# REAL AND IMAGINARY COMPONENTS OF ELECTROMAGNETIC LOGGING MEASUREMENTS

A method for making gain compensated electromagnetic logging measurements of a subterranean formation includes rotating an electromagnetic logging tool in a subterranean wellbore. The logging tool includes a transmitter having at least one transmitting antenna axially spaced apart from a receiver having at least one receiving antenna. Electromagnetic waves are transmitted into the subterranean wellbore using the at least one transmitting antenna. Voltage measurements corresponding to the transmitted electromagnetic waves are received at the receiving antenna. The voltage measurements are processed to compute real and imaginary directional resistivity measurement quantities.

**Description**

**CROSS REFERENCE TO RELATED APPLICATIONS**

The present application claims priority to U.S. Provisional Application 62/250,662 filed Nov. 4, 2015, the entirety of which is incorporated by reference.

**FIELD OF THE INVENTION**

Disclosed embodiments relate generally to downhole electromagnetic logging methods and more particularly to a logging tool and methods for computing real and imaginary components of electromagnetic logging 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.

Directional resistivity measurements are generally complex quantities, containing information about the signal attenuation and phase shift induced by the external environment (e.g., the subterranean formation). As such, directional resistivity measurements are commonly processed to compute a corresponding attenuation and phase shift. While such measurements are useful there remains room for further improvement, particularly in noisy measurement environments employing gain compensation.

**SUMMARY**

A method for making gain compensated electromagnetic logging measurements of a subterranean formation is disclosed. An electromagnetic logging tool is rotated in a subterranean wellbore. The logging tool includes a transmitter having at least one transmitting antenna axially spaced apart from a receiver having at least one receiving antenna. Electromagnetic waves are transmitted into the subterranean formation using the at least one transmitting antenna. Voltage measurements corresponding to the transmitted electromagnetic waves are received at the receiving antenna. The voltage measurements are processed to compute real and imaginary directional resistivity measurements such as gain compensated, real and imaginary, symmetrized and antisymmetrized measurement quantities.

The disclosed embodiments may provide various technical advantages. For example, the disclosed methodology may provide electromagnetic measurement quantities that highly robust to both systematic and incoherent random noise. Such measurements thus may provide for more accurate inversion for subterranean formation properties.

This summary is provided to introduce a selection of concepts that are further described below in the detailed description. This summary is not intended to identify key or essential features of the claimed subject matter, nor is it intended to be used as an aid in limiting the scope of the claimed subject matter.

**BRIEF DESCRIPTION OF THE DRAWINGS**

For a more complete understanding of the disclosed subject matter, and advantages thereof, reference is now made to the following descriptions taken in conjunction with the accompanying drawings, in which:

_{CSX }(_{CSX }(

_{CSY }(_{CSY }(

_{CAX }(_{CAX }(

_{CAY }(_{CAY }(

**DETAILED DESCRIPTION**

**10** suitable for employing various method embodiments disclosed herein. A semisubmersible drilling platform **12** is positioned over an oil or gas formation (not shown) disposed below the sea floor **16**. A subsea conduit **18** extends from deck **20** of platform **12** to a wellhead installation **22**. The platform may include a derrick and a hoisting apparatus for raising and lowering a drill string **30**, which, as shown, extends into borehole **40** and includes a drill bit **32** deployed at the lower end of a bottom hole assembly (BHA) that further includes an electromagnetic measurement tool **50** configured to make directional electromagnetic logging measurements. As described in more detail below the electromagnetic measurement tool **50** may include multi-axial antennas deployed on a logging while drilling tool body.

It will be understood that the deployment illustrated on **30** may include substantially any suitable downhole tool components, for example, including a steering tool such as a rotary steerable tool, a downhole telemetry system, and one or more MWD or LWD tools including various sensors for sensing downhole characteristics of the borehole and the surrounding formation. The disclosed embodiments are by no means limited to any particular drill string configuration.

