# Patent

## Title
Gain compensated directional propagation measurements

## Abstract

A method for obtaining gain compensated electromagnetic logging while drilling propagation measurements includes rotating an electromagnetic logging while drilling tool having first and second transverse transmitter antennas whose moments are orthogonal to one another and first and second transverse receiver antennas whose moments are orthogonal to one another in a subterranean wellbore. Electromagnetic voltage measurements are acquired from the first and second transverse receiver antennas while rotating. The acquired voltage measurements are processed to compute harmonic voltage coefficients. Ratios of selected ones of the harmonic voltage coefficients are in turn processed to compute a gain compensated quantity including a transverse term.

## Background
CROSS REFERENCE TO RELATED APPLICATIONS
This application claims the benefit of the following four U.S. Provisional Patent Applications (each of which was filed on Mar. 29, 2014): (i) Ser. No. 61/972,287 entitled Fully Gain Compensated Tensor Propagation Cross-Term Measurements with Orthogonal Antennas, (ii) Ser. No. 61/972,288 entitled Improved Symmetrized and Anti-symmetrized Measurements with Orthogonal Antennas, (iii) Ser. No. 61/972,289 entitled Compensated Directional Measurements using Tilted Moments that are Independent of Tilt Angle Parameter, and (iv) Ser. No. 61/972,290 entitled Compensated Array Configurations with Orthogonal Antennas.


FIELD OF THE DISCLOSURE
Disclosed embodiments relate generally to downhole electromagnetic logging methods and more particularly to a logging tool and a method for making gain compensated directional propagation measurements, such as phase shift and attenuation measurements, using orthogonal antennas.
BACKGROUND INFORMATION
The use of electromagnetic measurements in prior art downhole applications, such as logging while drilling (LWD) and wireline logging applications is well known. Such techniques may be utilized to determine a subterranean formation resistivity, which, along with formation porosity measurements, is often used to indicate the presence of hydrocarbons in the formation. Moreover, azimuthally sensitive directional resistivity measurements are commonly employed e.g., in pay-zone steering applications, to provide information upon which steering decisions may be made.
Downhole electromagnetic measurements are commonly inverted at the surface using a formation model to obtain various formation parameters, for example, including vertical resistivity, horizontal resistivity, distance to a remote bed, resistivity of the remote bed, dip angle, and the like. One challenge in utilizing directional electromagnetic resistivity measurements, is obtaining a sufficient quantity of data to perform a reliable inversion. The actual formation structure is frequently significantly more complex than the formation models used in the inversion. The use of a three-dimensional matrix of propagation measurements may enable a full three-dimensional measurement of the formation properties to be obtained as well as improve formation imaging and electromagnetic look ahead measurements. However, there are no known methods for providing a fully gain compensated tri-axial propagation measurement.
SUMMARY
A method for obtaining gain compensated electromagnetic logging while drilling propagation measurements is disclosed. An electromagnetic logging while drilling tool having first and second transverse transmitter antennas whose moments are orthogonal to one another and first and second transverse receiver antennas whose moments are orthogonal to one another is rotated in a subterranean wellbore. Electromagnetic voltage measurements are acquired from the first and second transverse receiver antennas while rotating. The acquired voltage measurements are processed to compute harmonic voltage coefficients. Ratios of selected ones of the harmonic voltage coefficients are in turn processed to compute a gain compensated quantity including a transverse term.
The disclosed methodology provides a method for obtaining a gain compensated three-dimensional matrix of measurements using orthogonal antennas. The acquired measurements are fully gain compensated and independent of antenna tilt angle variation. Moreover, the disclosed method and apparatus tends to be insensitive to bending and alignment angle errors.
This summary is provided to introduce a selection of concepts that are further described below in the detailed description. This summary is not intended to identify key or essential features of the claimed subject matter, nor is it intended to be used as an aid in limiting the scope of the claimed subject matter.



BRIEF DESCRIPTION OF THE DRAWINGS
For a more complete understanding of the disclosed subject matter, and advantages thereof, reference is now made to the following descriptions taken in conjunction with the accompanying drawings, in which:
FIG. 1 depicts one example of a drilling rig on which the disclosed electromagnetic logging methods may be utilized.
FIG. 2A depicts one example of the electromagnetic logging tool shown on FIG. 1.
FIG. 2B schematically depicts the antenna moments in an electromagnetic logging tool including triaxial transmitters and receivers.
FIGS. 3A, 3B, 3C, and 3D (collectively FIG. 3) depict the antenna moments for various example transmitter and receiver configurations for obtaining gain compensated axial cross term quantities.
FIG. 4 depicts a flow chart of one disclosed method embodiment for obtaining gain compensated axial cross term quantities.
FIGS. 5A and 5B (collectively FIG. 5) depict the antenna moments for various example transmitter and receiver configurations for obtaining gain compensated symmetrized and anti-symmetrized quantities.
FIG. 6 depicts a flow chart of one disclosed method embodiment for obtaining gain compensated symmetrized and anti-symmetrized quantities.
FIG. 7 depicts a three layer formation model used to evaluate the directional response of disclosed symmetrized and anti-symmetrized measurements.
FIGS. 8A and 8B depict symmetrized and anti-symmetrized phase shift and attenuation versus total vertical depth at 30 degrees relative dip.
FIGS. 9A and 9B depict symmetrized and anti-symmetrized phase shift and attenuation versus total vertical depth at 70 degrees relative dip.
FIGS. 10A and 10B depict symmetrized and anti-symmetrized phase shift and attenuation versus total vertical depth at 88 degrees relative dip.
FIGS. 11A, 11B, 11C, and 11D (collectively FIG. 11) depict the antenna moments for various example transmitter and receiver configurations for obtaining gain compensated transverse coupling and cross-coupling quantities.
FIG. 12 depicts a flow chart of one disclosed method embodiment 140 for obtaining gain compensated transverse term quantities.
FIG. 13 depicts the antenna moments for an electromagnetic logging tool including tilted transmitters.



DETAILED DESCRIPTION
FIG. 1 depicts an example drilling rig 10 suitable for employing various method embodiments disclosed herein. A semisubmersible drilling platform 12 is positioned over an oil or gas formation (not shown) disposed below the sea floor 16. A subsea conduit 18 extends from deck 20 of platform 12 to a wellhead installation 22. The platform may include a derrick and a hoisting apparatus for raising and lowering a drill string 30, which, as shown, extends into borehole 40 and includes a drill bit 32 deployed at the lower end of a bottom hole assembly (BHA) that further includes an electromagnetic measurement tool 50 configured to make directional electromagnetic logging measurements. As described in more detail below the electromagnetic measurement tool 50 may include multiple orthogonal antennas deployed on a logging while drilling tool body.
It will be understood that the deployment illustrated on FIG. 1 is merely an example. Drill string 30 may include substantially any suitable downhole tool components, for example, including a steering tool such as a rotary steerable tool, a downhole telemetry system, and one or more MWD or LWD tools including various sensors for sensing downhole characteristics of the borehole and the surrounding formation. The disclosed embodiments are by no means limited to any particular drill string configuration.
It will be further understood that the disclosed embodiments are not limited to use with a semisubmersible platform 12 as illustrated on FIG. 1. The disclosed embodiments are equally well suited for use with either onshore or offshore subterranean operations.
FIG. 2A depicts one example of an electromagnetic measurement tool 50. In the depicted embodiment measurement tool 50 includes first and second axially spaced transmitters 52 and 54 and first and second axially spaced receivers 56 and 58 deployed on a logging while drilling tool body 51, with the receivers 56 and 58 being deployed axially between the transmitters 52 and 54. As described in more detail below, each of the transmitters 52 and 54 and receivers 56 and 58 includes at least one transverse antenna and may further include an axial antenna. For example, the transmitters and receivers may include a bi-axial antenna arrangement including an axial antenna and a transverse (cross-axial) antenna. In another embodiment, the transmitters and receivers may include a tri-axial antenna arrangement including an axial antenna and first and second transverse antennas that are orthogonal to one another. As is known to those of ordinary skill in the art, an axial antenna is one whose moment is substantially parallel with the longitudinal axis of the tool. Axial antennas are commonly wound about the circumference of the logging tool such that the plane of the antenna is substantially orthogonal to the tool axis. A transverse antenna is one whose moment is substantially perpendicular to the longitudinal axis of the tool. A transverse antenna may include, for example, a saddle coil (e.g., as disclosed in U.S. Patent Publications 2011/0074427 and 2011/0238312 each of which is incorporated by reference herein).
FIG. 2B depicts the moments (magnetic dipoles) of one embodiment of measurement tool 50 in which the transmitters 52, 54 and receivers 56, 58 each include a tri-axial antenna arrangement. Each of the transmitters 52, 54 includes an axial antenna T1z and T2z and first and second transverse antennas T1x, T1y and T2x, T2y. Likewise, each of the receivers 56, 58 includes an axial antenna R1z and R2z and first and second transverse antennas R1x, R1y and R2x, R2y. It will be understood that the disclosed embodiments are not limited to a tri-axial antenna configuration such as that depicted on FIG. 2B.
As is known to those of ordinary skill in the art, a time varying electric current (an alternating current) in a transmitting antenna produces a corresponding time varying magnetic field in the local environment (e.g., the tool collar and the formation). The magnetic field in turn induces electrical currents (eddy currents) in the conductive formation. These eddy currents further produce secondary magnetic fields which may produce a voltage response in a receiving antenna. The measured voltage in the receiving antennae can be processed, as is known to those of ordinary skill in the art, to obtain one or more properties of the formation.
In general the earth is anisotropic such that its electrical properties may be expressed as a three-dimensional tensor that contains information on formation resistivity anisotropy, dip, bed boundaries and other aspects of formation geometry. It will be understood by those of ordinary skill in the art that the mutual couplings between the tri-axial transmitter antennas and the tri-axial receiver antennas depicted on FIG. 2B form a three-dimensional matrix and thus may have sensitivity to a full three-dimensional formation impedance tensor. For example, a three-dimensional matrix of measured voltages V may be expressed as follows:




  
    
      
        
          V
          ij
        
        =
        
          
            [
            
              
                
                  
                    V
                    ijxx
                  
                
                
                  
                    V
                    ijxy
                  
                
                
                  
                    V
                    ijxz
                  
                
              
              
                
                  
                    V
                    ijyx
                  
                
                
                  
                    V
                    ijyy
                  
                
                
                  
                    V
                    ijyz
                  
                
              
              
                
                  
                    V
                    ijzx
                  
                
                
                  
                    V
                    ijzy
                  
                
                
                  
                    V
                    ijzz
                  
                
              
            
            ]
          
          =
          
            
              
                I
                i
              
              ⁢
              
                Z
                ij
              
            
            =
            
              
                [
                
                  
                    
                      
                        I
                        ix
                      
                    
                    
                      0
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      
                        I
                        iy
                      
                    
                    
                      0
                    
                  
                  
                    
                      0
                    
                    
                      0
                    
                    
                      
                        I
                        iz
                      
                    
                  
                
                ]
              
              ⁡
              
                [
                
                  
                    
                      
                        Z
                        ijxx
                      
                    
                    
                      
                        Z
                        ijxy
                      
                    
                    
                      
                        Z
                        ijxz
                      
                    
                  
                  
                    
                      
                        Z
                        ijyx
                      
                    
                    
                      
                        Z
                        ijyy
                      
                    
                    
                      
                        Z
                        ijyz
                      
                    
                  
                  
                    
                      
                        Z
                        ijzx
                      
                    
                    
                      
                        Z
                        ijzy
                      
                    
                    
                      
                        Z
                        ijzz
                      
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        (
        1
        )
      
    
  




where Vij represent the three-dimensional matrix of measured voltages with i indicating the corresponding transmitter triad (e.g., T1 or T2) and j indicating the corresponding receiver triad (e.g., R1 or R2), Ii represent the transmitter currents, and Zij represent the transfer impedances which depend on the electrical and magnetic properties of the environment surrounding the antenna pair in addition to the frequency, geometry, and spacing of the antennas. The third and fourth subscripts indicate the axial orientation of the transmitter and receiver antennas. For example, V12xy represents a voltage measurement on the y-axis antenna of receiver R2 from a firing of the x-axis antenna of transmitter T1.
When bending of the measurement tool is negligible (e.g., less than about 10 degrees), the three dimensional voltage matrix may be modeled mathematically, for example, as follows:

Vij=GTi(RθtZijRθ)GRj  (2)

where Zij represent the transfer impedances as described above, GTi and GRj are diagonal matrices representing the transmitter and receiver gains, Rθ represents the rotation matrix about the z-axis through angle θ, and the superscript t represents the transpose of the corresponding matrix. The gain and rotation matrices in Equation 2 may be given, for example, as follows:




  
    
      
        
          G
          Ti
        
        =
        
          [
          
            
              
                
                  g
                  Tix
                
              
              
                0
              
              
                0
              
            
            
              
                0
              
              
                
                  g
                  Tiy
                
              
              
                0
              
            
            
              
                0
              
              
                0
              
              
                
                  g
                  Tiz
                
              
            
          
          ]
        
      
    
    
      
        (
        3
        )
      
    
  
  
    
      
        
          G
          Rj
        
        =
        
          [
          
            
              
                
                  g
                  Rjx
                
              
              
                0
              
              
                0
              
            
            
              
                0
              
              
                
                  g
                  Rjy
                
              
              
                0
              
            
            
              
                0
              
              
                0
              
              
                
                  g
                  Rjz
                
              
            
          
          ]
        
      
    
    
      
        (
        4
        )
      
    
  
  
    
      
        
          R
          θ
        
        =
        
          [
          
            
              
                
                  cos
                  ⁡
                  
                    (
                    θ
                    )
                  
                
              
              
                
                  -
                  
                    sin
                    ⁡
                    
                      (
                      θ
                      )
                    
                  
                
              
              
                0
              
            
            
              
                
                  sin
                  ⁡
                  
                    (
                    θ
                    )
                  
                
              
              
                
                  cos
                  ⁡
                  
                    (
                    θ
                    )
                  
                
              
              
                0
              
            
            
              
                0
              
              
                0
              
              
                1
              
            
          
          ]
        
      
    
    
      
        (
        5
        )
      
    
  




The rotated couplings (shown in the parentheses in Equation 2) may be expressed mathematically in harmonic form, for example, as follows:

