Offset-domain pseudoscreen prestack depth migration

Transcription

1 GEOPHYSICS, VOL. 67, NO. 6 (NOVEMBER-DECEMBER 2002); P , 7 FIGS. 0.90/ Offset-domain pseudoscreen prestack depth migration Shengwen Jin, Charles C. Mosher, and Ru-Shan Wu ABSTRACT The double square root equation for laterally varying media in midpoint-offset coordinates provides a convenient framework for developing efficient 3-D prestack wave-equation depth migrations with screen propagators. Offset-domain pseudoscreen prestack depth migration downward continues the source and receiver wavefields simultaneously in midpoint-offset coordinates. Wavefield extrapolation is performed with a wavenumber-domain phase shift in a constant background medium followed by a phase correction in the space domain that accommodates smooth lateral velocity variations. An extra wide-angle compensation term is also applied to enhance steep dips in the presence of strong velocity contrasts. The algorithm is implemented using fast Fourier transforms and tri-diagonal matrix solvers, resulting in a computationally efficient implementation. Combined with the common-azimuth approximation, 3-D pseudoscreen migration provides a fast wavefield extrapolation for 3-D marine streamer data. Migration of the 2-D Marmousi model shows that offset domain pseudoscreen migration provides a significant improvement over first-arrival Kirchhoff migration for steeply dipping events in strong contrast heterogeneous media. For the 3-D SEG-EAGE C3 Narrow Angle synthetic dataset, image quality from offset-domain pseudoscreen migration is comparable to shot-record finitedifference migration results, but with computation times more than 00 times faster for full aperture imaging of the same data volume. INTRODUCTION Three-dimensional prestack depth migration is an important and effective tool in identifying potential targets for oil and gas exploration. Strong-contrast heterogeneous media and complex structures present a great challenge to migration methods. Many standard methods, which work well in weak-contrast media, often fail in strong-contrast media. Ray-based Kirchhoff migration is the most commonly used method for 3-D prestack depth migration due to its high efficiency and flexibility in handling 3-D prestack data geometry (Ratcliff et al., 994; Audebert et al., 997). Migration accuracy of this approach, however, relies on the high-frequency asymptotic ray approximation. Gray and May (994) confirmed that Kirchhoff migration algorithms based on eikonal equation solvers do not yield results of the quality obtained with wave-equation algorithms in strong contrast media. More accurate ray-tracing algorithms that include all arrival traveltimes and their associated amplitudes become both more complicated to implement and computationally expensive. Attempts have been made to use wave-equation-based imaging technologies. Wave-equation migrations are accurate but usually expensive, prohibiting their use in many production applications. Methods implemented in the frequencywavenumber domain or in the frequency-wavenumber-space domain can handle steep dips up to 90 and have reasonably efficient implementations based on the fast Fourier transform (FFT). Phase-shift migration (Gazdag, 978) is accurate for V (z) media. Split-step Fourier (SSF) migration (Stoffa et al., 990) can handle low-contrast smooth lateral variations. In order to treat strong lateral velocity variations, several extensions of the phase-shift and SSF methods have been developed. These methods include interpolation-based techniques, such as the phase shift plus interpolation (PSPI) method (Gazdag and Sguazzero, 984) and the multiple reference velocity splitstep Fourier method (Kessinger, 992; Huang et al., 999). Interpolation techniques, however, can be costly since they require forward-inverse transforms over the space axes at every depth level for each reference velocity. If the number of reference velocities used is too small, inaccuracies associated with the interpolation process arises. Margrave and Ferguson (999) used nonstationary filter to derive the generalization of PSPI and nonstationary phase shift (NSPS) integral operators from the Helmhotz equation for laterally varying media. This approach, however, loses the efficiency of FFTs, resulting in Manuscript received by the Editor May 8, 200; revised manuscript received April 26, Screen Imaging Technology, Inc., 7322 SW Freeway, Suite 508, Houston, Texas jin@screenimaging.com. Chevron Texaco Exploration and Production Technology Company, 600 Bollinger Canyon Road, San Ramon, California 94583; cmosher@chevrontexaco.com. University of California, IGPP, Modeling and Imaging Laboratory, 56 High Street, Santa Cruz, California wrs@es.ucsc.edu. c 2002 Society of Exploration Geophysicists. All rights reserved. 895