It will be further understood that the disclosed embodiments are not limited to use with a semisubmersible platform **12** as illustrated on

**50**. In the depicted embodiment measurement tool **50** includes first and second axially spaced transmitters **52** and **54** and first and second axially spaced receivers **56** and **58** deployed on a logging while drilling tool body **51**, with the receivers **56** and **58** being deployed axially between the transmitters **52** and **54**. To obtain directional measurements, each of the transmitters **52** and **54** and receivers **56** and **58** generally includes at least one transverse antenna and may further include an axial antenna. For example, the transmitters and receivers may include a bi-axial antenna arrangement including an axial antenna and a transverse (cross-axial) antenna. In another embodiment, the transmitters and receivers may include a tri-axial antenna arrangement including an axial antenna and first and second transverse antennas that are orthogonal to one another. As is known to those of ordinary skill in the art, an axial antenna is one whose moment is substantially parallel with the longitudinal axis of the tool. Axial antennas are commonly wound about the circumference of the logging tool such that the plane of the antenna is substantially orthogonal to the tool axis. A transverse antenna is one whose moment is substantially perpendicular to the longitudinal axis of the tool. A transverse antenna may include, for example, a saddle coil (e.g., as disclosed in U.S. Patent Publications 2011/0074427 and 2011/0238312 each of which is incorporated by reference herein).

While not depicted on **52** and **54** and the receivers **56** and **58** may include a tilted antenna. Tilted antennas are commonly used to make directional resistivity measurements. As is known to those of ordinary skill in the art, a tilted antenna is one whose moment is angularly offset (tilted) with respect to the tool axis and is neither parallel with nor orthogonal to the tool axis.

**50** in which the transmitters **52**, **54** and receivers **56**, **58** each include a tri-axial antenna arrangement. Each of the transmitters **52**, **54** includes an axial transmitting antenna T**1**_{z }and T**2**_{z }and first and second transverse transmitting antennas T**1**_{x}, T**1**_{y }and T**2**_{x}, T**2**_{y}. Likewise, each of the receivers **56**, **58** includes an axial receiving antenna R**1**_{z }and R**2**_{z }and first and second transverse receiving antennas R**1**_{x}, R**1**_{y }and R2_{x}, R**2**_{y}. It will be understood that the disclosed embodiments are not limited to a tri-axial antenna configuration such as that depicted on

**50**′ in which the first and second transmitters are deployed on corresponding first and second subs **61** and **62** that are free to rotate with respect to one another (e.g., in an embodiment in which a drilling motor **65** is deployed therebetween). As in tool embodiment **50**, each of the transmitters T**1** and T**2** and receivers R**1** and R**1** may include a tri-axial antenna arrangement. In the example embodiment depicted the moment of R**1**_{z }is aligned with the moment of T**1**_{z }(and the z-azis) while the moments of R**1**_{x }and R**1**_{y }are rotationally offset from the moments of T**1**_{x }and T**1**_{y }by an offset angle a (e.g., 45 degrees in the depicted embodiment). The moment of R**2**_{z }is aligned with the moment of T**2**_{z }while the moments of R**2**_{x }and R**2**_{y }are rotationally offset from the moments of T**2**_{x }and T**2**_{y}by a (e.g., 45 degrees). The disclosed embodiments are, of course, not limited in these regards.

As stated above, the first and second subs **61** and **62** may rotate with respect to one another such that the moments of the x- and y-axis transmitting and receiving antennas are misaligned and rotate with respect to one another (i.e., the misalignment angle between the subs varies with time). Using the notation shown on **61** (the T**1**_{x }direction) is θ_{1 }with respect to an arbitrary ‘global’ (or wellbore) x-direction. Likewise, at the same instant in time, the orientation angle of the x-axis on sub **62** (the T**2**_{x }direction) is θ_{2 }with respect to the global x-direction. It will thus be understood that the moments of the x- and y-transmitting and receiving antennas T**1** and T**2** and R**1** and R**2** are misaligned by a misalignment angle y=θ_{1 }θ_{2}. It will be understood that θ_{1 }and θ_{2 }may be referred to as toolface angles of the first and second subs in that they define the rotational orientation of the subs with respect to a global reference direction. Since θ_{1 }and θ_{2 }are variable with time (owing to the rotation of the subs) and since the subs rotate at different rates the misalignment angle y also varies with time.

