Figure 1 shows single-well reflection imaging using a LWD dipole tool. Compared with the wireline logging scenario, the drill collar occupies a large portion of the borehole and has substantial influence on the wavefield characteristics. The LWD borehole model consists of the fluid inside the drill collar, the drill collar, the fluid ring between the drill collar and the formation, and the formation, whose outer radii are r ₁, r ₂, r ₃, and infinity, respectively. Assuming that the drill collar is in the center of the borehole, an annular dipole source is placed on the rim of the drill collar and pointed in the direction of the x-axis, with the radius of the source (modeled as a fluid ring) being the outer radius of the drill collar (i.e., r ₀ = r ₂), as shown in Figure 1B. The model shown in Figure 1 contains three fluid–solid boundaries at the inner and outer collar and the borehole–formation interfaces. The fluid–solid boundary condition requires the continuity of radial displacement and stress and vanishing of the azimuthal and axial shear stresses. The wave is actuated by assigning the radial displacement generated by the source to the drill collar at the source location. Combining the boundary conditions at the abovementioned interfaces yields a matrix equation with 12 unknown coefficients (Tang and Cheng, 2004): H × A f i n , A d c , B d c , C d c , D d c , E d c , F d c , A f o u t , B f o u t , B f m , D f m , F f m T = e , (1) where H is a 12 × 12 matrix and e is a 12 × 1 vector, the latter resulting from the direct contribution of the dipole source (the detailed expressions of matrix H and vector e are given in Supplementary Appendix A1). The superscripts fin, dc, and fout correspond to the fluid inside the drill collar, the drill collar, and the fluid ring, respectively, and the respective coefficients govern the strength of the guided waves inside the borehole. The amplitude coefficients with superscript fm are related to the radiated waves for imaging geological structures in the formation. For far-field radiation, the asymptotic solutions of the displacement potential function of the radiated P-, SH-, and SV-waves from the LWD dipole source are (Tang and Wei, 2012). ϕ f m = − B f m ω , k p 0 t I 1 f p 0 r 0 cos ⁡ φ e i ω R / α f m − i π / 4 4 π R χ f m = − D f m ω , k s 0 t I 1 f s 0 r 0 sin ⁡ φ e i ω R / β f m − i π / 4 4 π R Γ f m = − F f m ω , k s 0 t I 1 f s 0 r 0 cos ⁡ φ e i ω R / β f m − i π / 4 4 π R , (2) where ω is the circular frequency; α f m and β f m are the formation P- and S-wave velocities, respectively; k p 0 t = ω / α f m cos ⁡ θ t and k s 0 t = ω / β f m cos ⁡ θ t are the steepest–descent solutions of the radiated P- and S-wave wavenumbers, respectively; f p 0 = k p 0 t 2 − ω / α f 2 and f s 0 = k s 0 t 2 − ω / α f 2 denote the fluid radial wavenumbers calculated from k p 0 t and k s 0 t , respectively; α f is the fluid velocity; I ₁ represents the first kind of variant Bessel function with one order; θ t is the angle between the radiated-wave direction and the borehole z-axis; φ is the angle between the projection of the radiated-wave direction on the horizontal plane and the source orientation; and R stands for the distance from the source to the field point. The amplitude coefficients B f m , D f m , and F f m are found by solving Equation 1.

Equation 2 shows that the radiated far-field wave is the product of two terms, one of which controls the spatial distribution of the wavefield and is called the radiation function of the borehole source: R D P = B f m ω , k p 0 t I 1 f p 0 r 0 cos ⁡ φ R D S H = D f m ω , k s 0 t I 1 f s 0 r 0 sin ⁡ φ R D S V = F f m ω , k s 0 t I 1 f s 0 r 0 cos ⁡ φ . (3)

The common factor − e − i π / 4 / 4 π is omitted from Equation 3.

