GEOPHYSICS, VOL. 69, NO. 5 (SEPTEMBER-OCTOBER 2004); P. 1311–1319, 14 FIGS. 10.1190/1.1801947
Fast migration/inversion with multivalued rayfields: Part 1—Method, validation test, and application in 2D to Marmousi
Sheng Xu1 and Gilles Lambare´ 2 migration, taking into account amplitude and multivalued traveltimes. The computation of multivalued traveltime and amplitude maps can be done by wavefront construction (Vinje et al., 1993; Vinje et al., 1996a,b). The first numerical implementations of preserved amplitude migration (i.e., migration/inversion) only considered a single arrival (Tura et al., 1998; Thierry et al., 1999a,c), and the extension of migration/inversion to multivalued traveltimes required both theoretical and numerical studies. The theory of migration/inversion in complex media has been studied by ten Kroode et al. (1998) and Nolan and Symes (1996). They demonstrate that migration/inversion can be extended to account for multipathing, provided an imaging condition (or traveltime injectivity condition) is satisfied. This condition involves both the velocity macromodel and the acquisition geometry. When shot and receiver positions independently and fully cover the whole acquisition (three dimensions of data for a 2D survey and five dimensions for a 3D survey), it is generally satisfied in complex velocity macromodels, except for some tortuous configurations that generally can be neglected in real applications. Another important conclusion has been reached concerning the common image gathers (CIGs) used for migrationbased velocity analysis (Jin and Madariaga, 1993, 1994; Symes, 1993; Chauris and Noble, 2001) or for amplitude versus offset (AVO) analysis (Beydoun et al., 1993; Tura et al., 1998). In complex structure imaging, unexpected artifacts can be observed in CIGs in the offset domain (Duquet et al., 1994). Theoretical analysis of migration (Nolan and Symes, 1996) reveals these artifacts result from the violation of the imaging condition. To remedy this, Xu et al. (2001) proposed computing CIGs in the diffraction angle domain. Their solution appeared attractive and in practice seemed to significantly reduce the artifacts in the CIGs. However, a recent mathematical analysis of the approach demonstrates that ray-based angle
ABSTRACT We address the problem of building an efficient algorithm for ray-based migration/inversion in complex media characterized by multivalued traveltime. Since the multivalued ray maps can be computed by wavefront construction, the key point lies in the interpolation of these multivalued coarse maps. For this goal we propose two new strategies. The first one is based on an interpolation of the rayfield after identifying the ray branches on the elementary cells of the coarse grid; the second is based on an extrapolation of the arrivals from the coarse grid points. Both strategies are evaluated in two dimensions with the Marmousi data set, demonstrating the CPU efficiency and accuracy of both approaches. Since they both consider multiarrivals locally, the associated overhead cost with respect to a single-arrival implementation is quite low (twice the computing time of the Marmousi test case). The application also reveals that the new interpolation strategy is more efficient in terms of CPU time compared to the extrapolation strategy but at the expense of a more complex numerical implementation and less accurate results.
INTRODUCTION Three-dimensional prestack depth imaging became tractable by the beginning of the ’90s because of the development of efficient first-arrival traveltime solvers (Vidale, 1988). Rapidly, the limitations of such Kirchhoff migration codes appeared in complex media characterized by multivalued traveltimes (Geoltrain and Brac, 1993). At that time 3D wave equation migration remained unaffordable, and efforts have been made toward the improvement of Kirchhoff
Manuscript received by the Editor January 16, 2001; revised manuscript received January 21, 2004. 1 ´ Formerly Ecole des Mines de Paris, Fontainebleau, France; presently Veritas Geophysical Corp., 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: gilles.lambare@ geophy.ensmp.fr. ° c 2004 Society of Exploration Geophysicists. All rights reserved. 1311
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domain imaging still violates the imaging condition (Stolk and Symes, 2002). These first studies were essentially demonstrations of the capacities of ray-based migration/inversion for complex media (Operto et al., 2000), but numerically no attention was paid to implementing the theory efficiently. In two dimensions efficiency is not essential, but in three dimensions there is no way to avoid optimization of the migration code. Our paper presents efficient strategies for migration with multivalued traveltime and amplitude maps. Two different approaches are presented and evaluated in two dimensions using the synthetic Marmousi data set (Bourgeois et al., 1991). These strategies make 3D migration/inversion accounting for all arrivals tractable. In a companion paper (Xu et al., 2004) we extend the algorithm to three dimensions and apply it to the 3D SEG/EAGE salt model (Aminzadeh et al., 1997). First, we briefly recall the migration/inversion formulas in the presence of multivalued traveltimes. We then discuss the difficulties encountered with interpolating multivalued ray parameter maps for the ray shooting positions at surface and with interpolating the folded migration operator in the target. We present two different strategies adapted to multivaluedness, with an application to migration/inversion in the Marmousi model. The accuracy of the images is estimated by the comparison with the images obtained without interpolation. Finally, we discuss the overhead cost associated with the presence of multiple arrivals in migration/inversion.