**100** for computing real and imaginary measurement quantities. An electromagnetic measurement tool (e.g., one of the measurement tools depicted on **102** (e.g., while drilling the wellbore). One or more transmitters are sequentially fired at **104** thereby transmitting an electromagnetic wave into the subterranean formation while rotating in **102**. Voltage signals corresponding to the transmitted electromagnetic waves are received at **106** by a plurality of the receivers. The received voltage signals may then be processed at **108** to compute real and imaginary resistivity measurement components (e.g., real and imaginary gain compensated directional resistivity measurements). These real and imaginary components may then be further processed to compute one or more properties of the subterranean formation.

**120** for computing real and imaginary gain compensated measurement quantities. An electromagnetic measurement tool (e.g., one of the measurement tools depicted on **122** (e.g., while drilling the wellbore). Electromagnetic measurements are acquired at **124** while the tool is rotating and processed to obtain harmonic voltage coefficients. For example, one or more transmitters may be sequentially fired so as to transmit an electromagnetic wave into the subterranean formation. Corresponding voltage signals may be received by a plurality of the receivers and processed to compute the harmonic voltage coefficients. Ratios of selected harmonic voltage coefficients may then be processed to obtain gain compensated measurement quantities at **126**. The gain compensated measurement quantities may then be further processed to compute corresponding real and imaginary components at **128**. An inversion may be optionally processed at **130** using the real and imaginary gain compensated measurement quantities to compute one or more formation parameters (e.g., a formation resistivity, a dip angle, a distance to a remote bed boundary, and the like).

With continued reference to

In general, earth formations are anisotropic such that their electrical properties may be expressed as a 3×3 tensor that contains information on formation resistivity anisotropy, dip, bed boundaries and other aspects of formation geometry. It will be understood by those of ordinary skill in the art that the mutual couplings between the tri-axial transmitter antennas and the tri-axial receiver antennas depicted on _{ij }may be expressed as follows:

where V_{ij }represent the 3×3 matrix of measured voltages with i indicating the corresponding transmitter triad (e.g., T**1** or T**2**) and j indicating the corresponding receiver triad (e.g., R**1** or R**2**), I_{i }represent the transmitter currents, and Z_{ij }represent the transfer impedances which depend on the electrical and magnetic properties of the environment surrounding the antenna pair in addition to the frequency, geometry, and spacing of the antennas. The third and fourth subscripts indicate the axial orientation of the transmitter and receiver antennas. For example, V_{12xy }represents a voltage measurement on the y-axis antenna of receiver R**2** resulting from a firing of the x-axis antenna of transmitter T**1**.

When bending of the measurement tool is negligible (e.g., less than about 10 degrees), the measured voltages may be modeled mathematically, for example, as follows:

V_{ij}=G_{Ti}m_{Ti}^{t}(R_{θt}^{t}Z_{ij}R_{θr})m_{Rj}G_{Rj} (2)

where Z_{ij }are matrices representing the triaxial tensor couplings (impedances) between the locations of transmitter i and receiver j, G_{Ti }and G_{Rj }are diagonal matrices representing the transmitter and receiver gains, R_{θt }and R_{θr }represent the rotation matrices for rotating the transmitter and receiver about the z-axis through angles θ_{t }and θ_{r}, m_{Ti }and m_{Rj }represent the matrices of the direction cosines for the transmitter and receiver moments at θ=0, and the superscript t represents the transpose of the corresponding matrix. The matrices in Equation 2 may be given, for example, as follows:

