                      
                                                        m
                                                        A
                                                      
                                                    
                                                    -
                                                    
                                                      X
                                                      22
                                                      
                                                        n
                                                        A
                                                      
                                                    
                                                  
                                                  
                                                
                                                2
                                              
                                              +
                                            
                                          
                                        
                                        
                                          
                                            
                                              
                                                
                                                  
                                                  
                                                    
                                                      X
                                                      11
                                                      
                                                        m
                                                        A
                                                      
                                                    
                                                    -
                                                    
                                                      X
                                                      22
                                                      
                                                        m
                                                        A
                                                      
                                                    
                                                  
                                                  
                                                
                                                2
                                              
                                              +
                                              
                                                
                                                  
                                                  
                                                    
                                                      X
                                                      11
                                                      
                                                        n
                                                        A
                                                      
                                                    
                                                    -
                                                    
                                                      X
                                                      22
                                                      
                                                        n
                                                        A
                                                      
                                                    
                                                  
                                                  
                                                
                                                2
                                              
                                              +
                                            
                                          
                                        
                                        
                                          
                                            
                                              
                                                
                                                  
                                                  
                                                    
                                                      X
                                                      12
                                                      
                                                        m
                                                        A
                                                      
                                                    
                                                    +
                                                    
                                                      X
                                                      21
                                                      
                                                        m
                                                        A
                                                      
                                                    
                                                  
                                                  
                                                
                                                2
                                              
                                              +
                                              
                                                
                                                  
                                                  
                                                    
                                                      X
                                                      12
                                                      
                                                        n
                                                        A
                                                      
                                                    
                                                    +
                                                    
                                                      X
                                                      21
                                                      
                                                        n
                                                        A
                                                      
                                                    
                                                  
                                                  
                                                
                                                2
                                              
                                            
                                          
                                        
                                      
                                      )
                                    
                                  
                                  -
                                
                              
                            
                            
                              
                                
                                  
                                    
                                      c
                                      
                                        2
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        θ
                                      
                                      2
                                    
                                    ⁢
                                    
                                      Re
                                      ⁡
                                      
                                        [
                                        
                                          
                                            
                                              (
                                              
                                                
                                                  X
                                                  11
                                                  
                                                    m
                                                    A
                                                  
                                                
                                                -
                                                
                                                  X
                                                  22
                                                  
                                                    m
                                                    A
                                                  
                                                
                                              
                                              )
                                            
                                            *
                                          
                                          ⁢
                                          
                                            (
                                            
                                              
                                                X
                                                11
                                                
                                                  n
                                                  A
                                                
                                              
                                              -
                                              
                                                X
                                                22
                                                
                                                  m
                                                  A
                                                
                                              
                                            
                                            )
                                          
                                        
                                        ]
                                      
                                    
                                  
                                  -
                                
                              
                            
                            
                              
                                
                                  
                                    
                                      s
                                      
                                        2
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        θ
                                      
                                      2
                                    
                                    ⁢
                                    
                                      Re
                                      ⁡
                                      
                                        [
                                        
                                          
                                            
                                              (
                                              
                                                
                                                  X
                                                  12
                                                  
                                                    m
                                                    A
                                                  
                                                
                                                +
                                                
                                                  X
                                                  21
                                                  
                                                    m
                                                    A
                                                  
                                                
                                              
                                              )
                                            
                                            *
                                          
                                          ⁢
                                          
                                            (
                                            
                                              
                                                X
                                                12
                                                
                                                  n
                                                  A
                                                
                                              
                                              +
                                              
                                                X
                                                21
                                                
                                                  n
                                                  A
                                                
                                              
                                            
                                            )
                                          
                                        
                                        ]
                                      
                                    
                                  
                                  +
                                
                              
                            
                            
                              
                                
                                  
                                    s
                                    
                                      2
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      θ
                                    
                                  
                                  ⁢
                                  
                                    c
                                    
                                      2
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      θ
                                    
                                  
                                  ⁢
                                  
                                    Re
                                    [
                                    
                                      
                                        
                                          
                                            (
                                            
                                              
                                                X
                                                11
                                                
                                                  m
                                                  A
                                                
                                              
                                              -
                                              
                                                X
                                                22
                                                
                                                  m
                                                  A
                                                
                                              
                                            
                                            )
                                          
                                          *
                                        
                                        ⁢
                                        
                                          (
                                          
                                            
                                              X
                                              12
                                              
                                                n
                                                A
                                              
                                            
                                            +
                                            
                                              X
                                              21
                                              
                                                n
                                                A
                                              
                                            
                                          
                                          )
                                        
                                      
                                      +
                                    
                                  
                                
                              
                            
                            
                              
                                
                                  
                                    
                                      
                                        (
                                        
                                          
                                            X
                                            11
                                            
                                              n
                                              A
                                            
                                          
                                          -
                                          
                                            X
                                            22
                                            
                                              n
                                              A
                                            
                                          
                                        
                                        )
                                      
                                      *
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          X
                                          12
                                          
                                            m
                                            A
                                          
                                        
                                        +
                                        
                                          X
                                          21
                                          
                                            m
                                            A
                                          
                                        
                                      
                                      )
                                    
                                  
                                  ]
                                
                              
                            
                          
                          }
                        
                        .
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        H76
        )
      
    
  




Differentiating gives





  
    
      
        0
        =
        
          
            ∑
            
              
                j
                n
              
              
                
                  m
                  A
                
                ≠
                
                  n
                  A
                
              
            
            
              B
              ⁡
              
                (
                
                  
                    ω
                    ST
                  
                  ,
                  W
                
                )
              
            
          
          ⁢
          
            
              {
              
                
                  
                    
                      
                        
                          s
                          
                            4
                            ⁢
                            θ
                          
                        
                        ⁢
                        
                          Re
                          ⁡
                          
                            [
                            
                              
                                
                                  
                                    (
                                    
                                      
                                        X
                                        11
                                        
                                          m
                                          A
                                        
                                      
                                      -
                                      
                                        X
                                        22
                                        
                                          m
                                          A
                                        
                                      
                                    
                                    )
                                  
                                  *
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      X
                                      11
                                      
                                        n
                                        A
                                      
                                    
                                    -
                                    
                                      X
                                      22
                                      
                                        n
                                        A
                                      
                                    
                                  
                                  )
                                
                              
                              -
                              
                                
                                  
                                    (
                                    
                                      
                                        X
                                        12
                                        
                                          m
                                          A
                                        
                                      
                                      +
                                      
                                        X
                                        21
                                        
                                          m
                                          A
                                        
                                      
                                    
                                    )
                                  