RAY+BORN MIGRATION/INVERSION IN COMPLEX MEDIA Formulas Migration/inversion is based on the inversion of the ray+Born or ray+Kirchhoff approximations. Both approximations are linearizations of the relation connecting data perturbation to model perturbation. They differ by the way the model perturbation is introduced (perturbation of acoustic parameters for the Born approximation or reflectivity function for the Kirchhoff approximation). Formally, ray+Born and ray+Kirchhoff approximations and the corresponding implementations are very close. Our work is based on the Born approximation for the constant density acoustic equation. The migration/inversion formula for the perturbation of the square slowness δm(x) (x is the position in the image) can be adapted to the case of multiarrivals (Operto et al., 2000) (all arrivals are taken into account) and can be expressed with a decomposition into common diffraction angle images δm(x, θ ) (Thierry et al., 1999b; Xu et al., 2001):
δm(x) =
1 max Cmin (x)
Z
θmax
dθ θmin
δm(x,θ ),
(1)
max where Cmin (x) = θmax (x) − θmin (x) denotes the difference between the maximum and minimum angular apertures of diffraction angle θ , available in the acquisition geometry, and δm(x, θ) denotes the individual common angle images given by
δm(x, θ ) =
(r,x,s) X NX Ä
δ(θ − θn (r, x, s))Mn [δGobs ](r, x, s),
n=1
(2)
where Ä denotes all the traces in the acquisition geometry, δGobs denotes the preprocessed data (i.e., considered as containing only correctly deconvolved primary reflections.), N (r, x, s) denotes the number of multiarrival cross-contributions (i.e., the number of arrivals from the shotpoint times the number of arrivals from the receiver), and Mn is the kernel of the migration/inversion operator associated to the n th ray trajectory r → x → s (r and s denote, respectively, the receiver and shot positions). The term Mn is defined by α(r,x,s)
Mn [δGobs ](r, x, s) = En (r, x, s)Hilb
[δGobs (r, s, t)]|t=Tn (r,x,s) ,
where T is the sum of traveltimes, E is the amplitude weight of the migration/inversion operator, and H ilb denotes the Hilbert transform, in which α is the order of Hilbert transform. The term α differs for different dimension, α2D = 1 − αn and α3D = −αn with αn is the sum of the KMAH indices (Maslov, 1972; Chapman, 1985) of the nth cross-contribution for the rays r → x and s → x.
Amplitude weights for acquisition geometries For 2D surface surveys, the amplitude weight of the migration/inversion operator En (r, x, s) can be expressed as
¯¯ ¯ ¶¯ θn ¯¯ ∂βs ¯¯¯¯ ∂βr ¯¯ 2 cos 1s1r 2 ¯ ∂s ¯¯ ∂r ¯ En (r, x, s) = , π c02 (x)An (r, x, s) µ
2
(3)
where An (r, x, s) is the product of the amplitudes for the rays s → x and r → x, βs and βr are dip angles of these two rays at the imaging point, and 1s and 1r denote the spacing of shot and receiver sampling in acquisition geometry. For 3D surveys with marine acquisition, the traces are parameterized by their line, shot, and receiver numbers, L , S, R. Given in Thierry et al. (1999a), the amplitude weight En is defined by
¯ ¯ ¯¯ 1 |qn |1L1S1R ¯¯ ∂|qn | ¯¯¯¯ ∂(qn ) ¯¯ En (r, x, s) = , (2π )2 An (r, x, s) ¯ ∂θn ¯¯ ∂(L , S, R) ¯ (4)
where 1L, 1S, and 1R denote, respectively, the increment in line, shot, and receiver numbers and where vector q = pr + ps is the sum of the slowness vectors. The amplitude of one ray pair of the migration operator E compensates for the two-way amplitude (1/A) and accounts for the acquisition geometry through the Jacobians |∂|q|/∂θ||∂(q)/∂(L , S, R)|. The second Jacobian can be written
¯ ¯ ¯ ¯ ¯ ∂(q) ¯ ¯ ∂(pr + ps ) ∂(r, s) ¯ ¯=¯ ¯ ¯ ¯ ∂(L , S, R) ¯ ¯ ∂(r, s) ∂(L , S, R) ¯,
(5)
where the paraxial quantities [(∂(pr )/∂r), (∂(ps )/∂s)] are computed using paraxial ray tracing (Farra and Madariaga, 1987; Lucio et al., 1996) and the matrix [∂(r, s)/∂(L , S, R)] is inferred from the acquisition geometry (Jousset et al., 2000).