Using the T**1**x antenna direction as a reference direction for the first sub and the T**2**x antenna direction as a reference direction for the second sub, the matrices of the direction cosines of the transmitter and receiver moments may be given, for example, as follows:

m_{T1}=I

m_{R1}=R_{a }

m_{T2}=R_{y }

m_{R2}=R_{a}R_{y } (8)

where I represents the identity matrix, R_{a }represents the rotation matrix about the z-axis through the angle a, and R_{y }represents the rotation matrix about the z-axis through the angle y. It will be understood that Equations 2-8 are written for a general embodiment (such as shown on **61** and **62** are free to rotate with respect to one another (but are applicable to other configurations).

In an embodiment in which the transmitters and receivers are deployed on a common tool body (such that there is no misalignment as in _{t}=θ_{r }such that V_{ij}=G_{Ti}(R_{θ}^{t}Z_{ij}R_{θ})G_{Rj}. It will be understood that the disclosed embodiments are not limited in regard to the relative rotation of the transmitters and receivers. Gain compensated quantities may be computed with or without relative rotation between the transmitters and receivers. For example, commonly assigned U.S. patent application Ser. No. 14/549,396 (which is fully incorporated by reference herein) discloses methods for obtaining gain compensated measurements with differential rotation of the first transmitter and receiver with respect to the second transmitter and receiver (e.g., in an embodiment similar to that depicted on

The receiving antenna voltages may be measured while the tool rotates in the borehole. The measured voltages may be expressed mathematically in terms of their harmonic voltage coefficients, for example, as follows thereby enabling harmonic voltage coefficients to be obtained:

*V*_{ij}*=V*_{DC}_{_}_{ij}*V*_{FHC}_{_}_{ij }cos(θ)*V*_{FHS}_{_}_{ij }sin(θ)+*V*_{SHS}_{_}_{ij }cos (2θ)+*V*_{SHS}_{_}_{ij }sin(2θ) (9)

where V_{DC}_{_}_{ij }represents a DC voltage coefficient, V_{FHD}_{_}_{ij }and V_{FHS}_{_}_{ij }represent first order harmonic cosine and first order harmonic sine voltage coefficients (also referred to herein as first harmonic cosine and first harmonic sine voltage coefficients), and V_{SHC}_{_}_{ij }and V_{SHS}_{_}_{ij }represent second order harmonic cosine and second order harmonic sine voltage coefficients (also referred to herein as second harmonic cosine and second harmonic sine voltage coefficients) of the ij transmitter receiver couplings.

It will be understood that collocated tri-axial transmitter and receiver embodiments (e.g., as depicted on

It will be further understood that tilted antennas may be used to obtain many of the coupling and cross-coupling components described herein. For example, tilted antennas may be readily used to obtain the axial cross terms.

Ratios between the DC xx and yy voltage measurements or the second harmonic xx and yy voltage measurements may optionally be computed and allow a gain ratio of the x to y transmitter and gain ratio of the x to y receiver to be obtained. The voltage measurements may also be rotated mathematically to simulate rotation of the x and y antennas in the R**1** and R**2** receivers and the T**2** transmitter such that they are rotationally aligned with the x and y antennas in the T**1** transmitter. Such rotation removes the effect of the offset angle a and misalignment angle y on the measurements. Such computations are disclosed, for example, in U.S. patent application Ser. No. 14/549,396 which is incorporated by reference herein in its entirety.

The following tensor terms (and terms related to tensor terms) may be obtained from the back rotated coefficients (similar terms may also be obtained in embodiments in which back rotation is unnecessary):

The quantities in Equations 10-18 contain only x and z transmitter and receiver gains. These gains may be canceled out via computing various ones of the following ratios. The following term by term (TBT) compensation operators may be defined for any measurement X obtained between transmitter i and receiver j, for example, as follows:

where X_{ij}, X_{ji}, X_{ii}, and X_{jj }may include the measurement terms defined above with respect to

Various gain compensated quantities may be computed following the form of Equation 19. For example, only:

where CXX, CYY, and CZZ represent gain compensated xx, yy, and zz couplings (the tensor diagonal terms) and CXXplusYY represents a gain compensated quantity related to the sum of the xx and yy couplings. It will be understood that the disclosed embodiments are not limited to the above defined gain compensated measurement quantities. Other suitable gain compensated measurement quantities are disclosed in commonly assigned, co-pending U.S. application Ser. Nos. 14/285,581; 14/285,588; 14/339,959; 14/325,797; and 14/549,396 each of which is incorporated by reference herein in its entirety.