                                  *
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      X
                                      12
                                      
                                        n
                                        A
                                      
                                    
                                    +
                                    
                                      X
                                      21
                                      
                                        n
                                        A
                                      
                                    
                                  
                                  )
                                
                              
                            
                            ]
                          
                        
                      
                      +
                    
                  
                
                
                  
                    
                      
                        c
                        
                          4
                          ⁢
                          
                            
                          
                          ⁢
                          θ
                        
                      
                      ⁢
                      
                        Re
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  (
                                  
                                    
                                      X
                                      11
                                      
                                        m
                                        A
                                      
                                    
                                    -
                                    
                                      X
                                      22
                                      
                                        m
                                        A
                                      
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    X
                                    12
                                    
                                      n
                                      A
                                    
                                  
                                  +
                                  
                                    X
                                    21
                                    
                                      n
                                      A
                                    
                                  
                                
                                )
                              
                            
                            +
                            
                              
                                
                                  (
                                  
                                    
                                      X
                                      11
                                      
                                        n
                                        A
                                      
                                    
                                    -
                                    
                                      X
                                      22
                                      
                                        n
                                        A
                                      
                                    
                                  
                                  )
                                
                                *
                              
                              ⁢
                              
                                (
                                
                                  
                                    X
                                    12
                                    
                                      m
                                      A
                                    
                                  
                                  +
                                  
                                    X
                                    21
                                    
                                      m
                                      A
                                    
                                  
                                
                                )
                              
                            
                          
                          ]
                        
                      
                    
                  
                
              
              }
            
            .
          
        
      
    
    
      
        (
        H77
        )
      
    
  




The solution is





  
    
      
        
          tan
          ⁡
          
            (
            
              4
              ⁢
              
                
              
              ⁢
              φ
            
            )
          
        
        =
        
          -
          
            
              
                Re
                ⁡
                
                  [
                  
                    
                      
                        
                          a
                          ⇀
                        
                        *
                      
                      ·
                      
                        d
                        ⇀
                      
                    
                    +
                    
                      
                        
                          b
                          ⇀
                        
                        *
                      
                      ·
                      
                        c
                        ⇀
                      
                    
                  
                  ]
                
              
              
                Re
                ⁡
                
                  [
                  
                    
                      
                        
                          a
                          ⇀
                        
                        *
                      
                      ·
                      
                        b
                        ⇀
                      
                    
                    -
                    
                      
                        
                          c
                          ⇀
                        
                        *
                      
                      ·
                      
                        d
                        ⇀
                      
                    
                  
                  ]
                
              
            
            .
          
        
      
    
    
      
        (
        H78
        )
      
    
  




where

aj=X11mA−X22mA,

bj=X11nA−X22nA,

cj=X12mA+X21mA,

dj=X12nA+X21nA.  (H79)

and j={mA, nA, jn}. See discussion immediately following equation (H43) for handling ambiguity.

HTI Appendix





  
    O
    ⁡
    
      (
      
        θ
        ,
        
          s
          1
        
        ,
        
          δ
          ⁢
          
            
          
          ⁢
          s
        
      
      )
    
  
  =
  
    
      
        ∑
        
          k
          ,
          m
          ,
          n
        
      
      ⁢
      
        {
        
          
            
              
                4
                [
                
                  
                    c
                    θ
                    2
                  
                  (
                  
                    
                      YY
                      
                        j
                        S
                      
                    
                    -
                    
                      XX
                      
                        j
                        F
                      
                    
                    +
                    
                      
                        s
                        θ
                        2
                      
                      ⁡
                      
                        (
                        
                          
                            XX
                            
                              j
                              S
                            
                          
                          -
                          
                            YY
                            
                              j
                              F
                            
                          
                        
                        )
                      
                    
                    +
                  
                
              
            
          
          
            
              
                
                  
                    
                      
                        s
                        θ
                      
                      ⁢
                      
                        
                          c
                          θ
                        
                        ⁡
                        
                          (
                          
                            
                              XY
                              
                                j
                                S
                              
                            
                            +
                            
                              YX
                              
                                j
                                S
                              
                            
                            +
                            
                              XY
                              
                                j
                                F
                              
                            
                            +
                            
                              YX
                              
                                j
                                F
                              
                            
                          
                          )
                        
                      
                    
                    ]
                  
                  2
                
                +
              
            
          
          
            
              
                
                  
                    [
                    
                      
                        
                          s
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            θ
                          
                        
                        ⁡
                        
                          (
                          
                            
                              XX
                              
                                j
                                S
                              
                            
                            -
                            
                              YY
                              
                                j
                                S
                              
                            
                          
                          )
                        
                      
                      +
                      
                        
                          c
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            θ
                          
                        
                        ⁡
                        
                          (
                          
                            
                              XY
                              
                                j
                                S
                              
                            
                            +
                            
                              YX
                              
                                j
                                S
                              
                            
                          
                          )
                        
                      
                    
                    ]
                  
                  2
                
                +
              
            
          
          
            
              
                
                  [
                  
                    
                      
                        s
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          θ
                        
                      
                      ⁡
                      
                        (
                        
                          
                            XX
                            
                              j
                              F
                            
                          
                          -
                          
                            YY
                            
                              j
                              F
                            
                          
                        
                        )
                      
                    
                    +
                    
                      
                        c
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          θ
                        
                      
                      ⁡
                      
                        (
                        
                          
                            XY
                            
                              j
                              F
                            
                          
                          +
                          
                            YX
                            
                              j
                              F
                            
                          
                        
                        )
                      
                    
                  
                  ]
                
                2
              
            
          
        
        }
      
    
    =
    
      
        
          ∑
          
            k
            ,
            m
            ,
            n
          
        
        ⁢
        
          {
          
            
              
                
                  
                    
                      (
                      
                        1
                        +
                        
                          c
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            θ
                          
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        
                          YY
                          
                            j
                            S
                          
                        
                        -
                        
                          XX
                          
                            j
                            F
                          
                        
                      
                      )
                    
                  
                  +
                  
                    
                      (
                      
                        1
                        -
                        
                          c
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            θ
                          
                        
                      
                      )
                    
                    ⁢
                    
                      (
                      
                        
                          XX
                          
                            j
                            S
                          
                        
                        -
                        
                          YY
                          
                            j
                            F
                          
                        
                      
                      )
                    
                  
                  +
                
              
            
            
              
                
                  
                    
                      
                        
                          s
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            θ
                          
                        
                        ⁡
                        
                          (
                          
                            
                              XY
                              
                                j
                                S
                              
                            
                            +
                            
                              YX
                              
                                j
                                S
                              
                            
                            +
                            
                              XY
                              
                                j
                                F
                              
                            
                            +
                            
                              YX
                              
                                j
                                F
                              
                            
                          
