GEOPHYSICS, VOL. 64, NO. 1 (JANUARY-FEBRUARY 1999); P. 222–229, 12 FIGS.

3-D preserved amplitude prestack depth migration on a workstation Philippe Thierry∗ , Gilles Lambare´ ∗ , Pascal Podvin∗ , and Mark S. Noble∗ is incorporated in this kinematic approach, migration-based amplitude variation with offset (AVO) analysis cannot be performed (Beydoun et al., 1993; Tura et al., 1997). Moreover, complex structure imaging is often unsuccessful when based only on first-arrival traveltimes (Geoltrain and Brac, 1993). Until now, the extension of eikonal solvers to amplitude and multipath computation have not been successful. On the contrary, ray tracing does not suffer such limitations. It is now possible to improve 3-D depth migration since the last generation of dynamic ray-tracing algorithms based on the wavefront construction (WFC) method (Vinje et al., 1993, 1996a, b; Lambare´ et al., 1996; Lucio et al., 1996) is perfectly adapted to this purpose. In addition to computational efficiency, the possibility of obtaining quantitative information on reflectors (namely, reflection coefficient or impedance contrast) is another major advantage of ray-based migration methods. Theoretically, migration is reformulated in the frame of linearized seismic inversion. In particular, it was shown that ray+Born (Beylkin, 1985) or ray+Kirchhoff (Bleistein, 1987) linearized inversion translated into rather simple quantitative migration formulas. Jin et al. (1992) reconciled such direct inversion approaches with the general stochastic approach (Tarantola, 1987; Beydoun and Mendes, 1989) and introduced an iterative minimization together with extension to multiparameter inversion (Forgues and Lambare, ´ 1997). Here, we define preserved amplitude prestack depth migration (PAPsDM) as the result of a single iteration of ray+Born inversion for a single parameter (Thierry and Lambare, ´ 1995). We already demonstrated that PAPsDM could reasonably recover the amplitude of reflectors (Thierry et al., 1996; Operto et al., 1997a, b). We present an algorithm for 3-D PAPsDM together with an application to real data. We show that 3-D PAPsDM can be done with a relatively low computational cost (all results presented in this paper were computed on a workstation). Such a performance is attained with the use of embedded interpolations that were discussed and calibrated for the 2-D and 3-D cases in Operto et al. (1997a, b) and Thierry et al. (1999).

ABSTRACT

We present an algorithm based on the ray+Born approximation for 3-D preserved amplitude prestack depth migration (PAPsDM) of seismic reflection data. This ray+Born inversion scheme allows the quantitative recovery of model perturbations. The Green’s functions are estimated by dynamic ray tracing in 3-D heterogeneous smooth velocity fields with a wavefront construction (WFC) method. The PAPsDM algorithm was implemented on a single-processor Sun SPARC 20 workstation. Special attention was paid to CPU efficiency and memory requirements. We present an application on a 3-D real marine data set (13 Gbytes). About one week of CPU time is needed to obtain a migrated image of 7 × 1 × 1 km.

INTRODUCTION

Three-dimensional prestack depth migration (PsDM) is certainly the most accurate approach for imaging laterally heterogeneous media. Because of extensive CPU and memory requirements for 3-D seismic applications, most of the present 3-D PsDM algorithms are based on ray theory (Kirchhoff migration). In fact, it is now accepted that ray theory provides an excellent compromise between precision and computational efficiency in 3-D heterogeneous media. The main difficulty of ray-based migration is the computation of traveltimes in the target for all shot and receiver positions. In this perspective, a finite-difference solution of the eikonal equation for computing first-arrival traveltimes (Vidale, 1988; Podvin and Lecomte, 1991) was an early significant breakthrough. It provides extremely fast algorithms for kinematic migration (Reshef, 1991; Mufti et al., 1996; Noble et al., 1996). Yet, since no reliable amplitude information