2 896 Jin et al. much higher computational burdens. Ristow and Ruhl (994) developed a Fourier finite-difference (FFD) migration method, which cascades phase-shift and finite-difference operators for downward continuation. In these approaches, phase-shift steps are taken in a v(z) reference medium, followed by a finitedifference residual operator in the space domain to account for lateral velocity variations. A single reference velocity is used in FFD. Biondi (200) optimized FFD and derived an unconditionally stable implicit finite-difference operator even when the reference velocity is higher than the medium velocity and in presence of the sharp contrasts in velocities. He further proposed an FFD plus interpolation (FFDPI) to improve the accuracy. An alternative approach was used by Li (99), which consists of cascading a finite-difference operator in the space domain, followed by a phase-shift correction operator in the wavenumber domain to correct for dispersion and splitting errors in the FD operators. Migration with generalized screen propagators (GSP) (Wu, 994, 996; Huang & Wu, 996; Xie & Wu, 998; Jin et al., 999; Huang et al., 999; de Hoop et al., 2000) keeps the essential features of the split-step Fourier migration method but includes wide-angle corrections to improve accuracy for steeply dipping events in strong-contrast velocity media. The above mentioned mixed domain methods operate on either shot-record or stack data. An alternative approach is to operate on common-midpoint data using the double square root (DSR) equation (Yilmaz and Claerbout, 980). In shot-record migration, the operator is applied independently to extrapolate the source and receiver wavefields. An image at each depth level is produced by correlating source and receiver wavefields. Migration with the DSR operator first downward continues the wavefield based on receiver locations. Reciprocity is then invoked, and the wavefield is downward continued based on source locations. The image at each level is produced by extracting the data at zero time and zero offset. Popovici (996) developed a split-step DSR migration in midpoint-offset coordinates and applied it to the migration of the Marmousi data set. Tanis et al. (998) extended the method to prestack migration in the source-offset domain. Jin and Wu (999) proposed a framework of offset-domain screen propagators for the migration of midpoint-offset data and extended the method to 3-D prestack depth migration (Jin et al., 2000). Alkhalifah (2000) presented an approach to the challenging task of migrating separate offsets with a prestack phase-shift method by evaluating the offset-wavenumber integral using the stationaryphase approximation. For 3-D marine streamer data, commonazimuth migration (Biondi and Palacharla, 996; Mosher and Foster, 998; Etgen, 998) can reduce the dimensionality of the migration operator and provides fast wavefield extrapolation. Application of azimuthal moveout (Biondi and Chemingui, 994) increases the azimuth range of acquisition geometries that can accommodate common azimuth wavefield extrapolation. In this paper, we present the offset-domain pseudoscreen propagator and propose a hybrid implementation of wideangle correction by implicit finite-difference algorithm in the midpoint-offset coordinates. Performance of the method is illustrated for 2-D offset-domain migration of the Marmousi model, and for 3-D common-azimuth migration of SEG- EAGE C3 Narrow Angle dataset. OFFSET-DOMAIN SCREEN PROPAGATORS We begin with Claerbout s (985) DSR equation for a laterally varying velocity medium in time-space domain: { ( ) z p(x t 2 s, x g, z, t) = v 2 (x s, z) x s + v 2 (x g, z) ( t x g ) 2 } t p(x s, x g, z, t), () where v(x s, z) and v(x g, z) correspond to the velocities at the source location x s and receiver location x g at depth z, respectively; p(x s, x g, z, t) is the wavefield expressed as a function of survey geometric parameters x s and x g, depth z, and the arrival time t which is given by a single valued t(x s, xg,z). By changing variables from shot-receiver coordinates (x