Stanford Exploration Project, Report 113, July 8, 2003, pages 349 363 Amplitude and kinematic corrections of migrated images for non-unitary imaging operators Antoine Guitton 1 ABSTRACT Obtaining true-amplitude migrated images remains a challenging problem. One possible solution to address it is iterative inversion. However, inversion is an expensive process that can be rather difficult and expensive to apply, especially with 3-D data. In this paper, I propose computing an image that is close to the least-squares inverse image by approximating the Hessian, thus avoiding the need for iterative inversion. The Hessian is approximated with non-stationary matching filters. These filters are estimated from two images: one is the migration result (m 1 ) and the second is the migration result of the remodeled data computed from m 1. Tests on the Marmousi dataset show that this filtering approach gives results similar to iterative least-squares inversion at a lower cost. In addition, because no regularization was used in the inversion process, the filtering method produces an image with fewer artifacts. Applying this method in the angle domain yields similar conclusions. INTRODUCTION It is well known that most of our mathematical operators for seismic processing are not unitary. This means that for any linear transformation L, L L = I where ( ) stands for the adjoint and I is the identity matrix. Having non-unitary operators often results from approximations we make when building those operators. For example, we might not take the irregular and finite acquisition geometry of seismic surveys into account. For migration, different approaches exist to correct for these defects of our operators. Bleistein (1987), based on earlier works from Beylkin (1985), derived an inverse operator for Kirchhoff migration. A similar path is followed by Thierry et al. (1999) with the addition of non-linear inversion with approximated Hessian. Alternatively, least-squares migration with regularization has proven efficient with incomplete surface data, e.g., Nemeth et al. (1999) and illumination problems due to complex structures, e.g., Prucha et al. (1999); Kuehl and Sacchi (2001). Hu et al. (2001) introduce a deconvolution operator that approximates the inverse od the Hessian in the least-squares estimate of the migrated image. However, this method assumes a v(z) medium which means that the deconvolution operators are horizontally invariant. Recently, Rickett (2001) proposed estimating weighting functions from reference 1 email: antoine@sep.stanford.edu 349
350 Guitton SEP 113 images to compensate illumination effects for finite-frequency depth migration. This method corrects for amplitude effects only and requires some smoothing that can be rather difficult to estimate. In this paper, I propose a new strategy for approximating the inverse of the Hessian. This approach aims to estimate a bank of non-stationary matching filters (Rickett et al., 2001) between two migrated images that theoretically embed the effects of the Hessian. This approach is implemented after migration and is relatively cheap to apply. It can be applied on images migrated at zero-offset or in the angle domain. I illustrate this method with the Marmousi dataset. I demonstrate that this approach can effectively recover amplitudes similar to the one obtained with least-squares inversion. In addition, these filtered images have less artifacts than the least-squares result without regularization and at a much reduced cost. THEORY In this section, I show how the least-squares estimate of a migrated image can be approximated using non-stationary matching filters. In terms of cost, this approach is comparable to one and a half iterations of a conjugate-gradient method (CG), the first half iteration being the migration. The cost of estimating the non-stationary filters is negligible compared to the total cost of migration. First, given seismic data d and a migration operator L, we seek a model m such that This goal can be rewritten in the following form Lm = d. (1) 0 r d = Lm d (2) and is called the fitting goal. For migration, a model styling goal (regularization) is necessary to compensate for irregular geometry artifacts and uneven illumination (Prucha et al., 1999). I omit this term in my derivations and focus on the data fitting goal only. By estimating m in a least-squares sense, we want to minimize the objective function The least-squares estimate of the model is given by f (m) = r d 2 = Lm d 2. (3) ˆm = (L L) 1 L d (4) where L L is the Hessian of the transformation. My goal in this paper is to approximate the effects of the Hessian L L using non-stationary matching filters. Approximating the Hessian In equation (4), I define L d as the migrated image m 1 after a first migration such that ˆm = (L L) 1 m 1. (5)