RθtZijRθ=ZDC_ij+ZFHC_ij cos(θ)+ZFHS_ij sin(θ)+ZSHC_ij cos(2θ)+ZSHS_ij sin(2θ)  (6)

where ZDC_ij represents a DC (average) coupling coefficient, ZFHC_ij and ZFHS_ij represent first order harmonic cosine and first order harmonic sine coefficients, and and ZSHS_ij represent second order harmonic cosine and second order harmonic sine coefficients of the ij transmitter receiver couplings. These coefficients are shown below:




  
    
      
        
          
            Z
            DC_ij
          
          =
          
            [
            
              
                
                  
                    
                      
                        Z
                        ijxx
                      
                      +
                      
                        Z
                        ijyy
                      
                    
                    2
                  
                
                
                  
                    
                      (
                      
                        
                          Z
                          ijxy
                        
                        -
                        
                          Z
                          ijyx
                        
                      
                      )
                    
                    2
                  
                
                
                  0
                
              
              
                
                  
                    -
                    
                      
                        (
                        
                          
                            Z
                            ijxy
                          
                          -
                          
                            Z
                            ijyx
                          
                        
                        )
                      
                      2
                    
                  
                
                
                  
                    
                      
                        Z
                        ijxx
                      
                      +
                      
                        Z
                        ijyy
                      
                    
                    2
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  
                    Z
                    ijzz
                  
                
              
            
            ]
          
        
        ⁢
        
          

        
        ⁢
        
          
            Z
            FHC_ij
          
          =
          
            [
            
              
                
                  0
                
                
                  0
                
                
                  
                    Z
                    ijxz
                  
                
              
              
                
                  0
                
                
                  0
                
                
                  
                    Z
                    ijyz
                  
                
              
              
                
                  
                    Z
                    ijzx
                  
                
                
                  
                    Z
                    ijzy
                  
                
                
                  0
                
              
            
            ]
          
        
        ⁢
        
          

        
        ⁢
        
          
            Z
            FHS_ij
          
          =
          
            [
            
              
                
                  0
                
                
                  0
                
                
                  
                    Z
                    ijyz
                  
                
              
              
                
                  0
                
                
                  0
                
                
                  
                    -
                    
                      Z
                      ijxz
                    
                  
                
              
              
                
                  
                    Z
                    ijzy
                  
                
                
                  
                    -
                    
                      Z
                      ijzx
                    
                  
                
                
                  0
                
              
            
            ]
          
        
        ⁢
        
          

        
        ⁢
        
          
            Z
            SHC_ij
          
          =
          
            [
            
              
                
                  
                    
                      
                        Z
                        ijxx
                      
                      -
                      
                        Z
                        ijyy
                      
                    
                    2
                  
                
                
                  
                    
                      (
                      
                        
                          Z
                          ijxy
                        
                        +
                        
                          Z
                          ijyx
                        
                      
                      )
                    
                    2
                  
                
                
                  0
                
              
              
                
                  
                    
                      (
                      
                        
                          Z
                          ijxy
                        
                        +
                        
                          Z
                          ijyx
                        
                      
                      )
                    
                    2
                  
                
                
                  
                    -
                    
                      
                        (
                        
                          
                            Z
                            ijxx
                          
                          +
                          
                            Z
                            ijyy
                          
                        
                        )
                      
                      2
                    
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  0
                
              
            
            ]
          
        
        ⁢
        
          

        
        ⁢
        
          
            Z
            SHS_ij
          
          =
          
            [
            
              
                
                  
                    
                      (
                      
                        
                          Z
                          ijxy
                        
                        +
                        
                          Z
                          ijyx
                        
                      
                      )
                    
                    2
                  
                
                
                  
                    -
                    
                      
                        (
                        
                          
                            Z
                            ijxx
                          
                          -
                          
                            Z
                            ijyy
                          
                        
                        )
                      
                      2
                    
                  
                
                
                  0
                
              
              
                
                  
                    -
                    
                      
                        (
                        
                          
                            Z
                            ijxx
                          
                          -
                          
                            Z
                            ijyy
                          
                        
                        )
                      
                      2
                    
                  
                
                
                  
                    -
                    
                      
                        (
                        
                          
                            Z
                            ijxy
                          
                          +
                          
                            Z
                            ijyx
                          
                        
                        )
                      
                      2
                    
                  
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  0
                
              
            
            ]
          
        
      
    
    
      
        (
        7
        )
      
    
  




In general, the receiving antenna voltages are measured while the tool rotates in the borehole. Following the form of Equation 6, the measured voltages may be expressed mathematically in terms of its harmonic voltage coefficients, for example, as follows thereby enabling the harmonic voltage coefficients to be obtained:

Vij=VDC_ij+VFHC_ij cos(θ)+VFHS_ij sin(θ)+VSHC_ij cos(2θ)+VSHS_ij sin(2θ)  (8)

wherein where VDC_ij represents a DC voltage coefficient, VFHC_ij and VFHS_ij represent first order harmonic cosine and first order harmonic sine voltage coefficients (also referred to herein as first harmonic cosine and first harmonic sine voltage coefficients), and VSHC_ij and VSHS_ij represent second order harmonic cosine and second order harmonic sine voltage coefficients (also referred to herein as second harmonic cosine and second harmonic sine voltage coefficients) of the ij transmitter receiver couplings.
Gain Compensated Axial Cross Terms
It will be understood that collocated tri-axial transmitter and receiver embodiments (e.g., as depicted on FIG. 2B) are not required to gain compensate certain of the three-dimensional matrix components. For example, the axial cross terms (i.e., the xz, zx, yz, and zy terms) may be gain compensated using any tool embodiment that includes an axial transmitter antenna, a transverse (cross-axial) transmitter antenna, an axial receiver antenna, and a transverse receiver antenna deployed on the tool body. These transmitter and receiver antennas may be distributed along the tool body with substantially any suitable spacing and order. Moreover, the transmitter antennas and/or the receiver antennas may be collocated. The disclosed embodiments are not limited to any particular transmitter and receiver antenna configuration so long as the tool includes at least one axial transmitter antenna, at least one transverse transmitter antenna, at least one axial receiver antenna, and at least one transverse receiver antenna.
FIGS. 3A, 3B, 3C, and 3D (collectively FIG. 3) depict the antenna moments for various example transmitter and receiver configurations for obtaining gain compensated axial cross terms (also referred to herein as axial cross coupling impedances). FIG. 3A depicts an example tool embodiment 60 including a transmitter T axially spaced apart from a receiver R. The transmitter T includes collocated axial and transverse transmitting antennas having moments Tz and Tx. The receiver R includes collocated axial and transverse receiving antennas having moments Rz and Rx.
FIG. 3B depicts an alternative tool embodiment 65 that is similar to tool embodiment 60 in that it also includes a transmitter T including axial and transverse transmitting antennas having moments Tz and Tx. Tool embodiment 65 differs from tool embodiment 60 in that the axial and transverse receiving antennas Rz and Rx are not collocated, but are axially spaced apart from one another on the tool body.
FIG. 3C depicts another alternative tool embodiment 70 that is similar to tool embodiment 60 in that it also includes a receiver R including axial and transverse receiving antennas having moments Rz and Rx. Tool embodiment 70 differs from tool embodiment 60 in that the axial and transverse transmitting antennas Tz and Tx are not collocated, but are axially spaced apart from one another on the tool body.
FIG. 3D depicts still another alternative tool embodiment 75 including axial and transverse transmitting antennas and axial and transverse receiving antennas Tz and Tx and Rz and Rx. Tool embodiment 75 differs from tool embodiment 60 in that neither the transmitting antennas nor the receiving antennas are collocated, but are axially spaced apart on the tool body. It will be understood that receiver antennas are not necessarily deployed between the transmitter antennas as depicted (TRRT), but may be axially distributed in substantially any order, for example, (i) with the transmitter antennas between the receiver antennas (RTTR), (ii) with the transmitter antennas alternating with the receiver antennas (TRTR or RTRT), or (iii) with the transmitter antennas on one side and the receiver antennas on the other (TTRR or RRTT). It will thus be understood that the disclosed embodiments are not limited to collocation or non-collocation of the axial and transverse transmitting and/or receiving antennas or to any particular spacing and location thereof along the tool body.
It will further be understood that one or more of the transmitters and/or receivers in tool embodiments 60, 65, 70, and 75 may optionally further include a second transverse antenna such that the transmitter and/or receiver includes a triaxial antenna arrangement having three antennas that are arranged to be mutually independent (e.g., as in FIG. 2B).
FIG. 4 depicts a flow chart of one disclosed method embodiment 100 for obtaining one or more gain compensated axial cross terms. An electromagnetic measurement tool (e.g., one of the measurement tools depicted on FIG. 2B or FIG. 3) is rotated in a subterranean wellbore at 102. Electromagnetic measurements are acquired at 104 while the tool is rotating and processed to obtain harmonic voltage coefficients. Ratios of selected harmonic voltage coefficients may then be processed to obtain the gain compensated axial cross terms at 106.
The electromagnetic measurements may be acquired and processed to obtain harmonic voltage coefficients, for example, as describe above with respect to Equations 1 through 8. As described above, gain compensated axial cross terms may be obtained using a measurement tool including an axial transmitter antenna, a transverse transmitter antenna, an axial receiver antenna, and a transverse receiver antenna (each of the tool embodiments depicted on FIGS. 2B and 3A-3D include such axial and transverse transmitter and receiver antennas).
The measured voltages may be related to the impedances between the transmitter and receiver antennas as described above. The DC, first harmonic cosine, and first harmonic sine voltage coefficients may be expressed, for example, as follows in terms of the couplings and the respective transmitter and receiver gains:




  
    
      
        
          
            V
            DC_xx
          
          =
          
            
              g
              Tx
            
            ⁢
            
              g
              Rx
            
            ⁢
            
              
                
                  Z
                  xx
                
                +
                
                  Z
                  yy
                
              
              2
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            V
            DC_zz
          
          =
          
            
              g
              Tz
            
            ⁢
            
              g
              Rz
            
            ⁢
            
              Z
              zz
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            V
            
              FH
              ⁢
              C_xz
            
          
          =
          
            
              g
              Tx
            
            ⁢
            
              g
              Rz
            
            ⁢
            
              Z
              xz
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            V
            
              FH
              ⁢
              C_zx
            
          
          =
          
            
              g
              Tz
            
            ⁢
            
              g
              Rx
            
            ⁢
            
              Z
              zx
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            V
            
              FHS
              ⁢
              _xz
            
          
          =
          
            
              g
              Tx
            
            ⁢
            
              g
              Rz
            
            ⁢
            
              Z
              yz
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            V
            
              FHS
              ⁢
              _zx
            
          
          =
          
            
              g
              Tz
            
            ⁢
            
              g
              Rx
            
            ⁢
            
              Z
              zy
            
          
        
      
    
    
      
        (
        9
        )
      
    
  




where gTz and gTx represent the gains of the axial and transverse transmitter antennas, gRz and gRx represent the gains of the axial and transverse receiver antennas, VDC_xx is the DC voltage obtained from the x directed receiver when the x directed transmitter fires, VDC_zz is the DC voltage obtained from the z directed receiver when the z directed transmitter fires, VFHC_xz (VFHS_xz) is the first harmonic cosine (sine) voltage obtained from the z directed receiver when the x directed transmitter fires, and VFHC_zx (VFHS_zx) is the first harmonic cosine (sine) voltage obtained from the x directed receiver when the z directed transmitter fires.
Selected ratios of the DC, first harmonic cosine, and first harmonic sine voltage coefficients given in Equation 9 may be processed at 106 to compute the gain compensated axial cross terms. For example, a gain compensated quantity (ratio) related to the xz and/or the zx cross coupling impedances may be computed by processing a ratio of a product of the first harmonic cosine coefficients of the cross-coupling terms to a product of the DC coefficients of the direct coupling terms. Likewise, a gain compensated quantity (ratio) related to the yz and/or the zy cross coupling impedances may be computed by processing a ratio of a product of the first harmonic sine coefficients of the cross-coupling terms to a product of the DC coefficients of the direct coupling terms. It will be understood that the xz, zx, yz, and zy cross coupling impedances are also referred to herein as couplings. Such ratios may be expressed mathematically in general terms (for example for the configuration shown on FIG. 3A) as follows:




  
    
      
        
          CRxz
          =
          
            
              
                
                  V
                  FHC_zx
                
                ·
                
                  V
                  FHC_xz
                
              
              
                
                  V
                  
                    D
                    ⁢
                    C_xx
                  
                
                ·
                
                  V
                  
                    D
                    ⁢
                    C_zz
                  
                
              
            
            =
            
              
                2
                ⁢
                
                  
                
                ⁢
                
                  Z
                  zx
                
                ⁢
                
                  Z
                  xz
                
              
              
                
                  Z
                  zz
                
                ⁡
                
                  (
                  
                    
                      Z
                      xx
                    
                    +
                    
                      Z
                      yy
                    
                  
                  )
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          CRyz
          =
          
            
              
                
                  V
                  FHS_zx
                
                ·
                
                  V
                  FHS_xz
                
              
              
                
                  V
                  
                    D
                    ⁢
                    C_xx
                  
                
                ·
                
                  V
                  
                    D
                    ⁢
                    C_zz
                  
                
              
            
            =
            
              
                2
                ⁢
                
                  
                
                ⁢
                
                  Z
                  zy
                
                ⁢
                
                  Z
                  yz
                
              
              
                
                  Z
                  zz
                
                ⁡
                
                  (
                  
                    
                      Z
                      xx
                    
                    +
                    
                      Z
                      yy
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        10
        )
      
    
  




where CRxz and CRyz represent the gain compensated quantities (ratios). Note that the transmitter and receiver gains are fully canceled in Equation 10 resulting in the computed quantities being fully gain compensated.
The following discussion makes use of the notation and the antenna spacing described above with respect to FIG. 2B, although it will be understood that the disclosed embodiments are not so limited. As described above, the axial cross term voltages (the xz, zx, yz, and zy terms) may be fully gain compensated using any tool embodiment that includes an axial transmitter antenna, a transverse transmitter antenna, an axial receiver antenna, and a transverse receiver antenna.
It will be understood that in general ZTR=ZTRt. For example, the impedances Z12zx and Z21xz are identically equal in a homogeneous anisotropic medium. These impedances are only approximately equal in a heterogeneous medium (e.g., in the presence of bed boundaries) since the transmitter-receiver pairs 12zx and 21xz are not exactly located at the same points in space. A gain compensated quantity CZX that has the characteristics of a zx tensor coupling element may be obtained, for example, as given in the following equations:




  
    
      
        
          CZX
          =
          
            
              
                
                  
                    V
                    
                      FHC_
                      ⁢
                      12
                      ⁢
                      
                        
                      