Another term of the radiated wavefield is the typical spherical wave propagation factor e i ω R / v / R , where v represents the formation S- or P-wave velocity. Consequently, the far-field P-, SH-, and SV-waves can be written as u P = − i ρ f m α f m ω B f m ω , k p 0 t I 1 f p 0 r 0 cos ⁡ φ e i ω R / α f m − i π / 4 4 π ρ f m α f m 2 R u S H = i ρ f m β f m ω ⁡ sin ⁡ θ t D f m ω , k s 0 t I 1 f s 0 r 0 sin ⁡ φ e i ω R / β f m − i π / 4 4 π ρ f m β f m 2 R u S V = ρ f m ω 2 ⁡ sin ⁡ θ t F f m ω , k s 0 t I 1 f s 0 r 0 cos ⁡ φ e i ω R / β f m − i π / 4 4 π ρ f m β f m 2 R , (4) where ρ f m is the formation density. The expression in the square bracket of Equation 4 yields the far-field radiation directivity of the P-, SH-, and SV-waves, respectively, as R P ω ; θ t , φ = − i ρ f m α f m ω B f m ω , k p 0 t I 1 f p 0 r 0 cos ⁡ φ R S H ω ; θ t , φ = i ρ f m β f m ω ⁡ sin ⁡ θ t D f m ω , k s 0 t I 1 f s 0 r 0 sin ⁡ φ R S V ω ; θ t , φ = ρ f m ω 2 ⁡ sin ⁡ θ t F f m ω , k s 0 t I 1 f s 0 r 0 cos ⁡ φ . (5)

When the radiated waves represented by Equation 4 encounter a reflector in formation (assume that the reflector size is large compared to wavelength), they are reflected back and recorded by the receiver on the LWD tool (see Figure 1B). The received wave can generally be written as (Tang and Wei, 2012). I W V ω = S ω ⋅ R D ω ⋅ R F ω ⋅ e − ω D / 2 Q v 4 π ρ f m v 2 ⋅ e i ω D / v D , (6) where Q is the quality factor of the formation for the S- or P-waves; S is the source function; RD is the far-field radiation function given by Equation 3; RF denotes the reflection coefficient of the wave at the reflector, which can be calculated using the Zoeppritz equation (Aki and Richards, 1980). The form of RD and RF depends on the type of the incident wave considered (i.e., P-, SH-, or SV-waves); and D = R t + R i is the total traveled distance from source to reflector and back to borehole, as shown in Figure 1A.

In the LWD measurement, however, it is important to note that the take-off azimuth at the wave radiation may be significantly different from the incident azimuth at the wave reception. For the model shown in Figure 1A, the azimuth of the reflector relative to the tool x-axis is φ 0 ′ at the wave takeoff; due to the tool rotation, the reflector azimuth changes to φ 0 ′ + δ when the wave is reflected back to the borehole, where δ is the offset angle of the tool during wave propagation in formation (assuming that tool rotation is clockwise). By treating the reflected wave incidence as spherical wave radiation from the virtual source located at the mirror point of the source on the reflector outside (see Figure 1A), the propagation distance D in Equation 6 is equated to a straight line linking the virtual source and borehole, as given by D = r 2 + r ′ 0 2 − 2 rr 0 ′ ⁡ cos φ − φ 0 ′ − δ + z − z ′ 0 2 . (7)

As a result, we can express the spherical wave propagation factor in Equation 6 as the summation of multipole cylindrical waves: e i ω D / v D = 1 π ∫ − ∞ + ∞ ∑ n = 0 + ∞ ε n I n k v r K n k v r 0 ′ cos n φ − φ 0 ′ + δ e i k z − z ′ 0 d k , (8) where k v = k 2 − ω 2 / v 2 ; I n ( K n ) is the n-order first (second)-kind modified Bessel function; and the Neumann factor ε n is taken as 1 for n = 0 and 2 for n > 0 . The substitution of Equation 8 into Equation 6 yields the displacement potential function for the incident P-, SH-, and SV-waves in the frequency–wavenumber domain: ϕ i = S ω ⋅ R D P ω ⋅ R F P ω 4 π ρ f m α f m 2 ∑ n = 0 + ∞ ε n I n pr K n p r 0 ′ cos ⁡ n φ ⋅ cos ⁡ n φ 0 ′ + δ + sin ⁡ n φ ⋅ sin ⁡ n φ 0 ′ + δ χ i = S ω ⋅ R D S H ω ⋅ R F S H ω 4 π ρ f m β f m 2 ∑ n = 0 + ∞ ε n I n s r K n s r 0 ′ cos ⁡ n φ ⋅ cos ⁡ n φ 0 ′ + δ + sin ⁡ n φ ⋅ sin ⁡ n φ 0 ′ + δ Γ i = S ω ⋅ R D S V ω ⋅ R F S V ω 4 π ρ f m β f m 2 ∑ n = 0 + ∞ ε n I n s r K n s r 0 ′ cos ⁡ n φ ⋅ cos ⁡ n φ 0 ′ + δ + sin ⁡ n φ ⋅ sin ⁡ n φ 0 ′ + δ , (9) where the superscript i denotes the incident wave from the virtual source; the subscripts P, SH, and SV mean that the radiation function RD and reflection coefficient RF should correspond to the P-, SH-, and SV-waves, respectively. The common factor e − ω D / 2 Q v is omitted from Equation 9.