s, x g ) to midpoint-offset coordinates (m, h) using m = 2 (x g + x s ), h = 2 (x g x s ), (2) where m is the midpoint and h is the half offset between the source and receiver, we obtain the DSR equation in midpointoffset coordinates: z p(m, h, z, t) = ( s + g ) p(m, h, z, t), (3) t where s = g = v 2 (m h, z) 4 v 2 (m + h, z) 4 ( t m t ) 2, (4) ( t m + t ) 2, (5) where v(m h, z) and v(m + h, z) correspond to the velocities at the source and receiver locations at depth z. Under the screen approximation to equation (3) discussed by Jin and Wu (999) and Jin et al. (2000), the velocity is decomposed into a background v 0 and the corresponding perturbation v: v(m h, z) = v 0 (z) + v(m h, z), v(m + h, z) = v 0 (z) + v(m + h, z), (6) and total wavefield p is decomposed into a primary wavefield p 0 and a scattered wavefield p s : p(m, h, z, t) = p 0 (m, h, z, t) + p s (m, h, z, t). (7) For numerical implementation, the square root operators (4) and (5) need to be expanded into series, such as Taylor and Padé series. Different expansions with different orders have different wide-angle accuracies. We adopt the pseudoscreen approach (Wu and de Hoop, 996; Huang and Wu, 996;

3 Pseudoscreen Prestack Depth Migration 897 Jin et al., 999) for its efficiency and wide-angle accuracy. Operators (4) and (5) are expanded as small perturbation series, and only the first-order term of each is kept. We know that this first-order approximation is same as the Born approximation, which has accumulated phase error in the forward direction and large errors at wider angles. The pseudoscreen approach applys the phase matching in the forward direction followed by hybrid wide-angle corrections. The first-order approximation of operators (4) and (5) after phase matching can be written as (Jin and Wu, 999) s = g = v ( t m t ) 2 + v0 2 ( t 4 m + t ) 2 + v 2 0 v 2 0 S s v0 2 ( t 4 m t ), 2 (8) S g v0 2 ( t 4 m + t ), 2 where S s and S g are the relative slowness perturbations v 0 (9) S s = v(m h, z), v 0 S g =. (0) v(m + h, z) Note that in the exact forward direction (i.e., for the vertical propagation) s = /v(m h, z) and g = /v(m + h, z). Substituting equations (8) and (9) into equation (3) and taking Fourier transforms with respect to midpoint, offset, and time, yields [ ] z + ik z(k m, K h,z,ω) P(K m,k h,z,ω) = F(K m,k h,z,ω), () where K m and K h are the midpoint and offset wavenumbers, respectively; k z (K m, K h, z,ω) is the vertical wavenumber used for double downward continuations of the wavefields in a background medium: with k z = k zs + k zg (2) k zs = k0 2 4 (K m K h ) 2, k zg = k0 2 4 (K m + K h ) 2, (3) and F(K m, K h, z,ω) acts like a source term, F = ik2 0 k zs FT m,h { S s p(m,h,z,ω)} + ik2 0 k zg FT m,h { S g p(m,h,z,ω)}, (4) where FT m,h { } represents a 2-D Fourier transform with respect to midpoint m and half offset h, and k 0 = ω/v 0 (z) isthe background wavenumber. During the downward continuation, the primary wavefield P 0 (K m, K h, z i+,ω) at depth z i+ in the wavenumber domain is calculated from the wavefield at z i by a free propagation in the background medium: P 0 (K m, K h, z i+,ω)=e ik z z P 0 (K m, K h,z i,ω). (5) The scattered wavefield P s, which is generated by the velocity perturbations, is given by P s (K m, K h, z i+,ω)=e ik z z F(K m, K h,z i,ω) z. (6) The total extrapolated wavefield P(K m, K h, z i+,ω) is then obtained by summing the primary and scattered wavefields: P(K m, K h, z i+,ω) = P 0 (K m,k h,z i+,ω) + P s (K m,k h,z i+,ω). (7) Equation (7) is a basic equation of the pseudoscreen propagator in midpoint-offset coordinates. We can see that most computational costs contribute to the FFT. The addition of term P s in equation (7), however, increases the amplitude, leading to instabilities in recursive calculation. In the following sections, we propose a phase-screen approximation and a hybrid pseudoscreen propagator to resolve this problem. Offset-domain phase-screen migration Using a small-angle approximation in equation (4) (i.e., k 0 /k zs and k 0 /k zg ), equation (7) can be simplified as P(K m, K h, z i+,ω)=e ik z z FT m,h {[ + ik 0 ( S s + S g ) z]p(m,h,z i,ω)}. (8) Invoking a Rytov transform (Huang and Wu, 996) results in the dual-domain expression of phase-screen propagator for this case: P(K m, K h, z i+,ω)=e ik z z FT m,h {e ik 0 ( S s+ Sg) z p(m,h,z i,ω)}. (9) Rytov transform is a renormalization procedure, which sums up the multiple forward-scattering series and put it