                      ⁢
                      zx
                    
                  
                  ·
                  
                    V
                    
                      FHC_
                      ⁢
                      21
                      ⁢
                      
                        
                      
                      ⁢
                      xz
                    
                  
                
                
                  
                    V
                    
                      
                        D
                        ⁢
                        C_
                      
                      ⁢
                      
                        2
                        ⁢
                        2
                      
                      ⁢
                      
                        
                      
                      ⁢
                      xx
                    
                  
                  ·
                  
                    V
                    
                      
                        D
                        ⁢
                        C_
                      
                      ⁢
                      
                        1
                        ⁢
                        1
                      
                      ⁢
                      
                        
                      
                      ⁢
                      zz
                    
                  
                
              
            
            ≈
            
              
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Z
                    zx
                  
                  ⁢
                  
                    Z
                    xz
                  
                
                
                  
                    Z
                    zz
                  
                  ⁡
                  
                    (
                    
                      
                        Z
                        xx
                      
                      +
                      
                        Z
                        yy
                      
                    
                    )
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          CZX
          =
          
            
              
                
                  
                    V
                    
                      FHC_
                      ⁢
                      11
                      ⁢
                      
                        
                      
                      ⁢
                      zx
                    
                  
                  ·
                  
                    V
                    
                      FHC_
                      ⁢
                      22
                      ⁢
                      
                        
                      
                      ⁢
                      xz
                    
                  
                
                
                  
                    V
                    
                      
                        D
                        ⁢
                        C_
                      
                      ⁢
                      21
                      ⁢
                      
                        
                      
                      ⁢
                      xx
                    
                  
                  ·
                  
                    V
                    
                      
                        D
                        ⁢
                        C_
                      
                      ⁢
                      12
                      ⁢
                      
                        
                      
                      ⁢
                      zz
                    
                  
                
              
            
            ≈
            
              
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Z
                    zx
                  
                  ⁢
                  
                    Z
                    xz
                  
                
                
                  
                    Z
                    zz
                  
                  ⁡
                  
                    (
                    
                      
                        Z
                        xx
                      
                      +
                      
                        Z
                        yy
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




Likewise a gain compensated quantity CZY that has the characteristics of a zy tensor coupling element may be obtained, for example, as given in the following equations:




  
    
      
        
          CZY
          =
          
            
              
                
                  
                    V
                    
                      FHS_
                      ⁢
                      12
                      ⁢
                      
                        
                      
                      ⁢
                      zx
                    
                  
                  ·
                  
                    V
                    
                      FHS_
                      ⁢
                      21
                      ⁢
                      
                        
                      
                      ⁢
                      xz
                    
                  
                
                
                  
                    V
                    
                      
                        D
                        ⁢
                        C_
                      
                      ⁢
                      
                        2
                        ⁢
                        2
                      
                      ⁢
                      
                        
                      
                      ⁢
                      xx
                    
                  
                  ·
                  
                    V
                    
                      
                        D
                        ⁢
                        C_
                      
                      ⁢
                      
                        1
                        ⁢
                        1
                      
                      ⁢
                      
                        
                      
                      ⁢
                      zz
                    
                  
                
              
            
            ≈
            
              
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Z
                    zy
                  
                  ⁢
                  
                    Z
                    zy
                  
                
                
                  
                    Z
                    zz
                  
                  ⁡
                  
                    (
                    
                      
                        Z
                        xx
                      
                      +
                      
                        Z
                        yy
                      
                    
                    )
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          CZY
          =
          
            
              
                
                  
                    V
                    
                      FHS_
                      ⁢
                      11
                      ⁢
                      
                        
                      
                      ⁢
                      zx
                    
                  
                  ·
                  
                    V
                    
                      FHS_
                      ⁢
                      22
                      ⁢
                      
                        
                      
                      ⁢
                      xz
                    
                  
                
                
                  
                    V
                    
                      
                        D
                        ⁢
                        C_
                      
                      ⁢
                      21
                      ⁢
                      
                        
                      
                      ⁢
                      xx
                    
                  
                  ·
                  
                    V
                    
                      
                        D
                        ⁢
                        C_
                      
                      ⁢
                      12
                      ⁢
                      
                        
                      
                      ⁢
                      zz
                    
                  
                
              
            
            ≈
            
              
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Z
                    zy
                  
                  ⁢
                  
                    Z
                    zy
                  
                
                
                  
                    Z
                    zz
                  
                  ⁡
                  
                    (
                    
                      
                        Z
                        xx
                      
                      +
                      
                        Z
                        yy
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        12
        )
      
    
  




A gain compensated quantity CXZ that has the characteristics of a xz tensor coupling element may be obtained, for example, as given in the following equations:




  
    
      
        
          CXZ
          =
          
            
              
                
                  
                    V
                    
                      FHC_
                      ⁢
                      21
                      ⁢
                      
                        
                      
                      ⁢
                      zx
                    
                  
                  ·
                  
                    V
                    
                      FHC_
                      ⁢
                      12
                      ⁢
                      
                        
                      
                      ⁢
                      xz
                    
                  
                
                
                  
                    V
                    
                      
                        D
                        ⁢
                        C_
                      
                      ⁢
                      
                        1
                        ⁢
                        1
                      
                      ⁢
                      
                        
                      
                      ⁢
                      xx
                    
                  
                  ·
                  
                    V
                    
                      
                        D
                        ⁢
                        C_
                      
                      ⁢
                      22
                      ⁢
                      
                        
                      
                      ⁢
                      zz
                    
                  
                
              
            
            ≈
            
              
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Z
                    xz
                  
                  ⁢
                  
                    Z
                    xz
                  
                
                
                  
                    Z
                    zz
                  
                  ⁡
                  
                    (
                    
                      
                        Z
                        xx
                      
                      +
                      
                        Z
                        yy
                      
                    
                    )
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          CXZ
          =
          
            
              
                
                  
                    V
                    
                      FHC_
                      ⁢
                      22
                      ⁢
                      
                        
                      
                      ⁢
                      zx
                    
                  
                  ·
                  
                    V
                    
                      FHC_
                      ⁢
                      11
                      ⁢
                      
                        
                      
                      ⁢
                      xz
                    
                  
                
                
                  
                    V
                    
                      
                        D
                        ⁢
                        C_
                      
                      ⁢
                      12
                      ⁢
                      
                        
                      
                      ⁢
                      xx
                    
                  
                  ·
                  
                    V
                    
                      
                        D
                        ⁢
                        C_
                      
                      ⁢
                      21
                      ⁢
                      
                        
                      
                      ⁢
                      zz
                    
                  
                
              
            
            ≈
            
              
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Z
                    xz
                  
                  ⁢
                  
                    Z
                    xz
                  
                
                
                  
                    Z
                    zz
                  
                  ⁡
                  
                    (
                    
                      
                        Z
                        xx
                      
                      +
                      
                        Z
                        yy
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        13
        )
      
    
  




Likewise a gain compensated quantity CYZ that has the characteristics of a yz tensor coupling element may be obtained, for example, as given in the following equations:




  
    
      
        
          CYZ
          =
          
            
              
                
                  
                    V
                    
                      FHS_
                      ⁢
                      21
                      ⁢
                      
                        
                      
                      ⁢
                      zx
                    
                  
                  ·
                  
                    V
                    
                      FHS_
                      ⁢
                      12
                      ⁢
                      
                        
                      
                      ⁢
                      xz
                    
                  
                
                
                  
                    V
                    
                      
                        D
                        ⁢
                        C_
                      
                      ⁢
                      
                        1
                        ⁢
                        1
                      
                      ⁢
                      
                        
                      
                      ⁢
                      xx
                    
                  
                  ·
                  
                    V
                    
                      
                        D
                        ⁢
                        C_
                      
                      ⁢
                      22
                      ⁢
                      
                        
                      
                      ⁢
                      zz
                    
                  
                
              
            
            ≈
            
              
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Z
                    yz
                  
                  ⁢
                  
                    Z
                    yz
                  
                
                
                  
                    Z
                    zz
                  
                  ⁡
                  
                    (
                    
                      
                        Z
                        xx
                      
                      +
                      
                        Z
                        yy
                      
                    
                    )
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          CYZ
          =
          
            
              
                
                  
                    V
                    
                      FHS_
                      ⁢
                      11
                      ⁢
                      
                        
                      
                      ⁢
                      zx
                    
                  
                  ·
                  
                    V
                    
                      FHS_
                      ⁢
                      22
                      ⁢
                      
                        
                      
                      ⁢
                      xz
                    
                  
                
                
                  
                    V
                    
                      
                        D
                        ⁢
                        C_
                      
                      ⁢
                      12
                      ⁢
                      
                        
                      
                      ⁢
                      xx
                    
                  
                  ·
                  
                    V
                    
                      
                        D
                        ⁢
                        C_
                      
                      ⁢
                      21
                      ⁢
                      
                        
                      
                      ⁢
                      zz
                    
                  
                
              
            
            ≈
            
              
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Z
                    zy
                  
                  ⁢
                  
                    Z
                    zy
                  
                
                
                  
                    Z
                    zz
                  
                  ⁡
                  
                    (
                    
                      
                        Z
                        xx
                      
                      +
                      
                        Z
                        yy
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        14
        )
      
    
  




Gain compensated quantities may also be computed that are proportional to a product of the xz and zx terms as well a product of the yz and zy terms. For example, continuing to make use of the notation and the antenna spacing described above with respect to FIG. 2B, the gain compensated quantities CXZZX and CYZZY which are proportional to a product of the xz and zx terms and a product of the yz and zy terms may be obtained as follows:




  
    
      
        
          CXZZX
          =
          
            
              
                
                  
                    V
                    
                      FHS
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      ijzx
                    
                  
                  ·
                  
                    V
                    
                      FHS
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      ijxz
                    
                  
                
                
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      ijxx
                    
                  
                  ·
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      ijzz
                    
                  
                
              
            
            ≈
            
              
                
                  2
                  ⁢
                  
                    Z
                    xz
                  
                  ⁢
                  
                    Z
                    zx
                  
                
                
                  
                    Z
                    zz
                  
                  ⁡
                  
                    (
                    
                      
                        Z
                        xx
                      
                      +
                      
                        Z
                        yy
                      
                    
                    )
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          CYZZY
          =
          
            
              
                
                  
                    
                      V
                      
                        FHS
                        ⁢
                        
                          
                        
                        ⁢
                        _
                        ⁢
                        
                          
                        
                        ⁢
                        ijzx
                      
                    
                    ·
                    
                      V
                      
                        FHS
                        ⁢
                        
                          
                        
                        ⁢
                        _
                        ⁢
                        
                          
                        
                        ⁢
                        ijxz
                      
                    
                  
                  
                    
                      V
                      
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        C
                        ⁢
                        
                          
                        
                        ⁢
                        _
                        ⁢
                        
                          
                        
                        ⁢
                        ijxx
                      
                    
                    ·
                    
                      V
                      
                        D
                        ⁢
                        
                          
                        
                        ⁢
                        C
                        ⁢
                        
                          
                        
                        ⁢
                        _
                        ⁢
                        
                          
                        
                        ⁢
                        ijzz
                      
                    
                  
                
                ⁢
                
                  
                
              
            
            ≈
            
              
                
                  2
                  ⁢
                  
                    Z
                    yz
                  
                  ⁢
                  
                    Z
                    zy
                  
                
                
                  
                    Z
                    zz
                  
                  ⁡
                  
                    (
                    
                      
                        Z
                        xx
                      
                      +
                      
                        Z
                        yy
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        15
        )
      
    
  




Gain compensated quantities which are related to an xz zx product and a yz zy product may further be obtained as follows:




  
    
      
        
          CXZZX
          =
          
            
              
                
                  
                    V
                    
                      FHC
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      ijzx
                    
                  
                  ·
                  
                    V
                    
                      FHC
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      iixz
                    
                  
                
                
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      ijxx
                    
                  
                  ·
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      iizz
                    
                  
                
              
            
            ≈
            
              
                
                  2
                  ⁢
                  
                    Z
                    xz
                  
                  ⁢
                  
                    Z
                    zx
                  
                
                
                  
                    Z
                    zz
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        Z
                        xx
                      
                      +
                      
                        Z
                        yy
                      
                    
                    )
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          CYZZY
          =
          
            
              
                
                  
                    V
                    
                      FHS
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      ijzx
                    
                  
                  ·
                  
                    V
                    
                      FHS
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      iixz
                    
                  
                
                
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      ijxx
                    
                  
                  ·
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      iizz
                    
                  
                
              
            
            ≈
            
              
                
                  2
                  ⁢
                  
                    Z
                    yz
                  
                  ⁢
                  
                    Z
                    zy
                  
                
                
                  
                    Z
                    zz
                  
                  ⁡
                  
                    (
                    
                      
                        Z
                        xx
                      
                      +
                      
                        Z
                        yy
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        16
        )
      
    
  




and:




  
    
      
        
          CXZZX
          =
          
            
              
                
                  
                    V
                    
                      FHC
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      iizx
                    
                  
                  ·
                  
                    V
                    
                      FHC
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      jixz
                    
                  
                
                
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      jixx
                    
                  
                  ·
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      iizz
                    
                  
                
              
            
            ≈
            
              
                
                  2
                  ⁢
                  
                    Z
                    xz
                  
                  ⁢
                  
                    Z
                    zx
                  
                
                
                  
                    Z
                    zz
                  
                  ⁡
                  
                    (
                    
                      
                        Z
                        xx
                      
                      +
                      
                        Z
                        yy
                      
                    
                    )
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          CYZZY
          =
          
            
              
                
                  
                    V
                    
                      FHS
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      iizx
                    
                  
                  ·
                  
                    V
                    
                      FHS
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      jixz
                    
                  
                
                
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      jixx
                    
                  
                  ·
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      iizz
                    
                  
                
              
            
            ≈
            
              
                
                  2
                  ⁢
                  
                    Z
                    yz
                  
                  ⁢
                  
                    Z
                    zy
                  
                
                
                  
                    Z
                    zz
                  
                  ⁡
                  
                    (
                    
                      
                        Z
                        xx
                      
                      +
                      
                        Z
                        yy
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        17
        )
      
    
  




It will be understood that when using the notation and antenna spacing described above with respect to FIG. 2B, (i) Equation 15 relates to a tool embodiment including collocated axial and transverse transmitter antennas and collocated axial and transverse receiver antennas, (ii) Equation 16 relates to a tool embodiment including collocated axial and transverse transmitter antennas and non-collocated axial and transverse receiver antennas, and (iii) Equation 17 relates to a tool embodiment including non-collocated axial and transverse transmitter antennas and collocated axial and transverse receiver antennas.
Table 1 lists various antenna configurations from which gain compensated axial cross terms may be obtained. For example the top entry in Column 1 has the z transmitter in the left-most position, a z receiver in the next position, an x receiver, then an x transmitter in the right-most position. Using the (approximate) symmetry ZTR=ZTRt, each of these configurations, may have the character of a tensor component related to one of the following products: zx·zx, zx·xz, xz·zx, or xz·xz. Column 1 lists antenna combinations from which a gain compensated quantity related to the zx·zx product may be obtained. Column 2 lists antenna combinations from which a gain compensated quantity related to the xz·xz product may be obtained. Column 3 lists antenna combinations from which a gain compensated quantity related to the xz·zx product may be obtained. And column 4 lists antenna combinations from which a gain compensated quantity related to zx·xz product may be obtained. As noted above, gain compensated axial cross terms may be obtained using any tool configuration including an axial transmitter antenna, a transverse (cross-axial) transmitter antenna, an axial receiver antenna, and a transverse receiver antenna.
In Table 1 the transmitter and receiver antennas are listed from uphole to downhole positions on the tool (from left to right in the table). For example, Tz Rz Rx Tx indicates an axial transmitter antenna Tz located above an axial receiver antenna Rz and a transverse receiver antenna Rx located above a transverse transmitter antenna Tx. Adjacent transmitter antennas or receiver antennas may be collocated or non-collocated. In the above example, the axial receiver antenna Rz and the transverse receiver antenna Rx may be collocated or non-collocated such that the axial receiver antenna is above the transverse receiver antenna.












