GEOPHYSICS, VOL. 69, NO. 5 (SEPTEMBER-OCTOBER 2004); P. 1320–1328, 13 FIGS. 10.1190/1.1801948

Fast migration/inversion with multivalued rayfields: Part 2—Applications to the 3D SEG/EAGE salt model

Sheng Xu1 , Gilles Lambare´ 2 , and Henri Calandra3 sition of the migration/inversion operator in elementary regular patches. A patch is defined by an individual cell of the coarse grids and by an individual ray branch. The definition of the patches requires the identification of the ray branches over the cells. It is done via a set of criteria based on the KMAH index and on a traveltime continuity condition (Xu and Lambare, ´ 2004). For each patch the contributions to the migration/inversion operator are ultimately obtained by linear interpolation of the amplitude and phase of the kernel of the migration/inversion operator. The extrapolation strategy was proposed for reconstructing the migration/inversion operator from the coarsely sampled maps. For each arrival at a coarse gridpoint in the target, the contribution to the migration/inversion operator is extrapolated using a local plane function. The continuity of the extrapolated functions ensures the continuity of the image, and the results have better quality when compared to those obtained with our interpolation strategy. The simplicity of the numerical implementation of the extrapolation strategy is an additional advantage, counterbalanced in two dimensions by a computational time increased by a factor of two. In this paper we extend these strategies to three dimensions. It was the great motivation for our work, since complex models are generally three dimensional and alternatives to raybased migrations in three dimensions still remain expensive. The strategies can be directly applied in three dimensions (refer to Part I for details). The extension to three dimensions, however, involves some numerical specifics that we focus on here. Concerning the numerical implementation, attention has to be paid to the structure of the algorithm. Even though the final images were obtained using an SGI Origin 2000 machine with eight processors at 295 MHz, the development and testing of the codes was fully done on single-processor Sun Sparc workstations. This limitation forced us to adopt specific numerical solutions described in detail in the first paper.

ABSTRACT Three-dimensional prestack depth migration is the convenient approach for seismic imaging in the case of strong lateral variations of the velocity. Because of computing limitations, it has been limited to single-arrival kinematic Kirchhoff migration until recently. This approach fails in the case of complex media characterized by multiarrival traveltimes. We present numerical strategies for extending in three dimensions first-arrival kinematic Kirchhoff migration to multiarrival quantitative ray-based migration (preserved amplitude migration). We rely on wavefront construction in a smooth velocity model to compute the multivalued traveltime and amplitude maps, and the CPU efficiency of migration itself is ensured by efficient and robust interpolation or extrapolation strategies. We present an application to the synthetic 3D SEG/EAGE salt model. Taking into account multiarrivals clearly improves subsalt imaging at the price of quite limited computing costs (a 20% increase in our case, with respect to a preserved-amplitude single-arrival migration).

INTRODUCTION This paper is the second of a pair addressing efficient implementation of ray-based migration/inversion in the case of complex velocity macromodels characterized by multivalued travel time maps. In the first paper (Xu and Lambare, ´ 2004), hereafter referred to as Part I, we present two kinds of strategies (an interpolation strategy and an extrapolation strategy), accompanied by 2D tests with the Marmousi model and data set. The strategies rely on a coarse sampling of the migration/inversion operator both at the acquisition surface and in the target. The interpolation strategy is based on the decompo-

Manuscript received by the Editor January 16, 2001; revised manuscript received December 4, 2003. 1 ´ Formerly Ecole des Mines de Paris, Fontainebleau, France; presently Veritas Geophysical Corporation, 10300 Town Park Drive, Houston, Texas 77072. E-mail: sheng [email protected]. 2 Ecole des Mines de Paris, Centre de Recherche en Geophysique, ´ 35 rue Saint Honore, ´ 77 305 Fontainebleau Cedex, ´ France. E-mail: [email protected]. 3 TotalFinaElf, 800 Gessner, Suite 700, Houston, Texas 77025. E-mail : [email protected]. ° c 2004 Society of Exploration Geophysicists. All rights reserved. 1320