as an exponential term (see Ishimaru, 978, chapter 7). It also makes the operator unitary, so that the phase-screen propagator is uncoditionally stable. In equation (9), two phase-shift operators are implemented in the mixed space-wavenumber domain to speed up the computation. It is equivalent to the split-step DSR migration operator (Popovici, 996). Offset-domain pseudoscreen propagator When k zs or k zg approaches zero in equation (4), it will lead to a singularity. To avoid the singularity and have more accurate phase correction for wide-angle waves, we replace the filter function k 0 /k zg and k 0 /k zs with a first-order Padé expansion. Let R = K T /k 0, then we set f (R) = k 0 /k zg + ar2 br2, (20)

4 898 Jin et al. where the coefficients a and b are dependent on the approximation accuracy. For a = b = 0, we get the zeroth-order expansion operator, which is the same as the phase-screen propagator; a = 0.25, b = 0 yields the first-order operator; and a = b = 0.25 yields the second-order operator, which is used in the migration test. The coefficients a and b can also be adjusted to optimize the wide-angle correction (Ristow and Ruhl, 994; Xie and Wu, 998). The filters are then replaced by k 0 = + s, k 0 = + g, (2) k zs k zg where s = a[(k m K h )/k 0 ] 2 b[(k m K h )/k 0 ] 2, g = a[(k m + K h )/k 0 ] 2 b[(k m + K h )/k 0 ] 2. (22) Then, we obtain the offset domain pseudoscreen propagator: P(K m, K h, z i+,ω) = e ik z z FT m,h { e ik 0 ( Ss+ Sg) z p(m,h,z i,ω) } + P a (K m,k h,z i+,ω), (23) where P a is the wide-angle compensation term: P a = ik 0 ze ikz z { s FT m,h [ S s p(m,h,z i,ω)] + g FT m,h [ S g p(m,h,z i,ω)]}. (24) In equation (23), the first term is the phase-screen propagator. The second term P a enhances the accuracy for wider angles with large velocity perturbations by an extra phase correction. Implementation Note that the norm of the propagator in equation (23) is greater than unity, resulting in an unstable recursive calculation. By inversely transforming into the space domain, the second term in equation (23) is solved using an implicit finitedifference implemenmtation: p(m, h, z,ω) z = ik 0a( m h ) 2 k 2 0 +b( m h ) 2[ S s p(m,h,z,ω)] + ik 0a( m + h ) 2 k 2 0 +b( m + h ) 2[ S gp(m,h,z,ω)], (25) where m and h represent the first-order partial differentiation with respect to midpoint m and half offset h, respectively. This expression is calculated by a multiway splitting implicit finitedifference scheme to ensure the stability of the procedure and to minimize the numerical anisotropy (Ristow and Ruhl, 997). This implementation is similar to FFD, but in midpoint-offset coordinates. accuracy with large velocity perturbation, we choose v/v 0 =.5. The inner circle is the accurate dispersion curve; the outer circle is the dispersion curve from background velocity (i.e., the phase-shift solution). The phase-screen curve gives correct phase in the vertical direction but has large errors at wider angles. The pseudoscreen curve has a better solution and is closer to the accurate one. Impulse responses are also tested to investigate the dip effect of phase-screen and pseudoscreen propagators. Figure 2a shows the accurate impulse response of migration with offset 900 m within a constant velocity medium of 2000 m/s. In order to see the accuracy of these propagators with a velocity contrast, 333 m/s is used as a background velocity, resulting in a velocity ratio v/v 0 =.5. The phase-screen response is only accurate for small-angle waves, as shown in Figure 2b. Figure 2c shows the pseudoscreen impulse response. The pseudoscreen propagator significantly improves the steep-dip performance in the presence of realistic velocity perturbations. 3-D common-azimuth screen migration For 3-D prestack migration, the vertical wavenumber k z still can be expressed as a DSR operator, k z (K mx, K my, K hx, K hy,z,ω) = k0 2 [ (Kmx K hx ) (K my K hy ) 2] + k 2 0 4[ (Kmx + K hx ) 2 + (K my + K hy ) 2], (26) where K mx is the inline common-midpoint (CMP) wavenumber, K my is the crossline CMP wavenumber, K hx is the inline offset wavenumber, and K hy is the crossline offset wavenumber. Accuracy analysis Figure compares the dispersion curves from different propagators. For simplicity, we assume K h = 0 in this test. To show FIG.. Comparison of dispersions from different propagators. Velocity ratio v/v 0 =.5. The horizontal and vertical axes correspond to horizontal and vertical wavenumbers, respectively.