TABLE 1







Compensated 
Compensated 
Compensated 
Compensated 



ZX
XZ
XZ · ZX
ZX · XZ









Tz Rz Rx Tx
Rx Tz Tx Rz 
Rz Rx Tz Tx 
Tz Rx Tx Rz 



Tz Rz Tx Rx
Rx Tx Tz Rz 
Rz Rx Tx Tz 
Tz Tx Rz Rx 



Tz Rx Rz Tx
Rx Tx Rz Tz 
Rz Tx Rx Tz 
Tz Tx Rx Rz 



Rz Tz Rx Tx
Tx Rz Rx Tz
Rx Tz Rz Tx 
Tx Tz Rz Rx 



Rz Tz Tx Rx 
Tx Rx Tz Rz
Rx Rz Tz Tx 
Tx Tz Rx Rz 



Rz Tx Tz Rx 
Tx Rx Rz Tz
Rx Rz Tx Tz 
Tx Rz Tz Rx










It will be understood that since computation of the compensated quantities in Equations 11-17 involves taking a square root, there may be a 180 degree phase ambiguity (i.e., a sign ambiguity). As such, the gain ratios of the receivers may not be arbitrary, but should be controlled such that they are less than 180 degrees (i.e., the antenna wires should be connected to the electronics in the same way). For un-tuned receiving antennas, the electronic and antenna gain/phase mismatch (assuming the antenna wires are not flipped from one receiver to another) may generally be controlled to within about 30 degrees (particularly at the lower frequencies used for deep measurements). This is well within 180 degrees (even at elevated temperatures where the mismatch may be at its greatest).
A phase shift and attenuation may be computed for the compensated quantities listed above, for example, as follows:




  
    
      
        
          PS
          ⁢
          
            =
            def
          
          ⁢
          
            
              180
              π
            
            ⁢
            
              angle
              ⁡
              
                (
                
                  1
                  +
                  CQ
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          AT
          ⁢
          
            =
            def
          
          ⁢
          
            20
            ⁢
            log
            ⁢
            
              
            
            ⁢
            10
            ⁢
            
              (
              
                1
                +
                CQ
              
              )
            
          
        
      
    
    
      
        (
        18
        )
      
    
  




where PS represents the phase shift, AT represents attenuation, and CQ represents the compensated quantity (e.g., one of the quantities computed in Equations 11-17). These quantities may be equal to zero in simple formations. Thus, the phase shift and attenuation were computed by adding one to CQ in Equation 18.
Gain Compensated Symmetrized and Anti-Symmetrized Quantities
Symmetrized and anti-symmetrized directional resistivity quantities have been disclosed in U.S. Pat. Nos. 6,969,994 and 7,536,261 which are incorporated by reference herein in their entireties. In general, the symmetrized quantity is taken to be proportional to a difference between the xz and zx terms while the anti-symmetrized quantity is taken to be proportional to a sum of the xz and zx terms. The symmetrized measurement tends to be sensitive to bed boundaries and less sensitive to anisotropy and dip while the anti-symmetrized measurement tends to be sensitive to anisotropy and dip and less sensitive to bed boundaries.
It will be understood that a tool configuration including collocated tri-axial transmitter and receiver embodiments (e.g., as depicted on FIG. 2B) is not required to obtain gain compensated symmetrized and anti-symmetrized quantities. Logging tool embodiments similar to those described above for obtaining gain compensated axial cross terms may be utilized. For example, FIGS. 5A and 5B (collectively FIG. 5) depict antenna moments for various example transmitter and receiver configurations for obtaining gain compensated symmetrized and anti-symmetrized quantities. FIG. 5A depicts an embodiment including an axial receiver antenna Rz and a transverse receiver antenna Rx deployed axially between first and second transmitters T1 and T2. Each of the transmitters includes an axial transmitter antenna T1z and T2z and a transverse transmitter antenna T1x and T2x. FIG. 5B depicts an embodiment including an axial transmitter antenna Tz and a transverse transmitter antenna Tx deployed axially between first and second receivers R1 and R2. Each of the receivers includes an axial receiver antenna R1z and R2z and a transverse receiver antenna R1x and R2x. While the embodiments depicted on FIG. 5 include non-collocated antennas it will be understood that the axial and transverse transmitter and receiver antennas may optionally be collocated.
FIG. 6 depicts a flow chart of one disclosed method embodiment 120 for obtaining gain compensated symmetrized and anti-symmetrized quantities. An electromagnetic measurement tool (e.g., one of the measurement tools depicted on FIG. 2B or FIG. 5) is rotated in a subterranean wellbore at 122. Electromagnetic measurements are acquired at 124 while the tool is rotating and processed to obtain harmonic voltage coefficients. Ratios of selected harmonic voltage coefficients may then be processed to obtain the gain compensated symmetrized and anti-symmetrized quantities at 126. It will be understood that the harmonic voltage coefficients may be rotated to substantially any suitable reference angle prior to computing the gain compensated quantities at 126.
A gain compensated symmetrized measurement may be obtained, for example, via subtracting CZX from CXZ (e.g., as given in Equations 11 and 13) or via subtracting CZY from CYZ (e.g., as given in Equation 12 and 14). Likewise a gain compensated anti-symmetrized measurement may be obtained, for example, via adding CZX to CXZ (e.g., as given in Equations 11 and 13) or via adding CZY to CYZ (e.g., as given in Equation 12 and 14).
Symmetrized and anti-symmetrized coupling quantities S and A may further be expressed as combinations of products of the cross terms, for example, as follows:

S=√{square root over (Zxz2+Zzx2−2ZxzZzx)}

A=√{square root over (Zxz2+Zzx2+2ZxzZzx)}  (19)

Recognizing that CZX is proportional to Zzx (from Equation 11), CXZ is proportional to Zxz (from Equation 13), and CXZZX is proportional to the square root of Zxz·Zzx, gain compensated symmetrized Sc and anti-symmetrized Ac quantities may be given, for example, as follows:

Sc=√{square root over (CXZ2+CZX2−2CXZZX2)}

Ac=√{square root over (CXZ2+CZX2+2CXZZX2)}  (20)

where CXZ, CZX, and CXZZX may be obtained for example as described above with respect to Equations 10 through 17. Equation 20 may optionally further include a scaling factor to ensure that Sc is equal to zero in a homogeneous anisoptropic medium.
Following the notation and antenna spacing described above with respect to FIG. 2B, one particular embodiment of the symmetrized and anti-symmetrized quantities may be obtained by taking the following ratios:




  
    
      
        
          
            R
            zx
          
          =
          
            
              
                
                  V
                  
                    FHC
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    12
                    ⁢
                    zx
                  
                
                ·
                
                  V
                  
                    FHC
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    21
                    ⁢
                    xz
                  
                
              
              
                
                  V
                  
                    D
                    ⁢
                    
                      
                    
                    ⁢
                    C
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    22
                    ⁢
                    xx
                  
                
                ·
                
                  V
                  
                    D
                    ⁢
                    
                      
                    
                    ⁢
                    C
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    11
                    ⁢
                    zz
                  
                
              
            
            ~
            
              Z
              zx
              2
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            R
            xz
          
          =
          
            
              
                
                  V
                  
                    FHC
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    12
                    ⁢
                    xz
                  
                
                ·
                
                  V
                  
                    FHC
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    21
                    ⁢
                    zx
                  
                
              
              
                
                  V
                  
                    D
                    ⁢
                    
                      
                    
                    ⁢
                    C
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    11
                    ⁢
                    xx
                  
                
                ·
                
                  V
                  
                    D
                    ⁢
                    
                      
                    
                    ⁢
                    C
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    22
                    ⁢
                    zz
                  
                
              
            
            ~
            
              Z
              xz
              2
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            R
            ⁢
            
              
            
            ⁢
            
              1
              xzzx
            
          
          =
          
            
              
                
                  
                    V
                    
                      FHC
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      12
                      ⁢
                      zx
                    
                  
                  ·
                  
                    V
                    
                      FHC
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      12
                      ⁢
                      xz
                    
                  
                
                
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      12
                      ⁢
                      xx
                    
                  
                  ·
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      12
                      ⁢
                      zz
                    
                  
                
              
              ~
              
                Z
                xz
              
            
            ⁢
            
              Z
              zx
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            R
            ⁢
            
              
            
            ⁢
            
              2
              xzzx
            
          
          =
          
            
              
                
                  
                    V
                    
                      FHC
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      21
                      ⁢
                      zx
                    
                  
                  ·
                  
                    V
                    
                      FHC
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      21
                      ⁢
                      xz
                    
                  
                
                
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      21
                      ⁢
                      xx
                    
                  
                  ·
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      21
                      ⁢
                      zz
                    
                  
                
              
              ~
              
                Z
                xz
              
            
            ⁢
            
              Z
              zx
            
          
        
      
    
    
      
        (
        21
        )
      
    
  




It will be readily apparent that the ratios in Equation 21 are fully gain compensated and similar to the gain compensated quantities presented above with respect to Equations 11-17. It will be understood that corresponding ratios Rzy, Ryz, R1yzzy, and R2yzzy may be computed by replacing the first harmonic cosine coefficients with corresponding first harmonic sine coefficients. These ratios may be equivalently utilized to obtain the symmetrized and anti-symmetrized quantities.
To combine the quantities in Equation 21 such that the symmetric result is zero in a homogeneous anisotropic formation may require a scaling factor. Such a scaling factor may be obtained, for example, as follows:




  
    
      
        scale
        =
        
          
            
              V
              
                D
                ⁢
                
                  
                
                ⁢
                C
                ⁢
                
                  
                
                ⁢
                _
                ⁢
                
                  
                
                ⁢
                12
                ⁢
                zz
              
            
            ⁢
            
              V
              
                D
                ⁢
                
                  
                
                ⁢
                C
                ⁢
                
                  
                
                ⁢
                _
                ⁢
                
                  
                
                ⁢
                21
                ⁢
                zz
              
            
            ⁢
            
              V
              
                D
                ⁢
                
                  
                
                ⁢
                C
                ⁢
                
                  
                
                ⁢
                _
                ⁢
                
                  
                
                ⁢
                12
                ⁢
                xx
              
            
            ⁢
            
              V
              
                D
                ⁢
                
                  
                
                ⁢
                C
                ⁢
                
                  
                
                ⁢
                _
                ⁢
                
                  
                
                ⁢
                21
                ⁢
                xx
              
            
          
          
            
              V
              
                D
                ⁢
                
                  
                
                ⁢
                C
                ⁢
                
                  
                
                ⁢
                _
                ⁢
                
                  
                
                ⁢
                11
                ⁢
                zz
              
            
            ⁢
            
              V
              
                D
                ⁢
                
                  
                
                ⁢
                C
                ⁢
                
                  
                
                ⁢
                _
                ⁢
                
                  
                
                ⁢
                22
                ⁢
                zz
              
            
            ⁢
            
              V
              
                D
                ⁢
                
                  
                
                ⁢
                C
                ⁢
                
                  
                
                ⁢
                _
                ⁢
                
                  
                
                ⁢
                11
                ⁢
                xx
              
            
            ⁢
            
              V
              
                D
                ⁢
                
                  
                
                ⁢
                C
                ⁢
                
                  
                
                ⁢
                _
                ⁢
                
                  
                
                ⁢
                22
                ⁢
                xx
              
            
          
        
      
    
    
      
        (
        22
        )
      
    
  




such that the fully gain compensated symmetrized and anti-symmetrized quantities may be expressed as follows:

Sc=√{square root over (Rxz+Rzx−scale(R1xzzx+R2xzzx))}

Ac=√{square root over (Ryz+Rzx+scale(R1xzzx+R2xzzx))}  (23)

As described above with respect to Equations 11-17, taking the square root of a quantity can introduce a sign (or phase) ambiguity. Even with careful unwrapping of the phase in Equation 23, a symmetrized directional measurement Sc may have the same sign whether an approaching bed is above or below the measurement tool. The correct sign may be selected, for example, via selecting the sign of the phase or attenuation of the following relation:

TSD=√{square root over (Rzx)}−√{square root over (Rxz)}  (24)

where Rzx and Rxz are given in Equation 21. Similarly the anti-symmetrized directional measurement Ac in Equation 23 has the same sign whether the dip azimuth of the anisotropy is less than 180 degrees or greater than 180 degrees. This sign ambiguity may be resolved, for example, by taking the sign of the phase or attenuation of the following relation.

TAD=√{square root over (Rzx)}+√{square root over (Rxz)}  (25)

The symmetrized and anti-symmetrized measurements may now be re-defined, for example, as follows to eliminate the sign ambiguity.