                          )
                        
                      
                      ]
                    
                    2
                  
                  +
                
              
            
            
              
                
                  
                    
                      [
                      
                        
                          
                            s
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                          
                          ⁡
                          
                            (
                            
                              
                                XX
                                
                                  j
                                  S
                                
                              
                              -
                              
                                YY
                                
                                  j
                                  S
                                
                              
                            
                            )
                          
                        
                        +
                        
                          
                            c
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                          
                          ⁡
                          
                            (
                            
                              
                                XY
                                
                                  j
                                  S
                                
                              
                              +
                              
                                YX
                                
                                  j
                                  S
                                
                              
                            
                            )
                          
                        
                      
                      ]
                    
                    2
                  
                  +
                
              
            
            
              
                
                  
                    [
                    
                      
                        
                          s
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            θ
                          
                        
                        ⁡
                        
                          (
                          
                            
                              XX
                              
                                j
                                F
                              
                            
                            -
                            
                              YY
                              
                                j
                                F
                              
                            
                          
                          )
                        
                      
                      +
                      
                        
                          c
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            θ
                          
                        
                        ⁡
                        
                          (
                          
                            
                              XY
                              
                                j
                                F
                              
                            
                            +
                            
                              YX
                              
                                j
                                F
                              
                            
                          
                          )
                        
                      
                    
                    ]
                  
                  2
                
              
            
          
          }
        
      
      =
      
        
          
            ∑
            
              k
              ,
              m
              ,
              n
            
          
          ⁢
          
            {
            
              
                
                  
                    
                      (
                      
                        
                          YY
                          
                            j
                            S
                          
                        
                        -
                        
                          XX
                          
                            j
                            F
                          
                        
                        +
                        
                          XX
                          
                            j
                            S
                          
                        
                        -
                        
                          YY
                          
                            j
                            F
                          
                        
                      
                      )
                    
                    +
                  
                
              
              
                
                  
                    
                      
                        c
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          θ
                        
                      
                      ⁡
                      
                        (
                        
                          
                            YY
                            
                              j
                              S
                            
                          
                          -
                          
                            XX
                            
                              j
                              F
                            
                          
                          +
                          
                            XX
                            
                              j
                              S
                            
                          
                          -
                          
                            YY
                            
                              j
                              F
                            
                          
                        
                        )
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                        
                          
                            s
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                          
                          ⁡
                          
                            (
                            
                              
                                XY
                                
                                  j
                                  S
                                
                              
                              +
                              
                                YX
                                
                                  j
                                  S
                                
                              
                              +
                              
                                XY
                                
                                  j
                                  F
                                
                              
                              +
                              
                                YX
                                
                                  j
                                  F
                                
                              
                            
                            )
                          
                        
                        ]
                      
                      2
                    
                    +
                  
                
              
              
                
                  
                    
                      
                        
                          
                            
                              [
                              
                                
                                  
                                    s
                                    
                                      2
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      θ
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        XX
                                        
                                          j
                                          S
                                        
                                      
                                      -
                                      
                                        YY
                                        
                                          j
                                          S
                                        
                                      
                                    
                                    )
                                  
                                
                                +
                                
                                  
                                    c
                                    
                                      2
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      θ
                                    
                                  
                                  ⁡
                                  
                                    (
                                    
                                      
                                        XY
                                        
                                          j
                                          S
                                        
                                      
                                      +
                                      
                                        YX
                                        
                                          j
                                          S
                                        
                                      
                                    
                                    )
                                  
                                
                              
                              ]
                            
                            2
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            [
                            
                              
                                
                                  s
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      XX
                                      
                                        j
                                        F
                                      
                                    
                                    -
                                    
                                      YY
                                      
                                        j
                                        F
                                      
                                    
                                  
                                  )
                                
                              
                              +
                              
                                
                                  c
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      XY
                                      
                                        j
                                        F
                                      
                                    
                                    +
                                    
                                      YX
                                      
                                        j
                                        F
                                      
                                    
                                  
                                  )
                                
                              
                            
                            ]
                          
                          2
                        
                      
                    
                  
                
              
            
            }
          
        
        =
        
          
            
              ∑
              
                k
                ,
                m
                ,
                n
              
            
            ⁢
            
              {
              
                
                  
                    
                      
                        
                          (
                          
                            
                              YY
                              
                                j
                                S
                              
                            
                            -
                            
                              XX
                              
                                j
                                F
                              
                            
                            +
                            
                              XX
                              
                                j
                                S
                              
                            
                            -
                            
                              YY
                              
                                j
                                F
                              
                            
                          
                          )
                        
                        2
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          
                            c
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                            2
                          
                          ⁡
                          
                            (
                            
                              
                                YY
                                
                                  j
                                  S
                                
                              
                              -
                              
                                XX
                                
                                  j
                                  F
                                
                              
                              -
                              
                                XX
                                
                                  j
                                  S
                                
                              
                              +
                              
                                YY
                                
                                  j
                                  F
                                
                              
                            
                            )
                          
                        
                        2
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          
                            s
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                            2
                          
                          ⁡
                          
                            (
                            
                              
                                XY
                                
                                  j
                                  S
                                
                              
                              +
                              
                                YX
                                
                                  j
                                  S
                                
                              
                              +
                              
                                XY
                                
                                  j
                                  F
                                
                              
                              +
                              
                                YX
                                
                                  j
                                  F
                                
                              
                            
                            )
                          
                        
                        2
                      
                      +
                    
                  
                
                
                  
                    
                      2
                      ⁢
                      
                        (
                        
                          
                            YY
                            
                              j
                              S
                            
                          
                          -
                          
                            XX
                            
                              j
                              F
                            
                          
                          +
                          
                            XX
                            
                              j
                              S
                            
                          
                          -
                          
                            YY
                            
                              j
                              F
                            
                          
                        
                        )
                      
                    
                  
                
                
                  
                    
                      [
                      
                        
                          
                            c
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                          
                          ⁡
                          
                            (
                            
                              
                                YY
                                
                                  j
                                  S
                                
                              
                              -
                              
                                XX
                                
                                  j
                                  F
                                
                              
                              -
                              
                                XX
                                
                                  j
                                  S
                                
                              
                              +
                              
                                YY
                                
                                  j
                                  F
                                
                              
                            
                            )
                          
                        
                        +
                      
                    
                  
                
                
                  
                    
                      
                        
                          
                            s
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                          
                          ⁡
                          