5 Pseudoscreen Prestack Depth Migration 899 FIG. 2. Impulse responses of 2-D offset-domain screen migration. The input data set contains a band limited impulse at t = 800 ms for the constant-offset section of 900 m within a constant velocity medium 2000 m/s. (a) An accurate response with the true velocity, (b) the phase screen response with a velocity perturbation, and (c) the pseudoscreen impulse response. In both (b) and (c), 333 m/s is used as a reference velocity, resulting in the velocity, ratio v/v 0 =.5. As discussed by Biondi and Palacharla (996), we can take advantage of the limited azimuthal range of 3-D marine data to reduce significantly the computational cost by applying a downward continuation operator only along the offset-azimuth plane by stationary phase approximation. This approximation reduces the dimensionality of the 3-D DSR operator from 5 to 4, and significantly cuts the computational cost of 3-D common-azimuth migration. Figure 3 compares the impulse responses of commonazimuth migration with and without stationary-phase approximation. Figure 3a shows a depth slice assuming the crossline offset wavenumber K hy in equation (26) is set to zero for downward continuation. The maxium error is oriented at 45 with respect to the inline direction (Biondi, 2000). Figure 3b is the same depth slice using stationary-phase path K hy.it agrees quite well with the contour lines of the analytic impulse response. MIGRATION EXAMPLES We now illustrate the screen migration method with 2-D and 3-D synthetic data sets. The first example is applied to the 2-D Marmousi data set (Bourgeois et al., 99). The velocity model is shown in Figure 4a. It contains complicated geological features, especially shallow steep faults and an underlying high-velocity lateral salt body intrusion. In addition, the model contains salt structure related traps and a reservoir structure beneath this complex geology. During the migration, we use the minimum velocity as a background velocity at each depth step, because dips at low velocities with higher background velocities could be in the evanescent region (Kessinger, 992). Forty-eight offset sections with full offset range between 200 and 2550 m were used in the migration. Figure 4b shows the depth image by the pseudoscreen propagator. This algorithm images the shallow steep faults and a reservoir structure well enough to identify the complicated geological features. As a comparison, Figure 4c shows the result obtained by Kirchhoff migration using finite-difference eikonal traveltimes. As expected for a technique based on first arrivals, the multiple arrivals generated by this model result in mispositioning of reflections in complicated velocity regions. FIG. 3. Impulse response of 3-D common-azimuth migration. (a) A depth slice assuming K hy =0. (b) The depth slice with stationary-phase approximation. The solid contour lines correspond to the exact spreading surface.

6 900 Jin et al. As a second example, we show results from 3-D commonazimuth pseudoscreen migration of the 3-D SEG-EAEG salt model (Aminzadeh et al., 997). In this test, the input data are azimuthal-moveout (AMO) processed data from the C3 Narrow Angle (C3NA) subset of the SEG-EAEG model dataset. The result of AMO processing is a series of regularly sampled 3-D common-offset, common-azimuth volumes with a 20-m CMP spacing in both inline and crossline directions and with 80-m sampling along the inline offset. In the C3NA subset, there are 400 inlines, 400 crosslines, and 32 offsets (2400-m maximum offset). Three-dimensional pseudoscreen programs take common offset volumes as input, and produce a 3-D zero-time, zero-offset image volume as output. Sample results are shown in Figures 5 7. Figure 5 compares images of inline section 479 (X = 9560 m). Figure 5a shows the velocity model. Figure 5b and 5c are the images obtained by 3-D common-azimuth pseudoscreen and phase screen migrations, respectively. The pseudoscreen implementation shows significant improvements. The salt body is clearly defined, and the fault is also well imaged. Figure 6 shows a depth slice at 260 m. Figure 6a is the corresponding velocity model. Figure 6b is the image result obtained by 3-D common-azimuth pseudoscreen migration. Most features are consistent with the velocity model. As a comparison, the phase-screen migration result is shown in Figure 6c. Image quality is significantly degraded in the subsalt areas. To evaluate the impact of the common azimuth approximation, we compare results from a finite-difference shot-record migration of the same data volume (Ober et al., 997) shown in Figure 7. The pseudoscreen image of inline 479 is shown in Figure 7a, and the same inline from a shot migration program FIG. 4. Comparison of migration on Marmousi model between offset-domain pseudoscreen migration and Kirchhoff migration. (a) The velocity model. (b) The depth image obtained by offset-domain pseudoscreen migration. (c) Obtained by Kirchhoff migration with first-arrival traveltimes. FIG. 5. Three-dimensional prestack depth migration of SEG-EAGE C3NA salt model. (a) The velocity model for inline 479. (b) The image obtained by 3-D pseudoscreen migration. (c) The image obtained by phase-screen migration. The crossline interval is 20 m.

