Patent Publication Number: US-9835753-B2

Title: Gain compensated tensor propagation measurements using collocated antennas

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims priority as a Patent Cooperation Treaty patent application of U.S. Provisional patent application Ser. No. 61/868,451 filed Aug. 21, 2013 with the same title. 
    
    
     TECHNICAL FIELD 
     Disclosed embodiments relate generally to downhole electromagnetic logging methods and more particularly to a method for making full tensor, gain compensated propagation measurements such as phase shift and attenuation measurements. 
     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 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 much more complex than the formation models used in the inversion. The use of full tensor propagation measurements may enable a full tensor measurement of the formation properties to be obtained. However, finding reliable techniques for providing an accurate gain compensated full tensor measurement has been a challenge for the industry. 
     SUMMARY 
     A method for obtaining full tensor gain compensated propagation measurements is disclosed. The method includes rotating an electromagnetic logging tool in a subterranean borehole. The logging tool includes at least first and second axially spaced transmitters and at least first and second axially spaced receivers. The transmitters or receivers may include at least one antenna having a tilted moment with respect to the longitudinal axis of the logging tool. A plurality of voltage measurements is acquired while rotating the tool in the borehole. The voltage measurements may be fit to a harmonic expression to obtain harmonic coefficients. The voltage measurements are processed to construct a full tensor voltage measurement which is in turn further processed to obtain a full tensor gain compensated quantity. 
     In accordance with one embodiment, a method for obtaining full tensor gain compensated propagation measurements includes (a) rotating an electromagnetic logging tool in a subterranean borehole, the logging tool including at least first and second axially spaced transmitters and at least first and second axially spaced receivers, each of the receivers including a set of collocated linearly independent antennas, each of the transmitters including first and second collocated antennas, and with at least the first transmitter antenna having a tilted moment with respect to a longitudinal axis of the tool. The method further includes (b) acquiring a plurality of voltage measurements while rotating the tool in (a), the plurality of voltage measurements being obtained using a corresponding plurality of pairs of the transmitters and receivers, (c) fitting the voltage measurements to a harmonic expression to obtain harmonic coefficients, (d) causing a processor to process the harmonic coefficients to construct a full tensor voltage measurement, and (e) causing the processor to process the full tensor voltage measurement to obtain a fully gain compensated tensor quantity. 
     In accordance with another embodiment, a system includes an electromagnetic logging tool having at least first and second axially spaced transmitters and at least first and second axially spaced receivers, each of the receivers including a set of collocated linearly independent antennas, each of the transmitters including first and second collocated antennas, at least the first transmitter antenna having a tilted moment with respect to a longitudinal axis of the tool. The electromagnetic logging tool acquires a plurality of voltage measurements while rotating the electromagnetic logging tool in a borehole formed in a subterranean formation, with the plurality of voltage measurements being obtained using a corresponding plurality of pairs of the transmitters and receivers. The system further includes a processor that fits the voltage measurements to a harmonic expression to obtain harmonic coefficients, processes the harmonic coefficients to construct a full tensor voltage measurement, and processes the full tensor voltage measurement to obtain a fully gain compensated tensor quantity. 
     In accordance with another embodiment, a method for obtaining gain compensated full tensor electromagnetic antenna measurements includes (a) rotating an electromagnetic logging tool in a subterranean borehole, the logging tool including at least first and second axially spaced transmitters and at least first and second axially spaced receivers, each of the receivers including a set of collocated linearly independent antennas, each of the transmitters including first and second collocated calibration antennas having transverse moments with respect to one another and with respect to the longitudinal axis of the tool, and each of the transmitters further including a tilted antenna not collocated with the first and second calibration antennas and having a tilted moment with respect to a longitudinal axis of the tool. The method further includes (b) acquiring a plurality of voltage measurements while rotating the tool in (a), the plurality of voltage measurements obtained using a corresponding plurality of pairs of the tilted transmitter antennas and the triaxial receiver antennas, (c) fitting the voltage measurements to a harmonic expression to obtain harmonic coefficients, (d) causing a processor to process the harmonic coefficients to construct a full tensor voltage measurement, and (e) causing the processor to process the full tensor voltage measurement to obtain a fully gain compensated tensor quantity. 
     In accordance with yet another embodiment, a system includes an electromagnetic logging tool having at least first and second axially spaced transmitters and at least first and second axially spaced receivers, each of the receivers including a set of collocated linearly independent antennas, each of the transmitters including first and second collocated calibration antennas having transverse moments with respect to one another and with respect to the longitudinal axis of the tool, each of the transmitters further including a tilted antenna not collocated with the first and second calibration antennas and having a tilted moment with respect to a longitudinal axis of the tool. The electromagnetic logging tool acquires a plurality voltage measurements while rotating in a subterranean borehole, the plurality of voltage measurements being obtained using a corresponding plurality of pairs of the tilted transmitter antennas and the triaxial receiver antennas. The system also includes a processor that fits the voltage measurements to a harmonic expression to obtain harmonic coefficients, processes the harmonic coefficients to construct a full tensor voltage measurement, and processes the full tensor voltage measurement to obtain a fully gain compensated tensor quantity. 
     The disclosed embodiments may provide various technical advantages. For example, the disclosed methodology provides for full tensor gain compensated propagation measurements. The measurements are sensitive to vertical and horizontal formation resistivity (anisotropy) as well the presence of a remote bed boundary at all dip angles. The full tensor measurements may therefore be utilized in an inversion to obtain the vertical and horizontal resistivity of local and remote beds, as well as the distance and dip angle to the boundary. Moreover, the full tensor gain compensated propagation measurements are sensitive to the full three dimensional conductivity tensor and therefore may provide information on the three dimensional structure of the formation, for example, including faults, fractures, folded layers, and unconformities. 
     This summary is provided to introduce a selection of concepts that are further described below in the detailed description and is intended to familiarize the reader with certain aspects and contexts of embodiments of the present disclosure without limitation to 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 rig on which electromagnetic logging tools may be utilized. 
         FIG. 2A  depicts one example of the electromagnetic logging tool shown on  FIG. 1 . 
         FIG. 2B  schematically depicts an electromagnetic logging tool including collocated triaxial transmitters and receivers. 
         FIG. 2C  schematically depicts an electromagnetic logging tool including collocated linearly independent transmitter and receiver triads in which the a, b, and c antenna are not necessarily aligned with the x, y, and z axis of the logging tool. 
         FIG. 3  depicts a transmitter receiver pair deployed in an arbitrary global X, Y, Z reference frame. 
         FIG. 4  depicts a flow chart of a disclosed method embodiment. 
         FIG. 5  depicts a flow chart of another disclosed method embodiment. 
         FIG. 6  schematically depicts an electromagnetic logging tool suitable for use with the method embodiment depicted on  FIG. 5 . 
         FIG. 7  depicts a flow chart of still another disclosed method embodiment. 
         FIG. 8  schematically depicts an electromagnetic logging tool suitable for use with the method embodiment depicted on  FIG. 7 . 
     
    
    
     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 full three dimensional tensor electromagnetic logging measurements. 
     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 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 electromagnetic measurement tool  50 . In the depicted embodiment measurement tool  50  includes a logging-while-drilling (LWD) tool having first and second triaxial transmitters T 1  and T 2  depicted at  52  and  54  and first and second receivers R 1  and R 2  depicted at  56  and  58  spaced axially along LWD tool body  51 . In the depicted embodiment, each of the transmitters  52 ,  54  and receivers  56 ,  58  includes a collocated linearly independent antenna arrangement. 
     Electromagnetic logging tools commonly use axial, transverse, and/or tilted antennas. 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 a saddle coil (e.g., as disclosed in U.S. Patent Publications 2011/0074427 and 2011/0238312, incorporated herein by reference). A tilted antenna is one whose moment is neither parallel nor perpendicular to the longitudinal axis of the tool. Tilted antennas generate a mixed mode radiation pattern (i.e., a radiation pattern in which the moment is neither parallel nor perpendicular with the tool axis). It will be understood that a tilted antenna is not necessarily tilted in the sense that a plane of the antenna is tilted with respect to the tool axis. By tilted it is meant that the antenna has a tilted moment with respect to the axis. 
     As stated above with respect to  FIG. 2A , the transmitters  52 ,  54  and receivers  56 ,  58  each include a collocated linearly independent antenna arrangement (one example arrangement of which is depicted schematically on  FIG. 2B ). A triaxial antenna arrangement (also referred to as a triaxial transmitter, receiver, or transceiver) is one example of a linearly independent antenna arrangement in which two or three antennas (i.e., up to three distinct antenna coils) are arranged to be mutually independent. By mutually independent it is meant that the moment of any one of the antennas does not lie in the plane formed by the moments of the other antennas. Three tilted antennas is one common example of a triaxial antenna sensor. Three collocated orthogonal antennas, with one antenna axial and the other two transverse, is another common example of a triaxial antenna sensor. 
       FIG. 2B  depicts the moments of triaxial transmitters  52 ,  54  and receivers  56 ,  58 . Each of the transmitters  52 ,  54  includes an axial antenna T 1   z  and T 2   z  and first and second transverse antennas T 1   x , T 1   y  and T 2   x , T 2   y . Likewise, each of the receivers  56 ,  58  includes an axial antenna R 1   z  and R 2   z  and first and second transverse antennas R 1   x , R 1   y  and R 2   x , R 2   y . In the depicted embodiment, the moments of the transmitter and receiver antennas are mutually orthogonal and aligned with the x, y, and z axes as indicated in a conventional borehole reference frame in which the z-axis is coincident with the axis of the tool. It will be understood that the disclosed embodiments are expressly not limited in this regard. 
       FIG. 2C  depicts the moments of a more generalized antenna arrangement including first and second axially spaced transmitter and receiver triads in which the moments of each of the transmitter and receiver antennas define three linearly independent directions a, b, and c. It will be understood that the a, b, and c directions are not necessarily mutually orthogonal. Nor are they necessarily aligned with the x, y, and z axes of a borehole reference frame or any other reference frame. Moreover, the a, b, and c directions of the moments of any one transmitter or receiver are not necessarily aligned with the a, b, and c directions of the moments of any of the other transmitters or receivers. 
     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. 
     Full Tensor Coupling with Rotation and Bending 
     From Ampere&#39;s law, the relationship between the induced magnetic field and the current flow {right arrow over (J)} and displacement current ∂D due to an electric field {right arrow over (E)} applied to a material with conductivity  σ  and dielectric constant  ∈  is not necessarily in the same direction as the applied electric field.
 