Presented at the 66th Annual International Meeting, Society of Exploration Geophysicists. Published on Geophysics Online, October 8, 1998. Manuscript received by the Editor August 4, 1997; revised manuscript received April 9, 1998. ∗ Ecoles des Mines de Paris, ARMINES-GEOPHY, 35 rue Saint Honore, 77305 Fontainebleau Cedex, France. E-mail: philippe.thierry@geophy. ensmp.fr, [email protected], [email protected], [email protected]. ° c 1999 Society of Exploration Geophysicists. All rights reserved. 222

3-D PAPsDM

In this paper, we first recall the preserved amplitude migration approach in the specific case of a 3-D marine acquisition. Practical aspects of the PAPsDM algorithm, including interpolation strategies, are then detailed in the context of our application. Three million seismic traces were migrated into a small target (7 × 1 × 1 km). All parameters, including computation times, are provided for future comparisons and improvements. We show a residual data section, namely, the difference between an observed shot gather and a synthetic one computed with 3-D ray+Born modeling. Finally, we show that 3-D PAPsDM improves the image significantly when compared to 2.5-D PAPsDM. THEORETICAL ASPECTS

Preserved amplitude migration is based on the ray+Born approximation (Cohen et al., 1986). In this approximation, the model is split into a reference model (background) and a perturbation model δm. The perturbation of Green’s function (δG) corresponding to the reflected scalar wavefield is related linearly to the model perturbation by

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The solution δm minimizing the cost function (2) is well known as

δm = (B † QB)−1 B † QδGobs ,

where † denotes a transposed conjugated operator. The operator (B † QB) is called the Hessian. In classical seismic inversion (Tarantola, 1984; Beydoun and Mendes, 1989), this operator is discretized and becomes a huge matrix that cannot be inverted numerically. Consequently, iterative gradient minimization generally is introduced. To improve this local minimization, Jin et al. (1992) proposed to diagonalize the Hessian by choosing a judicious local weighting function Q in the cost function (2). Such an approximate Hessian allows for an efficient quasi-Newton iterative minimization of the cost function (Lambare´ et al., 1992; Thierry and Lambare, ´ 1995). The first iteration is equivalent to the Beylkin migration formula (Beylkin, 1985). For our marine case, the final 3-D PAPsDM expression is

δm(x0 ) =

Z

δGcal (r, ω, s) = ω2

dx δm(x)A(r, x, s) eiωT (r,x,s) , (1)

where s, r, and x respectively denote the source, receiver, and diffractor point positions, ω is the angular frequency, and A(r, x, s) = A(r, x)A(x, s) and T (r, x, s) = T (r, x) + T (x, s) are, respectively, the amplitude of the Born operator and the two-way traveltime computed in the reference model (Figure 1). The linear relation (1) can be inverted in the frame of the stochastic inverse problem theory (Tarantola, 1987). Consider a standard 3-D marine data set (δGobs ) with (ideally) parallel acquisition lines where each line consists of a number of shots recorded with moving streamers (Figure 1). Let us introduce the weighted `2 cost function C associated with a model perturbation δm(x),

C(δm) =

XZ

1 2 LSR

ω

dωQ|δGobs − δGcal (δm)|2 ,

(2)

where L , S, R are, respectively, the line, shot, and receiver numbers, and Q is a weighting function equivalent to a covariance matrix in the data space.

(3)

XXX 1 max [|q(x0 )|]min L S R × E(r, x0 , s)δGobs (r, T (r, x0 , s), s),

(4)

¯ ¯ 1 |q|1L1S1R ¯¯ ∂(q) ¯¯ E(r, x0 , s) = , (2π )2 A(r, x0 , s) ¯ ∂(L , S, R) ¯

(5)

with

where 1L, 1S, and 1R denote, respectively, the increments in line, shot, and receiver numbers, and vector q = pr + ps is the sum of the slowness vectors (Figure 1). The migration operator E compensates for the two-way amplitude (1/A), accounts for the acquisition geometry through the Jacobian |∂(q)/∂(L , S, R)|. The Jacobian operator can be written as

¯ ¯ ¯ ¯ ¯ ∂(q) ¯ ¯ ∂(pr + ps ) ∂(r, s) ¯ ¯=¯ ¯ ¯ ¯ ∂(L , S, R) ¯ ¯ ∂(r, s) ∂(L , S, R) ¯,