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Here, we detail how we implemented 3D multiarrival migration/inversion. Finally, we present an application to the 3D SEG/EAGE salt model (Aminzadeh et al., 1997). The application is described precisely (preprocessing, velocity macromodel, etc.), and the results are compared to single-arrival migration/inversion results. Interpolation and extrapolation strategies are compared in terms of image quality and CPU efficiency.

arrivals are considered sequentially point after point, resulting into a very simple algorithm. The overlapping of the extrapolation areas produces results in extra cost with respect to the interpolation strategy. However, memory management is much simpler since in the target the arrivals are considered sequentially, independently of the neighboring points.

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APPLICATION TO SEG/EAGE SALT MODEL

1) computation of multivalued maps from a coarse grid at the surface and 2) 3D migration/inversion slice after slice.

The synthetic 3D SEG/EAGE salt model (Aminzadeh et al., 1997) offers an interesting opportunity for testing our 3D migration/inversion strategies. It is an acoustic model with constant density. It contains thin sand layers and a salt body with a complex topology, both superimposed on a smooth velocity model (Figure 1). The complexity of the topology of the top of the salt body makes subsalt imaging a challenge. A dense data set was computed by finite differences with a central frequency of 20 Hz, and several classical data sets were extracted and are made available (Aminzadeh et al., 1997). For our test we used the classical 3C-NA(narrow-angle acquisition) data set which is a typical 3D marine acquisition. The acquisition geometry is described in Figure 2.

To compute multivalued maps (eleven ray parameters for the migration/inversion amplitude weight and three for arrivals identification in the interpolation strategy), we use the 3D wavefront construction code developed by Lucio et al. (1996). This code uses a uniform ray density criterion that ensures an accurate and efficient sampling of the rayfield even in triplicated areas. The wavefront construction code requires a smooth velocity model defined by cubic cardinal B-splines (de Boor, 1978). When the velocity macromodel is provided as a dense grid, a smoothing procedure is applied, and the smooth model is projected onto a cubic cardinal B-splines basis (Operto et al., 2003). For efficiency, upgoing rays are eliminated and ray-shooting aperture is limited. Numerical efficiency may be strongly affected by disk access. The access to the ray-computed maps was a serious bottleneck in previous 3D migration/inversion codes (Thierry et al., 1999a). Ideally, all multivalued maps should be stored in core memory, but their size exceeds currently available memory sizes, so compromises had to be introduced. We propose organizing the migration with an outer loop over the in-line slice in the target. This new organization requires that the data set be reread for each slice in the target, but the final gain in terms of total time remains favorable. First, we reorganize the traveltime maps according to the slice index. Since migration is done slice after slice, we have to read the maps that we store in the core memory only once. For each slice in the target, the traces are migrated sequentially using the interpolation or extrapolation strategies described in Part I. The interpolation strategy is used systematically to interpolate the maps for shooting position at the surface. For the interpolation for migration in the target, we use alternatively the interpolation or extrapolation strategies. An optimal management of the core memory and disk access ensures CPU efficiency. From a numerical implementation point of view, the extrapolation strategy has the great advantage of simplicity: the

Figure 1. Velocity cubes in the 3D SEG/EAGE salt model.