{right arrow over (∇)} x{right arrow over (H)}={right arrow over (J)}+∂D= σ {right arrow over (E)} =( σ − iω ∈   ) {right arrow over (E)}= σ ′{right arrow over (E)}   (1)
 
     In general the earth is anisotropic such that its electrical properties may be expressed as a tensor which contains information on formation resistivity anisotropy, dip, bed boundaries and other aspects of formation geometry. Thus the three dimensional current flow {right arrow over (J)} may be expressed as follows:
 
 J   x   +∂D   x =σ xx   ′E   x +σ xy   ′E   y +σ xz   ′E   z   (2)
 
 J   y   +∂D   y =σ yx   ′E   x +σ yy   ′E   y +σ yz   ′E   z   (3)
 
 J   z   +∂D   z =σ zx   ′E   x +σ zy   ′E   y +σ zz   ′E   z   (4)
 
     where the full (three dimensional) conductivity tensor may be given as follows: 
     
       
         
           
             
               
                 
                   
                     σ 
                     ′ 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             σ 
                             xx 
                             ′ 
                           
                         
                         
                           
                             σ 
                             xy 
                             ′ 
                           
                         
                         
                           
                             σ 
                             xz 
                             ′ 
                           
                         
                       
                       
                         
                           
                             σ 
                             yx 
                             ′ 
                           
                         
                         
                           
                             σ 
                             yy 
                             ′ 
                           
                         
                         
                           
                             σ 
                             yz 
                             ′ 
                           
                         
                       
                       
                         
                           
                             σ 
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                             ′ 
                           
                         
                         
                           
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                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     The mutual couplings between the collocated triaxial transmitter coils and the collocated triaxial receiver coils depicted on  FIGS. 2A and 2B  form a full tensor and have sensitivity to the full conductivity tensor given in Equation 5. The measured voltage V may be expressed as a full tensor as follows: 
     
       
         
           
             
               
                 
                   V 
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               V 
                               xx 
                             
                           
                           
                             
                               V 
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                         ⁡ 
                         
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                   ( 
                   6 
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     where V represents the measured voltage tensor in the receiver coils, I represents the transmitter currents, and Z represents the transfer impedance which depends on the electrical and magnetic properties of the environment surrounding the coil pair in addition to the frequency, coil geometry, and coil spacing. The first letter in the subscript in the V and Z tensors corresponds to the direction of the transmitter while the second corresponds to the direction of the receiver. For example Z xx  represents the mutual coupling between a transmitter firing with current I x  and aligned with the x axis and the receiver aligned with the x-axis, Z yx  represents the mutual coupling between the y transmitter firing with current I y  and the x-axis receiver, and so on. 
     With continued reference to  FIGS. 2A, 2B, and 2C , the measured voltage of any particular antenna coil (for a given transmitter current) can be related to a number of factors, such as the induced voltage in a subterranean formation, the direct coupling voltage on the coil, the induced voltage from the collar, as well as the transmitter and receiver gains. 
     
       
         
           
             
               
                 
                   V 
                   = 
                   
                     
                       
                         G 
                         T 
                       
                       ⁢ 
                       
                         ZG 
                         R 
                       
                     
                     = 
                     
                       
                         
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                                 g 
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                                 Rz 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     where G T  represents a diagonal matrix of the transmitter gains g Tx , g Ty , and g Tz  and G R  represents a diagonal matrix of the receiver gains g Rx , g Ry , and g Rz . It will be understood that in Equation 7, the transmitter currents I are included in the generalized transmitter gains. If the magnetic field produced by the transmitter coil is approximately constant in magnitude and direction across the receiver coil, then the mutual inductive coupling scales with the number of turns in the antenna coil and the effective coil areas of the transmitter and receiver. However, as described above with respect to  FIG. 2C , the antenna moments are not necessarily perfectly aligned with the x, y, and z coordinate axes, nor are they necessarily mutually orthogonal. Thus Equation 7 may be written in a such way that the gains include a generalized gain times a unit vector that points in the direction normal to the area enclosed by the antenna coil, for example, as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
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                         = 
                         
                           
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                             T 
                           
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                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     where t represents the transpose of the corresponding matrix. The subscripts a, b, and c refer to the antenna triad moment directions and define three linearly independent directions. It will be understood that a, b, and c are not necessarily mutually orthogonal. The matrix terms m Tax , m Tay , and m Taz  represent projections of a unit vector {circumflex over (m)} Ta  that is in the direction of the ‘a’ transmitter moment on the x, y, and z coordinate axes; m Tbx , m Tby , and m Tbz  represent projections of a unit vector {circumflex over (m)} Tb  that is in the direction of the ‘b’ transmitter moment on the x, y, and z coordinate axes; and m Tcx , m Tcy , and m Tcz  represent projections of a unit vector {circumflex over (m)} Tc  that is in the direction of the ‘c’ transmitter moment on the x, y, and z coordinate axes. Similarly, m Rax , m Ray , and m Raz  represent projections of a unit vector {circumflex over (m)} Ra  that is in the direction of the ‘a’ receiver moment on the x, y, and z coordinate axes; m Rbx , m Rby , and m Rbz  represent projections of a unit vector {circumflex over (m)} Rb  that is in the direction of the ‘b’ receiver moment on the x, y, and z coordinate axes; and m Rcx , m Rcy , and m Rcz  represent projections of a unit vector {circumflex over (m)} Rc  that is in the direction of the ‘c’ receiver moment on the x, y, and z coordinate axes. 
     The transfer impedance tensor, Z, is a function of the subterranean formation properties, for example, as expressed below:
 
 Z=ƒ (σ h ,σ v ,∈ h ,∈ v   ,L ,dip angle,dip azimuth angle,bed thickness)  (9)
 
     where σ h  represents the horizontal conductivity, σ v  represents the vertical conductivity, ∈ h  represents the horizontal dielectric constant, ∈ v  represents the vertical dielectric constant, and L represents the distance to a remote bed. The apparent dip angle is generally defined as the angle between and the tool axis and the normal vector of the bed. The apparent dip azimuth angle is generally defined as the angle between the xz plane of the tool and the horizontal component of the bed normal vector. A bed boundary is defined by two adjacent beds with different conductivities. 
       FIG. 3  depicts a transmitter receiver pair deployed in an arbitrary global X, Y, Z reference frame. Local x, y, z reference frames are depicted at each of the transmitter and receiver locations. In the depicted example, transmitter T 1  is deployed on a first sub  62  and R 1  is deployed on a second sub  64  such that BHA makeup (axial rotation) and bending (cross axial rotation) may rotate the transmitter and receiver triads (as well as the local reference frames) with respect to one another. In the following analysis, BHA makeup and bending are considered sequentially (although the disclosed embodiments are not limited in this regard). The tool is first considered to be unbent and aligned with the global Z axis. Since the rotational orientation of each sub may be considered to be random, the transmitter and receiver moments may be rotated about their corresponding local z axis, for example, as follows:
 
 R   za   m   R  and  R   zγ   m   T  
 
     where R zα  represents the rotation matrix of an axial rotation about angle α, R zγ  represents the rotation matrix of an axial rotation about angle γ, and m R  and m T  are matrices representing the magnetic moments of the receiver and transmitter triads. After the axial rotations given above, the angular offset between the local x, y, z reference frames may be given as γ−α. Tool bending may then be considered as a separate rotation about an arbitrary cross axial rotation axis, for example, as follows:
 
 {acute over (m)}   R   =R   Rbend   R   zα   m   R   =R   R   m   R  
 
 {acute over (m)}   T   =R   Tbend   R   zγ   m   T   =R   T   m   T   (10)
 
     where {acute over (m)} R  and {acute over (m)} T  represent the rotated receiver and transmitter moments (rotated both axially and via a tool bend). 
     With reference again to  FIG. 2C , a propagation tool may be considered having first and second transmitter triads and first and second receiver triads in which the magnetic moments of each transmitter and receiver are aligned along arbitrary linearly independent directions. By linearly independent, it is meant that none of the moment directions in the triad may be written as a linear combination of one or both of the other moment directions in the triad. 
     Voltages measured on the receiver R 2  triad induced by currents in the transmitter T 1  triad may be expressed as a 3×3 tensor, for example, as follows: 
     
       
         
           
             
               
                 
                   
                     V 
                     12 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             V 
                             
                               12 
                               ⁢ 
                               xx 
                             
                           
                         
                         
                           
                             V 
                             
                               12 
                               ⁢ 
                               xy 
                             
                           
                         
                         
                           
                             V 
                             
                               12 
                               ⁢ 
                               xz 
                             
                           
                         
                       
                       
                         
                           
                             V 
                             
                               12 
                               ⁢ 
                               yx 
                             
                           
                         
                         
                           
                             V 
                             
                               12 
                               ⁢ 
                               yy 
                             
                           
                         
                         
                           
                             V 
                             
                               12 
                               ⁢ 
                               yz 
                             
                           
                         
                       
                       
                         
                           
                             V 
                             
                               12 
                               ⁢ 
                               zx 
                             
                           
                         
                         
                           
                             V 
                             
                               12 
                               ⁢ 
                               zy 
                             
                           
                         
                         
                           
                             V 
                             
                               12 
                               ⁢ 
                               zz 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     As in Equations 7 and 8, the voltages V 12  may be expressed in terms of the electronic gains and rotated moments. Assuming no other coupling between the transmitter and receiver triads (i.e., that capacitive coupling, crosstalk, and noise are negligible) leads to the following tensor model:
 
 V   12   =G   T1   m   T1   t   Z   12   m   R2   G   R2   =G′   T1   Z   12   G′   R2   (12)
 
     Where G T1  represents the transmitter gains, G R2  represents the receiver gains, m T1   t  represents the transpose of the unit vector projections of transmitter T 1 , m R2  represents the unit vector projections of receiver R 2 , G′ T1 =G T1 m T1   t , and G′ R2 =m R2 G R2 . 
     The gains from receiver R 1  may be eliminated by taking the following combination of voltages and computing the quantity M 21 , for example, as follows:
 
 M   21   =V   11   V   21   −1   =G′   T1   Z   11   G′   R1   G′   R1   −1   Z   21   G′   T2   −1   =G′   T1   Z   11   Z   21   −1   G′   T2   −1    (13)
 
     It will be understood that Equation 13 assumes that G T2  and G R1  are invertible. These generalized gain matrices are each products of a diagonal gain matrix, a rotation matrix (for which the transpose is its inverse), and the moment matrix. Since the diagonal gain matrix and rotation matrices are invertible, invertibility of G′ T2  and G′ R1  depends on the invertibility of m T2  and m R1 . The gains from receiver R 2  may be similarly eliminated by computing the quantity M 12 , for example, as follows:
 