(6)

where the paraxial quantities (∂(pr )/∂(r), ∂(ps )/∂(s)) are computed during ray tracing (Farra and Madariaga, 1987; Lambare´ et al., 1996) and the matrix (∂(r, s)/∂(L , S, R)) is inferred from the acquisition geometry. It can be evaluated by finite differencing r(L, S , R) and s(L, S , R) (Clochard et al., 1997) or more simply approximated by constant mean values, as was done in our application. PRACTICAL ASPECTS

FIG. 1. Ray+Born approximation. Vector q is the sum of the slowness vectors pr = ∇x0 T (x0 , r) and ps = ∇x0 T (x0 , s).

Because of the size of a 3-D data set and target, 3-D prestack depth migration is one of the most expensive processing steps. Practical implementation requires the use of simplifications. As discussed in Thierry and Lambare´ (1995), we use embedded interpolations to limit the number of rays traced from the surface and the number of operations per image point in the target. Accounting for the preserved amplitude term, our 3-D PAPsDM algorithm [equation (4)] requires 11 parameters [the traveltime T , the amplitude A, the slowness vector p, the Jacobian operator ∂(p)/∂(x), where x denotes the surface position of the ray shooting point]. The computation of these 11 parameters is done in a smooth background velocity field with the dynamic ray-tracing algorithm developed by Lucio et al. (1996)

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and based on the WFC method proposed by Vinje et al. (1993). This algorithm uses a uniform ray-density criterion, providing an optimal compromise between computational efficiency, accuracy, and robustness, even in the case of multipathing and caustics. When the reference velocity field does not exhibit very strong lateral variations, surface interpolation of Green’s functions may be reasonably proposed to reduce the number of maps computed by ray tracing. We chose to compute all the maps during a preliminary stage and to store them on disk (Thierry and Lambare, ´ 1995). The required maps are read during the migration and partially stored in memory to take advantage of the vicinity of receiver positions to limit the number of input/output operations. Computation of the migration operator is an expensive operation. If T and E are supposed to have slow variations over the target zone, it appears reasonable to calculate them over a sparse regular target grid and then use linear interpolation onto the fine target grid. Finally, the contribution of each trace to the final migrated image is limited to a region around the common-midpoint position (Lambare´ et al., 1992; Thierry and Lambare, ´ 1995). The precision of all these approximations was discussed and calibrated in structurally simple cases (as in our 3-D application) and also in case of complex synthetic models (Operto et al., 1997a; Thierry et al., 1999).

Preprocessing and data management Data preprocessing (including spiking deconvolution, waterbottom multiple removal, and filtering) and velocity analysis were done by Elf GRC. In order to deal with limited disk resources and to manage this 13 Gbyte data set, we developed software for a very simple

FIG. 3. The 29 navigation lines of the 3-D real data set. Note the irregularity in the acquisition.

APPLICATION

We present an application of the 3-D PAPsDM scheme to a real 3-D marine data set acquired in the North Sea and provided by Norsk Hydro. This data set consists of 29 survey lines with about 500 shots per line (see the common first offset section in Figure 2). The acquisition consisted of one source and two streamers. Each streamer had 120 receivers spaced 25 m apart (about 3 million traces). The mean distance between cables was 100 m. Figure 3 shows the distribution of shot positions for the 29 navigation lines and Figure 4 shows receiver positions for the first shot of lines 25 and 26. This real acquisition geometry, substantially different from an ideally periodic pattern, is accounted for in the migration scheme.

FIG. 4. Streamers positions for the first shot of the lines 25 and 26. The feathering goes up to 500 m.

FIG. 2. Common first offset section for line 16 after preserved amplitude preprocessing. In accordance with our approach, no trace equalization was performed.