Singularities dumping In classical ray theory, singularities of the wavefield appear in the form of infinite amplitudes at caustics because of a vanishing geometrical spreading, J = det |∂x/∂(σ, φ)| (σ is the curvilinear abscissa along the ray and φ is the take-off angle
Fast Multiarrival Migration/Inversion I
at the ray shooting point). For example, when considering the amplitude of the 2D migration/inversion operator [equation (3)], the geometrical spreading first appears in the denominator with the amplitude in classical ray theory (it is proportional to the inverse square root of the geometrical spreading), but it is also in the numerator with the Jacobians |∂βsn /∂s| and |∂βrn /∂r| (proportional to the inverse of geometrical spreading). Consequently, at a caustic point the amplitude of the migration/inversion operator will be infinite (because of the Jacobians) and may produce erroneous spikes in the migrated image (Thierry et al., 1999c; Xu et al., 1999; Operto et al., 2000). Such artifacts are well known when forward modeling using the classical ray theory. They have motivated numerous extensions of the high-frequency asymptotic theory such as Maslov (Chapman, 1985) or Gaussian beams summations (Popov, 1982). Introduction of such extensions in the migration/inversion theory is possible. It has already been tested (Xu and Lambare, ´ 1998) and may provide very interesting results (Hill, 1990, 2001). We do not consider these types of approaches here, and, as in former studies (Xu et al., 1999; Operto et al., 2000), we add a damping term ² to the geometrical spreadings J damp ≈ J + ². We use J damp to compute the Jacobians and the classical ray theory amplitudes in equation (3). In practice, we took ² = 0.1 m/rad; it heuristically proved to damp locally exploding arrivals and provided reasonable results.
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face for the exact source and receiver positions. The kernel of the migration/inversion operator is then computed on the coarse target grid. Finally, the migration/inversion operator is interpolated to the fine grids in the target, and the contribution of the trace is added to the image. Our problem is to generalize such an interpolation scheme in the case of multivalued traveltime and amplitude maps.
Interpolation strategy for multivalued rayfields Let’s first look at some multivalued maps using a smoothed version of the 2D Marmousi model (Versteeg, 1993; Thierry et al., 1999c). We computed multivalued maps by wavefront construction (upgoing rays were eliminated) for ray-shooting positions at the surface (x = 6.5 km and x = 7.5 km). Figures 1 and 2 show the number of arrivals obtained at each point in the image. The triplications are localized in the deeper part of the Marmousi model. With such multivalued maps, the critical point for interpolation is to identify the ray branches.
FAST MIGRATION WITH MULTIPLE ARRIVALS From previous works, we already had a brute force migration/inversion code. Running this program with the Marmousi data set took about 23 hours on a Sun Sparc 20 workstation (Operto et al., 2000; Xu, 2001). But because efficiency is essential for 3D implementation, we also had to develop a fast algorithm. As a starting point, let us recall the strategy proposed by Thierry et al. (1999a,b) for fast implementation of single-arrival migration/inversion. In this strategy, maps are computed by ray tracing from a coarsely sampled regular grid at the surface into the coarsely sampled regular grid in the target. For a given trace, maps (amplitude, traveltime, slowness, and some other paraxial derivatives, required for computing the amplitude of the migration/inversion operator) are interpolated at the sur-
Figure 1. The number of arrivals for a ray-shooting position at x = 6.5 km at the surface. There are up to five arrivals. The rayfield is triplicated in the deeper part of the model, and we get multiple arrivals in the reservoir zone. Upgoing rays are eliminated.