 M   12   =V   22   V   12   −1   =G′   T2   Z   22   G′   R2   G′   R2   −1   Z   12   G′   T1   −1   =G′   T2   Z   22   Z   12   −1   G′   T1   −1    (14)
 
     Equation 14 assumes that G T1  and G R2  are invertible along with m T1  and m R2 . Combining M 21  and M 12 , for example, as follows results in a quantity    21  that depends only on the generalized gains of the transmitter T 1 . 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           21 
                         
                         = 
                         
                           
                             
                               M 
                               21 
                             
                             ⁢ 
                             
                               M 
                               12 
                             
                           
                           = 
                           
                             
                               
                                 G 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 Z 
                                 11 
                               
                               ⁢ 
                               
                                 Z 
                                 21 
                                 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 G 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 
                                   ′ 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 G 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 Z 
                                 22 
                               
                               ⁢ 
                               
                                 Z 
                                 12 
                                 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 G 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                                 
                                   ′ 
                                   - 
                                   1 
                                 
                               
                             
                             = 
                             
                               
                                 G 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 U 
                                 21 
                               
                               ⁢ 
                               
                                 G 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                                 
                                   ′ 
                                   - 
                                   1 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               G 
                               
                                 T 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                             ⁢ 
                             
                               m 
                               
                                 T 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                               t 
                             
                             ⁢ 
                             
                               R 
                               
                                 T 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                               t 
                             
                             ⁢ 
                             
                               U 
                               21 
                             
                             ⁢ 
                             
                               R 
                               
                                 T 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                             ⁢ 
                             
                               m 
                               
                                 T 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                               
                                 - 
                                 t 
                               
                             
                             ⁢ 
                             
                               G 
                               
                                 T 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                               
                                 - 
                                 1 
                               
                             
                           
                           = 
                           
                             
                               G 
                               
                                 T 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                             ⁢ 
                             
                               m 
                               
                                 T 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                               t 
                             
                             ⁢ 
                             
                               U 
                               21 
                               ′ 
                             
                             ⁢ 
                             
                               m 
                               
                                 T 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                               
                                 - 
                                 t 
                               
                             
                             ⁢ 
                             
                               G 
                               
                                 T 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                               
                                 - 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     Note that in the quantity    21  all receiver gains have been removed (e.g., only the transmitter T 1  gains remain). Moreover, since U′ 21 =R T1   t U 21 R T1  is the rotation of U 21  from the global reference frame to the local T 1  reference frame,    21  only depends on the gains and moment directions at the location of the T 1  transmitter. It is independent of the gains, moment directions, and orientations (including bending and alignment) of the other transmitter and receiver triads. 
     Similarly, M 21  and M 12  may also be combined, for example, to obtain a quantity    12  that depends only on the generalized gains of the transmitter T 2 . 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           12 
                         
                         = 
                         
                           
                             
                               M 
                               12 
                             
                             ⁢ 
                             
                               M 
                               21 
                             
                           
                           = 
                           
                             
                               
                                 G 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 Z 
                                 22 
                               
                               ⁢ 
                               
                                 Z 
                                 12 
                                 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 G 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                                 
                                   ′ 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 G 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 Z 
                                 11 
                               
                               ⁢ 
                               
                                 Z 
                                 21 
                                 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 G 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 
                                   ′ 
                                   - 
                                   1 
                                 
                               
                             
                             = 
                             
                               
                                 G 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 ′ 
                               
                               ⁢ 
                               
                                 U 
                                 12 
                               
                               ⁢ 
                               
                                 G 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                                 
                                   ′ 
                                   - 
                                   1 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               G 
                               
                                 T 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                             ⁢ 
                             
                               m 
                               
                                 T 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                               t 
                             
                             ⁢ 
                             
                               R 
                               
                                 T 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                               t 
                             
                             ⁢ 
                             
                               U 
                               12 
                             
                             ⁢ 
                             
                               R 
                               
                                 T 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                             ⁢ 
                             
                               m 
                               
                                 T 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                               
                                 - 
                                 t 
                               
                             
                             ⁢ 
                             
                               G 
                               
                                 T 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                               
                                 - 
                                 1 
                               
                             
                           
                           = 
                           
                             
                               G 
                               
                                 T 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                             ⁢ 
                             
                               m 
                               
                                 T 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                               t 
                             
                             ⁢ 
                             
                               U 
                               12 
                               ′ 
                             
                             ⁢ 
                             
                               m 
                               
                                 T 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                               
                                 - 
                                 t 
                               
                             
                             ⁢ 
                             
                               G 
                               
                                 T 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                               
                                 - 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     Note that in the quantity    12  all receiver gains have been removed (e.g., only the transmitter T 2  gains remain). Moreover, the resulting quantity    12  only depends on the gains and moment directions at the location of the T 1  transmitter. It is independent of the gains, moment directions, and orientations (including bending and alignment) of the other transmitter and receiver triads. Other expressions may be obtained by changing the order of the individual terms so that similar quantities are obtained depending only on the receiver R 1  and R 2  gains and rotations. 
     Consider now the special case in which the moments of the a, b, and c transmitter coils of T 1  are aligned with the x, y, and z axes. In this special case, the gain matrices are diagonal and the tensor quantity    21  may be expressed as follows: 
     
       
         
           
             
               
                 
                   
                     21 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             Ú 
                             
                               21 
                               ⁢ 
                               xx 
                             
                           
                         
                         
                           
                             
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   a 
                                 
                               
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   b 
                                 
                               
                             
                             ⁢ 
                             
                               Ú 
                               
                                 21 
                                 ⁢ 
                                 xy 
                               
                             
                           
                         
                         
                           
                             
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   a 
                                 
                               
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   c 
                                 
                               
                             
                             ⁢ 
                             
                               Ú 
                               
                                 21 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 xz 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   b 
                                 
                               
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   a 
                                 
                               
                             
                             ⁢ 
                             
                               Ú 
                               
                                 21 
                                 ⁢ 
                                 yx 
                               
                             
                           
                         
                         
                           
                             Ú 
                             
                               21 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               yy 
                             
                           
                         
                         
                           
                             
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   b 
                                 
                               
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   c 
                                 
                               
                             
                             ⁢ 
                             
                               Ú 
                               
                                 21 
                                 ⁢ 
                                 yz 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   c 
                                 
                               
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   a 
                                 
                               
                             
                             ⁢ 
                             
                               Ú 
                               
                                 21 
                                 ⁢ 
                                 zx 
                               
                             
                           
                         
                         
                           
                             
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   c 
                                 
                               
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   b 
                                 
                               
                             
                             ⁢ 
                             
                               Ú 
                               
                                 21 
                                 ⁢ 
                                 zy 
                               
                             
                           
                         
                         
                           
                             Ú 
                             
                               21 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               zz 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     Note that the diagonal terms of    21  are fully gain compensated while the cross terms are dependent on certain ratios of the transmitter gains on transmitter T 1 . Each of the tensor terms is also dependent on the rotation at the location of the T 1  transmitter since U′ 21 =R T1   t U 21 R T1  (see Equation 15). Taking the generalization of the standard case further, the phase shift and attenuation of    21  may further be computed. To find the natural logarithm of    21  the matrix is first diagonalized, for example, as follows:
 
 ′ 21   =P   −1     21   P   (18)
 
     where P is a matrix of eigenvectors of    21  (each column of P is an eigenvector of M 21 ) and  ′ 21  is a diagonal matrix whose diagonal elements are eigenvalues of    21 . Replacing each diagonal element of  ′ 21  by its natural log to obtain ln( ′ 21 ) yields:
 
ln(   21 )= P     ′   21   P   −1   (19)
 
     The phase shift and attenuation may then be expressed as follows: 
     
       
         
           
             
               
                 
                   
                     PS 
                     21 
                   
                   = 
                   
                     
                       180 
                       π 
                     
                     ⁢ 
                     
                       Im 
                       ⁡ 
                       
                         [ 
                         
                           ln 
                           ⁡ 
                           
                             ( 
                             
                               21 
                             
                             ) 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   
                     AD 
                     21 
                   
                   = 
                   
                     
                       20 
                       
                         log 
                         ⁡ 
                         
                           ( 
                           10 
                           ) 
                         
                       
                     
                     ⁢ 
                     
                       Re 
                       ⁡ 
                       
                         [ 
                         
                           ln 
                           ⁡ 
                           
                             ( 
                             
                               21 
                             
                             ) 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     where the phase shift PS 21  is given in degrees, the attenuation AD 21  is given in decibels, Im[ln(   21 )] represents the imaginary portion of ln(   21 ), and Re[ln(   21 )] represents the real portion of ln(   21 ). 
     It will be understood that the phase shift and attenuation tensors given in Equations 20 and 21 have a similar form to the tensor quantity    21  given in Equation 17 in that the diagonal elements are gain compensated and that the off diagonal elements (the cross terms) are equal to the gain compensated cross term multiplied by a gain ratio. The remaining gain error on each of the cross terms tends to be small (fractional) since each gain error is a ratio of the two transmitter gains. As a result, when employing the presently disclosed techniques, gain calibrations do not have to be as stringent as when compared to the prior art. 
     BHA Bending and Rotation 
       FIG. 4  depicts a flow chart of one example method embodiment  100  for obtaining a full tensor gain compensated propagation measurement. An electromagnetic logging tool is rotated in a subterranean borehole at  102 . The logging tool includes at least first and second axially spaced transmitters and at least first and second axially spaced receivers, each of the transmitters and each of the receivers including a set of three collocated, linearly independent antenna moments. At least one of the transmitters and/or receivers is non-triaxial. A plurality of full tensor voltage measurements is acquired at  104  while rotating at  102 . The full tensor voltage measurements are obtained from a corresponding plurality of pairs of the transmitters and receivers. A processor, such as a downhole processor, processes the full tensor voltage measurements to obtain a partially gain compensated full tensor quantity at  106 . As shown in the examples above and further below, a partially gain compensated full tensor (e.g., nine terms) may refer to one in which some of the tensor terms are fully gain compensated (e.g., diagonal terms in Equation 17), and some are not fully gain compensated (e.g., the cross terms of Equation 17 are dependent on certain ratios of the transmitter gains). 
     In logging while drilling operations, measurements are made while the logging tool rotates in the borehole. Such rotation may be included in the foregoing model. Consider a first transmitter receiver pair. Rotation of the drill string causes the logging tool to rotate about its z-axis such that the transmitter and receiver moments each rotate through a common angle θ about their local z-axis. Moreover, during directional drilling operations, the drill string typically bends to accommodate the changing borehole direction. Following Equation 10, the transmitter and receiver moments may be expressed as follows taking into account drill string rotation, bending, and relative rotation of the transmitter with respect to the receiver.
 