3-D PAPsDM

data compression/decompression algorithm. Our goal was to store data in a moderately compressed form (a compression rate on the order of three was sufficient) that would still allow fast, flexible, and direct access from the migration/inversion codes. A compressed seismic data format was adapted from the standard IEEE-754 32 bits floating point encoding norm, simply using fewer bits for both mantissa and exponent. For our data set, we adopted a (very conservative) encoding on 15 bits (exponent on 5 bits, mantissa on 9 bits), ensuring a relative precision better than 0.1%, whereas the dynamic range was on the order of 4.0e + 9. Last but not least, simply accounting for the applied mute yielded an almost supplementary 30% gain in file size. As a result, each seismic line originally occupying about 500 Mbytes was compressed onto a single file roughly one-third times smaller. Both computational and precision penalties caused by the decompression filter was found to be insignificant.

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done in a coarsely sampled moving cube (5.0 × 2.6 × 2.0 km) centered around each shotpoint, with its top at 1490 m and sampled at 101 m in the three directions. Computation time for each ray tracing was 290 s on a Sun SPARC 20 workstation. These initial maps (317 Mbytes) were stored on disk, and their computation took a total time of 20 hours. The second step consisted of a linear interpolation of the initial 500 × 500 m surface grid onto one at 100 × 100 m (Figure 7a). Interpolations were performed at constant offset to be sensitive only to lateral variations of the velocity field. Finally ray parameter maps were interpolated by cardinal cubic B-splines onto the desired target in about 4 hours

Velocity model and ray tracing A velocity model was provided in the form of a velocity grid with 500 m in-line (i.e., in the east-west direction), 25 m crossline (in the north-south direction), and 200 m vertically. In the ray-tracing code used for 3-D PAPsDM, model properties must be represented by cardinal cubic B-spline weights (Lucio et al., 1996). The original model was converted into B-spline representation by fitting the slowness field (Operto et al., 1997b). As there were only gentle cross-line velocity variations, the spacing of the B-spline nodes was chosen to be 500 m in both in-line and cross-line directions, while the original vertical spacing was kept unchanged (Figure 5). Because of the gentle structure of the macro model, we decided to compute the ray parameters maps using three embedded interpolation steps. First, shooting was performed from a coarse surface grid (Figure 6) with 36 × 7 points and a spacing of 500 m in both the x- and y-directions. Each ray shooting was

FIG. 6. Initial ray shotpoint grid. The spacing of this surface grid was 500 m in both the x- and y-directions, and there were 36 × 7 points.

FIG. 5. Several slices in the 3-D heterogeneous velocity model. This very simple model was provided by Elf GRC after conventional velocity analysis.

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(Figure 7b). The advantage of this three-step procedure is that once the initial ray tracing has been done, ray parameter maps can be interpolated easily and rapidly onto any desired target.

a)

Results

b)

FIG. 7. (a) Linear interpolation of maps from a 500 × 500 m surface grid onto a 100 × 100 m surface grid. Interpolations were performed at constant offset in order to be sensitive only to lateral variations of the velocity field. (b) The ray parameter maps were interpolated by cubic B-splines onto the desired coarsely sampled target.

As a first application, we imaged a 7 × 1 × 1 km volume using all the 29 navigation lines. The bin size was 25 × 25 × 5 m. This application took one week on a Sun SPARC 20 workstation using only 48 Mbytes of RAM. Figure 8 shows a vertical slice where tilted blocks appear separated by faults. Figures 9 and 10 allow us to see the lateral variations of the structure making it easier to understand the 3-D geometry of the target. From the migrated image, we can compute synthetic traces using the 3-D ray+Born expression (1). Figure 11 shows the real, the calculated, and the residual data for shot 201 of navigation line 16. In calculated data, major reflected events were recovered with a slight time mismatch because an imperfect velocity model was used. Thus this imperfection essentially explains the high energy in residual data. As a second application, we imaged a vertical in-line section using seven seismic lines (numbers 22–28) with the 3-D algorithm, and only one line (number 25) with the 2.5-D algorithm. Figure 12 show the 3-D and 2.5-D migrated sections of the velocity perturbation. Computation times on a Sun SPARC 20 workstation were, respectively, 4 hours and 12 minutes, including the Green’s function computation. Profiles are plotted at the same scale. Three-dimensional PAPsDM greatly improved the general quality of the image. The in-plane hypothesis fails both because of the strong lateral drift of the streamers (Figure 4) and the lateral variation of the geological structure (the major faults are not oriented orthogonally to the lines). CONCLUSION

We have demonstrated that 3-D PAPsDM for a small target volume is affordable even on standard workstations if some

FIG. 8. Vertical slice in the 3-D migrated cube. Some majors faults appear in the tilted block area.