                            (
                            
                              
                                XY
                                
                                  j
                                  S
                                
                              
                              +
                              
                                YX
                                
                                  j
                                  S
                                
                              
                              +
                              
                                XY
                                
                                  j
                                  F
                                
                              
                              +
                              
                                YX
                                
                                  j
                                  F
                                
                              
                            
                            )
                          
                        
                        ]
                      
                      +
                    
                  
                
                
                  
                    
                      2
                      ⁢
                      
                        s
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          θ
                        
                      
                      ⁢
                      
                        
                          c
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            θ
                          
                        
                        ⁡
                        
                          (
                          
                            
                              YY
                              
                                j
                                S
                              
                            
                            -
                            
                              XX
                              
                                j
                                F
                              
                            
                            -
                            
                              XX
                              
                                j
                                S
                              
                            
                            +
                            
                              YY
                              
                                j
                                F
                              
                            
                          
                          )
                        
                      
                    
                  
                
                
                  
                    
                      
                        (
                        
                          
                            XY
                            
                              j
                              S
                            
                          
                          +
                          
                            YX
                            
                              j
                              S
                            
                          
                          +
                          
                            XY
                            
                              j
                              F
                            
                          
                          +
                          
                            YX
                            
                              j
                              F
                            
                          
                        
                        )
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          
                            s
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                            2
                          
                          ⁡
                          
                            (
                            
                              
                                XX
                                
                                  j
                                  S
                                
                              
                              -
                              
                                YY
                                
                                  j
                                  S
                                
                              
                            
                            )
                          
                        
                        2
                      
                      +
                      
                        
                          
                            c
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                            2
                          
                          ⁡
                          
                            (
                            
                              
                                XY
                                
                                  j
                                  S
                                
                              
                              +
                              
                                YX
                                
                                  j
                                  S
                                
                              
                            
                            )
                          
                        
                        2
                      
                      +
                    
                  
                
                
                  
                    
                      
                        2
                        ⁢
                        
                          s
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            θ
                          
                        
                        ⁢
                        
                          
                            c
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                          
                          ⁡
                          
                            (
                            
                              
                                XX
                                
                                  j
                                  S
                                
                              
                              -
                              
                                YY
                                
                                  j
                                  S
                                
                              
                            
                            )
                          
                        
                        ⁢
                        
                          (
                          
                            
                              XY
                              
                                j
                                S
                              
                            
                            +
                            
                              YX
                              
                                j
                                S
                              
                            
                          
                          )
                        
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          
                            
                              
                                
                                  s
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                  2
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      XX
                                      
                                        j
                                        F
                                      
                                    
                                    -
                                    
                                      YY
                                      
                                        j
                                        F
                                      
                                    
                                  
                                  )
                                
                              
                              2
                            
                            +
                            
                              
                                
                                  c
                                  
                                    2
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    θ
                                  
                                  2
                                
                                ⁡
                                
                                  (
                                  
                                    
                                      XY
                                      
                                        j
                                        F
                                      
                                    
                                    +
                                    
                                      YX
                                      
                                        j
                                        F
                                      
                                    
                                  
                                  )
                                
                              
                              2
                            
                            +
                          
                        
                      
                      
                        
                          
                            2
                            ⁢
                            
                              s
                              
                                2
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                            
                            ⁢
                            
                              
                                c
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  θ
                                
                              
                              ⁡
                              
                                (
                                
                                  
                                    XX
                                    
                                      j
                                      F
                                    
                                  
                                  -
                                  
                                    YY
                                    
                                      j
                                      F
                                    
                                  
                                
                                )
                              
                            
                            ⁢
                            
                              (
                              
                                
                                  XY
                                  
                                    j
                                    F
                                  
                                
                                +
                                
                                  YX
                                  
                                    j
                                    F
                                  
                                
                              
                              )
                            
                          
                        
                      
                    
                  
                
              
              }
            
          
          =
          
            
              ∑
              
                k
                ,
                m
                ,
                n
              
            
            ⁢
            
              {
              
                
                  
                    
                      
                        
                          (
                          
                            
                              YY
                              
                                j
                                S
                              
                            
                            -
                            
                              XX
                              
                                j
                                F
                              
                            
                            +
                            
                              XX
                              
                                j
                                S
                              
                            
                            -
                            
                              YY
                              
                                j
                                F
                              
                            
                          
                          )
                        
                        2
                      
                      +
                    
                  
                
                
                  
                    
                      
                        c
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          θ
                        
                        2
                      
                      [
                      
                        
                          
                            (
                            
                              
                                YY
                                
                                  j
                                  S
                                
                              
                              -
                              
                                XX
                                
                                  j
                                  F
                                
                              
                              -
                              
                                XX
                                
                                  j
                                  S
                                
                              
                              +
                              
                                YY
                                
                                  j
                                  F
                                
                              
                            
                            )
                          
                          2
                        
                        +
                      
                    
                  
                
                
                  
                    
                      
                        
                          
                            
                              (
                              
                                
                                  XY
                                  
                                    j
                                    S
                                  
                                
                                +
                                
                                  YX
                                  
                                    j
                                    S
                                  
                                
                              
                              )
                            
                            2
                          
                          +
                          
                            
                              (
                              
                                
                                  XY
                                  
                                    j
                                    F
                                  
                                
                                +
                                
                                  YX
                                  
                                    j
                                    F
                                  
                                
                              
                              )
                            
                            2
                          
                        
                        ]
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          
                            
                              s
                              
                                2
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                              
                              2
                            
                            [
                            
                              
                                
                                  (
                                  
                                    
                                      XY
                                      
                                        j
                                        S
                                      
                                    
                                    +
                                    
                                      YX
                                      
                                        j
                                        S
                                      
                                    
                                    +
                                    
                                      XY
                                      
                                        j
                                        F
                                      
                                    
                                    +
                                    
                                      YX
                                      
                                        j
                                        F
                                      
                                    
                                  
                                  )
                                
                                2
                              
                              +
                            
                          
                        
                      
                      
                        
                          
                            
                              
                                
                                  
                                    (
                                    
                                      
                                        XX
                                        
                                          j
                                          S
                                        
                                      
                                      -
                                      
                                        YY
                                        
                                          j
                                          S
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                                +
                                
                                  
                                    (
                                    
                                      
                                        XX
                                        
                                          j
                                          F
                                        
                                      
                                      -
                                      
                                        YY
                                        
                                          j
                                          F
                                        
                                      
                                    
                                    )
                                  
                                  2
                                
                              
                              ]
                            
                            +
                          
                        
                      
                    
                  
                
                
                  
                    