Numerical implementation of prestack migration/inversion with multivalued rayfields requires a revision of the strategy used in the single–valued migration/inversion algorithms (Tura et al., 1998; Thierry et al., 1999a,b). These algorithms rely on interpolations of the amplitude and phase of the kernel of the migration/inversion operator from coarse grids at the surface and in the target. In Part I we propose strategies adapted to multiarrivals; here, we address specifics of the 3D algorithms. Our 3D migration/inversion codes involve two main steps:

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Preprocessing of SEG/EAGE salt data set Preprocessing of seismic data is very important for the quality of the results of migration/inversion. Ideally the multiples

and the refracted waves should be removed and a zero-phase deconvolution applied for the far-field signature. Preprocessing in a 3D complex model is a challenge and beyond the scope of this paper, where we apply a rather naive procedure. The source signature introduced in the finite-differences code was provided with the data set. In a first step we did a deterministic minimum-phase deconvolution of the source signature convolved with the shot and receiver ghosts. Then a predictive spiking deconvolution was applied to remove waterbottom reverberations (we used Seismic Unix software). The length of the predictive filter was adapted to the thickness of the water layer [maxlag = 2 × (ws + wr )/1500 + 0.15 s, where ws and wr denote, respectively, the water depth at shot and at receiver in meters]. Finally, a band-pass filter (5, 10, 24, 32 Hz) was applied to reduce the dispersion phenomena in the wavefield. Figure 3 shows some common shot gathers before and after the preprocessing. Our preprocessing is adapted to the structurally simple part of the model, but it is certainly insufficient in the complex parts of the model. For example, as we will see later on the migrated images, strong multiples of the top of the salt were not eliminated.

Velocity macromodel

Figure 2. The C3-NA acquisition in the SEG/EAGE salt model. The data set consists of 50 lines with 97 shots per line. There are 8 streamers per shot, with a maximum of 68 receivers per streamer. The spacing of the shot positions are, respectively, 160 m and 80 m in the cross-line and in-line directions. The spacing of receivers is 40 m in the in-line and cross-line directions. The red squares indicate the shot positions zone, and the white box indicates the position of the migrated/inverted volume.

Figure 3. Data for shot X = 3.86 km and Y = 3.18 km and for the second streamer: (a) raw data; (b) preprocessed data. Data for shot X = 5.78 km and Y = 5.58 km for the seventh streamer: (c) raw data; (d) preprocessed data.

The problem of the estimation of the velocity macromodel was clearly not addressed in this study. Fortunately, we had the exact velocity model used for the finite-difference modeling. It is a velocity grid with 20 m spacing in the three directions. For the migration/inversion our velocity macromodel is obtained by smoothing the exact model in slowness with a normalized 3D spatial Gaussian filter (Operto et al., 2000) (Figure 4). We use a 3D wavefront construction code with a uniform ray density

Figure 4. The 2D ray fields superimposed on vertical 2D sections of the 3D velocity macromodel. (a) In-line slice at X = 6.0 km; (b) cross-line slice at Y = 6.0 km.

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criterion to compute the multivalued maps. The initial fan of rays at the source is ±45◦ with respect to the vertical in the in-line and cross-line directions (Lucio et al., 1996). The maps are computed in a 4 × 5-km centered box (cross-line × in-line) for each ray-shooting position. The six strongest arrivals were computed. Our ray tracing is fully three dimensional; however, to get some idea of the complexity of the wave propagation in the smooth velocity macromodel, Figure 4 shows various 2D rayfields superimposed on 2D sections of the 3D velocity macromodel. Triplications develop where there are strong lateral variations of the velocity model, i.e., at the edge of the salt body and below sharp changes in the topography of the top or bottom of the salt body. The sampling of the coarse grids at the surface and in the target is 100 m (Thierry et al., 1999b), and the coarse ray-shooting grid at the surface covers X ∈ [3, 12] km and Y ∈ [0., 9.] km. Storing the maps on disk requires 40 Gbytes. Computing the maps took two days on an eight-processor SGI machine; while 2 hours were needed to sort the maps slice by slice.

Single arrival versus multiarrival migration/inversion Figures 5 and 6 show views of the images cubes obtained by multiple arrival 3D migration/inversion using, respectively, our interpolation and our extrapolation strategies. The sampling of the target was 25 m in the X - and Y -directions and 10 m

Figure 5. Views of the migrated/inverted cube in the 3D SEG/ EAGE salt model obtained with our interpolation strategy. Compare to Figure 1.