The incident waves, in turn, will induce wave motions in and outside the borehole, and the induced wave displacement potential function of the P- and S-waves in the frequency–wavenumber domain has the same form as in Equation 8: ϕ f = ∑ n = 0 + ∞ A n f ⁡ cos n φ + A ′ n f ⁡ sin n φ I n f r + B n f ⁡ cos n φ + B ′ n f ⁡ sin n φ K n f r K n k v r 0 ′ , ϕ = ∑ n = 0 + ∞ A n ⁡ cos n φ + A n ′ ⁡ sin n φ I n pr + B n ⁡ cos n φ + B n ′ ⁡ sin n φ K n pr K n k v r 0 ′ , χ = ∑ n = 0 + ∞ C n ⁡ sin n φ − C n ′ ⁡ cos n φ I n s r + D n ⁡ sin n φ − D n ′ ⁡ cos n φ K n s r K n k v r 0 ′ , Γ = ∑ n = 0 + ∞ E n ⁡ cos n φ + E n ′ ⁡ sin n φ I n s r + F n ⁡ cos n φ + F n ′ ⁡ sin n φ K n s r K n k v r 0 ′ , (10) where ϕ f is the function of the fluid P-wave inside the drill collar or fluid ring; ϕ , χ , and Γ are for the P-, SH-, and SV-waves in the drill collar or formation; f = k 2 − ω 2 / α f 2 , p = k 2 − ω 2 / α 2 , and s = k 2 − ω 2 / β 2 are the fluid P-, solid P-, and S-wave radial wavenumbers, respectively; α f , α , and β denote fluid P-, solid P-, and S-wave velocities, respectively; and A n f ∼ F n and A ′ n f ∼ F n ′ are the amplitude coefficients of the respective waves. For the wavefield inside the collar (outside borehole), we must set the coefficients B n f i n and B ′ n f i n ( A n f m , A ′ n f m , C n f m , C ′ n f m , E n f m , and E ′ n f m ) to 0 to prevent the wavefield from going to infinity.

Combining Equations 9, 10 in conjunction with the boundary conditions for the three interfaces of the LWD model (see Figure 1), we obtain two matrix equations for the amplitude coefficients in the abovementioned equations and  H × A n f i n , A n d c , B n d c , C n d c , D n d c , E n d c , F n d c , A n f o u t , B n f o u t , B n f m , D n f m , F n f m T = e , H × A ′ n f i n , A ′ n d c , B ′ n d c , C ′ n d c , D ′ n d c , E ′ n d c , F ′ n d c , A ′ n f o u t , B ′ n f o u t , B ′ n f m , D ′ n f m , F ′ n f m T = e ′ , (11) where the superscripts of the coefficient vectors have the same meaning as in Equation 1 and the matrix H is the same as in Equation 1. The vectors e and e ′ result from the P-, SH-, and SV-wave incidence from the virtual source. The detailed expressions of the vectors are given in Supplementary Appendix A2. Note that the tool rotation effect between wave radiation and incidence is included in the expressions.

The radial distance r 0 ′ of the virtual source considered for the reflection imaging problem is generally large compared to wavelength such that the far-field condition k v r 0 ′ ≫ 1 holds true, and the steepest-descent method can be used to calculate the wavenumber integral in Equation 10 (Tang and Patterson, 2009).