                      2
                      ⁢
                      
                        (
                        
                          
                            YY
                            
                              j
                              S
                            
                          
                          -
                          
                            XX
                            
                              j
                              F
                            
                          
                          +
                          
                            XX
                            
                              j
                              S
                            
                          
                          -
                          
                            YY
                            
                              j
                              F
                            
                          
                        
                        )
                      
                    
                  
                
                
                  
                    
                      [
                      
                        
                          
                            c
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                          
                          ⁡
                          
                            (
                            
                              
                                YY
                                
                                  j
                                  S
                                
                              
                              -
                              
                                XX
                                
                                  j
                                  F
                                
                              
                              -
                              
                                XX
                                
                                  j
                                  S
                                
                              
                              +
                              
                                YY
                                
                                  j
                                  F
                                
                              
                            
                            )
                          
                        
                        +
                      
                    
                  
                
                
                  
                    
                      
                        
                          
                            s
                            
                              2
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                            
                          
                          ⁡
                          
                            (
                            
                              
                                XY
                                
                                  j
                                  S
                                
                              
                              +
                              
                                YX
                                
                                  j
                                  S
                                
                              
                              +
                              
                                XY
                                
                                  j
                                  F
                                
                              
                              +
                              
                                YX
                                
                                  j
                                  F
                                
                              
                            
                            )
                          
                        
                        ]
                      
                      +
                    
                  
                
                
                  
                    
                      2
                      ⁢
                      
                        s
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          θ
                        
                      
                      ⁢
                      
                        
                          c
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            θ
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  (
                                  
                                    
                                      YY
                                      
                                        j
                                        S
                                      
                                    
                                    -
                                    
                                      XX
                                      
                                        j
                                        F
                                      
                                    
                                    -
                                    
                                      XX
                                      
                                        j
                                        S
                                      
                                    
                                    +
                                    
                                      YY
                                      
                                        j
                                        F
                                      
                                    
                                  
                                  )
                                
                              
                            
                            
                              
                                
                                  
                                    (
                                    
                                      
                                        XY
                                        
                                          j
                                          S
                                        
                                      
                                      +
                                      
                                        YX
                                        
                                          j
                                          S
                                        
                                      
                                      +
                                      
                                        XY
                                        
                                          j
                                          F
                                        
                                      
                                      +
                                      
                                        YX
                                        
                                          j
                                          F
                                        
                                      
                                    
                                    )
                                  
                                  +
                                
                              
                            
                            
                              
                                
                                  
                                    
                                      (
                                      
                                        
                                          XX
                                          
                                            j
                                            S
                                          
                                        
                                        -
                                        
                                          YY
                                          
                                            j
                                            S
                                          
                                        
                                      
                                      )
                                    
                                    ⁢
                                    
                                      (
                                      
                                        
                                          XY
                                          
                                            j
                                            S
                                          
                                        
                                        +
                                        
                                          YX
                                          
                                            j
                                            S
                                          
                                        
                                      
                                      )
                                    
                                  
                                  +
                                
                              
                            
                            
                              
                                
                                  
                                    (
                                    
                                      
                                        XX
                                        
                                          j
                                          F
                                        
                                      
                                      -
                                      
                                        YY
                                        
                                          j
                                          F
                                        
                                      
                                    
                                    )
                                  
                                  ⁢
                                  
                                    (
                                    
                                      
                                        XY
                                        
                                          j
                                          F
                                        
                                      
                                      +
                                      
                                        YX
                                        
                                          j
                                          F
                                        
                                      
                                    
                                    )
                                  
                                
                              
                            
                          
                          ]
                        
                      
                    
                  
                
              
              }
            
          
        
      
    
  




Define





  
    
      a
      
        S
        ,
        j
      
    
    =
    
      
        YY
        
          j
          S
        
      
      +
      
        XX
        
          j
          S
        
      
    
  
  ,
  
    
      a
      
        F
        ,
        j
      
    
    =
    
      
        YY
        
          j
          F
        
      
      +
      
        XX
        
          j
          F
        
      
    
  
  ,
  
    

  
  ⁢
  
    
      b
      
        S
        ,
        j
      
    
    =
    
      
        YY
        
          j
          S
        
      
      -
      
        XX
        
          j
          S
        
      
    
  
  ,
  
    
      b
      
        F
        ,
        j
      
    
    =
    
      
        YY
        
          j
          F
        
      
      -
      
        XX
        
          j
          F
        
      
    
  
  ,
  
    

  
  ⁢
  
    
      c
      
        S
        ,
        j
      
    
    =
    
      
        XY
        
          j
          S
        
      
      +
      
        XY
        
          j
          S
        
      
    
  
  ,
  
    
      c
      
        F
        ,
        j
      
    
    =
    
      
        
          XY
          
            j
            F
          
        
        +
        
          
            YX
            
              j
              F
            
          
          .
          
            

          
          ⁢
          
            a
            j
          
        
      
      =
      
        
          
            YY
            
              j
              S
            
          
          -
          
            XX
            
              j
              F
            
          
          +
          
            XX
            
              j
              S
            
          
          -
          
            YY
            
              j
              F
            
          
        
        =
        
          
            a
            
              S
              ,
              j
            
          
          -
          
            a
            
              F
              ,
              j
            
          
        
      
    
  
  ,
  
    

  
  ⁢
  
    
      b
      j
    
    =
    
      
        
          YY
          
            j
            S
          
        
        -
        
          XX
          
            j
            F
          
        
        -
        
          XX
          
            j
            S
          
        
        +
        
          YY
          
            j
            F
          
        
      
      =
      
        
          b
          
            S
            ,
            j
          
        
        -
        
          b
          
            F
            ,
            j
          
        
      
    
  
  ,
  
    

  
  ⁢
  
    
      c
      j
    
    =
    
      
        
          XY
          
            j
            S
          
        
        +
        
          YX
          
            j
            F
          
        
        +
        
          XY
          
            j
            S
          
        
        +
        
          YX
          
            j
            F
          
        
      
      =
      
        
          c
          
            S
            ,
            j
          
        
        -
        
          
            c
            
              F
              ,
              j
            
          
          .
        