Scdef2sign(angle(TSD))√{square root over (Rzx+Rxz−scale(R1xz_zx+R2xz_zx))}

Acdef2sign(angle(TSD))√{square root over (Rzx+Rxz−scale(R1xz_zx+R2xz_zx))}  (26)

Symmetrized directional phase shift and attenuation measurements TDSP and TDSA may then be defined, for example, as follows:




  
    
      
        
          TDSP
          ⁢
          
            =
            def
          
          ⁢
          
            
              180
              π
            
            ⁢
            
              angle
              ⁡
              
                (
                
                  1
                  +
                  Sc
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          TDSA
          ⁢
          
            =
            def
          
          ⁢
          
            20
            ⁢
            log
            ⁢
            
              
            
            ⁢
            10
            ⁢
            
              (
              
                1
                +
                Sc
              
              )
            
          
        
      
    
    
      
        (
        27
        )
      
    
  




Likewise, anti-symmetrized directional phase shift and attenuation TDAP and TDAA measurements may also be defined, for example, as follows:




  
    
      
        
          TDAP
          ⁢
          
            =
            def
          
          ⁢
          
            
              180
              π
            
            ⁢
            
              angle
              ⁡
              
                (
                
                  1
                  +
                  Ac
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          TDAA
          ⁢
          
            =
            def
          
          ⁢
          
            20
            ⁢
            log
            ⁢
            
              
            
            ⁢
            10
            ⁢
            
              (
              
                1
                +
                Ac
              
              )
            
          
        
      
    
    
      
        (
        28
        )
      
    
  




The disclosed embodiments are now described in further detail with respect to the following non-limiting examples in FIGS. 7, 8A, 8B, 9A, 9B, 10A, and 10B. These examples are analytical (mathematical) and were computed using Equations 26-28 via software code developed using a point dipole model.
FIG. 7 depicts a three layer formation model used to evaluate the response of the compensated symmetrized and anti-symmetrized measurement quantities described above with respect to Equations 27, 28, and 29. The upper layer had a horizontal resistivity of 2 ohm·m and a vertical resistivity of 5 ohm·m. The middle layer had a horizontal and vertical resistivities of 200 ohm·m while the lower layer had a horizontal resistivity of 5 ohm·m and a vertical resistivity of 10 ohm·m. The upper and lower boundaries of the middle layer were at −15 and +15 feet, respectively. The electromagnetic tool was inclined at a non-zero dip angle D. In the examples that follow (in FIGS. 8A though 10B), a tool model configuration similar to that shown on FIG. 2B was utilized. The receiver R1 and transmitter T1 were located at +13 and +40 inches with respect to the midpoint between receivers R1 and R2. The receiver R2 and the transmitter T2 were located at −13 and −40 inches. Zero depth was defined as the depth at which the midpoint between receivers R1 and R2 crossed the midpoint of the middle layer in the formation on FIG. 7.
FIGS. 8A and 8B depict symmetrized and anti-symmetrized phase shift and attenuation versus total vertical depth at 30 degrees relative dip. FIGS. 9A and 9B depict symmetrized and anti-symmetrized phase shift and attenuation versus total vertical depth at 70 degrees relative dip. FIGS. 10A and 10B depict symmetrized and anti-symmetrized phase shift and attenuation versus total vertical depth at 88 degrees relative dip. These figures illustrate that the symmetrized quantity is zero away from the bed boundary. Near the boundary, the sign changes depending on whether the bed is approached from above or below. The magnitude of the symmetrized response is substantially independent of anisotropy. The anti-symmetrized quantity is sensitive to anisotropy and dip with comparatively less sensitivity to the bed boundaries.
Gain Compensated Transverse Terms
It will be understood that collocated tri-axial transmitter and receiver embodiments (e.g., as depicted on FIG. 2B) are not required to obtain gain compensated transverse terms (i.e., the xx and yy direct coupling impedances and the xy and yx cross coupling impedances). The xx and yy direct coupling impedances and the xy and yx cross coupling impedances are also referred to herein as xx, yy, xy, and yx couplings. These terms may be gain compensated, for example, using any tool embodiment that includes x- and y-axis transmitter antennas and x- and y-axis receiver antennas deployed on the tool body. These transmitter and receiver antennas may be distributed along the tool body with substantially any suitable spacing and order. Moreover, the transmitter antennas and/or the receiver antennas may optionally be collocated. The disclosed embodiments are not limited to any particular transmitter and receiver antenna configuration so long as the tool includes at least x- and y-axis transmitter antennas and x- and y-axis receiver antennas.
FIGS. 11A and 11B depict the antenna moments for various example transmitter and receiver configurations suitable for obtaining gain compensated xy and yx couplings. FIG. 11A depicts an example tool embodiment 80 including a transmitter T axially spaced apart from a receiver R. The transmitter T includes collocated x- and y-axis transmitting antennas having moments Tx and Ty. The receiver R includes collocated x- and y-axis receiving antennas having moments Rx and Rz.
FIG. 11B depicts an alternative tool embodiment 85 including x- and y-axis transmitter antennas and x- and y-axis receiver antennas Tx and Ty and Rx and Ry. Tool embodiment 85 differs from tool embodiment 80 in that neither the transmitter antennas nor the receiver antennas are collocated, but are axially spaced apart on the tool body. It will be understood that the receiver antennas are not necessarily deployed between the transmitter antennas as depicted (TRRT), but may be axially distributed in substantially any order, for example, (i) with the transmitter antennas between the receiver antennas (RTTR), (ii) with the transmitter antennas alternating with the receiver antennas (TRTR or RTRT), or (iii) with the transmitter antennas on one side and the receiver antennas on the other (TTRR or RRTT). It will thus be understood that the disclosed embodiments are not limited to collocation or non-collocation of the axial and transverse transmitting and/or receiving antennas or to any particular spacing or location thereof along the tool body.
FIGS. 11C and 11D depict the antenna moments for various example transmitter and receiver configurations suitable for obtaining gain compensated xx and yy couplings and xy and yx couplings. FIG. 11C depicts another tool embodiment 90 that is similar to tool embodiment 50 shown on FIG. 2B in that it includes first and second transmitters T1 and T2 and first and second receivers R1 and R2. Tool embodiment 90 includes collocated x- and y-axis transmitters T1x and T1y and collocated x- and y-axis receivers R2x and R2y. Tool embodiment 80 further includes x-axis receiver R1x and x-axis transmitter T2x. FIG. 11D depicts still another alternative tool embodiment 95 including x-axis and y-axis transmitters and receivers. Tool embodiment 95 includes first and second x-axis receivers R1x and R2x deployed axially between first and second x-axis transmitters T1x and T2x. Tool embodiment 95 further includes a y-axis receiver R3y deployed between the x-axis receivers and a y-axis transmitter T4y.
With continued reference to FIG. 11, it will be understood that one or more or all of the transmitters and/or receivers depicted in tool embodiments 80, 85, 85, and 95 may further include axial (z-axis) antennas such that the transmitter and/or receiver includes a cross axial pair of antennas or a triaxial antenna arrangement. Moreover, one or more axial antennas may be located substantially anywhere in the depicted antenna arrays. The disclosed embodiments are not limited in regard to the inclusion or location of axial antennas.
FIG. 12 depicts a flow chart of one disclosed method embodiment 140 for obtaining gain compensated transverse term quantities. An electromagnetic measurement tool (e.g., one of the measurement tools depicted on FIG. 2B or FIG. 11) is rotated in a subterranean wellbore at 142. Electromagnetic measurements are acquired at 144 while the tool is rotating and processed to obtain harmonic voltage coefficients. Ratios of selected harmonic voltage coefficients may then be processed to obtain the gain compensated transverse terms (the xx and/or yy couplings and the xy and/or yx couplings).
The electromagnetic measurements may be acquired and processed to obtain harmonic coefficients, for example, as describe above with respect to Equations 1 through 8. Following Equations 3, 4, 7, and 8 and with respect to FIG. 2B, the DC and second harmonic voltages may be expressed, for example, as follows in terms of the couplings and the respective transmitter and receiver gains.




  
    
      
        
          
            
              
                V
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  C
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  xx
                
              
              =
              
                
                  g
                  Tx
                
                ⁢
                
                  g
                  Rx
                
                ⁢
                
                  
                    
                      z
                      xx
                    
                    +
                    
                      z
                      yy
                    
                  
                  2
                
              
            
          
          
            
              
                V
                
                  SHC
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  xx
                
              
              =
              
                
                  g
                  Tx
                
                ⁢
                
                  g
                  Rx
                
                ⁢
                
                  
                    
                      z
                      xx
                    
                    -
                    
                      z
                      yy
                    
                  
                  2
                
              
            
          
        
        
          
            
              
                V
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  C_
                  ⁢
                  
                    
                  
                  ⁢
                  xy
                
              
              =
              
                
                  g
                  Tx
                
                ⁢
                
                  g
                  Ry
                
                ⁢
                
                  
                    
                      z
                      xy
                    
                    -
                    
                      z
                      yx
                    
                  
                  2
                
              
            
          
          
            
              
                V
                
                  SHC
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  xy
                
              
              =
              
                
                  g
                  Tx
                
                ⁢
                
                  g
                  Ry
                
                ⁢
                
                  
                    
                      z
                      xy
                    
                    +
                    
                      z
                      yx
                    
                  
                  2
                
              
            
          
        
        
          
            
              
                V
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  C
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  yx
                
              
              =
              
                
                  g
                  Ty
                
                ⁢
                
                  g
                  Rx
                
                ⁢
                
                  
                    
                      z
                      yx
                    
                    -
                    
                      z
                      xy
                    
                  
                  2
                
              
            
          
          
            
              
                V
                
                  SHC
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  yx
                
              
              =
              
                
                  g
                  Ty
                
                ⁢
                
                  g
                  Rx
                
                ⁢
                
                  
                    
                      z
                      yx
                    
                    +
                    
                      z
                      xy
                    
                  
                  2
                
              
            
          
        
        
          
            
              
                V
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  C
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  yy
                
              
              =
              
                
                  g
                  Ty
                
                ⁢
                
                  g
                  Ry
                
                ⁢
                
                  
                    
                      z
                      xx
                    
                    +
                    
                      z
                      yy
                    
                  
                  2
                
              
            
          
          
            
              
                V
                
                  SHC
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  yy
                
              
              =
              
                
                  g
                  Ty
                
                ⁢
                
                  g
                  Ry
                
                ⁢
                
                  
                    
                      z
                      yy
                    
                    -
                    
                      z
                      xx
                    
                  
                  2
                
              
            
          
        
        
          
            
              
                V
                
                  SHS
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  xx
                
              
              =
              
                
                  g
                  Tx
                
                ⁢
                
                  g
                  Rx
                
                ⁢
                
                  
                    
                      z
                      xy
                    
                    +
                    
                      z
                      yx
                    
                  
                  2
                
              
            
          
          
            
              
                V
                
                  SHS
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  yx
                
              
              =
              
                
                  g
                  Ty
                
                ⁢
                
                  g
                  Rx
                
                ⁢
                
                  
                    
                      z
                      yy
                    
                    -
                    
                      z
                      xx
                    
                  
                  2
                
              
            
          
        
        
          
            
              
                V
                
                  SHS
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  xy
                
              
              =
              
                
                  g
                  TX
                
                ⁢
                
                  g
                  Ry
                
                ⁢
                
                  
                    
                      z
                      yy
                    
                    -
                    
                      z
                      xx
                    
                  
                  2
                
              
            
          
          
            
              
                V
                
                  SHS
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  yy
                
              
              =
              
                
                  g
                  Ty
                
                ⁢
                
                  g
                  Ry
                
                ⁢
                
                  
                    
                      -
                      
                        z
                        xy
                      
                    
                    -
                    
                      z
                      yx
                    
                  
                  2
                
              
            
          
        
      
    
    
      
        (
        29
        )
      
    
  




where gTx and gTy represent the gains of the x-axis and y-axis transmitter antennas and gRx and gRy represent the gains of the x-axis and y-axis receiver antennas.
Selected ratios of the DC and second harmonic voltages given in Equation 29 may be processed at 146 to compute the gain compensated quantities including the transverse terms. For example, following the notation and antenna spacing described above with respect to FIG. 2B, a gain compensated xx quantity may be computed from either the xx or yy voltage measurements as follows:




  
    
      
        
          
            CXX
            xx
          
          =
          
            
              
                
                  
                    (
                    
                      
                        V
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          C
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          12
                          ⁢
                          xx
                        
                      
                      +
                      
                        V
                        
                          SHC
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          12
                          ⁢
                          xx
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        V
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          C
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          21
                          ⁢
                          xx
                        
                      
                      +
                      
                        V
                        
                          SHC
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          21
                          ⁢
                          
                            
                          
                          ⁢
                          xx
                        
                      
                    
                    )
                  
                
                
                  
                    (
                    
                      
                        V
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          C
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          11
                          ⁢
                          xx
                        
                      
                      +
                      
                        V
                        
                          SHC
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          11
                          ⁢
                          
                            
                          
                          ⁢
                          xx
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        V
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          C
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          22
                          ⁢
                          xx
                        
                      
                      +
                      
                        V
                        
                          SHC
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          22
                          ⁢
                          xx
                        
                      
                    
                    )
                  
                
              
            
            =
            
              
                
                  
                    Z
                    
                      12
                      ⁢
                      xx
                    
                  
                  ⁢
                  
                    Z
                    
                      21
                      ⁢
                      xx
                    
                  
                
                
                  
                    Z
                    
                      11
                      ⁢
                      xx
                    
                  
                  ⁢
                  
                    Z
                    
                      11
                      ⁢
                      xx
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            CXX
            yy
          
          =
          
            
              
                
                  
                    (
                    
                      
                        V
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          C
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          12
                          ⁢
                          yy
                        
                      
                      ±
                      
                        V
                        
                          SHC
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          12
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        V
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          C
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          21
                          ⁢
                          yy
                        
                      
                      -
                      
                        V
                        
                          SHC
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          21
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                
                
                  
                    (
                    
                      
                        V
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          C_
                          ⁢
                          
                            
                          
                          ⁢
                          11
                          ⁢
                          yy
                        
                      
                      ±
                      
                        V
                        
                          SHC
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          11
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        V
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          C
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          22
                          ⁢
                          yy
                        
                      
                      ±
                      
                        V
                        
                          SHC
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          22
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                
              
            
            =
            
              
                
                  
                    Z
                    
                      12
                      ⁢
                      xx
                    
                  
                  ⁢
                  
                    Z
                    
                      21
                      ⁢
                      xx
                    
                  
                
                
                  
                    Z
                    
                      11
                      ⁢
                      xx
                    
                  
                  ⁢
                  
                    Z
                    
                      11
                      ⁢
                      xx
                    
                  
                
              
            
          
        
      
    
    
      
        (
        30
        )
      