 {acute over (m)}   R   _   rot   =R   R   _   BHA   _   rot   {acute over (m)}   R   =R   R   _   BHA   _   rot   R   Rbend   R   zα   m   R  
 
 {acute over (m)}   T   _   rot   =R   T   _   BHA   _   rot   {acute over (m)}   T   =R   T   _   BHA   _   rot   R   Tbend   R   zγ   m   T   (22)
 
     where {acute over (m)} R   _   rot  and é T   _   rot  represent the receiver and transmitter moments after drill string rotation and R R   _   BHA   _   rot  and R T   _   BHA   _   rot  represent the rotation matrices that rotate the receiver and transmitter moments about their respective tool axes (which are rotated with respect to one another owing to relative axial rotation BHA bending). Again, consider the case in which the moments of the a, b, and c antenna are aligned with the x, y, and z axes:
 
 {circumflex over ({acute over (m)})}   R   _   rot   _   a   =R   R   _   BHA   _   rot   {circumflex over ({acute over (m)})}   Ra   ={circumflex over ({acute over (m)})}   Ra  cos(θ)+ {circumflex over ({acute over (m)})}   Rb  sin(θ)  (23)
 
 {circumflex over ({acute over (m)})}   R   _   rot   _   b   =R   R   _   BHA   _   rot   {circumflex over ({acute over (m)})}   Rb   =−{circumflex over ({acute over (m)})}   Ra  sin(θ)+ {circumflex over ({acute over (m)})}   Rb  cos(θ)
 
{circumflex over ({acute over ( m )})} R   _   rot   _   c   =R   R   _   BHA   _   rot {circumflex over ({acute over ( m )})} Rc ={circumflex over ({acute over ( m )})} Rc  
 
 {circumflex over ({acute over (m)})}   T   _   rot   _   a   =R   T   _   BHA   _   rot   {circumflex over ({acute over (m)})}   Ta   ={circumflex over ({acute over (m)})}   Ta  cos(θ)+ {circumflex over ({acute over (m)})}   Tb  sin(θ)
 
 {circumflex over ({acute over (m)})}   T   _   rot   _   b   =R   T   _   BHA   _   rot   {circumflex over ({acute over (m)})}   Tb   =−{circumflex over ({acute over (m)})}   Ta  sin(θ)+ {circumflex over ({acute over (m)})}   Tb  cos(θ)
 
{circumflex over ({acute over ( m )})} T   _   rot   _   c   =R   T   _   BHA   _   rot {circumflex over ({acute over ( m )})} Tc ={circumflex over ({acute over ( m )})} Tc   (24)
 
     The voltage tensor V TR (θ) measured at any rotation angle θ may be expressed, for example, as follows:
 
 V   TR (θ)= G   T ( {acute over (m)}   T     rot   (θ)) t   Z{acute over (m)}   R     rot   (θ) G   R   (25)
 
     Following Equation 25, the voltage tensor V TR (0), at angle θ=0, may be expressed as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             V 
                             TR 
                           
                           ⁡ 
                           
                             ( 
                             0 
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             G 
                             T 
                           
                           ⁢ 
                           
                             T 
                             t 
                           
                           ⁢ 
                           Z 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             R 
                           
                           ⁢ 
                           
                             G 
                             R 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       g 
                                       Ta 
                                     
                                   
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     
                                       g 
                                       Tb 
                                     
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                   
                                     
                                       
                                         g 
                                         Tc 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     
                                       Tax 
                                     
                                   
                                   
                                     
                                       Tbx 
                                     
                                   
                                   
                                     
                                       Tcx 
                                     
                                   
                                 
                                 
                                   
                                     
                                       Tay 
                                     
                                   
                                   
                                     
                                       Tby 
                                     
                                   
                                   
                                     
                                       Tcy 
                                     
                                   
                                 
                                 
                                   
                                     
                                       Taz 
                                     
                                   
                                   
                                     
                                       Tbz 
                                     
                                   
                                   
                                     
                                       Tcz 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           t 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             Z 
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     
                                       Rax 
                                     
                                   
                                   
                                     
                                       Rbx 
                                     
                                   
                                   
                                     
                                       Rcx 
                                     
                                   
                                 
                                 
                                   
                                     
                                       Ray 
                                     
                                   
                                   
                                     
                                       Rby 
                                     
                                   
                                   
                                     
                                       Rcy 
                                     
                                   
                                 
                                 
                                   
                                     
                                       Raz 
                                     
                                   
                                   
                                     
                                       Rbz 
                                     
                                   
                                   
                                     
                                       Rcz 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   
                                     g 
                                     Ra 
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     g 
                                     Rb 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     
                                       g 
                                       Rc 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     g 
                                     Ta 
                                   
                                   ⁢ 
                                   
                                     g 
                                     Ta 
                                   
                                   ⁢ 
                                   
                                     Ta 
                                     t 
                                   
                                   ⁢ 
                                   Z 
                                   ⁢ 
                                   
                                     
                                       m 
                                       ^ 
                                     
                                     Ra 
                                   
                                 
                               
                               
                                 
                                   
                                     g 
                                     Ta 
                                   
                                   ⁢ 
                                   
                                     g 
                                     Rb 
                                   
                                   ⁢ 
                                   
                                     Ta 
                                     t 
                                   
                                   ⁢ 
                                   Z 
                                   ⁢ 
                                   
                                     
                                       m 
                                       ^ 
                                     
                                     Rb 
                                   
                                 
                               
                               
                                 
                                   
                                     g 
                                     Ta 
                                   
                                   ⁢ 
                                   
                                     g 
                                     Rc 
                                   
                                   ⁢ 
                                   
                                     Ta 
                                     t 
                                   
                                   ⁢ 
                                   Z 
                                   ⁢ 
                                   
                                     
                                       m 
                                       ^ 
                                     
                                     Rc 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     g 
                                     Tb 
                                   
                                   ⁢ 
                                   
                                     g 
                                     Ta 
                                   
                                   ⁢ 
                                   
                                     Tb 
                                     t 
                                   
                                   ⁢ 
                                   Z 
                                   ⁢ 
                                   
                                     
                                       m 
                                       ^ 
                                     
                                     Ra 
                                   
                                 
                               
                               
                                 
                                   
                                     g 
                                     Tb 
                                   
                                   ⁢ 
                                   
                                     g 
                                     Rb 
                                   
                                   ⁢ 
                                   
                                     Tb 
                                     t 
                                   
                                   ⁢ 
                                   Z 
                                   ⁢ 
                                   
                                     
                                       m 
                                       ^ 
                                     
                                     Rb 
                                   
                                 
                               
                               
                                 
                                   
                                     g 
                                     Tb 
                                   
                                   ⁢ 
                                   
                                     g 
                                     Rc 
                                   
                                   ⁢ 
                                   
                                     Tb 
                                     t 
                                   
                                   ⁢ 
                                   Z 
                                   ⁢ 
                                   
                                     
                                       m 
                                       ^ 
                                     
                                     Rc 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     g 
                                     Tc 
                                   
                                   ⁢ 
                                   
                                     g 
                                     Ta 
                                   
                                   ⁢ 
                                   
                                     Tc 
                                     t 
                                   
                                   ⁢ 
                                   Z 
                                   ⁢ 
                                   
                                     
                                       m 
                                       ^ 
                                     
                                     Ra 
                                   
                                 
                               
                               
                                 
                                   
                                     g 
                                     Tc 
                                   
                                   ⁢ 
                                   
                                     g 
                                     Rb 
                                   
                                   ⁢ 
                                   
                                     Tc 
                                     t 
                                   
                                   ⁢ 
                                   Z 
                                   ⁢ 
                                   
                                     
                                       m 
                                       ^ 
                                     
                                     Rb 
                                   
                                 
                               
                               
                                 
                                   
                                     g 
                                     Tc 
                                   
                                   ⁢ 
                                   
                                     g 
                                     Rc 
                                   
                                   ⁢ 
                                   
                                     Tc 
                                     t 
                                   
                                   ⁢ 
                                   Z 
                                   ⁢ 
                                   
                                     
                                       m 
                                       ^ 
                                     
                                     Rc 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     Rotating the BHA one quarter turn to 
             θ   =     π   2           
rotates the a antenna to a direction at which the b antenna was pointing at θ=0 and rotates the b antenna to a direction opposite that the direction at which the a antenna was pointing at θ=0. The direction of the c antenna remains unchanged (as it is coincident with the rotation axis of the BHA). Following Equation 25, the voltage tensor
 
               V   TR     ⁡     (     π   2     )           
may be expressed as follows:
 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       TR 
                     
                     ⁡ 
                     
                       ( 
                       
                         π 
                         2 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         G 
                         T 
                       
                       ⁢ 
                       
                         T 
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             π 
                             2 
                           
                           ) 
                         
                         t 
                       
                       ⁢ 
                       Z 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         R 
                       
                       ⁢ 
                       
                         ( 
                         
                           π 
                           2 
                         
                         ) 
                       
                       ⁢ 
                       
                         G 
                         R 
                       
                     
                     = 
                     
                       
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       g 
                                       Ta 
                                     
                                   
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     
                                       g 
                                       Tb 
                                     
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                   
                                     
                                       g 
                                       Tc 
                                     
                                   
                                 
                               
                               ] 
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     
                                       Tbx 
                                     
                                   
                                   
                                     
                                       - 
                                       
                                         Tax 
                                       
                                     
                                   
                                   
                                     
                                       Tcx 
                                     
                                   
                                 
                                 
                                   
                                     
                                       Tby 
                                     
                                   
                                   
                                     
                                       - 
                                       
                                         Tay 
                                       
                                     
                                   
                                   
                                     
                                       Tcy 
                                     
                                   
                                 
                                 
                                   
                                     
                                       Tbz 
                                     
                                   
                                   
                                     
                                       - 
                                       
                                         Taz 
                                       
                                     
                                   
                                   
                                     
                                       Tcz 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           t 
                         
                         ⁢ 
                         
                           
                             Z 
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     
                                       Rbx 
                                     
                                   
                                   
                                     
                                       - 
                                       
                                         Rax 
                                       
                                     
                                   
                                   
                                     
                                       Rcx 
                                     
                                   
                                 
                                 
                                   
                                     
                                       Rby 
                                     
                                   
                                   
                                     
                                       - 
                                       
                                         Ray 
                                       
                                     
                                   
                                   
                                     
                                       Rcy 
                                     
                                   
                                 
                                 
                                   
                                     
                                       Rbz 
                                     
                                   
                                   
                                     
                                       - 
                                       
                                         Raz 
                                       
                                     
                                   
                                   
                                     
                                       Rcz 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   
                                     g 
                                     Ra 
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     g 
                                     Rb 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     g 
                                     Rc 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                       = 
                       