      
    
  




Then





  O
  =
  
    
      
        a
        2
      
      +
      
        
          c
          
            2
            ⁢
            
              
            
            ⁢
            θ
          
          2
        
        ⁡
        
          (
          
            
              b
              2
            
            +
            
              c
              S
              2
            
            +
            
              c
              F
              2
            
          
          )
        
      
      +
      
        
          s
          
            2
            ⁢
            
              
            
            ⁢
            θ
          
          2
        
        ⁡
        
          (
          
            
              c
              2
            
            +
            
              b
              S
              2
            
            +
            
              b
              F
              2
            
          
          )
        
      
      +
      
        2
        ⁢
        
          c
          
            2
            ⁢
            
              
            
            ⁢
            θ
          
        
        ⁢
        
          
            a
            ⇀
          
          ·
          
            b
            ⇀
          
        
      
      +
      
        2
        ⁢
        
          s
          
            2
            ⁢
            
              
            
            ⁢
            θ
          
        
        ⁢
        
          
            a
            ⇀
          
          ·
          
            c
            ⇀
          
        
      
      +
      
        2
        ⁢
        
          s
          
            2
            ⁢
            
              
            
            ⁢
            θ
          
        
        ⁢
        
          
            c
            
              2
              ⁢
              
                
              
              ⁢
              θ
            
          
          ⁡
          
            (
            
              
                
                  b
                  ⇀
                
                ·
                
                  c
                  ⇀
                
              
              -
              
                
                  
                    b
                    ⇀
                  
                  S
                
                ·
                
                  
                    c
                    ⇀
                  
                  S
                
              
              -
              
                
                  
                    b
                    ⇀
                  
                  F
                
                ·
                
                  
                    c
                    ⇀
                  
                  F
                
              
            
            )
          
        
      
    
    =
    
      
        
          a
          2
        
        +
        
          
            c
            
              2
              ⁢
              
                
              
              ⁢
              θ
            
            2
          
          ⁡
          
            (
            
              
                b
                2
              
              +
              
                c
                2
              
              -
              
                2
                ⁢
                
                  
                    
                      c
                      ⇀
                    
                    S
                  
                  ·
                  
                    
                      c
                      ⇀
                    
                    F
                  
                
              
            
            )
          
        
        +
        
          
            s
            
              2
              ⁢
              
                
              
              ⁢
              θ
            
            2
          
          ⁡
          
            (
            
              
                c
                2
              
              +
              
                b
                2
              
              -
              
                2
                ⁢
                
                  
                    
                      b
                      ⇀
                    
                    S
                  
                  ·
                  
                    
                      b
                      ⇀
                    
                    F
                  
                
              
            
            )
          
        
        +
        
          2
          ⁢
          
            
          
          ⁢
          
            c
            
              2
              ⁢
              
                
              
              ⁢
              θ
            
          
          ⁢
          
            
              a
              ⇀
            
            ·
            
              b
              ⇀
            
          
        
        +
        
          2
          ⁢
          
            s
            
              2
              ⁢
              
                
              
              ⁢
              θ
            
          
          ⁢
          
            
              a
              ⇀
            
            ·
            
              c
              ⇀
            
          
        
        +
        
          2
          ⁢
          
            s
            
              2
              ⁢
              
                
              
              ⁢
              θ
            
          
          ⁢
          
            
              c
              
                2
                ⁢
                
                  
                
                ⁢
                θ
              
            
            ⁡
            
              (
              
                
                  
                    
                      b
                      ⇀
                    
                    S
                  
                  ·
                  
                    
                      c
                      ⇀
                    
                    F
                  
                
                +
                
                  
                    
                      b
                      ⇀
                    
                    F
                  
                  ·
                  
                    
                      c
                      ⇀
                    
                    S
                  
                
              
              )
            
          
        
      
      =
      
        
          a
          2
        
        +
        
          b
          2
        
        +
        
          c
          2
        
        -
        
          2
          ⁢
          
            
              c
              
                2
                ⁢
                θ
              
              2
            
            ⁡
            
              (
              
                
                  
                    c
                    ⇀
                  
                  S
                
                ·
                
                  
                    c
                    ⇀
                  
                  F
                
              
              )
            
          
        
        -
        
          2
          ⁢
          
            
              s
              
                2
                ⁢
                
                  
                
                ⁢
                θ
              
              2
            
            ⁡
            
              (
              
                
                  
                    b
                    ⇀
                  
                  S
                
                ·
                
                  
                    b
                    ⇀
                  
                  F
                
              
              )
            
          
        
        +
        
          2
          ⁢
          
            c
            
              2
              ⁢
              
                
              
              ⁢
              θ
            
          
          ⁢
          
            
              a
              ⇀
            
            ·
            
              b
              ⇀
            
          
        
        +
        
          2
          ⁢
          
            c
            
              2
              ⁢
              
                
              
              ⁢
              θ
            
          
          ⁢
          
            
              a
              ⇀
            
            ·
            
              c
              ⇀
            
          
        
        +
        
          2
          ⁢
          
            s
            
              2
              ⁢
              
                
              
              ⁢
              θ
            
          
          ⁢
          
            
              
                c
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  θ
                
              
              ⁡
              
                (
                
                  
                    
                      
                        b
                        ⇀
                      
                      S
                    
                    ·
                    
                      
                        c
                        ⇀
                      
                      F
                    
                  
                  +
                  
                    
                      
                        b
                        ⇀
                      
                      F
                    
                    ·
                    
                      
                        c
                        ⇀
                      
                      S
                    
                  
                
                )
              
            
            .
          
        
      
    
  




Differentiate,





  0
  =
  
    
      
        
          s
          
            4
            ⁢
            
              
            
            ⁢
            θ
          
        
        ⁡
        
          (
          
            
              
                
                  c
                  ⇀
                
                S
              
              ·
              
                
                  c
                  ⇀
                
                F
              
            
            -
            
              
                
                  b
                  ⇀
                
                S
              
              ·
              
                
                  b
                  ⇀
                
                F
              
            
          
          )
        
      
      +
      
        
          c
          
            4
            ⁢
            
              
            
            ⁢
            θ
          
        
        ⁡
        
          (
          
            
              
                
                  b
                  ⇀
                
                S
              
              ·
              
                
                  c
                  ⇀
                
                F
              
            
            +
            
              
                
                  b
                  ⇀
                
                F
              
              ·
              
                
                  c
                  ⇀
                
                S
              
            
          
          )
        
      
      -
      
        
          s
          
            2
            ⁢
            θ
          
        
        ⁢
        
          
            a
            ⇀
          
          ·
          
            b
            ⇀
          
        
      
      +
      
        
          c
          
            2
            ⁢
            
              
            
            ⁢
            θ
          
        
        ⁢
        
          
            a
            ⇀
          
          ·
          
            c
            ⇀
          
        
      
    
    =
    
      
        
          2
          ⁢
          
            s
            
              2
              ⁢
              
                
              
              ⁢
              θ
            
          
          ⁢
          
            
              c
              
                2
                ⁢
                
                  
                