For the reflected wave reception in LWD, the receivers are located on the rim of the drill collar, having the same radius r 0 as that of the ring source. Therefore, the received wavefield corresponds to that of the borehole fluid at the drill collar interface.

For fluid pressure ( p = ρ f ω 2 ϕ f o u t ), p ω = e i ω D / v − i π / 4 D ∑ n = 0 + ∞ A n f o u t ⁡ cos n φ + A ′ n f o u t ⁡ sin n φ I n f 0 r 0 + B n f o u t ⁡ cos n φ + B ′ n f o u t ⁡ sin n φ K n f 0 r 0 ρ f ω 2 . (12)

For radial fluid displacement ( u r f = ∂ ϕ f o u t / ∂ r ), u r f ω = e i ω D / v − i π / 4 D ∑ n = 0 + ∞ A n f o u t ⁡ cos n φ + A ′ n f o u t ⁡ sin n φ n I n f 0 r 0 / r 0 + f 0 I n + 1 f 0 r 0 + B n f o u t ⁡ cos n φ + B ′ n f o u t ⁡ sin n φ n K n f 0 r 0 / r 0 − f 0 K n + 1 f 0 r 0 , (13) where f 0 = k 0 i 2 − ω / α f 2 is the radial wavenumber of the fluid P-wave, k 0 i = ω / v cos π − θ i is the steepest–descent solution of the wavenumber of the incident wave, and θ i represents the angle between the incident direction of the wave and borehole axis; note that the wave travel distance D in Equations 6, 8 becomes D = r ′ 0 2 + z − z ′ 0 2 under the condition of r ≪ r 0 ′ . Fourier transform for Equations 12, 13 to the time domain yields the received waveform, which can be compared with waveform data from finite-difference numerical modeling.

To validate the result from the abovementioned analytical solution, we carried out 3D finite-difference modeling (Wei and Tang, 2012) for the model given in Figure 1 (see model parameters in Table 1). The reflector is an interface between formations 1 and 2. The reflector has an azimuth angle of 0° and an inclination angle of 45°. The borehole is surrounded by a fast formation (see Table 1), and the borehole-to-virtual source distance is 10 m. Inside the borehole, the receivers are at a 1-m distance above the source and azimuthally spaced at 45°, as shown in Figure 2A. Tan et al. (2016) indicated that the dominant frequency band of LWD dipole shear-wave remote detection is 1,800 Hz–2,500 Hz. Therefore, a 2,500-Hz Ricker wavelet is used for the source excitation here. For this numerical validation modeling, the tool is stationary, i.e., the δ angle in Equation 9 is set to 0.

Figure 2B compares the numerical (dots) and analytical (curve) waveforms at the azimuthally distributed receivers for the dipole source pointing to the x-axis direction. The comparison is in the time range from 5 ms to 9 ms, so that the waves traveling along the borehole, as shown in finite-difference modeling, are skipped, and only the reflected waves from the surrounding formation are compared. For validation purposes, Figure 2B shows only the comparison for the SV-waves for the inclined reflector model. The numerical and analytical waves agree very well, validating our asymptotic solution result.

The source–receiver configuration for the LWD measurement system is shown in Figure 1B, where the wave signals radiated from the dipole source are recorded by two receivers that are oppositely mounted on the collar rim along the dipole source direction. These two receivers form a receiving station, and the tool consists of eight or more such receiving stations. The receivers record two signals x x 1 m and x x 2 m , where the first (second) letter denotes the source (receiver) orientation and the superscript m indicates the number of times of source excitation and data acquisition. It should be noted that the time interval between the two adjacent source excitations should be long enough to ensure that the first received signal is not affected by the second excitation. The LWD single-component dipole waveform is obtained by x x m = x x 1 m − x x 2 m . (14)

For a centered tool in the borehole, Equation 14 produces a dipole signal as the n = 1 term is dominant the sum of the series of Equations 12, 13. The far-field wavefield of the centered annular dipole source is equivalent to that generated by a single force source on the well axis, and this simplified view is shown in Figure 4.