Figure 2. The number of arrivals for a ray-shooting position at 7.5 km at the surface. Triplications appear in the left part (x > 8. km) characterized by strong lateral variations. Upgoing rays are eliminated.
Figure 3. Interpolation in the target. The pink box shows an elementary cell in the coarse grid defined by four neighboring points (A, B, C, D). If there are multiple arrivals, the interpolation must be done independently on each branch. For example, among the arrivals in red or in green, three connections between arrivals must be identified (A → B, A → C, A → D). The amplitude, the two-way traveltime, and the diffracting angle are computed on the coarse grid, and the interpolation of the migration/inversion operator from the four points ( A, B, C, D) provides the final image on the fine grid.
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Since arrivals are not stored a priori according to ray branches, it is necessary to identify the ray branches a posteriori. First we consider the interpolation of the migration/inversion operator in the target. Consider four adjacent points (A, B, C, D) in the coarse target grid and suppose that two arrivals have been identified at points A, B, C, and D (Figure 3). Given that we have a criterion for determining if two arrivals at two different points belong to the same branch, interpolation will be done in the cell (A, B, C, D) if and only if the arrivals observed at the four points belong to the same branch. The problem of interpolating maps for ray-shooting positions at the surface (Figure 4), or in the target, in either two or three dimensions is very similar (only the number of points changes). All of the difficulties are attributable to the identification of the ray branches among arrivals at points. The identification of the ray branches is a local problem; the key point is to design a criterion to identify the arrivals from the same branch at two adjacent points. The multivalued maps computed by wavefront construction include traveltime, amplitude, slowness vector, KMAH index, take-off angle, and some other paraxial ray parameters, which can be used to identify the ray branches.
Consider two adjacent points with several arrivals. We propose to use a first criterion based on the KMAH index. If the KMAH indices corresponding to the arrivals at two points are different, they do not belong to the same branch. This is a necessary but insufficient condition. To avoid ambiguity, we propose using an additional criterion based on the continuity of traveltime. Consider two points A and B with, respectively, N A and N B arrivals and with the same KMAH index (Figure 5). Using the slowness vectors provided by the ray-tracing code, the traveltimes can be extrapolated linearly from A to B and from B to A for all arrivals. Certainly higher order extrapolations could be used with paraxial ray information, but using linear extrapolation ensures both CPU efficiency and consistency with the later interpolations. We propose to identify the ray branches by considering the misfit between the exact and the extrapolated traveltimes. Let TAn and PAn be, respectively, the traveltime and slowness vector for the nst arrival at point A (and, similarly, TBm and PBm at point B). Arrival n at point A will be connected to arrival m at point B if
° ¡ ¢° ° E = °T An − TBm + Pm B · (X B − X A ) ° m ¡ n ¢° + °TB − T A + PnA · (X A − X B ) °
(6)
is the minimum for m ∈ [1, N B ]. To avoid incorrect connections, we imposed the misfit min(E nm ) to be in any case smaller than a threshold ²0 (in real
Figure 4. Interpolation at the surface. The blue triangles represent the regularly spaced ray-shooting positions at the surface. The red triangle is the interpolated source position. Identification of the branches is necessary for interpolating the maps at the surface. The rays in red and green indicate respectively the two ray branches.
Figure 5. A traveltime continuity condition for identifying ray branches. Two arrivals at neighboring points will be connected if they correspond to a minimum misfit between the exact and linearly extrapolated traveltimes from one point to the other. The pink lines show the extrapolated traveltimes at point A from the three arrivals at point B. The slowness vectors at point B are used.
Figure 6. Synthetic seismograms computed by ray tracing in the smooth Marmousi model. The shot position at the surface is X = 6.5 km, and the receiver line is at 2.60 km in depth: (a) with no approximation; (b) with the multiarrival interpolation strategy at the surface and in the target; (c) with the interpolation strategy at the surface and the extrapolation strategy in the target.
Fast Multiarrival Migration/Inversion I
application we took ²0 = kX A − X B k × 10−4 s, where kX A −X B k is expressed in meters).