SEP 113 Approximating the Hessian 351 In equation (5), ˆm and L L are unknown. Since I am looking for an approximate of the Hessian, I need to find two known images that are related by the same expression as in equation (5). This can be easily achieved by remodeling the data from m 1 with L and remigrating them with L as follows: d 1 = Lm 1 (6) m 2 = L d 1 = L Lm 1. (7) Notice a similarity between equations (5) and (7) except that in equation (7), only L L is unknown. Notice that m 2 has a mathematical significance: it is a vector of the Krylov subspace for the model ˆm. Now, I assume that we can write the inverse Hessian as a linear operator B such that ˆm = Bm 1 (8) and m 1 = Bm 2. (9) Equation (9) can be approximated as a fitting goal for a matching filter estimation problem where B is the convolution matrix with a bank of non-stationary filters (Rickett et al., 2001). This choice is rather arbitrary but reflects the general idea that the Hessian is a transform operator between two similar images. My hope is not to perfectly represent the Hessian, but to improve the migrated image at a lower cost than least-squares migration. In addition in equations (8) and (9), the deconvolution process becomes a convolution, which makes it much more stable and easy to apply. Hence, I can rewrite equation (9) such that the matrix B becomes a vector and m 2 becomes a convolution matrix (Robinson and Treitel, 1980): The goal now is to minimize the residual m 1 = M 2 b. (10) 0 r m1 = m 1 M 2 b (11) in a least-squares sense. Because we have many unknown filter coefficients in b, I introduce a regularization term that penalizes differences between filters as follows: 0 r m1 = m 1 M 2 b 0 r b = Rb (12) where R is the Helix derivative (Claerbout, 1998). The objective function for equation (12) becomes f (b) = r m1 2 +ɛ 2 r b 2. (13) where ɛ is a constant. The least-squares inverse is thus given by Once ˆb is estimated, the final image is obtained by computing ˆb = (M 2 M 2 +ɛ 2 R R) 1 M 2 m 1. (14) ˆm = m 1 ˆb (15)
352 Guitton SEP 113 where ( ) is the convolution operator. Therefore, I propose computing first a migrated image m 1, then computing a migrated image m 2 (equation (7)), and finally estimating a bank of non-stationary matching filters b, e.g., equation (12). The final improved image is obtained by applying the matching filters to the first image m 1, e.g., equation (8). In the next section, I illustrate this idea with the Marmousi dataset. I show that an image similar to the least-squares migration image can be effectively obtained. MIGRATION RESULTS I illustrate the proposed algorithm with the Marmousi dataset. I use a prestack split-step migration method with one reference velocity. I demonstrate with zero-offset images and angle gathers that the approximation of the Hessian with adaptive filters gives a result comparable to least-squares migration with fewer artifacts. Zero-offset prestack migration results I first show in Figure 1 a comparison between the migration result of the Marmousi dataset (m 1 ) and the remodeled data (m 2 ). We notice that the migration of the remodeled data (Figure 1b) lowers the amplitudes in the upper part of the model. Therefore, we expect the filters to correct for this difference. Figure 2 displays few estimated filters for the Marmousi result. The filters are ten by ten with 40 patches in depth and 80 along the midpoint axis. I shown only a fifth of these filters in both axes. It is interesting to notice that these filters have their highest value at zero-lag, meaning that we have a strong amplitude correction with few kinematic changes. The zero-lag values are also larger at the top of the model, as anticipated. Looking more closely at these filters, we see that the coefficients follow the structure of the Marmousi model (upper right corner). Having estimated the filters b in equation (12), I apply them to m 1 to obtain an improved image. To validate this approach I show in Figure 3a the result of five conjugate gradient (CG) iterations with the Marmousi data. This results show higher amplitudes at the top but with inversion artifacts. This problem should be addressed with a proper regularization scheme (Prucha et al., 1999). In Figure 3b, I show the corrected image with the approximated Hessian B. The amplitude behavior is very similar to Figure 3a, without the inversion artifacts. Additionally, the cost is much lower. I show in Figure 4 the ratio of the envelope of Figure 3b and 1a. This Figure illustrates that the effects of the non-stationary filters, i.e, the Hessian, are stronger on the top of the model. Angle domain results Now, I apply the same method in the angle domain as opposed to the zero-offset image domain. The angle panels are created after migration from the offset panels directly (Stolt and Weglein,
SEP 113 Approximating the Hessian 353 Figure 1: (a) Migration result of the Marmousi dataset, i.e., m 1 in equation (9). (b) Migration result of the remodeled data, i.e., m 2 in equation (9). antoine3-m1.m2 [CR,M]
354 Guitton SEP 113 Figure 2: Each cell represents a non-stationary filter with its zero-lag coefficient in the middle. A fifth of the filters are actually shown in both directions. Each filter position corresponds roughly to a similar area in the model space (Figure 1a). After close inspection of the filter coefficients, these filters seem to follow the structure of the Marmousi model. They are also stronger at the top of the model, as expected. antoine3-filter-minv0 [CR]