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in depth. Computing time was 5 days and 4 days, respectively. In practice it appears that the easier management of memory in the extrapolation strategy allows one to fully compensate for the cost from overlapping the extrapolation functions with respect to the interpolation strategy. Results can be compared to the cube in the exact model (Figure 1). The salt body is well defined, but many artifacts appear in the subsalt area. A slight shift in depth appears between the two images, attributable to the different approximation of the migration/inversion operator. With the extrapolation strategy traveltimes are generally underestimated, while they are overestimated with the interpolation strategy. This is because of the main curvature of the wavefronts. A comparison against single-arrival, i.e., first or strongest, migration/inversion can be done. The comparison uses our interpolation strategy. Figures 7 and 8 shows 3D migrated/ inverted sections obtained using the strongest arrival and all of the arrivals. The image obtained using the extrapolation strategy is also given in Figure 7. Figures 9 and 10 also show the differences between the various migrated/inverted sections. Differences are localized at the bottom of the salt and in the subsalt area. Computing time for a single-arrival migration was 4 days, and we see that the cost for the multiarrivals is quite small because the triplications of travel arrivals are very local.

Figure 6. Views of the migrated/inverted cube in the 3D SEG/ EAGE salt model obtained with our extrapolation strategy. Compare to Figure 1.

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Figure 7. The 3D migrated/inverted in-line sections at X = 6.0 km. (a) Single arrival using first-arrival and interpolation strategy; (b) multiarrivals with interpolation; (c) single arrival using strongest arrival and interpolation strategy; (d) multiarrivals with extrapolation.

Figure 8. The 3D migrated/inverted cross-line sections at Y = 6.0 km. (a) Single arrival using first arrival; (b) multiarrivals with interpolation; (c) single arrival using strongest arrival and the interpolation strategy; (d) multiarrivals with extrapolation.

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Figure 9. Differences between 3D migrated/inverted in-line sections at X = 6.0 km. (a) Between single-arrival migration/inversion using first arrival and the strongest arrival; (b) between multiarrivals and strongest arrival migration/inversion; (c) between multiarrivals and first-arrival migration/inversion; (d) multiarrivals migration/inversion with our interpolation strategy.

Figure 10. Differences between 3D migrated/inverted cross-line sections at Y = 6.0 km. (a) Between single-arrival migration/inversion using first arrival and using the strongest arrival; (b) between multiarrivals and strongest arrival migration/inversion; (c) between multiarrivals and first-arrival migration/inversion; (d) multiarrivals migration/ inversion with our interpolation strategy.

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The origin of the artifacts in the images has to be analyzed. First, the preprocessing that we applied for water-bottom multiples reduction is not adapted to the case of complex geological structures and introduces many artifacts. In addition, it was not designed for eliminating longer-period multiples, and spurious organized events can indisputably be associated with multiple reflections from the top or bottom of the salt at the free surface. Second, when we look at cross-line sections, the distribution of artifacts clearly mimics the zones of triplication in the cross-line direction (see Figure 4b). They are particularly dense beneath changes in the topography of the top or bottom of the salt in the cross-line direction. We suspect these artifacts could in part result from the violation of the imaging condition expressed by Nolan and Symes (1996). The imaging condition states that any locally coherent event in the data set should be focused unambiguously in depth after depth migration. This condition is generally fulfilled when the acquisition geometry is such that shot and receiver positions cover the whole acquisition surface independently. When the acquisition is limited, as in classical 3D marine acquisition (there is no folding in the cross-line direction), violations of the imaging condition may appear from triplications in the cross-line direction. For this partial acquisition one lateral slope of the locally coherent events is missing, and the migrated image can be altered by spurious localized events. This can happen in case of the C3-NA experiment, which is a marine 3D classical acquisition.