Extrapolation strategy for multivalued rayfields The interpolation strategy proposed for multivalued rayfields has the great advantage of coping with multiarrivals locally. However, it also suffers some drawbacks. First, the limitation of the interpolation in the target and at the surface to areas where a ray branch can be identified at all points simultaneously is a rather restrictive condition. In practice, large areas effectively covered by ray branches are missing after the interpolation because the identification of the ray branches is missing at just one of the neighboring points. This is particularly true for points close to caustics which bear an important part of the propagated energy. Moreover, the abrupt truncations as-
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sociated with the boundaries of the interpolated patches may lead to discontinuities in the migrated images and still more in the CIGs, which benefit less from the migration summation. For all of these reasons, we propose an alternative strategy that remedies some of these drawbacks. We propose to regularize the truncations during migration processing. Rather than using complex theoretical methods based on Gaussian beams or Maslov summation (Hill, 1990, 2001; Xu and Lambare, ´ 1998), our method consists of smoothing the kernel of the migration/inversion operator in the target. This kernel also depends on shot and receiver positions at the surface, but, as is demonstrated in later applications, a smoothing in the target is sufficient for cleaning most of the artifacts while preserving computing efficiency. In the migration/inversion formulas (1) and (2), the key parameters are the amplitude, the phase, and the two-way traveltime of the asymptotic kernel—respectively, En , αn , and Tn . They are all sampled in the target on a coarsely sampled grid [xi j ]:
h
i N N N Ei j i j , αi j i j , Ti j i j ,
(7)
where Ni j denotes the number of cross-contributions at gridpoint xi j . To approximate the kernel of the migration/inversion in the target by a weighted summation of local plane functions, we propose B[p0 , α0 ](x; t):
M(r, x, s; t) ≈
Ni j XX ij
Einj
£ ¤¡ ¢ B ∇Ti nj , αinj x − xi j ; t − Ti nj ,
n=1
(8)
Figure 7. Traveltime curves associated with (a) the interpolation strategy, (b) the extrapolation strategy, and (c) the exact rayfield. The truncation of the caustic area appears clearly with the interpolation strategy, while with the extrapolation strategy the shadow zones are filled with some arrivals.
Figure 9. Image obtained after multiarrival migration/inversion with our interpolation strategy on the Marmousi data set (1 hour of CPU time).
Figure 8. Image obtained after multiarrival migration/inversion without interpolation or extrapolation on the Marmousi data set (23 hours of CPU time).
Figure 10. Image obtained after multiarrival migration/ inversion with our extrapolation strategy on the Marmousi data set (2 hours of CPU time).
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where
B[p0 , α0 ](x; t) = W(x)Hilb
(1−α0 )
[δ](t + p0 · x).
(9)
Here, p0 denotes the slope of the plane function, while α0 denotes its phase; W(x) is the interpolant weight. The fact that W(x) has a compact support is very important for CPU efficiency. Thus, we use the same weight as the first order of Bspline interpolant, which also equals to a bilinear interpolant weight in two dimensions. For a discussion of B-spline weighting function of higher order or in higher dimension, we refer to Unser et al. (1993a,b).
Since the construction of the migration/inversion operator is done by extrapolation from the points of the coarse grid, we call it extrapolation strategy.
VALIDATION TEST Our interpolation and extrapolation strategies were proposed to estimate the kernel of the migration/inversion operator in the target, but they can be applied to any asymptotic wavefield characterized by multivalued amplitudes, times, and phases. The computation of synthetic seismograms is
Figure 11. CIGs in the diffracting angle domain in the noncomplex part of the Marmousi model (X = 3.5 km), calculated (a) with no interpolation or extrapolation, (b) with our interpolation strategy at the surface and in the target, and (c) with our interpolation strategy at the surface and with our extrapolation strategy in the target.
Figure 12. CIGs in the diffracting angle domain in the complex part of the Marmousi model (X = 6.2 km), calculated (a) with no interpolation nor extrapolation, (b) with our interpolation strategy at the surface and in the target, and (c) with our interpolation strategy at the surface and with our extrapolation strategy in the target.