Relative to the reflection azimuth, the dipole source vector can be projected onto two orthogonal directions perpendicular and parallel to the reflector, giving rise to the SV (and P) and SH components in the reflection data. We now record the data using the x-oriented receivers of the measurement system. Considering the tool rotation, the recorded data are as follows: x x S φ 0 ′ = S H ⋅ sin ⁡ φ 0 ′ ⋅ sin φ 0 ′ + δ + S V ⋅ cos ⁡ φ 0 ′ ⋅ cos φ 0 ′ + δ , x x P φ 0 ′ = P ⋅ cos ⁡ φ 0 ′ ⋅ cos φ 0 ′ + δ , (15) where x x S and x x P are the dipole S- and P-wave components, respectively, the sum of which is the xx data in Equation 14. In field data processing, one transforms the rotation angle of the measured data from the tool coordinate system to the fixed coordinate system to eliminate the influence of tool rotation. In the field measurement, the angle between the x-axis of the tool coordinates and the x-axis of the earth coordinates (e.g., north) is always recorded, as shown in Figure 4. Using AZ, we obtain x x S A Z = S H ⋅ sin A Z − φ N ⋅ sin A Z − φ N + δ + S V ⋅ cos A Z − φ N ⋅ cos A Z − φ N + δ , x x P A Z = P ⋅ cos A Z − φ N ⋅ cos A Z − φ N + δ (16) where φ N is the angle between the reflector azimuth and coordinate north. Using the triangle function property, Equation 16 can also be written as x x S A Z = S V − S H 2 ⋅ cos ⁡ 2 A Z − φ N + δ 2 + S V + S H 2 ⋅ cos ⁡ δ , x x P A Z = P 2 ⋅ cos ⁡ 2 A Z − φ N + δ 2 + P 2 ⋅ cos ⁡ δ . (17)

Equation 17 indicates that relative to the case of no tool rotation (i.e., δ = 0 ), the reflector azimuth for a rotating tool, as determined from x x S or x x P , is offset by δ / 2 in the opposite direction of the tool rotation.

Using the analytical solution of Equation 12, the operation of the proposed one-dipole-source and two-receiver LWD measurement system can be accurately modeled and analyzed. In this modeling, we assume a 60°-dip reflector facing the north direction (i.e., φ N = 0 ) and a borehole-to-virtual source distance of 20 m. The source-to-receiver distance is 3 m; a 2,500-Hz Ricker wavelet is used for the source excitation. Table 1 shows the simulation parameters of the borehole and formation (the fast formation is used for formation 1). Figures 5, 6 show the modeled x x S and x x P data, respectively, as calculated using Equation 14, for the azimuth AZ ranging from 0° to 360°, where the azimuthal labels represent the azimuth and the radial label represents the wave arrival time. In the respective figures, the stationary (A, δ = 0 ) and rotating (B, δ = 20 o ) tool cases are modeled, and the variation patterns of the normalized wave amplitude x x S A Z and x x P A Z are compared in (C).

The results show the maximum of x x S and x x P points, respectively, toward the reflector strike and normal direction, as is the typical dipole characteristic. However, if the azimuth of the tool changes significantly during wave propagation in formation, the variation pattern of the x x S and x x P data will change accordingly. The modeling results given in Figures 5C, 6C indicate that the azimuth offset of δ = 20 o results in a 10° offset in the opposite direction of the tool rotation, which is consistent with Equation 17. Therefore, when using the amplitude information of dipole data x x to determine the reflector azimuth, the result needs to be shifted δ / 2 toward the rotation direction of the tool to correct the rotation effect. Based on the abovementioned analyses, the formula for determining the reflector azimuth using the LWD system is as follows.

For S-wave, φ N = max x x S A Z + δ / 2 ± π / 2 . (18)

For P-wave, φ N = max x x P A Z + δ / 2 + π / 2 ⋅ 1 ± 1 , (19) where the symbol max denotes seeking the value of AZ to maximize the function x x . The offset angle of the tool can be calculated from the rotation speed (revolutions per minute, RPM) and the arrival time (T ₀) of the reflected signal as δ = R P M ⋅ T 0 . The waveforms xx ^(m) in Equation 14 represent the recorded data at different times of source excitations, and the depth position of the source might change for different time stances. However, in the field logging scenario, the rate of penetration (ROP) of the bit is usually slow, which allows us to ignore the effect of the depth variation on the recorded data.