                ⁢
                θ
              
            
            ⁡
            
              (
              
                
                  
                    
                      c
                      ⇀
                    
                    S
                  
                  ·
                  
                    
                      c
                      ⇀
                    
                    F
                  
                
                -
                
                  
                    
                      b
                      ⇀
                    
                    S
                  
                  ·
                  
                    
                      b
                      ⇀
                    
                    F
                  
                
              
              )
            
          
        
        +
        
          
            (
            
              
                2
                ⁢
                
                  c
                  
                    2
                    ⁢
                    θ
                  
                  2
                
              
              -
              1
            
            )
          
          ⁢
          
            (
            
              
                
                  
                    b
                    ⇀
                  
                  S
                
                ·
                
                  
                    c
                    ⇀
                  
                  F
                
              
              +
              
                
                  
                    b
                    ⇀
                  
                  F
                
                ·
                
                  
                    c
                    ⇀
                  
                  S
                
              
            
            )
          
        
        -
        
          
            s
            
              2
              ⁢
              
                
              
              ⁢
              θ
            
          
          ⁢
          
            
              a
              ⇀
            
            ·
            
              b
              ⇀
            
          
        
        +
        
          
            c
            
              2
              ⁢
              θ
            
          
          ⁢
          
            
              a
              ⇀
            
            ·
            
              c
              ⇀
            
          
        
      
      =
      
        
          
            s
            
              2
              ⁢
              θ
            
          
          ⁡
          
            [
            
              
                2
                ⁢
                
                  
                    c
                    
                      2
                      ⁢
                      θ
                    
                  
                  ⁡
                  
                    (
                    
                      
                        
                          
                            c
                            ⇀
                          
                          S
                        
                        ·
                        
                          
                            c
                            ⇀
                          
                          F
                        
                      
                      -
                      
                        
                          
                            b
                            ⇀
                          
                          S
                        
                        ·
                        
                          
                            b
                            ⇀
                          
                          F
                        
                      
                    
                    )
                  
                
              
              -
              
                
                  a
                  ⇀
                
                ·
                
                  b
                  ⇀
                
              
            
            ]
          
        
        +
        
          
            (
            
              
                2
                ⁢
                
                  c
                  
                    2
                    ⁢
                    θ
                  
                  2
                
              
              -
              1
            
            )
          
          ⁢
          
            (
            
              
                
                  
                    b
                    ⇀
                  
                  S
                
                ·
                
                  
                    c
                    ⇀
                  
                  F
                
              
              +
              
                
                  
                    b
                    ⇀
                  
                  F
                
                ·
                
                  
                    c
                    ⇀
                  
                  S
                
              
            
            )
          
        
        +
        
          
            c
            
              2
              ⁢
              θ
            
          
          ⁢
          
            
              a
              ⇀
            
            ·
            
              
                c
                ⇀
              
              .
            
          
        
      
    
  




Square,





  
    
      
        s
        
          2
          ⁢
          
            
          
          ⁢
          θ
        
        2
      
      ⁡
      
        [
        
          
            2
            ⁢
            
              
                c
                
                  2
                  ⁢
                  θ
                
              
              ⁡
              
                (
                
                  
                    
                      
                        c
                        ⇀
                      
                      S
                    
                    ·
                    
                      
                        c
                        ⇀
                      
                      F
                    
                  
                  -
                  
                    
                      
                        b
                        ⇀
                      
                      S
                    
                    ·
                    
                      
                        b
                        ⇀
                      
                      F
                    
                  
                
                )
              
            
          
          -
          
            
              a
              ⇀
            
            ·
            
              b
              ⇀
            
          
        
        ]
      
    
    2
  
  =
  
    
      
        [
        
          
            
              (
              
                
                  2
                  ⁢
                  
                    c
                    
                      2
                      ⁢
                      θ
                    
                    2
                  
                
                -
                1
              
              )
            
            ⁢
            
              (
              
                
                  
                    
                      b
                      ⇀
                    
                    S
                  
                  ·
                  
                    
                      c
                      ⇀
                    
                    F
                  
                
                +
                
                  
                    
                      b
                      ⇀
                    
                    F
                  
                  ·
                  
                    
                      c
                      ⇀
                    
                    S
                  
                
              
              )
            
          
          +
          
            
              c
              
                2
                ⁢
                θ
              
            
            ⁢
            
              
                a
                ⇀
              
              ·
              
                c
                ⇀
              
            
          
        
        ]
      
      2
    
    .
  




Rearrange





  0
  =
  
    
      
        
          
            (
            
              
                c
                
                  2
                  ⁢
                  θ
                
                2
              
              -
              1
            
            )
          
          ⁡
          
            [
            
              
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Using the apparatus, systems, and methods disclosed herein, those in the petroleum recovery industry and other industries may now be able to more accurately and rapidly assess the properties of geologic formations, including stress induced anisotropy. Increased operational efficiency and client satisfaction may result.
Such embodiments of the inventive subject matter may be referred to herein, individually and/or collectively, by the term “invention” merely for convenience and without intending to voluntarily limit the scope of this application to any single invention or inventive concept if more than one is in fact disclosed. Thus, although specific embodiments have been illustrated and described herein, it will be appreciated by those of ordinary skill in the art that any arrangement that is calculated to achieve the same purpose may be substituted for the specific embodiments shown. Various embodiments use permutations and/or combinations of embodiments described herein. It is to be understood that the above description is intended to be illustrative, and not restrictive, and that the phraseology or terminology employed herein is for the purpose of description. Combinations of the above embodiments and other embodiments will be apparent to those of skill in the art upon studying the above description.

## Claims

1. An apparatus comprising:
a tool having a longitudinal axis;
a transmitter ring attached to the tool, the transmitter ring having orthogonal dipole transmitters;
a number of receiver rings, each receiver ring disposed along a length of the longitudinal axis at a different distance from the other receiver rings along the length, each receiver ring having a plurality of acoustic dipole receivers azimuthally disposed around the tool;
recording logic to record a plurality of acoustic waveforms corresponding to acoustic waves received at the acoustic dipole receivers when the tool is operated in a geological formation, the acoustic waves being generated by the orthogonal dipole transmitters; and
a signal processor to generate an objective function, based on ordered sets of the plurality of acoustic waveforms received at each receiver ring, with respect to an azimuth angle, wherein the azimuth angle corresponds to an orientation of the orthogonal dipole transmitters and the acoustic dipole receivers relative to fast and slow principal flexural wave axes, with the signal processor structured to estimate a global minimum of the objective function with respect to the azimuth angle and to determine an anisotropy angle of the geological formation based on the global minimum.