The gain compensated quantities defined above in Equations 20-23 may be further manipulated, for example, to compute the following measurement quantities:

where UHR and UHA represent gain compensated harmonic resistivity and harmonic anisotropy measurements and CS and CA represent gain compensated symmetrized and anti-symmetrized measurement quantities, and where:

It will be appreciated that the gain compensated measurement quantities described above with respect to Equations 20-27 are complex quantities and that these quantities may be represented as a corresponding attenuation and phase shift, for example, as follows:

where CQ represents the compensated quantity, for example, from Equations 20-27 and ATT and PS represent the attenuation and phase shift of the complex quantity.

While the above described compensated measurement quantities have wide potential applicability in electromagnetic logging operations, one aspect of the present disclosure was the realization that the accuracy of the attenuation and phase shift measurements can be questionable in certain noisy operating conditions. It was further realized that real and imaginary measurement quantities tend to be significantly more robust under the same noisy conditions. These features are described in more detail below with respect to the computational examples.

With reference again to

RCQ=Real(CQ)

ICQ=Imag(CQ) (29)

where RCQ and ICQ represent the real and imaginary components of the gain compensated measurement quantities.

In one particular embodiment, a complex geometric mean is computed to construct gain compensated xz and zx couplings. These may be obtained from R_{zx }and R_{xz }(which are also listed above), for example, as follows:

The average phase angle may be obtained from R_{zx }and R_{xz}, for example, as follows:

φ_{zx}=(φ_{zx1}+φ_{zx2})/2

φ_{xz}=(φ_{xz1}+φ_{xz 2})/2 (31)

where φ_{zx }and φ_{xz }represent the average phase angles of the quantities R_{zx }and R_{xz }and φ_{zx1}, φ_{zx2}, φ_{xz1}, and φ_{xz2 }represent the phase angles of the zx**1**, zx**2**, xz**1**, and xz**2** ratios listed in Equation 30. The phase angles φ_{zx1}, φ_{zx2}, φ_{xz1}, and φ_{xz2 }may be computed, for example, as follows:

φ_{zx1}=unwrap[angle(*zx*1)+*s ift]*

φ_{zx2}=unwrap[angle(*zx*2)*s ift]*

φ_{xz1}=unwrap[angle(*xz*1)+*s ift]*

φ_{xz2}=unwrap[angle(*xz*2)*s ift]* (32)

where unwrap [·] corrects a radian phase angle by adding multiples ±2π as necessary, angle(·) computes the phase angle of a complex quantity, and s ift represents an arbitrary phase shift (75 degrees was used in the examples that follow). Upon computing the average phase angles (in Equation 31), compensated zx and xz cross couplings may be constructed, for example, as follows:

C_{ZX}=√{square root over (|R_{zx}|)}e^{iφ}^{zx }

C_{XZ}=√{square root over (|R_{xz}|)}e^{iφ}^{xz } (33)

where C_{ZX }and C_{XZ }represent the compensated zx and xz cross couplings and |R_{zx}| and |R_{xz}| represent the magnitudes of R_{zx }and R_{xz }given in Equation 30. C_{ZY }and C_{YZ}, representing compensated zy and yz cross couplings, may be computed similarly, for example, as follows:

C_{ZY}=√{square root over (|R_{ZY}|)}e^{iφ}^{zy }

C_{YZ}=√{square root over (|R_{YZ}|)}e^{iφ}^{yz } (34)

where:

φ_{zy}=(φ_{zy1}+φ_{zy2})/2; φ_{yz}=(φ_{yz1}+φ_{yz2})/2 φ_{zy1}=unwrap[angle(zy**1**)+s ift]; φ_{yz1}=unwrap[angle(yz**1**)+s ift] φ_{zy2}=unwrap[angle(zy**2**) s ift]; φ_{yz2}=unwrap [angle(yz**2**) s ift]

Compensated symmetrized and antisymmetrized measurement quantities may then be computed from the compensated cross coupling components in Equations 33 and 34, for example, as follows:

C_{SX}=C_{ZX}C_{XZ }

C_{SY}=C_{ZY}C_{YZ }

C_{AX}=C_{ZX}+C_{XZ }

C_{AY}=C_{ZY}+C_{YZ } (35)

where C_{SX }and C_{SY }represent the compensated symmetrized x and y-axis quantities and C_{AX }and C_{AY }represent the compensated antisymmetrized x and y-axis quantities. The real and imaginary components of these complex quantities may then be computed, for example, as follows:

R_{CSX}=real(C_{SX}), I_{CSX}=imag(C_{SX})

R_{CSY}=real(C_{SY}), I_{CSY}=imag(C_{SY})

R_{CAX}=real(C_{AX}), I_{CAX}=imag(C_{AX})

R_{CAY}=real(C_{AY}), I_{CAY}=imag(C_{AY}) (36)

The disclosed embodiments are now described in further detail with respect to the following non-limiting examples in **1** and R**2**, and transmitter T**2** were located 7, 63, and 70 feet above transmitter T**1**. A two-layer formation model was used in which the upper bed had a horizontal resistivity of 1 ohm.m and a vertical resistivity of 1 ohm.m and the lower bed had a horizontal resistivity of 200 ohm.m and a vertical resistivity of 200 ohm.m. Zero depth was defined as the depth at which the transmitter T**1** crossed the bed boundary. The apparent dip angle between the bed boundary and the tool axis was 70 degrees.

In each example, the gain compensated measurement quantity of interest was simulated using three distinct error conditions; (i) no error, (ii) systematic mismatch error including ±10% gain and ±30% phase variation, and (iii) incoherent random noise error in which incoherent random noise including ±10% gain and ±30% phase variation was added to every dimension including depth points and toolface angles.

**202** (the incoherent random noise causes negative attenuation near the boundary). An amplitude mismatch is also evident in the phase shift simulation as depicted at **204** (the incoherent random noise causes about a 50% reduction in the peak phase shift near the boundary).

**206** (the incoherent random noise causes negative attenuation near the boundary). An amplitude mismatch is also evident in the phase shift simulation as depicted at **208** (the incoherent random noise causes about a 100% increase in the peak phase shift near the boundary).

_{CSX }(_{CSX }(

_{CSY }(_{CSY }(

_{CAX }(_{CAX }(

_{CAY }(_{CAY }(

It will be understood that the various methods disclosed herein for computing real and imaginary gain compensated measurement quantities may be implemented on a on a downhole processor. By downhole processor it is meant an electronic processor (e.g., a microprocessor or digital controller) deployed in the drill string (e.g., in the electromagnetic logging tool or elsewhere in the BHA). In such embodiments, the computed quantities may be stored in downhole memory and/or transmitted to the surface while drilling via known telemetry techniques (e.g., mud pulse telemetry or wired drill pipe). Whether transmitted to the surface or computed at the surface, the quantities may then be utilized in an inversion process (along with a formation model) to obtain various formation parameters as described above.

Although methods for making real and imaginary gain compensated electromagnetic logging measurements have been described in detail, it should be understood that various changes, substitutions and alternations can be made herein without departing from the spirit and scope of the disclosure as defined by the appended claims.