Fast Multiarrival Migration/Inversion I
interesting since it provides a physical understanding of the effects of our approximations. To validate the method we chose the Marmousi velocity model, low-pass filtered to σ < 150 m (Operto et al., 2000). The multivalued maps are computed by a 2D wavefront construction code (Lambare´ et al., 1996). Suppose we have coarsely
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sampled multiple arrival traveltime and amplitude maps for the ray-shooting positions (x, z) = (6.475, 0) km and (x, z) = (6.525, 0) km and the coarse grid in the target is sampled to 100 m in x and 50 m in depth. Our interpolation strategy is applied first at the surface for the ray-shooting position (x, z) = (6.5, 0) km. Then we apply alternately our interpolation and extrapolation strategies in the target for a receiver line at 2.9 km depth in the middle between two horizontal lines of coarse grid nodes (Figure 6b,c). The reference seismograms obtained with no interpolation and no extrapolation are shown in Figure 6a. The seismograms are quite similar with the three approaches. The amplitude, traveltime, and shape of arrivals have not been significantly altered, but differences appear in the vicinity of caustics. With our interpolation strategy, truncations appear at the boundaries of ray branches. They are the result of the sampling and interpolation on the coarse grids. With successive interpolations, significant parts of the curves disappear. With our extrapolation strategy, there is no abrupt truncation at caustics and some energy is spread into the shadow zones. Figure 7 explains this phenomenon. Certainly the extrapolation strategy does not rely on a strong theoretical background. It is not an extension of asymptotic ray theory in the sense that it is not asymptotically valid, but at finite frequencies it fits the classical ray theory quite well in the zones where the asymptotic wavefield has smooth variations. In the zones with strong variations or even with singularities, our approach provides a regular smooth solution heuristically closer to the band-limited solution. The regularity of the gathers of seismograms is an interesting property for the quality of migrated images. With this example one can also see that the use of linear interpolation or extrapolations associated with coarsely sampled maps to 100 m and 50 m appears to be a reasonable compromise between CPU efficiency and accuracy.
APPLICATION TO MARMOUSI MODEL
Figure 13. Logs for the relative impedence perturbation obtained at X = 3.7, 6.2, and 8.0 km with our extrapolation strategy (1X = (100, 50) m), with an interpolation strategy for multiple arrivals, a multiple arrival migration/ inversion with no interpolation or extrapolation, and finally obtained by band-pass filtering of the exact model.
The Marmousi model is a well-known 2D synthetic acoustic model (Bourgeois et al., 1991). Data were computed by finite differences of the acoustic equation, and the model was given by dense velocity and density grids sampled in X and Z to 4 m. A hydrocarbon trap is located just under the complex structure at X ∈ [5500, 7500] m and around Z = 2500 m. We use the same data preprocessing as in Operto et al. (2000). Operto et al. (2000) present a 2D migration/inversion result of the Marmousi model
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with full consideration of multivalued rayfields. It was obtained without interpolation or extrapolation in the migration/inversion code, i.e., ray tracing was done for each shot and receiver position, and the maps were computed directly at the sampling of the final image grid. Using multiple arrivals clearly improves the quality of the image with respect to singlearrival migration (even when choosing the strongest arrival). This result provides us with a reference for testing the accuracy of our fast 2D multiple-arrival migration/inversion code. Figure 8 shows the result without interpolation or extrapolation. About 23 hours on a Sun Sparc 20 workstation were required for computing the image, while only 30 minutes were needed when using the interpolation strategy proposed by Thierry et al. (1999a) for a single arrival (with a coarse sampling of 100 m at the surface and in the target). Figures 9 and 10 show the migrated images obtained with codes based on the interpolation and extrapolation strategies, respectively. Ray shooting was done with a coarse sampling of 50 m at the surface, from 0.4 to 9.0 km. Coarse target grid sampling was 100 m in x and 50 m in depth. Fine target grid sampling was 25 m in x and 5 m in depth. About 1 hour was needed to obtain the image on the Sun workstation with the interpolation strategy. With the extrapolation strategy, CPU time went up to 2 hours. The quality of the images is very close to the reference ones (Figure 8). The CIGs computed with the three approaches were calculated in the diffracting angle domain θ. In Figure 11, the CIGs were computed at x = 3.5 km in the part of the Marmousi model where the velocity structure is relatively simple and where, consequently, there are few caustics. The CIGs are very flat and are only affected by a few artifacts because of the limited aperture of acquisition. The CIGs obtained by the interpolation and extrapolation strategies are very similar to those obtained without interpolation. Some slight differences appear in the shallow part, where the CIGs are degraded because of bilinear interpolations applied to systematically strongly curved isochrons (from the vincinity of the source). Figure 12 shows CIGs in the complex part of the Marmousi model, calculated with the three approaches. There are many caustics, but the CIGs in the diffracting angle domain are quite regular. Some new artifacts appear. They are studied by Stolk
and Symes (2002) and could be associated with a violation of the imaging condition. The CIGs obtained with our interpolation and extrapolation strategies in the complex part of the Marmousi model are curiously less affected by artifacts. This may be because our interpolation strategy truncates the migration/inversion operator near caustics, an area where the artifacts should be much more enhanced. With the extrapolation strategy abrupt truncations have been smoothed, reducing artifacts in the CIGs. Figure 13 shows logs of the relative impedance perturbation obtained with these methods: the exact model bandpass filtered according to the source signature, multi-valued migration/ inversion using our reference code, and our interpolation and extrapolation strategies. Compared with the interpolation method, which loses some energy in the complex part of the Marmousi model, our extrapolation strategy recovers almost the same amplitude as the reference image.