2. The apparatus of claim 1, wherein the signal processor is structured to generate the objective function with respect to the azimuth angle and a set of auxiliary parameters.

3. The apparatus of claim 1, wherein the auxiliary parameters include fast and slow slowness values.

4. The apparatus of claim 1, wherein a receiver ring of the number of receiver rings has N dipole receivers and is operable to provide N(N−1)/2 independent 2×2 waveforms, N being an integer greater than two.

5. The apparatus of claim 1, wherein the signal processor is structured to diagonalize independent 2×2 waveforms, each independent 2×2 waveform being in-line and cross-line components according to two azimuth indices, the two azimuth indices not equal to each other, each azimuth index correlated to a different one of the acoustic dipole receivers of one of the receiver rings.

6. The apparatus of claim 1, wherein the signal processor is structured to remove existing ambiguities associated with the fast and slow principal flexural wave axes and/or objective function minima and maxima.

7. The apparatus of claim 1, wherein the tool comprises one of a wireline tool or a measurement while drilling tool.

8. A processor-implemented method of estimating an azimuth angle, executed by one or more processors that perform the method, the method comprising:
recording a plurality of acoustic waveforms corresponding to acoustic waves received at a number of receiver rings, each receiver ring disposed along a length of a longitudinal axis of a tool at a different distance from the other receiver rings along the length of the longitudinal axis, the tool surrounded by a geological formation, each receiver ring having a plurality of acoustic dipole receivers azimuthally disposed around the tool and receiving acoustic waves, the acoustic waves being generated by orthogonal dipole transmitters in a transmitter ring attached to the tool;
generating an objective function, based on ordered sets of the plurality of acoustic waveforms received at each receiver ring, with respect to an azimuth angle, wherein the azimuth angle corresponds to an orientation of the orthogonal dipole transmitters and the acoustic dipole receivers relative to fast and slow principal flexural wave axes;
estimating a global minimum of the objective function with respect to the azimuth angle; and
determining an anisotropy angle of the geological formation based on the global minimum.

9. The processor-implemented method of claim 8, wherein the method includes forming the ordered sets of the plurality of acoustic waveforms from a receiver ring of the number of receiver rings having N dipole receivers providing N(N−1)/2 independent 2×2 waveforms, N being an integer greater than two.

10. The processor-implemented method of claim 8, wherein the method includes diagonalizing independent 2×2 waveforms, each independent 2×2 waveform being in-line and cross-line components according to two azimuth indices, the two azimuth indices not equal to each other, each azimuth index correlated to a different one of the acoustic dipole receivers of one of the receiver rings.

11. The processor-implemented method of claim 8, wherein generating the objective function includes extending a first objective function to include generalized waveforms correlated to the plurality of acoustic dipole receivers azimuthally disposed around the tool in each ring, the first objective function generated with respect to in-line and cross-line waveforms from transmitter rings to receiver rings, each transmitter ring having only two dipole transmitters orthogonal to each other and each receiver ring having only two dipole receivers orthogonal to each other.

12. The processor-implemented method of claim 8, wherein estimating a global minimum of the objective function is accomplished analytically or by using a numerical search algorithm.

13. The processor-implemented method of claim 8, wherein determining the anisotropy angle of the geological formation includes implementing the generating, the estimating, and the determining for a data point space defined over multiple frequency bands having different starting frequencies or by multiple different time periods having different starting times.

14. The processor-implemented method of claim 8, wherein the method includes removing existing ambiguities associated with the fast and slow principal flexural wave axes and/or objective function minima and maxima.

15. The processor-implemented method of claim 14, wherein removing existing objective function minimum/maximum ambiguity includes substituting the azimuth angle into the objective function and substituting an ambiguous angle into the objective function, and selecting an angle that minimizes the objective function.

16. The processor-implemented method of claim 14, wherein removing existing fast slow principal axis ambiguity includes either
constraining auxiliary parameters a priori to eliminate the ambiguity before minimizing the objective function; or
eigenvalue alignment a priori to eliminate the ambiguity before minimizing the objective function; or
rotating cross-dipole waveforms of the ordered sets of the plurality of acoustic waveforms by min/max ambiguity resolved azimuth angle into the fast and slow waveforms and comparing slowness of the cross-dipole waveforms using either a time semblance algorithm or frequency semblance algorithm.

17. A non-transitory machine-readable storage device having instructions stored thereon, which, when performed by a machine, cause the machine to perform operations, the operations comprising:
recording a plurality of acoustic waveforms corresponding to acoustic waves received at a number of receiver rings, each receiver ring disposed along a length of a longitudinal axis of a tool at a different distance from the other receiver rings along the length of the longitudinal axis, the tool surrounded by a geological formation, each receiver ring having a plurality of acoustic dipole receivers azimuthally disposed around the tool and receiving acoustic waves, the acoustic waves being generated by orthogonal dipole transmitters in a transmitter ring attached to the tool;
generating an objective function, based on ordered sets of the plurality of acoustic waveforms received at each receiver ring, with respect to an azimuth angle, wherein the azimuth angle corresponds to an orientation of the orthogonal dipole transmitters and the acoustic dipole receivers relative to fast and slow principal flexural wave axes;
estimating a global minimum of the objective function with respect to the azimuth angle; and
determining an anisotropy angle of the geological formation based on the global minimum.

18. The non-transitory machine-readable storage device of claim 17, wherein generating the objective function includes generating the objective function with respect to the azimuth angle and a set of auxiliary parameters.

19. The non-transitory machine-readable storage device of claim 18, wherein the auxiliary parameters include fast and slow slowness values.

20. The non-transitory machine-readable storage device of claim 17, wherein the instructions include instructions to form the ordered sets of the plurality of acoustic waveforms from a receiver ring of the number of receiver rings having N dipole receivers providing N(N−1)/2 independent 2×2 waveforms, N being an integer greater than two.

21. The non-transitory machine-readable storage device of claim 17, wherein the instructions include instructions to diagonalize independent 2×2 waveforms, each independent 2×2 waveform being in-line and cross-line components according to two azimuth indices, the two azimuth indices not equal to each other, each azimuth index correlated to a different one of the acoustic dipole receivers of one of the receiver rings.

22. The non-transitory machine-readable storage device of claim 17, whereingenerating the objective function includes extending a first objective function to include generalized waveforms correlated to the plurality of acoustic dipole receivers azimuthally disposed around the tool in each ring, the first objective function generated with respect to in-line and cross-line waveforms from transmitter rings to receiver rings, each transmitter ring having only two dipole transmitters orthogonal to each other and each receiver ring having only two dipole receivers orthogonal to each other.