    
  




where CXXxx and CXXyy represent the gain compensated xx quantities computed from the xx and yy voltage measurements. Since these quantities are identical they may be combined (e.g., averaged) to improve the signal to noise ratio.
A gain compensated yy quantity may also be computed from either the xx or yy voltage measurements as follows:




  
    
      
        
          
            CYY
            xx
          
          =
          
            
              
                
                  
                    (
                    
                      
                        V
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          C
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          12
                          ⁢
                          xx
                        
                      
                      -
                      
                        V
                        
                          SHC
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          12
                          ⁢
                          
                            
                          
                          ⁢
                          xx
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        V
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          C
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          21
                          ⁢
                          
                            
                          
                          ⁢
                          xx
                        
                      
                      -
                      
                        V
                        
                          SHC
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          21
                          ⁢
                          xx
                        
                      
                    
                    )
                  
                
                
                  
                    (
                    
                      
                        V
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          C
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          11
                          ⁢
                          xx
                        
                      
                      -
                      
                        V
                        
                          SHC
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          11
                          ⁢
                          xx
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        V
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          C
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          22
                          ⁢
                          xx
                        
                      
                      -
                      
                        V
                        
                          SHC
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          22
                          ⁢
                          xx
                        
                      
                    
                    )
                  
                
              
            
            =
            
              
                
                  
                    Z
                    
                      12
                      ⁢
                      yy
                    
                  
                  ⁢
                  
                    Z
                    
                      21
                      ⁢
                      yy
                    
                  
                
                
                  
                    Z
                    
                      11
                      ⁢
                      y
                    
                  
                  ⁢
                  
                    Z
                    
                      11
                      ⁢
                      yy
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            CYY
            yy
          
          =
          
            
              
                
                  
                    (
                    
                      
                        V
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          C
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          12
                          ⁢
                          yy
                        
                      
                      ∓
                      
                        V
                        
                          SHC
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          12
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        V
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          C
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          21
                          ⁢
                          yy
                        
                      
                      ∓
                      
                        V
                        
                          SHC
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          21
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                
                
                  
                    (
                    
                      
                        V
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          C
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          11
                          ⁢
                          yy
                        
                      
                      ∓
                      
                        V
                        
                          SHC
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          11
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        V
                        
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          C_
                          ⁢
                          
                            
                          
                          ⁢
                          22
                          ⁢
                          yy
                        
                      
                      ∓
                      
                        V
                        
                          SHC
                          ⁢
                          
                            
                          
                          ⁢
                          _
                          ⁢
                          
                            
                          
                          ⁢
                          22
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                
              
            
            =
            
              
                
                  
                    Z
                    
                      12
                      ⁢
                      yy
                    
                  
                  ⁢
                  
                    Z
                    
                      21
                      ⁢
                      yy
                    
                  
                
                
                  
                    Z
                    
                      11
                      ⁢
                      yy
                    
                  
                  ⁢
                  
                    Z
                    
                      11
                      ⁢
                      y
                    
                  
                
              
            
          
        
      
    
    
      
        (
        31
        )
      
    
  




where CYYxx and CYYyy represent the gain compensated yy quantities computed from the xx and yy voltage measurements. These quantities are also identical and may be combined (e.g., averaged) to improve the signal to noise ratio.
Various gain compensated quantities that are combinations of the xx and yy couplings may also be computed from the xx and/or yy voltage measurements. A compensated quantity proportional to the sum of the xx and yy couplings may be computed from the xx and/or yy voltage measurements, for example, as follows:




  
    
      
        
          
            CXXplusYY
            xx
          
          =
          
            
              
                
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      12
                      ⁢
                      xx
                    
                  
                  ⁢
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      21
                      ⁢
                      xx
                    
                  
                
                
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      11
                      ⁢
                      xx
                    
                  
                  ⁢
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      22
                      ⁢
                      xx
                    
                  
                
              
            
            =
            
              
                
                  
                    (
                    
                      
                        Z
                        
                          12
                          ⁢
                          xx
                        
                      
                      +
                      
                        Z
                        
                          12
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        Z
                        
                          21
                          ⁢
                          xx
                        
                      
                      +
                      
                        Z
                        
                          21
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                
                
                  
                    (
                    
                      
                        Z
                        
                          11
                          ⁢
                          xx
                        
                      
                      +
                      
                        Z
                        
                          11
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        Z
                        
                          22
                          ⁢
                          xx
                        
                      
                      +
                      
                        Z
                        
                          22
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            CXXplusYY
            yy
          
          =
          
            
              
                
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      12
                      ⁢
                      yy
                    
                  
                  ⁢
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      21
                      ⁢
                      yy
                    
                  
                
                
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      11
                      ⁢
                      yy
                    
                  
                  ⁢
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      22
                      ⁢
                      yy
                    
                  
                
              
            
            =
            
              
                
                  
                    (
                    
                      
                        Z
                        
                          12
                          ⁢
                          xx
                        
                      
                      +
                      
                        Z
                        
                          12
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        Z
                        
                          21
                          ⁢
                          xx
                        
                      
                      +
                      
                        Z
                        
                          21
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                
                
                  
                    (
                    
                      
                        Z
                        
                          11
                          ⁢
                          xx
                        
                      
                      +
                      
                        Z
                        
                          11
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        Z
                        
                          22
                          ⁢
                          xx
                        
                      
                      +
                      
                        Z
                        
                          22
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        32
        )
      
    
  




where CXXplusYYxx and CXXplusYYyy represent gain compensated quantities computed from the xx and yy voltage measurements. A compensated quantity proportional to the difference between the xx and yy couplings may be computed from the xx and/or yy voltage measurements, for example, as follows:




  
    
      
        
          
            CXXminusYY
            xx
          
          =
          
            
              
                
                  
                    V
                    
                      SHC
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      12
                      ⁢
                      xx
                    
                  
                  ⁢
                  
                    V
                    
                      SHC
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      21
                      ⁢
                      xx
                    
                  
                
                
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      11
                      ⁢
                      xx
                    
                  
                  ⁢
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      22
                      ⁢
                      xx
                    
                  
                
              
            
            =
            
              
                
                  
                    (
                    
                      
                        Z
                        
                          12
                          ⁢
                          xx
                        
                      
                      -
                      
                        Z
                        
                          12
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        Z
                        
                          21
                          ⁢
                          xx
                        
                      
                      -
                      
                        Z
                        
                          21
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                
                
                  
                    (
                    
                      
                        Z
                        
                          11
                          ⁢
                          xx
                        
                      
                      +
                      
                        Z
                        
                          11
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        Z
                        
                          22
                          ⁢
                          xx
                        
                      
                      +
                      
                        Z
                        
                          22
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            CXXminusYY
            yy
          
          =
          
            
              
                
                  
                    V
                    
                      SHC
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      12
                      ⁢
                      yy
                    
                  
                  ⁢
                  
                    V
                    
                      SHC
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      21
                      ⁢
                      yy
                    
                  
                
                
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      11
                      ⁢
                      yy
                    
                  
                  ⁢
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      22
                      ⁢
                      yy
                    
                  
                
              
            
            =
            
              
                
                  
                    (
                    
                      
                        Z
                        
                          12
                          ⁢
                          xx
                        
                      
                      -
                      
                        Z
                        
                          12
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        Z
                        
                          21
                          ⁢
                          xx
                        
                      
                      -
                      
                        Z
                        
                          21
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                
                
                  
                    (
                    
                      
                        Z
                        
                          11
                          ⁢
                          xx
                        
                      
                      +
                      
                        Z
                        
                          11
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        Z
                        
                          22
                          ⁢
                          xx
                        
                      
                      +
                      
                        Z
                        
                          22
                          ⁢
                          yy
                        
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        33
        )
      
    
  




where CXXminusYYxx and CXXminusYYyy represent gain compensated quantities computed from the xx and yy voltage measurements. A compensated quantity proportional to the difference between the xx and yy components may also be computed from the xx and/or yy voltage measurements, for example, as follows:




  
    
      
        
          
            CXXminusYY
            ijxx
          
          =
          
            
              
                V
                
                  SHC
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  ijxx
                
              
              
                V
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  
                    C
                    ijxx
                  
                
              
            
            =
            
              
                (
                
                  
                    Z
                    ijxx
                  
                  -
                  
                    Z
                    ijyy
                  
                
                )
              
              
                (
                
                  
                    Z
                    ijxx
                  
                  +
                  
                    Z
                    ijyy
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            CXXminusYY
            ijyy
          
          =
          
            
              -
              
                
                  V
                  
                    SHC
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    ijyy
                  
                
                
                  V
                  
                    D
                    ⁢
                    
                      
                    
                    ⁢
                    C
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    ijyy
                  
                
              
            
            =
            
              
                (
                
                  
                    Z
                    ijxx
                  
                  -
                  
                    Z
                    ijyy
                  
                
                )
              
              
                (
                
                  
                    Z
                    ijxx
                  
                  +
                  
                    Z
                    ijyy
                  
                
                )
              
            
          
        
      
    
    
      
        (
        34
        )
      
    
  




where CXXminusYYijxx and CXXminusYYijyy represent gain compensated quantities computed from the xx and yy voltage measurements.
Gain compensated quantities that are combinations of the xy and yx couplings may also be computed from the xx and/or yy voltage measurements. For example, a compensated quantity proportional to the sum of the xy and yx couplings may be computed from the xx and/or yy voltage measurements as follows:




  
    
      
        
          
            CXYplusYX
            ijxx
          
          =
          
            
              -
              
                
                  V
                  
                    SHS
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    ijxx
                  
                
                
                  V
                  
                    D
                    ⁢
                    
                      
                    
                    ⁢
                    C
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    ijxx
                  
                
              
            
            =
            
              
                (
                
                  
                    Z
                    ijxy
                  
                  +
                  
                    Z
                    ijyx
                  
                
                )
              
              
                (
                
                  
                    Z
                    ijxx
                  
                  +
                  
                    Z
                    ijyy
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            CXYplusYX
            ijyy
          
          =
          
            
              -
              
                
                  V
                  
                    SHS
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    ijyy
                  
                
                
                  V
                  
                    D
                    ⁢
                    
                      
                    
                    ⁢
                    C
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    ijyy
                  
                
              
            
            =
            
              
                (
                
                  
                    Z
                    ijxy
                  
                  +
                  
                    Z
                    ijyx
                  
                
                )
              
              
                (
                
                  
                    Z
                    ijxx
                  
                  +
                  
                    Z
                    ijyy
                  
                
                )
              
            
          
        
      
    
    
      
        (
        35
        )
      
    
  




where CXYplusYXijxx and CXYplusYXijyy represent gain compensated quantities computed from the xx and yy voltage measurements. Since the first and second quantities in each of Equations 32, 33, 34, and 35 are identical they may be combined (e.g., averaged) to improve the signal to noise ratio as described above with respect to the quantities in Equations 30 and 31.
Gain compensated quantities that are combinations of the xy and yx couplings may also be computed from the xx, yy, xy, and yx voltage measurements. For example, a compensated quantity proportional to the difference between the xy and yx couplings may be computed from the xx, yy, xy, and yx voltage measurements as follows:




  
    
      
        
          CXYminusYX
          ij
        
        =
        
          
            
              -
              
                
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      ijxy
                    
                  
                  ⁢
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      ijyx
                    
                  
                
                
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      ijxx
                    
                  
                  ⁢
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      ijyy
                    
                  
                
              
            
          
          =
          
            
              (
              
                
                  Z
                  ijxy
                
                -
                
                  Z
                  ijyx
                
              
              )
            
            
              (
              
                
                  Z
                  ijxx
                
                +
                
                  Z
                  ijyy
                
              
              )
            
          
        
      
    
    
      
        (
        36
        )
      
    
  




where CXYminusYXij represents the gain compensated quantity. A compensated quantity proportional to the difference between the xy and yx couplings may be computed, for example, as follows:




  
    
      
        CXYminusYX
        =
        
          
            
              -
              
                
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      12
                      ⁢
                      xy
                    
                  
                  ⁢
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      21
                      ⁢
                      yx
                    
                  
                
                
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      11
                      ⁢
                      xx
                    
                  
                  ⁢
                  
                    V
                    
                      D
                      ⁢
                      
                        
                      
                      ⁢
                      C
                      ⁢
                      
                        
                      
                      ⁢
                      _
                      ⁢
                      
                        
                      
                      ⁢
                      22
                      ⁢
                      yy
                    
                  
                
              
            
          
          =
          
            
              
                
                  (
                  
                    
                      Z
                      
                        12
                        ⁢
                        xy
                      
                    
                    -
                    
                      Z
                      
                        12
                        ⁢
                        yx
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      Z
                      
                        21
                        ⁢
                        xy
                      
                    
                    -
                    
                      Z
                      
                        21
                        ⁢
                        yx
                      
                    
                  
                  )
                
              
              
                
                  (
                  
                    
                      Z
                      
                        11
                        ⁢
                        xx
                      
                    
                    +
                    
                      Z
                      
                        11
                        ⁢
                        yy
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      Z
                      
                        22
                        ⁢
                        xx
                      
                    
                    +
                    
                      Z
                      
                        22
                        ⁢
                        yy
                      
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        37
        )
      
    
  




The gain compensated quantities in Equations 35 and 36 may be combined to obtain gain compensated xy and yx quantities CXYj and CYXij, for example, as follows:




  
    
      
        
          
            CXY
            ij
          
          =
          
            
              
                
                  CXYplusYX
                  ijxx
                
                +
                
                  CXYminusYX
                  ij
                
              
              2
            
            =
            
              
                Z
                ijxy
              
              
                (
                
                  
                    Z
                    ijxx
                  
                  +
                  
                    Z
                    ijyy
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            CXY
            ij
          
          =
          
            
              
                
                  CXYplusYX
                  ijyy
                
                +
                
                  CXYminusYX
                  ij
                
              
              2
            
            =
            
              
                Z
                ijxy
              
              
                (
                
                  
                    Z
                    ijxx
                  
                  +
                  
                    Z
                    ijyy
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            CYX
            ij
          
          =
          
            
              
                
                  CXYplusYX
                  ijxx
                
                -
                
                  CXYminusYX
                  ij
                
              
              2
            
            =
            
              
                Z
                ijyx
              
              
                (
                
                  
                    Z
                    ijxx
                  
                  +
                  
                    Z
                    ijyy
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            CYX
            ij
          
          =
          
            
              
                
                  CYXplusYX
                  ijyy
                
                -
                
                  CXYminusYX
                  ij
                
              
              2
            
            =
            
              
                Z
                ijyx
              
              
                (
                
                  
                    Z
                    ijxx
                  
                  +
                  
                    Z
                    ijyy
                  
                
                )
              