## Claims

1. A method for making gain compensated electromagnetic logging measurements of a subterranean formation, the method comprising

- (a) rotating an electromagnetic logging tool in a subterranean wellbore, the logging tool including a transmitter axially spaced apart from a receiver, the transmitter including at least one transmitting antenna, the receiver including at least one receiving antenna;

- (b) causing the at least one transmitting antenna to transmit electromagnetic waves into the subterranean formation;

- (c) using the at least one receiving antenna to receive voltage measurements corresponding to the electromagnetic waves transmitted in (b); and

- (d) processing the voltage measurements received in (c) to compute real and imaginary directional resistivity measurements.

2. The method of claim 1, wherein:

- the at least one transmitting antenna comprises at least one axial transmitting antenna and at least one transverse transmitting antenna;

- the at least one receiving antenna comprises at least one axial receiving antenna and at least one transverse receiving antenna.

3. The method of claim 2, wherein (d) further comprises:

- (i) processing the voltage measurements received in (c) to compute harmonic voltage coefficients;

- (ii) processing ratios of selected ones of the harmonic voltage coefficients to compute a gain compensated measurement quantity; and

- (ii) processing the gain compensated measurement quantity to compute real and imaginary components thereof.

4. The method of claim 1, further comprising:

- (e) processing a mathematical inversion using the real and imaginary directional resistivity measurements to compute at least one property of the subterranean formation.

5. A method for making gain compensated electromagnetic logging measurements of a subterranean formation, the method comprising

- (a) rotating an electromagnetic logging tool in a subterranean wellbore, the logging tool including a transmitter axially spaced apart from a receiver, the transmitter including an axial transmitting antenna and at least one transverse transmitting antenna, the receiver including an axial receiving antenna and at least one transverse receiving antenna;

- (b) causing the axial transmitting antenna and the at least one transverse transmitting antenna to sequentially transmit corresponding electromagnetic waves into the subterranean formation;

- (c) using the axial receiving antenna and the at least one transverse receiving antenna to receive voltage measurements corresponding to the electromagnetic waves transmitted in (b); and

- (d) processing the voltage measurements received in (c) to compute harmonic voltage coefficients; and

- (e) processing ratios of selected ones of the harmonic voltage coefficients computed in (d) to compute real and imaginary components of a gain compensated measurement quantity.

6. The method of claim 5, wherein the real and imaginary components of a gain compensated measurement quantity computed in (e) comprise real and imaginary components of at least one of a symmetrized and an antisymmetrized gain compensated measurement quantity.

7. The method of claim 6, wherein (e) further comprises:

- (i) processing a first average phase angle from first and second ratios of selected ones of the harmonic voltage coefficients;

- (ii) processing a second average phase angle from third and fourth ratios of selected ones of the harmonic voltage coefficients;

- (iii) processing the first and second average phase angles to compute corresponding first and second gain compensated axial cross couplings; and

- (iv) processing the first and second gain compensated axial cross couplings to compute the real and imaginary components of the at least one of a symmetrized and an antisymmetrized gain compensated measurement quantity.

8. The method of claim 7, wherein the real and imaginary components of the symmetrized gain compensated measurement quantity are computed from a difference between the first and second gain compensated axial cross couplings.

9. The method of claim 7, wherein the real and imaginary components of the anti symmetrized gain compensated measurement quantity are computed from a sum of the first and second gain compensated axial cross couplings.

10. The method of claim 7, wherein the first and second average phase angles are computed using the following mathematical equations:

- φzx=(φzx1+φzx2)/2

- φxz=(φxz1+100 xz2)/2

- φzx1=unwrap[angle(zx1)+shift]

- φxz1=unwrap[angle(xz1)+shift]

- φzx2=unwrap[angle(zx2)−shift]

- φxz2=unwrap[angle(xz2)−shift]

- wherein φzx and φxz represent the first and second average phase angles and φzx1, φzx2, φzx1, φzx2 represent the first, second, third, and fourth ratios.