                           
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     g 
                                     Ta 
                                   
                                   ⁢ 
                                   
                                     g 
                                     Ta 
                                   
                                   ⁢ 
                                   
                                     Tb 
                                     t 
                                   
                                   ⁢ 
                                   Z 
                                   ⁢ 
                                   
                                     
                                       m 
                                       ^ 
                                     
                                     Rb 
                                   
                                 
                               
                               
                                 
                                   
                                     - 
                                     
                                       g 
                                       Ta 
                                     
                                   
                                   ⁢ 
                                   
                                     g 
                                     Rb 
                                   
                                   ⁢ 
                                   
                                     Tb 
                                     t 
                                   
                                   ⁢ 
                                   Z 
                                   ⁢ 
                                   
                                     
                                       m 
                                       ^ 
                                     
                                     Ra 
                                   
                                 
                               
                               
                                 
                                   
                                     g 
                                     Ta 
                                   
                                   ⁢ 
                                   
                                     g 
                                     Rc 
                                   
                                   ⁢ 
                                   
                                     Ta 
                                     t 
                                   
                                   ⁢ 
                                   Z 
                                   ⁢ 
                                   
                                     
                                       m 
                                       ^ 
                                     
                                     Rc 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     - 
                                     
                                       g 
                                       Tb 
                                     
                                   
                                   ⁢ 
                                   
                                     g 
                                     Ta 
                                   
                                   ⁢ 
                                   
                                     Ta 
                                     t 
                                   
                                   ⁢ 
                                   Z 
                                   ⁢ 
                                   
                                     
                                       m 
                                       ^ 
                                     
                                     Rb 
                                   
                                 
                               
                               
                                 
                                   
                                     g 
                                     Tb 
                                   
                                   ⁢ 
                                   
                                     g 
                                     Rb 
                                   
                                   ⁢ 
                                   
                                     Ta 
                                     t 
                                   
                                   ⁢ 
                                   Z 
                                   ⁢ 
                                   
                                     
                                       m 
                                       ^ 
                                     
                                     Ra 
                                   
                                 
                               
                               
                                 
                                   
                                     - 
                                     
                                       g 
                                       Tb 
                                     
                                   
                                   ⁢ 
                                   
                                     g 
                                     Rc 
                                   
                                   ⁢ 
                                   
                                     Tb 
                                     t 
                                   
                                   ⁢ 
                                   Z 
                                   ⁢ 
                                   
                                     
                                       m 
                                       ^ 
                                     
                                     Rc 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     g 
                                     Tc 
                                   
                                   ⁢ 
                                   
                                     g 
                                     Ta 
                                   
                                   ⁢ 
                                   
                                     Tc 
                                     t 
                                   
                                   ⁢ 
                                   Z 
                                   ⁢ 
                                   
                                     
                                       m 
                                       ^ 
                                     
                                     Rb 
                                   
                                 
                               
                               
                                 
                                   
                                     - 
                                     
                                       g 
                                       Tc 
                                     
                                   
                                   ⁢ 
                                   
                                     g 
                                     Rb 
                                   
                                   ⁢ 
                                   
                                     Tc 
                                     t 
                                   
                                   ⁢ 
                                   Z 
                                   ⁢ 
                                   
                                     
                                       m 
                                       ^ 
                                     
                                     Ra 
                                   
                                 
                               
                               
                                 
                                   
                                     g 
                                     Tc 
                                   
                                   ⁢ 
                                   
                                     g 
                                     Rc 
                                   
                                   ⁢ 
                                   
                                     Tc 
                                     t 
                                   
                                   ⁢ 
                                   Z 
                                   ⁢ 
                                   
                                     
                                       m 
                                       ^ 
                                     
                                     Rc 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     Taking the compensated combinations as in Equations 15 and 16 gives similar results: 
     
       
         
           
             
               
                 
                   
                     
                       12 
                     
                     ⁢ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             Ú 
                             
                               12 
                               ⁢ 
                               xx 
                             
                           
                         
                         
                           
                             
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   a 
                                 
                               
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   b 
                                 
                               
                             
                             ⁢ 
                             
                               Ú 
                               
                                 12 
                                 ⁢ 
                                 xy 
                               
                             
                           
                         
                         
                           
                             
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   a 
                                 
                               
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   c 
                                 
                               
                             
                             ⁢ 
                             
                               Ú 
                               
                                 12 
                                 ⁢ 
                                 xz 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   b 
                                 
                               
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   a 
                                 
                               
                             
                             ⁢ 
                             
                               Ú 
                               
                                 12 
                                 ⁢ 
                                 yx 
                               
                             
                           
                         
                         
                           
                             Ú 
                             
                               12 
                               ⁢ 
                               yy 
                             
                           
                         
                         
                           
                             
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   b 
                                 
                               
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   c 
                                 
                               
                             
                             ⁢ 
                             
                               Ú 
                               
                                 12 
                                 ⁢ 
                                 yz 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   c 
                                 
                               
                               
                                 g 
                                 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     a 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                 
                               
                             
                             ⁢ 
                             
                               Ú 
                               
                                 12 
                                 ⁢ 
                                 zx 
                               
                             
                           
                         
                         
                           
                             
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   c 
                                 
                               
                               
                                 g 
                                 
                                   T 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                   ⁢ 
                                   b 
                                 
                               
                             
                             ⁢ 
                             
                               Ú 
                               
                                 12 
                                 ⁢ 
                                 zy 
                               
                             
                           
                         
                         
                           
                             Ú 
                             
                               12 
                               ⁢ 
                               zz 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
     where    12 (0) represents the quantity    12  at θ=0. A similar combination may be obtained at 
                   θ   =     π   2                             
and then mathematically rotating the results by −90 degrees.
 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             0 
                           
                           
                             
                               - 
                               1 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     
                       12 
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           π 
                           2 
                         
                         ) 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                           
                           
                             
                               
                                 - 
                                 1 
                               
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                         
                           
                             
                               Ú 
                               
                                 12 
                                 ⁢ 
                                 xx 
                               
                             
                           
                           
                             
                               
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     b 
                                   
                                 
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     a 
                                   
                                 
                               
                               ⁢ 
                               
                                 Ú 
                                 
                                   12 
                                   ⁢ 
                                   xy 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     b 
                                   
                                 
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     c 
                                   
                                 
                               
                               ⁢ 
                               
                                 Ú 
                                 
                                   12 
                                   ⁢ 
                                   xz 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     a 
                                   
                                 
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     b 
                                   
                                 
                               
                               ⁢ 
                               
                                 Ú 
                                 
                                   12 
                                   ⁢ 
                                   yx 
                                 
                               
                             
                           
                           
                             
                               Ú 
                               
                                 12 
                                 ⁢ 
                                 yy 
                               
                             
                           
                           
                             
                               
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     a 
                                   
                                 
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     c 
                                   
                                 
                               
                               ⁢ 
                               
                                 Ú 
                                 
                                   12 
                                   ⁢ 
                                   yz 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     c 
                                   
                                 
                                 
                                   g 
                                   
                                     
                                       T 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                       ⁢ 
                                       b 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                   
                                 
                               
                               ⁢ 
                               
                                 Ú 
                                 
                                   12 
                                   ⁢ 
                                   zx 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     c 
                                   
                                 
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     a 
                                   
                                 
                               
                               ⁢ 
                               
                                 Ú 
                                 
                                   12 
                                   ⁢ 
                                   zy 
                                 
                               
                             
                           
                           
                             
                               Ú 
                               
                                 12 
                                 ⁢ 
                                 zz 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
     Note that Equation 29 is similar to Equation 28 except that a g T1a  and g T1b  have traded places. In principle, multiplying the xy and yx terms in Equations 28 and 29 may be used to eliminate the gain in these terms. However, such an approach does not eliminate the gain in the cross terms involving the c antenna (i.e., the xz, zx, yz, and zy terms). 
     Compensation Using BHA Rotation and Tilted Antennas 
       FIG. 5  depicts a flow chart of another example method embodiment  120  for obtaining a full tensor gain compensated propagation measurement. An electromagnetic logging tool is rotated in a subterranean borehole at  122 . The logging tool includes at least first and second axially spaced transmitters and at least first and second axially spaced receivers. Each of the receivers includes a set of collocated linearly independent antennas. Each of the transmitters includes first and second collocated antennas, at least the first transmitter antenna having a tilted moment with respect to a longitudinal axis of the tool. A plurality voltage measurements is acquired at  124  while rotating the tool at  122 . The plurality of voltage measurements is obtained using a corresponding plurality of pairs of the transmitters and receivers. The voltage measurements are fit to a harmonic expression at  126  to obtain harmonic coefficients. The harmonic coefficients are processed at  128  to construct a full tensor voltage measurement. The full tensor voltage measurement is further processed at  130  to obtain a fully gain compensated tensor quantity. 
       FIG. 6  depicts another embodiment of a tool configuration including a tilted transmitter arrangement. The depicted tool configuration includes first and second subs  72  and  74  as described above with respect to  FIG. 3 . First and second receivers are deployed on the corresponding subs. Each receiver includes a set of collocated linearly independent antennas (referred to herein as a receiver triad). First and second transmitters (T 1  and T 2 ) each of which includes an a antenna having a moment tilted with respect to the tool axis and a b antenna having a moment perpendicular to the tool axis are also deployed on the corresponding subs as depicted. The moment of the a antenna is tilted at an angle β with respect to the tool axis. 
     While  FIG. 6  depicts first and second subs  72  and  74 , each including a corresponding transmitter and a receiver, it will be understood that the disclosed embodiments are not limited to any particular collar configuration. For example, each of the transmitters and receivers may be deployed on distinct subs. In other embodiments, both transmitters and receivers may be deployed on a single sub. 
     It will be further understood, according to the principle of reciprocity, that the transmitting and receiving antennas may operate as either a transmitter or a receiver when coupled with the appropriate transmitter and/or receiver electronics such that the transmitters and receivers may be swapped without affecting the gain compensation methodology that follows. Therefore, in the embodiment depicted on  FIG. 6 , the transmitters T 1  and T 2  may be swapped with the receivers R 1  and R 2  such that each transmitter includes a set of collocated linearly independent antennas and each receiver includes an a antenna having a moment tilted with respect to the tool axis. 
     The moment {circumflex over ({acute over (m)})} Ta  of the a transmitter may be decomposed into moments parallel and perpendicular to the tool axis direction û T , for example, as follows:
 