The above mentioned analysis results also exhibit an inherent 180°-azimuth ambiguity of the dipole system, as apparent from Equations 16, 17 (and from Figures 5C, 6C). This ambiguity can be eliminated using the arrival time difference between the xx ₁ and xx ₂ data of the oppositely mounted receivers of the LWD system, as shown below.

For the case of δ = 0 shown in Figures 5, 6, Figure 7 compares the waveforms xx ₁ and xx ₂ of the S- and P-waves (normalized by their maximum amplitude) for the tool azimuth AZ value of 30°. For this configuration, receiver x ₁ (x ₂) faces forward (backward) to the reflector such that xx ₁ arrives earlier than xx ₂ for both the S-wave (Figure 7A) and P-wave (Figure 7B) cases. For a closer look at the time difference, the S-waveform of xx ₂ is flipped in Figure 7A (this flipping does not change the arrival time of the waveform). The arrival time difference between xx ₁ and xx ₂ can be utilized to provide a method for determining which side the reflector is at, front or back, thus eliminating the 180° ambiguity of the existing dipole acquisition system. It should be noted that AZ is the take-off azimuth of the tool at the source radiation. The tool azimuth at the data recording is ( A Z + δ ). Therefore, when using the arrival time difference between the reflected waves xx ₁ and xx ₂ to distinguish the reflector azimuth, it may be necessary to offset the tool azimuth by δ toward the rotation direction to compensate for the rotation-related arrival time changes.

We now use the synthetic LWD dipole acoustic reflection data to test the applicability of the proposed LWD system and method for the drilling rotation condition. For the model shown in Figure 8, the reflector intersects the borehole with an angle of 30° in the middle of the 140-m-depth section, with the azimuth angles of the reflector in the lower and upper part of the intersection being 60° and 240°, respectively. The dipole S-wave detection method is used for data acquisition and processing, the results of which are given in Figure 9. It is assumed that the drilling has an RPM of 100 and an ROP of 10 m per hour. The time interval of the source excitation is 5 s. The time sampling interval and sampling points of the wavefield are 36 μs and 1,024, respectively. The dipole source is located 3 m below the receivers, and the source frequency is 2,500 Hz. The parameters of the borehole and formation and reflector are shown in Table 1, where fast formation 1 is used for the simulation.

Panel 1 of Figure 9 shows the azimuthal curve of the tool relative to north, indicating that the tool rotates rapidly for this while drilling simulation. Note that for a clearer view, the results of panels 1, 6, and 7 are displayed at intervals of 20 depth points. The computed xx ₁ and xx ₂ waveform data are displayed as variable density images in panels 2 and 3, respectively. For comparison, we also computed the borehole flexural waves traveling directly along the borehole, as indicated in the panels. For the data analysis, we first separate the xx ₁ and xx ₂ waves from the direct waves in the data and then compute the dipole data xx using Equation 14. Next, taking a moving depth window (about 41 adjacent depths, enough to cover several rotation cycles), the energy (squared amplitude) of the xx data versus the azimuth was computed for each depth within the window. Moving the window along the entire depth zone yields an energy image map versus tool azimuth in panel 4. The trend of the maximum energy delineates the reflector azimuth, as shown by the black line in panel 4, which shows that the reflector strike is approximately 140° ∼ 150° or 320° ∼ 330°. Therefore, the reflector azimuth is approximately 50° ∼ 60° or 230° ∼ 240°. Note that the delineated azimuth trend shows a drift from the borehole–reflector intersection toward the top or bottom of the well. This drift is the result of the tool azimuth offset between wave radiation and reception, as discussed before. From the wave travel time and RMP of the simulation, the offset angle δ = R P M ⋅ T 0 is calculated and plotted in panel 5. Correcting the data of panel 4 using the δ data of panel 5, we obtain the reflector azimuth in panel 6, as shown by the two data trends (markers) of approximately 60° and 240°. The result, however, shows the typical 180° ambiguity in the reflector azimuth. The final step of the analysis is to eliminate this ambiguity, and the time difference between the xx ₁ and xx ₂ waves is evaluated. The result is color-coded using the data of panel 6 (red: the time difference is positive; blue: the time difference is negative). The time-difference information allows us to choose the correct reflector azimuth. The azimuth data in panel 7 are consistent with the model shown in Figure 8, which shows that the up-dip and down-dip components of the reflector are on the opposite sides of the well, with azimuth angles of approximately 240° and 60°, respectively. Detailed time-difference and tool-rotation correction analyses are discussed in the following section.