            
          
        
      
    
    
      
        (
        38
        )
      
    
  




As described above with respect to Equation 19, a phase shift and attenuation may be computed for the compensated quantities listed above, for example, as follows:




  
    
      
        
          PS
          ⁢
          
            =
            def
          
          ⁢
          
            
              180
              π
            
            ⁢
            
              angle
              ⁡
              
                (
                
                  1
                  +
                  CQ
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          AT
          ⁢
          
            =
            def
          
          ⁢
          
            20
            ⁢
            log
            ⁢
            
              
            
            ⁢
            10
            ⁢
            
              (
              
                1
                +
                CQ
              
              )
            
          
        
      
    
    
      
        (
        39
        )
      
    
  




where PS represents the phase shift, AT represents attenuation, and CQ represents the compensated quantity (e.g., one of the quantities computed in Equations 30-38).
With respect to the embodiment depicted on FIG. 11D and Equation 37, a gain compensated measurement CXYminusYX sensitive to xy-yx may be obtained, for example, as follows:




  
    
      
        CXYminusYX
        =
        
          
            
              
                
                  V
                  
                    D
                    ⁢
                    
                      
                    
                    ⁢
                    C
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    13
                    ⁢
                    xy
                  
                
                ⁢
                
                  V
                  
                    D
                    ⁢
                    
                      
                    
                    ⁢
                    C
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    41
                    ⁢
                    yx
                  
                
              
              
                
                  V
                  
                    D
                    ⁢
                    
                      
                    
                    ⁢
                    C
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    11
                    ⁢
                    xx
                  
                
                ⁢
                
                  V
                  
                    D
                    ⁢
                    
                      
                    
                    ⁢
                    C
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    43
                    ⁢
                    yy
                  
                
              
            
          
          =
          
            
              
                
                  (
                  
                    
                      Z
                      
                        13
                        ⁢
                        xy
                      
                    
                    -
                    
                      Z
                      
                        13
                        ⁢
                        yx
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      Z
                      
                        42
                        ⁢
                        xy
                      
                    
                    -
                    
                      Z
                      
                        42
                        ⁢
                        yx
                      
                    
                  
                  )
                
              
              
                
                  (
                  
                    
                      Z
                      
                        11
                        ⁢
                        xx
                      
                    
                    +
                    
                      Z
                      
                        11
                        ⁢
                        yy
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      Z
                      
                        43
                        ⁢
                        xx
                      
                    
                    +
                    
                      Z
                      
                        43
                        ⁢
                        yy
                      
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        40
        )
      
    
  




Likewise, a gain compensated measurement CXXminusYY sensitive to xx-yy may be obtained, for example, as follows:




  
    
      
        CXXminusYY
        =
        
          
            
              
                
                  V
                  
                    SHC
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    13
                    ⁢
                    xy
                  
                
                ⁢
                
                  V
                  
                    SHC
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    41
                    ⁢
                    yx
                  
                
              
              
                
                  V
                  
                    D
                    ⁢
                    
                      
                    
                    ⁢
                    C
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    11
                    ⁢
                    xx
                  
                
                ⁢
                
                  V
                  
                    D
                    ⁢
                    
                      
                    
                    ⁢
                    C
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    43
                    ⁢
                    zz
                  
                
              
            
          
          =
          
            
              
                
                  (
                  
                    
                      Z
                      
                        13
                        ⁢
                        xx
                      
                    
                    -
                    
                      Z
                      
                        13
                        ⁢
                        yy
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      Z
                      
                        42
                        ⁢
                        xx
                      
                    
                    -
                    
                      Z
                      
                        42
                        ⁢
                        yy
                      
                    
                  
                  )
                
              
              
                
                  (
                  
                    
                      Z
                      
                        11
                        ⁢
                        xx
                      
                    
                    +
                    
                      Z
                      
                        11
                        ⁢
                        yy
                      
                    
                  
                  )
                
                ⁢
                
                  (
                  
                    
                      Z
                      
                        43
                        ⁢
                        xx
                      
                    
                    +
                    
                      Z
                      
                        43
                        ⁢
                        yy
                      
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        41
        )
      
    
  




Gain Compensated Axial Term

Techniques are disclosed above for obtaining fully gain compensated quantities related to each of the eight non-axial three-dimensional impedances (i.e., the xz, zx, yz, zy, xx, yy, xy, and yx terms). A gain compensated zz (axial) coupling may be also obtained from the DC harmonic voltage coefficients, for example, as follows:




  
    
      
        CZZ
        =
        
          
            
              
                
                  V
                  
                    D
                    ⁢
                    
                      
                    
                    ⁢
                    C
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    12
                    ⁢
                    zz
                  
                
                ⁢
                
                  V
                  
                    D
                    ⁢
                    
                      
                    
                    ⁢
                    C
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    21
                    ⁢
                    xx
                  
                
              
              
                
                  V
                  
                    D
                    ⁢
                    
                      
                    
                    ⁢
                    C
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    11
                    ⁢
                    zz
                  
                
                ⁢
                
                  V
                  
                    D
                    ⁢
                    
                      
                    
                    ⁢
                    C
                    ⁢
                    
                      
                    
                    ⁢
                    _
                    ⁢
                    
                      
                    
                    ⁢
                    22
                    ⁢
                    xx
                  
                
              
            
          
          =
          
            
              
                
                  Z
                  
                    12
                    ⁢
                    zz
                  
                
                ⁢
                
                  Z
                  
                    21
                    ⁢
                    zz
                  
                
              
              
                
                  Z
                  
                    11
                    ⁢
                    zz
                  
                
                ⁢
                
                  Z
                  
                    22
                    ⁢
                    zz
                  
                
              
            
          
        
      
    
    
      
        (
        42
        )
      
    
  




where CZZ represents the compensated measurement (the zz direct coupling impedance). A phase shift and attenuation for CZZ may also be computed.
Gain Compensated Axial Cross Terms Using Tilted Moments
FIG. 13 depicts the antenna moments for an electromagnetic logging tool including tilted transmitters T1 and T2. The transmitters are deployed about axially spaced receivers R1 and R2, each of which includes collocated axial and transverse antennas R1z, R1x and R2z, R2x. The measurement tool depicted on FIG. 13 may be used to obtain gain compensated measurements, for example, as described above with respect to FIG. 4 and Equations 10-28. For example, with respect to Equation 12, the DC and first harmonic cosine voltage coefficients may be expressed as follows:




  
    
      
        
          
            V
            
              D
              ⁢
              
                
              
              ⁢
              C
              ⁢
              
                
              
              ⁢
              _
              ⁢
              
                
              
              ⁢
              xx
            
          
          =
          
            
              g
              
                T
                ⁢
                
                  
                
                ⁢
                2
              
            
            ⁢
            
              g
              
                R
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                x
              
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  β
                  
                    T
                    ⁢
                    
                      
                    
                    ⁢
                    2
                  
                
                )
              
            
            ⁢
            
              
                
                  z
                  xx
                
                +
                
                  z
                  yy
                
              
              2
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            V
            
              D
              ⁢
              
                
              
              ⁢
              C
              ⁢
              
                
              
              ⁢
              _
              ⁢
              
                
              
              ⁢
              zz
            
          
          =
          
            
              g
              
                T
                ⁢
                
                  
                
                ⁢
                1
              
            
            ⁢
            
              g
              
                
                  R
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  z
                
                ⁢
                
                  
                
              
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  β
                  
                    T
                    ⁢
                    
                      
                    
                    ⁢
                    1
                  
                
                )
              
            
            ⁢
            
              Z
              zz
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            V
            
              FHC
              ⁢
              
                
              
              ⁢
              _
              ⁢
              
                
              
              ⁢
              xz
            
          
          =
          
            
              g
              
                T
                ⁢
                
                  
                
                ⁢
                2
              
            
            ⁢
            
              g
              
                R
                ⁢
                
                  
                
                ⁢
                z
              
            
            ⁢
            
              sin
              ⁡
              
                (
                
                  β
                  
                    T
                    ⁢
                    
                      
                    
                    ⁢
                    2
                  
                
                )
              
            
            ⁢
            
              Z
              xz
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            V
            
              FHC
              ⁢
              
                
              
              ⁢
              _
              ⁢
              
                
              
              ⁢
              zx
            
          
          =
          
            
              g
              
                T
                ⁢
                
                  
                
                ⁢
                1
              
            
            ⁢
            
              g
              Rx
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  β
                  
                    T
                    ⁢
                    
                      
                    
                    ⁢
                    1
                  
                
                )
              
            
            ⁢
            
              Z
              zx
            
          
        
      
    
    
      
        (
        43
        )
      
    
  




where βT1 represents the tilt angle between the T1 antenna moment and the axis of the electromagnetic measurement tool. Notice that sin(βT2) and cos(βT1) may be lumped with the transmitter gains such that gT2x=gT2 sin(βT2) and gT1z=gT1 cos(βT1). The sine and cosine terms thus cancel in computing the aforementioned ratios in the same way that the transmitter gains cancel. In this way any of the compensated quantities described above with respect to Equations 10-28 may be computed using a measurement tool including tilted transmitters as depicted on FIG. 13.
It will be understood that the various methods disclosed herein for obtaining fully gain compensated quantities may be implemented on a on a downhole processor. By downhole processor it is meant an electronic processor (e.g., a microprocessor or digital controller) deployed in the drill string (e.g., in the electromagnetic logging tool or elsewhere in the BHA). In such embodiments, the fully gain compensated quantities may be stored in downhole memory and/or transmitted to the surface while drilling via known telemetry techniques (e.g., mud pulse telemetry or wired drill pipe). Alternatively, the harmonic fitting coefficients may be transmitted uphole and the compensated quantities may be computed at the surface using a surface processor. Whether transmitted to the surface or computed at the surface, the quantity may be utilized in an inversion process (along with a formation model) to obtain various formation parameters as described above.
Although gain compensated directional propagation measurements have been described in detail, it should be understood that various changes, substitutions and alternations can be made herein without departing from the spirit and scope of the disclosure as defined by the appended claims.

## Claims

1. A method for making downhole electromagnetic logging while drilling measurements, the method comprising
(a) rotating an electromagnetic logging while drilling tool in a subterranean wellbore, the logging tool including first and second transverse transmitter antennas whose moments are orthogonal to one another and first and second transverse receiver antennas whose moments are orthogonal to one another;
(b) acquiring a plurality of electromagnetic voltage measurements from the first and second transverse receiver antennas while rotating in (a);
(c) processing the voltage measurements acquired in (b) to compute harmonic voltage coefficients;
(d) processing a ratio of the harmonic voltage coefficients to compute a gain compensated quantity including a transverse term, wherein the transverse term is one an xx direct coupling impedance, a yy direct coupling impedance, an xy cross coupling impedance and yx cross coupling impedance or a combination thereof.

2. The method of claim 1, wherein the processing in (d) is performed by a downhole processor.

3. The method of claim 2, further comprising:
(e) transmitting the gain compensated quantity to a surface location; and
(f) causing a surface computer to invert the gain compensated quantity to obtain one or more properties of a subterranean formation.

4. The method of claim 1, further comprising:
(e) processing the gain compensated quantity to compute a gain compensated phase shift and a gain compensated attenuation.

5. The method of claim 1, wherein the harmonic voltage coefficients computed in (c) comprise DC, first order harmonic sine, first order harmonic cosine, second order harmonic sine, and second order harmonic cosine voltage coefficients.

6. The method of claim 1, wherein the gain compensated quantity computed in (d) comprises at least one proportional to an xx coupling or a yy coupling.

7. The method of claim 6, wherein the gain compensated quantity is computed using at least one of the following mathematical equations:



  
    
      CXX
      xx
    
    =
    
      
        
          
            (
            
              
                V
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  C
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  12
                  ⁢
                  xx
                
              
              +
              
                V
                
                  SHC
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  12
                  ⁢
                  xx
                
              
            
            )
          
          ⁢
          
            (
            
              
                V
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  C
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  21
                  ⁢
                  xx
                
              
              +
              
                V
                
                  SHC
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  21
                  ⁢
                  xx
                
              
            
            )
          
        
        
          
            (
            
              
                V
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  C_
                  ⁢
                  
                    
                  
                  ⁢
                  11
                  ⁢
                  xx
                
              
              +
              
                V
                
                  SHC
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  11
                  ⁢
                  xx
                
              
            
            )
          
          ⁢
          
            (
            
              
                V
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  C
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  22
                  ⁢
                  xx
                
              
              +
              
                V
                
                  SHC
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  22
                  ⁢
                  xx
                
              
            
            )
          
        
      
    
  




  
    
      CXX
      yy
    
    =
    
      
        
          
            (
            
              
                V
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  C
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  12
                  ⁢
                  yy
                
              
              +
              
                V
                
                  SHC
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  12
                  ⁢
                  yy
                
              
            
            )
          
          ⁢
          
            (
            
              
                V
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  C
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  21
                  ⁢
                  yy
                
              
              +
              
                V
                
                  SHC
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  21
                  ⁢
                  yy
                
              
            
            )
          
        
        
          
            (
            
              
                V
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  C
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  11
                  ⁢
                  yy
                
              
              +
              
                V
                
                  SHC
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  11
                  ⁢
                  yy
                
              
            
            )
          
          ⁢
          
            (
            
              
                V
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  C
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  22
                  ⁢
                  yy
                
              
              +
              
                V
                
                  SHC
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  22
                  ⁢
                  yy
                
              
            
            )
          
        
      
    
  




  
    
      CYY
      xx
    
    =
    
      
        
          
            (
            
              
                V
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  C
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  12
                  ⁢
                  xx
                
              
              -
              
                V
                
                  SHC
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  12
                  ⁢
                  xx
                
              
            
            )
          
          ⁢
          
            (
            
              
                V
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  C
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  21
                  ⁢
                  xx
                
              
              -
              
                V
                
                  SHC
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  21
                  ⁢
                  xx
                
              
            
            )
          
        
        
          
            (
            
              
                V
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  C
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  11
                  ⁢
                  xx
                
              
              -
              
                V
                
                  SHC
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  11
                  ⁢
                  xx
                
              
            
            )
          
          ⁢
          
            (
            
              
                V
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  C
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  22
                  ⁢
                  xx
                
              
              -
              
                V
                
                  SHC
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  22
                  ⁢
                  xx
                
              
            
            )
          
        
      
    
  




  
    
      CYY
      yy
    
    =
    
      
        
          
            (
            
              
                V
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  C
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  12
                  ⁢
                  yy
                
              
              -
              
                V
                
                  SHC
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  12
                  ⁢
                  yy
                
              
            
            )
          
          ⁢
          
            (
            
              
                V
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  C
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  21
                  ⁢
                  yy
                
              
              -
              
                V
                
                  SHC
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  21
                  ⁢
                  yy
                
              
            
            )
          
        
        
          
            (
            
              
                V
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  C
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  11
                  ⁢
                  yy
                
              
              -
              
                V
                
                  SHC
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  11
                  ⁢
                  yy
                
              
            
            )
          
          ⁢
          
            (
            
              
                V
                
                  D
                  ⁢
                  
                    
                  
                  ⁢
                  C
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  22
                  ⁢
                  yy
                
              
              -
              
                V
                
                  SHC
                  ⁢
                  
                    
                  
                  ⁢
                  _
                  ⁢
                  
                    
                  
                  ⁢
                  22
                  ⁢
                  yy
                
              
            
            )
          
        
      
    
  


wherein CXXxx and CXXyy represent the quantities proportional to the xx coupling, CYYxx and CYYyy represent the quantities proportional to the yy coupling, VDC_11xx, VDC_12xx, VDC_21xx, VDC_22xx, VDC_11yy, VDC_12yy, VDC_21yy, and VDC_22yy represent DC harmonic voltage coefficients, and VSHC_11xx, VSHC_12xx, VSHC_21xx, VSHC_22xx, VSHC_11yy, VSHC_12yy, VSHC_21yy, and VSHC_22yy represent second harmonic cosine voltage coefficients.