{circumflex over ({acute over ( m )})} Ta =({circumflex over ({acute over ( m )})} Ta −({circumflex over ({acute over ( m )})} Ta   ·û   T ))+({circumflex over ({acute over ( m )})} Ta   ·û   T ) {circumflex over ( u )} T =sin(β){circumflex over ({acute over ( m )})} Ta   _   perp +cos(β) û   T   (30)
 
     where {circumflex over ({acute over (m)})} Ta   _   perp  represents the component of the a transmitter antenna that is orthogonal to the tool axis û T . 
     As described above, rotation of the drill string causes the logging tool to rotate about its z-axis such that the transmitter and receiver moments each rotate through a common angle θ about their local z-axis. The transmitter and receiver moments may be expressed as functions of the angle θ, for example, as follows:
 
{circumflex over ({acute over ( m )})} T   _   rot   _   a   =R   T   _   BHA   _   rot {circumflex over ({acute over ( m )})} Ta =sin(β)({circumflex over ({acute over ( m )})} T   _   perp  cos(θ)+{circumflex over ({acute over ( m )})} Tb  sin(θ))+cos(β) û   T  
 
 {circumflex over ({acute over (m)})}   T   _   rot   _   b   =R   T   _   BHA   _   rot   {circumflex over ({acute over (m)})}   Tb =−sin(β) {circumflex over ({acute over (m)})}   Ta   _   perp  sin(θ)+ {circumflex over ({acute over (m)})}   Tb  cos(θ)  (31)
 
 {circumflex over ({acute over (m)})}   R   _   rot   _   a   =R   R   _   BHA   _   rot   {circumflex over ({acute over (m)})}   Ra   ={circumflex over ({acute over (m)})}   Ra  cos(θ)+ {circumflex over ({acute over (m)})}   Rb  sin(θ)
 
 {circumflex over ({acute over (m)})}   R   _   rot   _   b   =R   R   _   BHA   _   rot   {circumflex over ({acute over (m)})}   Rb   =−{circumflex over ({acute over (m)})}   Ra  sin(θ)+ {circumflex over ({acute over (m)})}   Rb  cos(θ)
 
{circumflex over ({acute over ( m )})} R   _   rot   _   c   =R   R   _   BHA   _   rot {circumflex over ({acute over ( m )})} Rc ={circumflex over ({acute over ( m )})} Rc   (32)
 
     where {circumflex over ({acute over (m)})} T   _   rot   _   a  and {circumflex over ({acute over (m)})} T   _   rot   _   b  represent the a and b transmitter moments with rotation and {circumflex over ({acute over (m)})} R   _   rot   _   a , {circumflex over ({acute over (m)})} R   _   rot   _   b , and {circumflex over ({acute over (m)})} R   _   rot   _   c  represent the a, b, and c receiver moments. 
     The voltage V TR (θ) measured for any particular transmitter receiver pair may be expressed as given above in Equation 25 where the transmitter gain matrix G T  is given as follows: 
     
       
         
           
             
               
                 
                   
                     G 
                     T 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               g 
                               Ta 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 β 
                                 ) 
                               
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             g 
                             Tb 
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             
                               g 
                               Ta 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 β 
                                 ) 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
     The voltage V TR (θ) may further be expressed in harmonic form, for example, as follows:
 
 V   TR (θ)= V   TR   _   DC   +V   TR   _   FHC  cos(θ)+ V   TR   _   FHS  sin(θ)+ . . .  V   TR   _   SHC  cos(2θ)+ V   TR   _   SHS  sin(2θ)  (34)
 
     wherein V TR   _   DC  represents the DC (or average), V TR   _   FHC  and V TR   _   FHS  represent the first harmonic cosine and sine, and V TR   _   SHC  and V TR   _   SHS  represent the second harmonic cosine and sine. For the tilted a transmitter antenna, the DC, second harmonic cosine, and second harmonic sine terms are dependent on {circumflex over ({acute over (m)})} Ta   _   perp  whereas the first harmonic cosine and first harmonic sine terms are dependent on û T . The harmonic terms may be obtained by fitting the measured voltages during rotation (as a function of tool face angle) to Equation 34. The contributions of the moment of transmitter antenna a that are parallel and perpendicular to the tool may be separated from one another using the harmonic terms. For example,
 
 V   TaRa   _   FHC   =g   Ta   g   Ra  cos(β) û   T   t   Z{circumflex over ({acute over (m)})}   Ra  
 
 V   TaRa   _   FHS   =g   Ta   g   Ra  cos(β) û   T   t   Z {circumflex over ({acute over ( m )})} Rb   (35)
 
     A full three dimensional voltage tensor may then be obtained by fitting each of the rotation dependent voltage measurements V TaRa (θ), V TaRb (θ), V TaRc (θ), V TbRa (θ), V TbRb (θ), and V TbRc (θ) to Equation 34 and solving for the corresponding harmonics. The harmonics may then be used to obtain the various voltage tensor terms. Following the procedure described above with respect to Equations 26 and 27, the voltage tensor V TR (0), at angle θ=0, may be constructed from the following combination of measured voltage harmonics: 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       TR 
                     
                     ⁡ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       V 
                                       
                                         TaRa 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         _ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         D 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         C 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     V 
                                     
                                       TaRa 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       _ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       SHC 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       V 
                                       
                                         TaRb 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         _ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         D 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         C 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     V 
                                     
                                       TaRb 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       _ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       SHC 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               V 
                               
                                 TaRc 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 FHC 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     
                                       V 
                                       
                                         TbRa 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         _ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         D 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         C 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     V 
                                     
                                       TaRa 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       _ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       SHC 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       V 
                                       
                                         TbRb 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         _ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         D 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         C 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     V 
                                     
                                       TbRb 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       _ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       SHC 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               V 
                               
                                 TcRc 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 FHC 
                               
                             
                           
                         
                         
                           
                             
                               V 
                               
                                 TaRa 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 FHC 
                               
                             
                           
                           
                             
                               V 
                               
                                 TaRb 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 FHC 
                               
                             
                           
                           
                             
                               V 
                               
                                 TaRc 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 D 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 C 
                               
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                         
                       
                         [ 
                         
                           
                             
                               
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     β 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ta 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ra 
                                 
                                 ⁢ 
                                 
                                   
                                     Ta 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     _ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     perp 
                                   
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Ra 
                                 
                               
                             
                             
                               
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     β 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ta 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Rb 
                                 
                                 ⁢ 
                                 
                                   
                                     Ta 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     _ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     perp 
                                   
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Rb 
                                 
                               
                             
                             
                               
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     β 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ta 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Rc 
                                 
                                 ⁢ 
                                 
                                   
                                     Ta 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     _ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     perp 
                                   
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Rc 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   g 
                                   Tb 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ra 
                                 
                                 ⁢ 
                                 
                                   
                                     Tb 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     _ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     perp 
                                   
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Ra 
                                 
                               
                             
                             
                               
                                 
                                   g 
                                   Tb 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Rb 
                                 
                                 ⁢ 
                                 
                                   
                                     Tb 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     _ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     perp 
                                   
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Rb 
                                 
                               
                             
                             
                               
                                 
                                   g 
                                   Tb 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Rc 
                                 
                                 ⁢ 
                                 
                                   
                                     Tb 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     _ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     perp 
                                   
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Rc 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     β 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ta 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ra 
                                 
                                 ⁢ 
                                 
                                   
                                     u 
                                     ^ 
                                   
                                   T 
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Ra 
                                 
                               
                             
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     β 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ta 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Rb 
                                 
                                 ⁢ 
                                 
                                   
                                     u 
                                     ^ 
                                   
                                   T 
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Rb 
                                 
                               
                             
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     β 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ta 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Rc 
                                 
                                 ⁢ 
                                 
                                   
                                     u 
                                     ^ 
                                   
                                   T 
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Rc 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
     Inspection of the last expression in Equation 37 reveals that it is equal to Equation 27 with the equivalent transmitter gains given in Equation 33. Similarly, the voltage tensor 
                 V   TR     ⁡     (     π   2     )       ,         
at angle
 
               θ   =     π   2       ,         
may be expressed as follows:
 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       TR 
                     
                     ⁡ 
                     
                       ( 
                       
                         π 
                         2 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       V 
                                       
                                         TaRa 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         _ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         D 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         C 
                                       
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     V 
                                     
                                       TaRa 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       _ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       SHC 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       V 
                                       
                                         TaRb 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         _ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         D 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         C 
                                       
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     V 
                                     
                                       TaRb 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       _ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       SHC 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               V 
                               
                                 TaRc 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 FHS 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     
                                       V 
                                       
                                         TbRa 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         _ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         D 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         C 
                                       
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     V 
                                     
                                       TaRa 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       _ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       SHC 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       V 
                                       
                                         TbRb 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         _ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         D 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         C 
                                       
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     V 
                                     
                                       TbRb 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       _ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       SHC 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               V 
                               
                                 TcRc 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 FHS 
                               
                             
                           
                         
                         
                           
                             
                               V 
                               
                                 TaRa 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 FHS 
                               
                             
                           
                           
                             
                               V 
                               
                                 TaRb 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 FHS 
                               
                             
                           
                           
                             
                               V 
                               
                                 TaRc 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 D 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 C 
                               
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                         
                       
                         [ 
                         
                           
                             
                               
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     β 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ta 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ta 
                                 
                                 ⁢ 
                                 
                                   Tb 
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Rb 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     sin 
                                     ⁡ 
                                     
                                       ( 
                                       β 
                                       ) 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ta 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Rb 
                                 
                                 ⁢ 
                                 
                                   Tb 
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Ra 
                                 
                               
                             
                             
                               
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     β 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ta 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Rc 
                                 
                                 ⁢ 
                                 
                                   Tb 
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Rc 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   - 
                                   
                                     g 
                                     Tb 
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ta 
                                 
                                 ⁢ 
                                 
                                   Ta 
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Rb 
                                 
                               
                             
                             
                               
                                 
                                   g 
                                   Tb 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Rb 
                                 
                                 ⁢ 
                                 
                                   Ta 
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Ra 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     g 
                                     Tb 
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   Rc 
                                 
                                 ⁢ 
                                 
                                   Ta 
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Rc 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     β 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   Tc 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ta 
                                 
                                 ⁢ 
                                 
                                   Tc 
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Rb 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     cos 
                                     ⁡ 
                                     
                                       ( 
                                       β 
                                       ) 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   Tc 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Rb 
                                 
                                 ⁢ 
                                 
                                   Tc 
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Ra 
                                 
                               
                             
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     β 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   Tc 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Rc 
                                 
                                 ⁢ 
                                 
                                   Tc 
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Rc 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
     The quantity    12 (0) may be computed, for example, as described above with respect to Equation 28. 
     