11. The method of claim 7, wherein the first and second gain compensated axial cross couplings are computed using the following equations:

- CZX=√{square root over (|Rzx|)}·eiφzx

- CXZ=√{square root over (|Rxz|)}·eiφxz

- wherein CZX and CXZ represent the first and second gain compensated axial cross couplings, Rzx represents a product of the first and second ratios, Rxz represents a product of the third and fourth ratios, and φzx and φxz represent the first and second average phase angles.

12. The method of claim 7, wherein the real and imaginary components of the symmetrized gain compensated measurement quantity is computed using the following equations:

- RCSX=real(CZX−CXZ), ICSX=imag(CZX−CXZ)

- wherein RCSX and ICSX represent the real and imaginary components of the symmetrized gain compensated measurement quantity and CZX−CXZ represents a difference between the first and second gain compensated axial cross couplings.

13. The method of claim 7, wherein the real and imaginary components of the antisymmetrized gain compensated measurement quantity is computed using the following equations:

- RCAX=real(CZX+CXZ), ICAX=imag(CZX−CXZ)

- wherein RCAX and ICAX represent the real and imaginary components of the antisymmetrized gain compensated measurement quantity and CZX+CXZ represents a sum of the first and second gain compensated axial cross couplings.

14. The method of claim 5, further comprising:

- (f) processing a mathematical inversion using the real and imaginary components of a gain compensated measurement quantity to compute at least one property of the subterranean formation.

15. A downhole logging while drilling tool comprising:

- a logging while drilling tool body;

- a transmitter including at least one transmitting antenna deployed on the tool body;

- a receiver including at least one receiving antenna deployed on the tool body, the receiver being axially spaced apart from the transmitter;

- a controller configured to (i) cause the at least one transmitting antenna to transmit electromagnetic waves; (ii) cause the at least one receiving antenna to receive voltage measurements corresponding to the electromagnetic waves transmitted in (i); and (iii) process the voltage measurements received in (ii) to compute real and imaginary directional resistivity measurements.

16. The downhole logging while drilling tool of claim 15, wherein the at least one transmitting antenna comprises at least one axial transmitting antenna and at least one transverse transmitting antenna.

17. The downhole logging while drilling tool of claim 15, wherein (iii) comprises:

- (a) process the voltage measurements received in (ii) to compute harmonic voltage coefficients;

- (b) process ratios of selected ones of the harmonic voltage coefficients to compute a gain compensated measurement quantity; and

- (c) process the gain compensated measurement quantity to compute real and imaginary components thereof.

18. The downhole logging while drilling tool of claim 17, wherein the real and imaginary components of a gain compensated measurement quantity computed in (c) comprises real and imaginary components of at least one of a symmetrized and an antisymmetrized gain compensated measurement quantity.

19. The downhole logging while drilling tool of claim 15, wherein the controller is further configured to (iv) process a mathematical inversion using the real and imaginary directional resistivity measurements to compute at least one property of a subterranean formation.

20. The downhole logging while drilling tool of claim 17, wherein (c) further comprises:

- (i) processing a first average phase angle from first and second ratios of selected ones of the harmonic voltage coefficients;

- (ii) processing a second average phase angle from third and fourth ratios of selected ones of the harmonic voltage coefficients;

- (iii) processing the first and second average phase angles to compute corresponding first and second gain compensated axial cross couplings; and

- (iv) processing the first and second gain compensated axial cross couplings to compute the real and imaginary components of the at least one of a symmetrized and an antisymmetrized gain compensated measurement quantity.

**Patent History**

**Publication number**: 20180321413

**Type:**Application

**Filed**: Oct 14, 2016

**Publication Date**: Nov 8, 2018

**Patent Grant number**: 10627536

**Inventors**: Helen Xiaoyan Zhong (Sugar Land, TX), Mark T. Frey (Sugar Land, TX), Peter T. Wu (Missouri City, TX)

**Application Number**: 15/773,638

**Classifications**

**International Classification**: G01V 3/28 (20060101); G01V 3/38 (20060101);