Figure 11 shows various common image gathers (CIGs) in the common angle domain, obtained using our interpolation and extrapolation strategies. All of the events are flat, including those associated with multiple reflections in the subsalt area. The quality of CIGs computed using the extrapolation strategy is clearly improved compared with those computed using the interpolation strategy. The images benefit from the regularity of the kernel of the migration/inversion operator. Figure 12 shows various common angle sections for several angle ranges. As expected, the image shows lower frequencies when we go to large apertures. Finally, Figure 13 shows various logs for the relative perturbation of acoustic impedance, compared to the logs extracted from the exact model (they were filtered acording to the bandwidth of the source signature). In the noncomplex part of the model [(X, Y ) = (7.1, 2.5) km], the amplitude of the perturbation is quite well recovered. However, there is a significant underestimation of amplitude for the first interface that results from the inaccuracy of our interpolation scheme in the case of highly curved isochrons. There is another significant underestimation for the fourth interface from the top as a result of our bad estimate of the aperture range (θmax − θmin ) in the case of dipping events (Operto et al., 2003). In the complex part of the model [(X, Y ) = (7.1, 6.1) km and (X, Y ) = (7.1, 7.5) km], the contrasts at the top and base of the salt are slightly shifted in depth because of the smoothing of the velocity model. We also see an underestimation of the amplitude of the velocity contrasts. This can be from the inaccuracy of the ray+Born approximation in the case of strong perturbations. We also see on these two logs that the amplitude of the contrasts associated with the salt are so dominant that proper imaging below the salt requires a serious improvement of preprocessing and acquisition.

CONCLUSIONS We have demonstrated with an application to the 3D SEG/EAGE salt model that 3D migration/inversion with all arrivals is affordable with only a limited additional cost. This limited cost is because the areas with multiarrivals are quite limited in the image. Contrary to the 2D case, the extrapolation strategy is more efficient than the interpolation strategy. This is the result of a simpler management of the core memory. Moreover, the extrapolation strategy is much simpler to implement numerically and provides more accurate results. Therefore, we recommend this strategy. Finally, concerning the quality of the images obtained by multiarrival migration/inversion, many artifacts remain in the images. Some of them are certainly from the limitations of ray-based migration, but many others are undisputably associated with the defectiveness of preprocessing and the inadequacy of the acquisition geometry. The 3D migration results obtained with the same data set using other migration codes are affected by similar artifacts (Jin et al., 2000; Mo and Young, 2000; Mosher and Foster, 2000), which in fact severely conceal the subsalt structure. Improving imaging in complex media therefore requires improving the full seismic imaging sequence from acquisition to processing. Figure 11. CIGs in diffracting angle domain obtained by migration/inversion in the 3D SEG/EAGE salt model obtained. (left) CIG at (7.1, 2.5) km and (right) at (7.1, 6.1) km. (Top) with interpolation strategy and (bottom) extrapolation strategy.

ACKNOWLEDGMENTS This work was partly funded by the European Commission within the JOULE project, 3D-Focus (contract JOF3-CT970029). We thank Reda Baina (Elf Pau and Ipedex) and Pascal

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Figure 12. In-line sections, at X = 7.1 km, obtained by 3D multiarrivals migration/inversion for various diffracting angle ranges. ´ Podvin (Ecole des Mines de Paris) for fruitful discussions. We also thank Paul Docherty (Fairfield Industries) for providing us with the synthetic classical data set and model. Finally, we thank Associate Editor Tamas Nemeth, Greg Waite, Hongbo Zhou, M. Wei Yang, and anonymous reviewers for helping us to improve the paper.

REFERENCES

Figure 13. Logs of the perturbation of velocity, obtained by 3D migration/inversion (dashed line) with the interpolation strategy and by smoothing the exact velocity model (solid line). (a) At (X, Y ) = (7.1, 2.5) km; (b) at (X, Y ) = (7.1, 6.1) km; and (c) at (X, Y ) = (7.1, 7.5) km.

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