       
         
           
             
               
                 
                   
                     
                       12 
                     
                     ⁢ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                   = 
                   
                       
                     
                       [ 
                       
                         
                           
                             
                               Ú 
                               
                                 12 
                                 ⁢ 
                                 xx 
                               
                             
                           
                           
                             
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     β 
                                     
                                       T 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     a 
                                   
                                 
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     b 
                                   
                                 
                               
                               ⁢ 
                               
                                 Ú 
                                 
                                   12 
                                   ⁢ 
                                   xy 
                                 
                               
                             
                           
                           
                             
                               
                                 tan 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     β 
                                     
                                       T 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 Ú 
                                 
                                   12 
                                   ⁢ 
                                   xz 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 1 
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       β 
                                       
                                         T 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         1 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     b 
                                   
                                 
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     a 
                                   
                                 
                               
                               ⁢ 
                               
                                 Ú 
                                 
                                   12 
                                   ⁢ 
                                   yx 
                                 
                               
                             
                           
                           
                             
                               Ú 
                               
                                 12 
                                 ⁢ 
                                 yy 
                               
                             
                           
                           
                             
                               
                                 1 
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       β 
                                       
                                         T 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         1 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     b 
                                   
                                 
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     a 
                                   
                                 
                               
                               ⁢ 
                               
                                 Ú 
                                 
                                   12 
                                   ⁢ 
                                   yz 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 cot 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     β 
                                     
                                       T 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 Ú 
                                 
                                   12 
                                   ⁢ 
                                   zx 
                                 
                               
                             
                           
                           
                             
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     β 
                                     
                                       T 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     a 
                                   
                                 
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     b 
                                   
                                 
                               
                               ⁢ 
                               
                                 Ú 
                                 
                                   12 
                                   ⁢ 
                                   zy 
                                 
                               
                             
                           
                           
                             
                               Ú 
                               
                                 
                                   12 
                                   ⁢ 
                                   zz 
                                 
                                 ⁢ 
                                 
                                     
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
     The tensor quantity    12 (0) includes five (out of nine) fully compensated tensor terms with the y-axis cross terms including a ratio of the tilted transmitter gains. Likewise, following Equation 29, a similar combination may be obtained at 
             θ   =     π   2           
and then mathematically rotating the results by −90 degrees.
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           12 
                         
                         ⁢ 
                         
                           ( 
                           
                             π 
                             2 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           
                             [ 
                             
                               
                                 
                                   0 
                                 
                                 
                                   
                                     - 
                                     1 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             12 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 π 
                                 2 
                               
                               ) 
                             
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     0 
                                   
                                   
                                     1 
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     
                                       - 
                                       1 
                                     
                                   
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                   
                                     1 
                                   
                                 
                               
                               ] 
                             
                           
                         
                         = 
                       
                     
                       
                   
                   ⁢ 
                   
                       
                     
                       [ 
                       
                         
                           
                             
                               Ú 
                               
                                 12 
                                 ⁢ 
                                 xx 
                               
                             
                           
                           
                             
                               
                                 1 
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       β 
                                       
                                         T 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         1 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     b 
                                   
                                 
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     a 
                                   
                                 
                               
                               ⁢ 
                               
                                 Ú 
                                 
                                   12 
                                   ⁢ 
                                   xy 
                                 
                               
                             
                           
                           
                             
                               
                                 1 
                                 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       β 
                                       
                                         T 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         1 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     b 
                                   
                                 
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     a 
                                   
                                 
                               
                               ⁢ 
                               
                                 Ú 
                                 
                                   12 
                                   ⁢ 
                                   xz 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     β 
                                     
                                       T 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     a 
                                   
                                 
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     b 
                                   
                                 
                               
                               ⁢ 
                               
                                 Ú 
                                 
                                   12 
                                   ⁢ 
                                   yx 
                                 
                               
                             
                           
                           
                             
                               Ú 
                               
                                 12 
                                 ⁢ 
                                 yy 
                               
                             
                           
                           
                             
                               
                                 tan 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     β 
                                     
                                       T 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 Ú 
                                 
                                   12 
                                   ⁢ 
                                   yz 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     β 
                                     
                                       T 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     a 
                                   
                                 
                                 
                                   g 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                     ⁢ 
                                     b 
                                   
                                 
                               
                               ⁢ 
                               
                                 Ú 
                                 
                                   12 
                                   ⁢ 
                                   zx 
                                 
                               
                             
                           
                           
                             
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     β 
                                     
                                       T 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 Ú 
                                 
                                   12 
                                   ⁢ 
                                   zy 
                                 
                               
                             
                           
                           
                             
                               Ú 
                               
                                 
                                   12 
                                   ⁢ 
                                   zz 
                                 
                                 ⁢ 
                                 
                                     
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     
                         
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
     The tensor quantity 
               12     ⁢     (     π   2     )           
includes five (out of nine) fully compensated tensor terms with the x-axis cross terms including a ratio of the tilted transmitter gains. It will be appreciated that the computed quantities    12 (0) and
 
               12     ⁢     (     π   2     )           
together contain sufficient information to compute a fully gain compensated tensor quantity (assuming that the tilt angle β T1  is known). By fully gain compensated, it is meant that each of the nine tensor terms is compensated with respect to transmitter and receiver gains. The fully compensated quantity    12C  may be computed tensor term by tensor term from    12 (0) and
 
                 12     ⁢     (     π   2     )       ,         
for example, as follows:
 
     
       
         
           
             
               
                 
                   
                     
                       12 
                       ⁢ 
                       C 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               
                                 12 
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                                 xx 
                               
                               ⁢ 
                               
                                 12 
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     π 
                                     2 
                                   
                                   ) 
                                 
                                 xx 
                               
                             
                           
                         
                         
                           
                             
                               
                                 12 
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                                 xy 
                               
                               ⁢ 
                               
                                 12 
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     π 
                                     2 
                                   
                                   ) 
                                 
                                 xy 
                               
                             
                           
                         
                         
                           
                             
                               cot 
                               ⁡ 
                               
                                 ( 
                                 
                                   β 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               12 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 0 
                                 ) 
                               
                               xz 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 12 
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                                 yx 
                               
                               ⁢ 
                               
                                 12 
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     π 
                                     2 
                                   
                                   ) 
                                 
                                 yx 
                               
                             
                           
                         
                         
                           
                             
                               
                                 12 
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                                 yy 
                               
                               ⁢ 
                               
                                 12 
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     π 
                                     2 
                                   
                                   ) 
                                 
                                 yy 
                               
                             
                           
                         
                         
                           
                             
                               cot 
                               ⁡ 
                               
                                 ( 
                                 
                                   β 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               12 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   π 
                                   2 
                                 
                                 ) 
                               
                               yz 
                             
                           
                         
                       
                       
                         
                           
                             
                               tan 
                               ⁡ 
                               
                                 ( 
                                 
                                   β 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 12 
                                 xz 
                               
                             
                             ⁢ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                         
                           
                             
                               tan 
                               ⁡ 
                               
                                 ( 
                                 
                                   β 
                                   
                                     T 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               12 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   π 
                                   2 
                                 
                                 ) 
                               
                               zy 
                             
                           
                         
                         
                           
                             
                               
                                 12 
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                                 zz 
                               
                               ⁢ 
                               
                                 12 
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     π 
                                     2 
                                   
                                   ) 
                                 
                                 zz 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
           
         
       
     
     which reduces to the following fully gain compensated tensor quantity: 
     
       
         
           
             
               
                 
                   
                     
                       12 
                       ⁢ 
                       C 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             Ú 
                             
                               12 
                               ⁢ 
                               xx 
                             
                           
                         
                         
                           
                             Ú 
                             
                               12 
                               ⁢ 
                               xy 
                             
                           
                         
                         
                           
                             Ú 
                             
                               12 
                               ⁢ 
                               xz 
                             
                           
                         
                       
                       
                         
                           
                             Ú 
                             
                               12 
                               ⁢ 
                               yx 
                             
                           
                         
                         
                           
                             Ú 
                             
                               12 
                               ⁢ 
                               yy 
                             
                           
                         
                         
                           
                             Ú 
                             
                               12 
                               ⁢ 
                               yz 
                             
                           
                         
                       
                       
                         
                           
                             Ú 
                             
                               12 
                               ⁢ 
                               zx 
                             
                           
                         
                         
                           
                             Ú 
                             
                               12 
                               ⁢ 
                               zy 
                             
                           
                         
                         
                           
                             Ú 
                             
                               12 
                               ⁢ 
                               zz 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
           
         
       
     
     While the fully gain compensated tensor quantity computed in Equation 41 has been described with respect to the antenna configuration depicted on  FIG. 4 , it will be understood that the disclosed methodology is not so limited. The disclosed methodology is more general and applied to collocated transmitter receiver combinations having arbitrary orientations so long as at least one antenna is tilted and the matrix made up of the moments is invertible. As described, axial rotation may be used to ‘swap’ antenna positions and to separate out the projection of the tilted antenna moments to form the equivalent of a triaxial antenna coupling. The matrix combination of two such transmitter receiver pairs results in a tensor quantity that depends only on the orientation of one colocated antenna set. Moreover, the transmitter and antenna roles may also be swapped and the same result obtained. 
     Compensation Using Calibration Antennas 
     A collocated transmitter (or receiver) having at least one tilted moment may be difficult to fabricate. Moreover, firing of a large number of transmitter antennas can lengthen the time it takes to acquire a given set of data. One alternative to the tilted antenna arrangement described above is to make use of a nearby set of calibration antennas to match some of the antenna gains. 
       FIG. 7  depicts a flow chart of still another example method embodiment  140  for obtaining a full tensor gain compensated propagation measurement. An electromagnetic logging tool is rotated in a subterranean borehole at  142 . The logging tool includes first and second axially spaced tilted transmitters, first and second axially spaced calibration transmitters, and first and second axially spaced receivers. Each of the receivers includes a set of collocated linearly independent antennas. Each of the calibration transmitters includes first and second collocated calibration antennas having transverse moments with respect to one another and with respect to the longitudinal axis of the tool. Each of the tilted transmitters includes a tilted antenna not collocated with the corresponding calibration antennas and having a tilted moment with respect to a longitudinal axis of the tool. A plurality of voltage measurements is acquired at  144  while rotating the tool at  142 . The plurality of voltage measurements is obtained using a corresponding plurality of pairs of the tilted transmitter antennas and the linearly independent receiver antennas. The voltage measurements are fit to a harmonic expression at  146  to obtain harmonic coefficients. The harmonic coefficients are processed at  148  to construct a full tensor voltage measurement which is in turn further processed at  150  to obtain a fully gain compensated tensor quantity. 
       FIG. 8  depicts an embodiment of a downhole BHA configuration including first and second subs  82  and  84  (although the disclosed embodiments are by no means limited in this regard) that can be used in accordance with method  140  of  FIG. 7 . In the depicted embodiment, each sub includes a receiver R 1  and R 2 , a calibration transmitter CT 1  and CT 2 , and a “deep” tilted transmitter T 1  and T 2 . On each sub, the receiver, calibration transmitter, and the tilted transmitter are axially spaced apart from one another. The receivers R 1  and R 2  include a set of collocated linearly independent antennas. The deep tilted transmitters T 1  and T 2  have large enough moments so that they may transmit a signal having sufficient strength to be received on the other sub and include an antenna having a moment tilted with respect to the longitudinal axis of the corresponding sub. The calibration transmitters CT 1  and CT 2  include first and second collocated, transverse calibration antennas that may be used to match the gains of the nearby a and b transverse receivers. The calibration antennas may be configured so that they generate sufficient signal strength so as to be received by the nearby receiver antennas (i.e., the receiver located on the same sub, but not receiver antennas on other subs). The calibration transmitters therefore need not be deep transmitters. 
     It will be understood, according to the principle of reciprocity, that the transmitting and receiving antennas may operate as either a transmitter or a receiver when coupled with the appropriate transmitter and/or receiver electronics such that the transmitters and receivers may be swapped without affecting the gain compensation methodology that follows. Therefore, in the embodiment depicted on  FIG. 8 , the transmitters T 1  and T 2  may be swapped with the receivers R 1  and R 2  such that each transmitter includes a set of collocated linearly independent antennas and each receiver includes a tilted antenna. Such an embodiment would also include first and second calibration receivers, each including first and second collocated, transverse antennas 
     To describe an example calibration procedure, consider the antennas on one of the subs. The d and e transmitting antennas are transverse to the local axial direction, perpendicular to one another, and oriented at an angle ψ (that is not 0 or 90 degrees) with respect to the local a and b receivers. The ratio of the DC terms of the voltage measured by the a and b receivers upon firing the d transmitter (during rotation of the drill string as indicated above in Equation 34) may be given as follows: 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       
                         TdRa 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         D 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         C 
                       