3-D PAPsDM

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FIG. 9. Horizontal section in the 3-D migrated cube, clearly showing the major faults in the tilted block area.

FIG. 10. Sections in the 3-D migrated cube showing the lateral variation of the structure. judicious simplifications are introduced. We have shown that application to 3-D real data greatly improves the 2.5-D results even in case of small variations in the velocity model. The interpolation effects as well as the quantitative recovery of model perturbations was detailed in Thierry and Lambare´ (1995) and Thierry et al. (1999). The main advantages of interpolations are in the significant decrease of the required memory space and in a slight decrease of the computation time. The model is certainly not very complex in terms of velocity field. However, it has been demonstrated that this method could be applied successfully to complex cases (Marmousi 2-D and Overthrust 3-D) (Operto et al., 1997a, b). With such an approach, computation time for PAPsDM is slightly greater than the one needed for kinematic PsDM (which cannot handle any reliable amplitude information), but it introduces a practical method for 3-D migration-based AVO analysis (Tura et al., 1997) since it can be transformed easily into ray+Kirchhoff common-offset migration.

This efficient algorithm could also be used in the near future as a basis for migration velocity analysis (Liu and Bleistein, 1995; Tieman, 1995). ACKNOWLEDGMENTS

This work was partly funded by the European Commission and Norwegian Research Council in the framework of the JOULE projects “3-D Asymptotic Seismic Imaging” (contract JOU2-CT93-0321) and “Reservoir Oriented Delineation Technology” (contract JOF3-CT95-0019). We thank Norsk Hydro for providing the 3-D data set, and C. Hanitzsch and A. Tura (Elf GRC) for preprocessing it. The paper benefited ´ from very fruitful discussions with Stephane Operto (Ecole des Mines de Paris), Jan Pajchel (Norsk Hydro), and Henri Calandra (Elf Aquitaine). REFERENCES Beydoun, W. B., and Mendes, M., 1989, Elastic ray-born `2 -migration/inversion: Geophys. J., 97, 151–160.

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FIG. 11. Real data (a), calculated data (b), and residuals (c) for one streamer of shot gather 201. Considering the low signal/noise ratio of data and the inacurracy of the velocity model used for migration, a low-pass filter ([0, 14, 20, 30] Hz) had to be applied to the data to allow for significant comparisons.

Beydoun, W. B., Hanitzsch, C., and Jin, S., 1993, Why migrate before AVO? A simple example: 55th Ann. Mtg., Eur. Assn. Expl. Geophys., Extended Abstracts, B044. Beylkin, G., 1985, Imaging of discontinuities in the inverse scaterring problem by inversion of a causal generalized radon transform: J. Math. Phys., 26, 99–108. Bleistein, N., 1987, On the imaging of reflectors in the earth: Geophysics, 52, 931–942. Clochard, V., Nicoletis, L., Svay-Lucas, J., Mendes, M., and Anjos, L., 1997, Interest of ray Born modeling and imaging for 3-D walka-