COST FOR IMAGING WITH MULTIPLE ARRIVALS The CPU time required for multiarrival migration/inversion based on the interpolation strategy is about a factor of two greater than that required for the single-arrival method of Thierry et al. (1999b). This may appear quite good, considering that up to five arrivals for the ray tracing from the source and the ray tracing from the receiver are taken into account. For understanding this performance, consider a trace with source at x = 7.5 km at the surface and the receiver at x = 6.5 km. Figures 1 and 2 show the number of arrivals obtained from the source and receiver, respectively, after eliminating the upgoing rayfield. There are up to five arrivals, and the triplications appear in the deep part of the model. In Figure 2 there is no caustic in the right side, where there are no strong lateral variations of the velocity. Figure 14 shows the number of cross-contributions taken into account for the migration/inversion for this trace. (When performing migration/inversion of the data set, the maximum number of cross-contributions observed was 25.) The high values are reached only locally; consequently, the overhead as a result of multivaluedness is reasonable (63% for this particular trace). Because of our interpolation strategy, which considers arrivals locally, the overhead on computing time is of the same order of magnitude. Globally, considering all of the traces and the cost of identifying the ray branches, our code only doubles the computing time with respect to a single arrival migration/inversion code. Finally, we can compare CPU time between our multiarrival interpolation and extrapolation strategies. The computing time with our extrapolation strategy is increased by a factor of two. Extrapolation strategy is also a local approach and provides much simpler memory management. But overlapping local functions increases the computing time.
CONCLUSION AND DISCUSSION Figure 14. Distribution of the number of cross-contributions [number of arrivals from the source (Figure 2) times number of arrivals from the receiver (Figure 3)] to the migration/inversion operator in the Marmousi model for the trace s = 6.5 km and r = 7.5 km.
We have demonstrated that fast and accurate migration/inversion algorithms can be implemented for multivalued rayfields. We have proposed an interpolation and an extrapolation strategy, both based on a coarse grid sampling of the kernel of the migration/inversion operator in the target and
Fast Multiarrival Migration/Inversion I
at the acquisition surface. The interpolation strategy requires the identification of the ray branches on multivalued ray parameter maps, while in the extrapolation strategy arrivals are considered individually. With the interpolation strategy, computing time in the Marmousi model only increases by a factor of two with respect to a single-arrival migration/inversion. The CPU time with the extrapolation strategy increases by a factor of two with respect to the multiarrival interpolation strategy. However, the images obtained with the interpolation strategy are both structurally and quantitatively better than those obtained with the interpolation strategy. For both approaches the extension to three dimensions is affordable and is developed in the companion paper (Xu et al., 2004).
ACKNOWLEDGMENTS This work was funded partly by the European Commission within the JOULE project, 3D-Focus (contract JOF3-CT970029). We thank Andreas Ehinger (IFP) for providing us with the Marmousi model and data set. We thank Henri Calandra (Elf), Reda Baina (IPEDEX and now OPERA), and Pascal ´ Podvin (Ecole des Mines de Paris) for fruitful discussions. We thank Associate Editor Tamas Nemeth, Greg Waite, Hongbo Zhou, M. Wei Yang, and the anonymous reviewers for helping us to improve the paper.
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