                     
                     
                       V 
                       
                         TdRb 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         D 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         C 
                       
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           g 
                           Ra 
                         
                         
                           g 
                           Rb 
                         
                       
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             
                               Z 
                               xx 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               Z 
                               yy 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               Z 
                               xy 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               Z 
                               yx 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                       
                         ( 
                         
                           
                             
                               Z 
                               xx 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               Z 
                               yy 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               Z 
                               yx 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               Z 
                               xy 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
           
         
       
     
     Likewise the ratio of the DC terms may also be obtained upon firing the e transmitter as follows: 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       
                         TeRa 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         D 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         C 
                       
                     
                     
                       V 
                       
                         TeRb 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         D 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         C 
                       
                     
                   
                   = 
                   
                     
                       
                         g 
                         Ra 
                       
                       
                         g 
                         
                           Rb 
                           ⁢ 
                           
                               
                           
                         
                       
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             
                               Z 
                               xx 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               Z 
                               yy 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               Z 
                               yx 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               Z 
                               xy 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                       
                         ( 
                         
                           
                             
                               Z 
                               xx 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               Z 
                               yy 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               Z 
                               xy 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               Z 
                               yx 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
     A gain ratio of the a to b receivers may then be obtained by combining Equations 43 and 44, for example as follows: 
     
       
         
           
             
               
                 
                   
                     g 
                     ratio 
                   
                   = 
                   
                     
                       
                         g 
                         Ra 
                       
                       
                         g 
                         Rb 
                       
                     
                     = 
                     
                       
                         - 
                         
                           
                             
                               V 
                               
                                 TdRa 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 D 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 C 
                               
                             
                             ⁢ 
                             
                               V 
                               
                                 TeRa 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 D 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 C 
                               
                             
                           
                           
                             
                               V 
                               
                                 TdRb 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 D 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 C 
                               
                             
                             ⁢ 
                             
                               V 
                               
                                 TeRb 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 D 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 C 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
     Using the gain ratio given in Equation 45, a fully gain compensated deep resistivity measurement may be obtained, for example, via firing the tilted transmitter antenna on the first sub and receiving the transmitted electromagnetic waves using the receiver antennas on the second sub. The measured voltage tensor may then be given, for example, as follows: 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       TR 
                     
                     ⁡ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       V 
                                       
                                         TaRa 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         _ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         D 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         C 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     V 
                                     
                                       TaRa 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       _ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       SHC 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 g 
                                 ratio 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         
                                           V 
                                           
                                             TaRb 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             _ 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             D 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             C 
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         V 
                                         
                                           TaRb 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           _ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           SHC 
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               V 
                               
                                 TaRc 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 FHC 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 
                                   g 
                                   ratio 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           
                                             V 
                                             
                                               TaRb 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               _ 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               D 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               C 
                                             
                                           
                                           - 
                                         
                                       
                                     
                                     
                                       
                                         
                                           V 
                                           
                                             TaRb 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             _ 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             SHC 
                                           
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       V 
                                       
                                         TaRa 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         _ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         D 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         C 
                                       
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     V 
                                     
                                       
                                         TaRa 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         _ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         SHC 
                                       
                                       ⁢ 
                                       
                                           
                                       
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               V 
                               
                                 TaRc 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 FHS 
                               
                             
                           
                         
                         
                           
                             
                               V 
                               
                                 TaRa 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 FHC 
                               
                             
                           
                           
                             
                               V 
                               
                                 TaRb 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 FHS 
                               
                             
                           
                           
                             
                               V 
                               
                                 TaRc 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 _ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 D 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 C 
                               
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                         
                       
                         [ 
                         
                           
                             
                               
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     β 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ta 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ra 
                                 
                                 ⁢ 
                                 
                                   
                                     Ta 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     _ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     perp 
                                   
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Ra 
                                 
                               
                             
                             
                               
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     β 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ta 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Rb 
                                 
                                 ⁢ 
                                 
                                   
                                     Ta 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     _ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     perp 
                                   
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Rb 
                                 
                               
                             
                             
                               
                                 
                                   g 
                                   Ta 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Rc 
                                 
                                 ⁢ 
                                 
                                   
                                     Ta 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     _ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     perp 
                                   
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Rc 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     β 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ta 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ra 
                                 
                                 ⁢ 
                                 
                                   
                                     Tb 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     _ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     perp 
                                   
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Ra 
                                 
                               
                             
                             
                               
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     β 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ta 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Rb 
                                 
                                 ⁢ 
                                 
                                   
                                     Tb 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     _ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     perp 
                                   
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Rb 
                                 
                               
                             
                             
                               
                                 
                                   g 
                                   Ta 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Rc 
                                 
                                 ⁢ 
                                 
                                   
                                     Tb 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     _ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     perp 
                                   
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Rc 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     β 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ta 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ra 
                                 
                                 ⁢ 
                                 
                                   
                                     u 
                                     ^ 
                                   
                                   T 
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Ra 
                                 
                               
                             
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     β 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ta 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Rb 
                                 
                                 ⁢ 
                                 
                                   
                                     u 
                                     ^ 
                                   
                                   T 
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Rb 
                                 
                               
                             
                             
                               
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     β 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   g 
                                   Ta 
                                 
                                 ⁢ 
                                 
                                   g 
                                   Rc 
                                 
                                 ⁢ 
                                 
                                   
                                     u 
                                     ^ 
                                   
                                   T 
                                   t 
                                 
                                 ⁢ 
                                 Z 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     m 
                                     ^ 
                                   
                                   Rc 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   46 
                   ) 
                 
               
             
           
         
       
     
     Using the equivalent transmitter and receiver gains given below: 
     
       
         
           
             
               
                 
                   
                     G 
                     T 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               g 
                               Ta 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 β 
                                 ) 
                               
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             
                               g 
                               Ta 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 β 
                                 ) 
                               
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             
                               g 
                               Ta 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 β 
                                 ) 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
             
               
                 
                   
                     G 
                     R 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             g 
                             Ra 
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             g 
                             Ra 
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             g 
                             Rc 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
           
         
       
     
     and computing the quantity described above with respect to Equation 39 yields the fully compensated tensor quantity    12 (0): 
     
       
         
           
             
               
                 
                   
                     
                       12 
                     
                     ⁢ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             Ú 
                             
                               12 
                               ⁢ 
                               xx 
                             
                           
                         
                         
                           
                             Ú 
                             
                               12 
                               ⁢ 
                               xy 
                             
                           
                         
                         
                           
                             
                               tan 
                               ⁡ 
                               
                                 ( 
                                 β 
                                 ) 
                               
                             
                             ⁢ 
                             
                               Ú 
                               
                                 12 
                                 ⁢ 
                                 xz 
                               
                             
                           
                         
                       
                       
                         
                           
                             Ú 
                             
                               12 
                               ⁢ 
                               yx 
                             
                           
                         
                         
                           
                             Ú 
                             
                               12 
                               ⁢ 
                               yy 
                             
                           
                         
                         
                           
                             
                               tan 
                               ⁡ 
                               
                                 ( 
                                 β 
                                 ) 
                               
                             
                             ⁢ 
                             
                               Ú 
                               
                                 12 
                                 ⁢ 
                                 yz 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               cot 
                               ⁡ 
                               
                                 ( 
                                 β 
                                 ) 
                               
                             
                             ⁢ 
                             
                               Ú 
                               
                                 12 
                                 ⁢ 
                                 zx 
                               
                             
                           
                         
                         
                           
                             
                               cot 
                               ⁡ 
                               
                                 ( 
                                 β 
                                 ) 
                               
                             
                             ⁢ 
                             
                               Ú 
                               
                                 12 
                                 ⁢ 
                                 zy 
                               
                             
                           
                         
                         
                           
                             Ú 
                             
                               12 
                               ⁢ 
                               zz 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   49 
                   ) 
                 
               
             
           
         
       
     
     The use of the aforementioned calibration transmitters may enable an electromagnetic logging tool to be constructed using fewer deep transmitters. Moreover, in the embodiments shown in  FIGS. 7 and 8 , the calibration transmitters and the tilted transmitter are not collocated, which may simplify the manufacture and fabrication of such logging tools. 
     It will be understood that the various methods disclosed herein for obtaining a full tensor gain compensated quantity may be implemented on a processor, which can include a downhole processor and/or a surface processor. By downhole processor, it is meant an electronic processor (e.g., a microprocessor or digital controller or the like) deployed in the drill string (e.g., in the electromagnetic logging tool or elsewhere in the BHA). In such embodiments, the full tensor gain compensated quantity 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). In some embodiments, the harmonic fitting coefficients may 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. In other embodiments, a surface processor can be used, wherein the electromagnetic measurement data is transmitted to the surface, and the surface processor processes the measurements to obtain full tensor gain compensated quantities. 
     Although full tensor gain compensated 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.