8. The method of claim 1, wherein the gain compensated quantity computed in (d) comprises at least one quantity proportional to an xx coupling plus a yy coupling or an xx coupling minus a yy coupling.

9. The method of claim 8, wherein processing the ratio in (d) comprises computing a ratio of a first product of DC voltage coefficients to a second product of DC voltage coefficients to compute the gain compensated quantity.

10. The method of claim 9, wherein the gain compensated quantity is computed using at least one of the following mathematical equations:



  
    
      CXXplusYY
      xx
    
    =
    
      
        
          
            V
            
              D
              ⁢
              
                
              
              ⁢
              C
              ⁢
              
                
              
              ⁢
              _
              ⁢
              
                
              
              ⁢
              12
              ⁢
              xx
            
          
          ⁢
          
            V
            
              D
              ⁢
              
                
              
              ⁢
              C
              ⁢
              
                
              
              ⁢
              _
              ⁢
              
                
              
              ⁢
              21
              ⁢
              xx
            
          
        
        
          
            V
            
              D
              ⁢
              
                
              
              ⁢
              C
              ⁢
              
                
              
              ⁢
              _
              ⁢
              
                
              
              ⁢
              11
              ⁢
              xx
            
          
          ⁢
          
            V
            
              D
              ⁢
              
                
              
              ⁢
              C
              ⁢
              
                
              
              ⁢
              _
              ⁢
              
                
              
              ⁢
              22
              ⁢
              xx
            
          
        
      
    
  




  
    
      CXXplusYY
      yy
    
    =
    
      
        
          
            V
            
              D
              ⁢
              
                
              
              ⁢
              C
              ⁢
              
                
              
              ⁢
              _
              ⁢
              
                
              
              ⁢
              12
              ⁢
              yy
            
          
          ⁢
          
            V
            
              D
              ⁢
              
                
              
              ⁢
              C
              ⁢
              
                
              
              ⁢
              _
              ⁢
              
                
              
              ⁢
              21
              ⁢
              yy
            
          
        
        
          
            V
            
              D
              ⁢
              
                
              
              ⁢
              C
              ⁢
              
                
              
              ⁢
              _
              ⁢
              
                
              
              ⁢
              11
              ⁢
              yy
            
          
          ⁢
          
            V
            
              D
              ⁢
              
                
              
              ⁢
              C
              ⁢
              
                
              
              ⁢
              _
              ⁢
              22
              ⁢
              yy
            
          
        
      
    
  


wherein CXXplusYYxx and CXXplusYYyy represent the quantities proportional to an xx coupling plus a yy coupling and VDC_11xx, VDC_12xx, VDC_21xx, VDC_22xx, VDC_11yy, VDC_12yy, VDC_21yy, and VDC_22yy represent the DC voltage coefficients.

11. The method of claim 8, wherein processing the ratio in (d) comprises computing a ratio of a first product of second harmonic cosine voltage coefficients to a second product of DC voltage coefficients to compute the gain compensated quantity.

12. The method of claim 11, wherein the gain compensated quantity is computed using at least one of the following mathematical equations:



  
    
      CXXminusYY
      xx
    
    =
    
      
        
          
            V
            
              SHC
              ⁢
              
                
              
              ⁢
              _
              ⁢
              
                
              
              ⁢
              12
              ⁢
              xx
            
          
          ⁢
          
            V
            
              SHC
              ⁢
              
                
              
              ⁢
              _
              ⁢
              
                
              
              ⁢
              21
              ⁢
              xx
            
          
        
        
          
            V
            
              D
              ⁢
              
                
              
              ⁢
              C
              ⁢
              
                
              
              ⁢
              _
              ⁢
              
                
              
              ⁢
              11
              ⁢
              xx
            
          
          ⁢
          
            V
            
              D
              ⁢
              
                
              
              ⁢
              C
              ⁢
              
                
              
              ⁢
              _
              ⁢
              
                
              
              ⁢
              22
              ⁢
              xx
            
          
        
      
    
  




  
    
      CXXminusYY
      yy
    
    =
    
      
        
          
            V
            
              SHC
              ⁢
              
                
              
              ⁢
              _
              ⁢
              
                
              
              ⁢
              12
              ⁢
              yy
            
          
          ⁢
          
            V
            
              SHC
              ⁢
              
                
              
              ⁢
              _
              ⁢
              
                
              
              ⁢
              21
              ⁢
              yy
            
          
        
        
          
            V
            
              D
              ⁢
              
                
              
              ⁢
              C
              ⁢
              
                
              
              ⁢
              _
              ⁢
              
                
              
              ⁢
              11
              ⁢
              yy
            
          
          ⁢
          
            V
            
              D
              ⁢
              
                
              
              ⁢
              C
              ⁢
              
                
              
              ⁢
              _
              ⁢
              22
              ⁢
              yy
            
          
        
      
    
  


wherein CXXminusYYxx and CXXminusYYyy represent the quantities proportional to an xx coupling minus a yy coupling, VDC_11xx, VDC_22XX, VDC_11yy, and VDC_22yy represent the DC voltage coefficients, and VSHC_12xx, VSHC_21xx, VSHC_12yy, and VSHC_21yy represent the second harmonic cosine voltage coefficients.

13. The method of claim 8, wherein processing the ratio in (d) comprises computing a ratio of a second harmonic cosine voltage coefficient to a DC voltage coefficient to compute the gain compensated quantity.

14. The method of claim 13, wherein the gain compensated quantity is computed using at least one of the following mathematical equations:



  
    
      CXXminusYY
      ijxx
    
    =
    
      
        V
        
          SHC
          ⁢
          
            
          
          ⁢
          _
          ⁢
          
            
          
          ⁢
          ijxx
        
      
      
        V
        
          D
          ⁢
          
            
          
          ⁢
          
            C
            ijxx
          
        
      
    
  




  
    
      CXXminusYY
      ijyy
    
    =
    
      -
      
        
          V
          
            SHC
            ⁢
            
              
            
            ⁢
            _
            ⁢
            
              
            
            ⁢
            ijyy
          
        
        
          V
          
            D
            ⁢
            
              
            
            ⁢
            C
            ⁢
            
              
            
            ⁢
            _
            ⁢
            
              
            
            ⁢
            ijyy
          
        
      
    
  


wherein CXXminusYYijxx and CXXminusYYijyy represent the quantities proportional to an xx coupling minus a yy coupling, VDC_ijxx and VDC_ijyy represent the DC voltage coefficients, and VSHC_ijxx and VSHC_ijyy represent the second harmonic cosine voltage coefficients.

15. The method of claim 1, wherein the gain compensated quantity computed in (d) comprises at least one measurement proportional to an xy coupling plus a yx coupling or an xy coupling minus a yx coupling.

16. The method of claim 15, wherein processing the ratio in (d) comprises computing a ratio of a second harmonic sine voltage coefficient to a DC voltage coefficient to compute the gain compensated quantity.

17. The method of claim 16, wherein the gain compensated quantity is computed using at least one of the following mathematical equations:



  
    
      CXYplusYX
      ijxx
    
    =
    
      -
      
        
          V
          
            SHC
            ⁢
            
              
            
            ⁢
            _
            ⁢
            
              
            
            ⁢
            ijxx
          
        
        
          V
          
            D
            ⁢
            
              
            
            ⁢
            C
            ⁢
            
              
            
            ⁢
            _
            ⁢
            
              
            
            ⁢
            ijxx
          
        
      
    
  




  
    
      CXYplusYX
      ijyy
    
    =
    
      -
      
        
          V
          
            SHC
            ijyy
          
        
        
          V
          
            D
            ⁢
            
              
            
            ⁢
            
              C
              ijyy
            
          
        
      
    
  


wherein CXYplusYXijxx and CXYplusYXijyy represent the quantities proportional to an xy coupling plus yx coupling, VDC_ijxx and VDC_ijyy represent the DC voltage coefficients, and VSHS_ijxx and VSHC_ijyy represent the second harmonic sine voltage coefficients.

18. The method of claim 15, wherein processing the ratio in (d) comprises computing a ratio of a first product of DC voltage coefficients to a second product of DC voltage coefficients to compute the gain compensated quantity.

19. The method of claim 18, wherein the gain compensated quantity is computed using at least one of the following mathematical equations:



  
    
      CXXminusYX
      ij
    
    =
    
      
        -
        
          
            
              V
              
                D
                ⁢
                
                  
                
                ⁢
                C
                ⁢
                
                  
                
                ⁢
                _
                ⁢
                
                  
                
                ⁢
                ijxy
              
            
            ⁢
            
              V
              
                D
                ⁢
                
                  
                
                ⁢
                C
                ⁢
                
                  
                
                ⁢
                _
                ⁢
                
                  
                
                ⁢
                ijyx
              
            
          
          
            
              V
              
                D
                ⁢
                
                  
                
                ⁢
                C
                ⁢
                
                  
                
                ⁢
                _
                ⁢
                
                  
                
                ⁢
                ijxx
              
            
            ⁢
            
              V
              
                D
                ⁢
                
                  
                
                ⁢
                C
                ⁢
                
                  
                
                ⁢
                _
                ⁢
                
                  
                
                ⁢
                ijyy
              
            
          
        
      
    
  




  
    CXXminusYX
    =
    
      
        -
        
          
            
              V
              
                D
                ⁢
                
                  
                
                ⁢
                C
                ⁢
                
                  
                
                ⁢
                _
                ⁢
                
                  
                
                ⁢
                12
                ⁢
                xy
              
            
            ⁢
            
              V
              
                D
                ⁢
                
                  
                
                ⁢
                C
                ⁢
                
                  
                
                ⁢
                _
                ⁢
                
                  
                
                ⁢
                21
                ⁢
                yx
              
            
          
          
            
              V
              
                D
                ⁢
                
                  
                
                ⁢
                C
                ⁢
                
                  
                
                ⁢
                _
                ⁢
                
                  
                
                ⁢
                11
                ⁢
                xx
              
            
            ⁢
            
              V
              
                D
                ⁢
                
                  
                
                ⁢
                C
                ⁢
                
                  
                
                ⁢
                _
                ⁢
                
                  
                
                ⁢
                22
                ⁢
                yy
              
            
          
        
      
    
  


wherein CXYminusYXij and CXYminusYX represent the quantities proportional to an xy coupling minus yx coupling and VDC_ijxy, VDC_ijyx, VDC_ijxx, VDC_ijyy, VDC_11xx, VDC_12xy, VDC_21yx, and VDC_22yy, represent the DC voltage coefficients.

20. The method of claim 15, wherein the gain compensated quantity computed in (d) further comprises at least one measurement proportional to a xy coupling or a yx coupling.

21. The method of claim 20, wherein the gain compensated quantity is computed using at least one of the following mathematical equations:



  
    
      CXY
      ij
    
    =
    
      
        
          CXYplusYX
          ijxx
        
        +
        
          CXYminusYX
          ij
        
      
      2
    
  




  
    
      CXY
      ij
    
    =
    
      
        
          CXYplusYX
          ijyy
        
        +
        
          CXYminusYX
          ij
        
      
      2
    
  




  
    
      CYX
      ij
    
    =
    
      
        
          CXYplusYX
          ijxx
        
        -
        
          CXYminusYX
          ij
        
      
      2
    
  




  
    
      CYX
      ij
    
    =
    
      
        
          CXYplusYX
          ijyy
        
        -
        
          CXYminusYX
          ij
        
      
      2
    
  


wherein CXYij and CYXij represent the quantities proportional to the xy coupling and the yx coupling, CXYplusYXijxx and CXYplusYXijyy represent quantities proportional to an xy coupling plus a yx coupling, and CXYminusYXij represents a quantity proportional to an xy coupling minus a yx coupling.

22. The logging while drilling tool of claim 20, wherein the first and second transmitter antennas are non-collocated and the first and second receiver antennas are non-collocated.

23. The logging while drilling tool of claim 20, wherein the first transmitter antenna has a moment that is rotationally aligned with a moment of the first receiver antenna.

24. The method of claim 1, wherein (d) further comprises computing a square root of the ratio.

25. An electromagnetic logging while drilling tool comprising:
a logging while drilling tool body;
first and second transverse transmitter antennas whose moments are orthogonal to one another deployed on the tool body;
first and second transverse receiver antennas whose moments are orthogonal to one another deployed on the tool body;
a controller configured to (i) cause the first and second transverse transmitter antennas antenna to transmit corresponding electromagnetic waves; (ii) acquire electromagnetic voltage measurements from the first and second transverse receiver antennas while the first and second transverse transmitter antennas are transmitting; (iii) process the electromagnetic voltage measurements to compute harmonic voltage coefficients; and (iv) process a ratio of the harmonic voltage coefficients to compute at least one gain compensated quantity including a transverse term, wherein the transverse term is one an xx direct coupling impedance, a yy direct coupling impedance, an xy cross coupling impedance and yx cross coupling impedance.

26. The logging while drilling tool of claim 25, wherein the first and second transmitter antennas are collocated with one another.

27. The logging while drilling tool of claim 25, wherein the first and second receiver antennas are collocated with one another.