7 Pseudoscreen Prestack Depth Migration is in Figure 7b. Image quality in the sediment areas is better for our results because of the improved steep-dip response of the pseudoscreen propagator compared to the finite-difference propagator used in the shot-record migration (Ober et al., 997). Subsalt image quality is comparable for the two results. 90 The moderately dipping fault in the sediments at the lower right side of the image is more visible on the pseudoscreen result, again because of the improved steep dip fidelity of the pseudoscreen operator. We conclude that the common azimuth approximation is robust, even for complex geologic structures with high contrast. In terms of the computational time, it takes about 8 hours to migrate the full volume on a 32-CPU 800-MHz Linux PC cluster, whereas shot-record 3-D finite-difference migration takes about 000 hours for the same image volume (4800 shots). Of course, the computation time for shot-record migration could be reduced by limiting aperture, the number of shots, and migrating multiple shots together, but at the expense of image quality. CONCLUSIONS We have presented a pseudoscreen prestack depthmigration method in midpoint-offset coordinates. The pseudoscreen propagator consists of a phase-screen propagator and a wide-angle compensation term. The phase-screen propagator is implemented in the mixed space-wavenumber domain. The wide-angle term is implemented by an implicit finitedifference scheme to stablize the recursive calculation. This term enhances the steep dips in strong contrast velocity media. The source and receiver wavefields are simultaneously downward continued with a zero-time and zero-offset imaging condition at each depth step. The algorithm has all the advantages of phase-shift and phase-screen migration or split-step DSR FIG. 6. Three-dimensional prestack depth migration of SEG-EAGE C3NA salt model. (a) The velocity slice at depth 260 m. (b) The depth image obtained by 3-D pseudoscreen migration. (c) The depth image obtained by phase-screen migration. Both inline and crossline interval are 20 m. FIG. 7. Comparison with 3-D shot migration of SEG-EAGE C3NA salt model. (a) The depth image of inline 479 by 3-D pseudoscreen migration. (b) The depth image obtained from Sandia Labs Salvo 3-D shot-record finite-difference migration.

8 902 Jin et al. migration, but also accommodates strong lateral velocity variations. The method is applicable to both 2-D offset-domain migration and 3-D common-azimuth migration. Migrations of the 2-D Marmousi and 3-D SEG-EAGE C3 Narrow Angle data sets show that the pseudoscreen propagator provides improved image quality over Kirchhoff implementations, and comparable quality to shot-record finite-difference migrations. Computational time is more than 00 times faster than shotrecord finite-difference migration for full-aperture imaging of the same data volume. ACKNOWLEDGMENTS The authors thank D. J. Foster and P. L. Stoffa for many valuable discussions on the migration in midpoint-offset coordinates, and M. V. de Hoop, L. J. Huang, M. C. Fehler, D. Walraven, X. B. Xie, C. Peng, R. C. Cook, and S. Xu for advice on the implementation of screen propagators. We also thank X. B. Xie for his help with plotting dispersions. We gratefully acknowledge Biondo Biondi and the Stanford Exploration Project for supplying the AMO-processed commonazimuth data set. We also thank Curtis Ober and Sandia National Laboratory for providing results from the Salvo shotrecord migration of the SEG data. Finally, we thank associate editor Tamas Nemeth, B. Biondi, P. Sava, and anonymous reviewers for many constructive comments. REFERENCES Alkhalifah, T., 2000, Prestack phase-shift migration of separate offsets: Geophysics, 65, Aminzadeh, F., Brac, J., and Kunz, T., 997, 3-D salt and overthrust models: Soc. Expl. Geophys. Audebert, F., Nichols, D., Rekdal, T., Biondi, B., and Urdaneta, H., 997, Imaging complex geologic structure with single-arrival Kirchhoff prestack depth migration: Geophysics, 62, Biondi, B., 2000, 3-D wave-equation prestack imaging under salt: 70th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, , Stable wide-angle Fourier finite-difference downward extrapolation of 3-D wavefields: 7st Ann. 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