ways: 59th Ann. Mtg., Eur. Assn. Geoscientists Engin., Expanded Abstracts, P035. Cohen, J. K., Hagin, F., and Bleistein, N., 1986, Three-dimensional Born inversion with an arbitrary reference: Geophysics, 51, 1552–1558. Farra, V., and Madariaga, R., 1987, Seismic waveform modeling in heterogeneous media by ray perturbation theory: J. Geophys. Res., 92, 2697–2712. Forgues, E., and Lambare, ´ G., 1997, Parametrization study for acoustic and elastic ray+born inversion.: J. Seis. Expl., 6, 253–277. Geoltrain, J., and Brac, S., 1993, Can we image complex structures with first-arrival traveltime?: Geophysics, 58, 564–575. Jin, S., Madariaga, R., Virieux, J., and Lambare, ´ G., 1992, Twodimensional asymptotic iterative elastic inversion: Geophys. J. Internat., 108, 575–588. Lambare, ´ G., Lucio, P. S., and Hanyga, A., 1996, Two-dimensional multivalued traveltime and amplitude maps by uniform sampling of ray field: Geophys. J. Internat., 125, 584–598. Lambare, ´ G., Virieux, J., Madariaga, R., and Jin, S., 1992, Iterative asymptotic inversion of seismic profiles in the acoustic approximation: Geophysics, 57, 1138–1154. Liu, Z., and Bleistein, N., 1995, Migration velocity analysis: Theory and an iterative algorithm: Geophysics, 60, 142–153. Lucio, P. S., Lambare, ´ G., and Hanyga, A., 1996, 3-D multivalued travel time and amplitude maps: Pageoph, 148, 449–479. Mufti, I. R., Pita, J. A., and Huntley, R. W., 1996, Finite-difference depth migration of exploration-scale 3-D seismic data: Geophysics, 61, 776–794. Noble, M., Marsset, B., Missiaen, T., and Versteeg, W., 1996, Near surface 2-D and 3-D data processing—beyond stack: 58th Ann. Mtg., Eur. Assn. Expl. Geophys, Expanded Abstracts, M033. Operto, S., Lambare, ´ G., Podvin, P., and Thierry, P., 1997a, 3-D preserved amplitude prestack imaging of the Overthrust model: 59th Ann. Mtg., Eur. Assn. Geoscientists Eng., Expanded Abstracts, A043. ——— 1997b, CPU efficient ray+Born inversion for complex velocity fields: 59th Ann. Mtg., Eur. Assn. Geoscientists Eng., Expanded Abstracts, P033. Podvin, P., and Lecomte, I., 1991, Finite difference computation of traveltimes in very contrasted velocity model: A massively parallel approach and its associated tools: Geophys. J. Internat., 105, 271–284. Reshef, M., 1991, Prestack depth imaging of three-dimensional shot gathers: Geophysics, 56, 1158–1163. Tarantola, A., 1984, Inversion of seismic reflexion data in the acoustic approximation: Geophysics, 49, 1259–1266. ——— 1987, Inverse problem theory: Methods for data fitting and model parameter estimation: Elsevier. Thierry, P., and Lambare, ´ G., 1995, 2.5D true amplitude migration on a workstation: 65th Ann. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 156–159. Thierry, P., Lambare, ´ G., Podvin, P., and Noble, M., 1996, 3D prestack preserved amplitude migration: Application to real data: 66th Ann. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 555–558. Thierry, P., Operto, S., and Lambare, ´ G., 1999, Fast 2-D ray+Born migration/inversion in complex media: Geophysics, 64, 162–181. Tieman, H. J., 1995, Migration velocity analysis: Accounting for the effects of lateral velocity variations: Geophysics, 60, 164–175. Tura, A., Hanitzsch, C., and Calandra, H., 1997, 3-D AVO migration/inversion of field data: J. Seis. Expl., 6, 117–125. Vidale, D., 1988, Finite-difference calculation of travel time: Bull. Seis. Soc. Am., 78, 2062–2076. Vinje, V., Iversen, E., Astebøl, K., and Gjøystdal, H., 1996a, Estimation of multivalued arrivals in 3-D models using wavefront construction—Part I: Geophys. Prosp., 44, 819–842. ——— 1996b, Estimation of multivalued arrivals in 3-D models using wavefront construction—Part II: Tracing and interpolation: Geophys. Prosp., 44, 843–858. Vinje, V., Iversen, E., and Gjøystdal, H., 1993, Traveltime and amplitude estimation using wavefront construction: Geophysics, 58, 1157–1166.

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FIG. 12. (a) 2.5-D PAPsDM section of line 25 (y = 5.81 km). (b) 3-D PAPsDM result for the same section using seven navigation lines (numbers 22–28). The two sections are plotted with the same gain.

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