To substantiate the abovementioned modeling and analysis results, we analyze the data at depths of 55.8464 m and 139.6063 m, which correspond to the up- and down-dip parts of the reflector, respectively. For the depth of 55.8464 m, 20 measurement points are selected above and below the depth, and the (normalized) amplitude of the wave x x versus the azimuth is shown in Figure 10A for 41 depths (markers). The red curve given in Figure 10A is from the linear fitting of the data. The amplitude data peak at approximately 150° or 330°, indicating that the reflector strike is approximately 150° or 330° and the reflector azimuth is approximately 60° or 240°. Figure 10B compares the waveforms xx ₁ and xx ₂ for this depth point, where the tool azimuth is 300°. The figure shows that xx ₁ is ahead of xx ₂, which indicates that receiver x ₁ (x ₂) is at the same (opposite) side of the reflector. The true azimuth of the reflector is therefore 240°.

For the down-dip depth of 139.6063 m, Figure 11A shows that the amplitude data peak at approximately 140° or 320° and the reflector azimuth is approximately 50° or 230°. As shown in panel 5 of Figure 9, at this depth point, the tool rotation offset angle is 17.6°; after the rotation correction, the reflector azimuth is approximately 58.8° or 238.8°. The tool azimuth for this depth point is 173°. According to the rotation angle of the tool, the true tool azimuth at the data recording should be 190.6°. The waveforms measured at this depth show that xx ₂ is ahead of xx ₁ (Figure 11B), indicating that receiver x ₂ (x ₁) is at the same (opposite) side of the reflector. The true azimuth of the reflector is approximately 58.8°. We notice that the azimuth result at 55.8464 m is obtained without the tool rotation correction. This is because the reflector is close to the borehole at this depth, and the angle-offset value due to the rotation is small and can thus be ignored. This synthetic modeling example validates the concept of the LWD system and the analysis results.

Encouraged by the success of the synthetic LWD datatest and results, we use a dipole dataset from wireline logging for a further test. Figure 12 shows the azimuthal imaging results of dipole shear-wave measurement data within a 20-m interval of a well. The azimuth curve in panel 1 shows that the tool rotates continuously in this well section. First, the acquired four-component dipole data (xx, xy, yx, and yy) are processed through filtering, wavefield separation, and migration imaging, which yields the reflection signal from the reflector outside the borehole. In addition, the four-component reflection data are rotated to obtain the imaging of the well section in the 0°–150° azimuth (relative to the North Pole), as shown in panels 2–7. It can be seen that there is a high inclination angle reflector approximately 15 m away from the well, and the reflector is most clearly imaged in the 60° azimuth, whereas it is not imaged in the 150° azimuth, which indicates that the reflector strike is approximately 60° north by east (or 60° south by west).

Furthermore, only the xx component is used for azimuth recognition. For the depth of 799 m, 15 measurements are selected above and below the depth point, respectively. Figures 13A, B show the normalized azimuth amplitude results in the form of polar and Cartesian coordinate systems, respectively. The results show that at the depth range corresponding to these 31 measurement points, the tool rotated continuously by approximately 70°, and the signal amplitude reaches its maximum near 240°. Combined with the theoretical results shown in Figure 5, it is observed that the reflector strike is approximately 240° (i.e., 60° south by west), which is consistent with the orientation identification results obtained using the four-component rotation shown in Figure 12. The field example preliminarily confirms the practicability of the method proposed in this paper.