SEP 113 Approximating the Hessian 355 Figure 3: (a) Model estimated after five iterations of CG. The model is noisy because no regularization has been applied. (b) Model estimated after applying the adaptive filters to m 1. The amplitude behavior is similar to (a) without the artifacts and with fewer iterations. antoine3-inv.ampcorr [CR,M]
356 Guitton SEP 113 Figure 4: Ratio of the envelopes of Figures 1a and 3b. Brighter colors correspond to higher values. The main effect of the filters is clearly visible at the top. antoine3-amplitude0 [CR] 1985; Weglein and Stolt, 1999; Sava and Fomel, 2000; Sava et al., 2001). I show in Figures 5 and 6 the angle gathers at two locations of the model (3050 and 5550 m) for the migration result (m1), the filtering result (ˆb m 1 ) and the inversion result after five CG iterations. Again, the angle gather after filtering has less artifacts than with inversion with a similar amplitude pattern throughout the section. In Figure 6, we notice that the filtering approach improves the continuity of some reflectors (between 500 and 600 m.) and that the amplitude increases for large angles. This behavior is similar to what Sava et al. (2001) observed for wave equation migration with amplitude corrections. I display in Figure 7 the estimated filters for the angle gather in Figure 5. Again, we see that we have higher amplitudes for filters at large aperture angles. In addition, we notice a smoothing effects of the filters along the angle axis. Finally I show in Figure 8 a comparison of stacked images across angle for m 1 and m 2. The bottom reflectors are stronger than in Figure 1 with fewer migration artifacts, as expected from the stack. The differences between the two stacked-data panels in Figure 8 are similar to what we observe in Figure 1, i.e, stronger amplitudes at the top of the image for m 1 and fewer migration artifacts for m 2. In Figure 9, I display a comparison between the inversion and filtering approaches for the stacked data. Again, the amplitude pattern is the same for both images with far less artifacts in the filtering result (Figure 9b).
SEP 113 Approximating the Hessian 357 DISCUSSION In this paper, I presented a method that intends to correct migrated images by approximating the Hessian of the imaging operator with non-stationary matching filters. These filters are estimated from two migration results. One migrated image, m 1, corresponds to the first migration result. The second image, m 2, is computed by re-modeling the data from m 1 and then by re-migrating it. It turns out that the relationship between m 1 and m 2 is similar to the relationship that exists between the least-squares inverse ˆm and m 1. In the proposed approach, this relationship is simply captured by matching filters. I demonstrate with the Marmousi dataset that this approach gives a better image than does least-squares without regularization and at a lower cost. In addition this approach can be used on images migrated at zero-offset or in the angle domain. As opposed to Hu et al. (2001), the correction in the image is completely data driven, does not depend on the velocity, and can be applied with any migration operator. It also works in the poststack or prestack domain without any extra effort. Providing the data and the ability to run at least two migrations to estimate m 2, this method would be easy to apply with 3-D migrated images. Compared to Rickett (2001), this proposed approach does not need reference images. In addition, the multi-dimensional filters offer more degrees of freedom for the correction than does a simple zero-lag weight: in that we might also correct for kinematic changes and move energy locally in the image. In the limit case where we choose one filter per point and only one coefficient (zero-lag) per filter, the matching filter approach would theoretically perform better than Rickett s method because the weights would be optimal in a least-squares sense without ad-hoc smoothing. In the future, it would be valuable to go beyond 2-D filters by extending them to 3-D and to test this 3-D approach with more field data. In addition, these filters could be used as preconditioning operators providing faster convergence for iterative inversion. The stability of the inverse filters still needs to be addressed, however. ACKNOWLEDGMENTS I would like to thank Paul Sava for providing the migration codes. REFERENCES Beylkin, G., 1985, Imaging of discontinuities in the inverse scattering 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, no. 07, 931 942. Claerbout, J., 1998, Multidimensional recursive filters via a helix: Geophysics, 63, no. 05, 1532 1541.
358 Guitton SEP 113 Figure 5: The cmp location is 3050 m. (a) Migration result. (b) Inversion result. (c) Filtering result (ˆb m 1 ). The filtering result is comparable in amplitude to the inversion panel with fewer artifacts. antoine3-comp.ang.100 [CR,M] Hu, J., Schuster, G. T., and Valasek, P., 2001, Poststack migration deconvolution: Geophysics, 66, no. 3, 939 952. Kuehl, H., and Sacchi, M., 2001, Generalized least-squares DSR migration using a common angle imaging condtion: 71st Ann. Internat. Mtg, Soc. Expl. Geophys., Expanded Abstracts, 1025 1028. Nemeth, T., Wu, C., and Schuster, G. T., 1999, Least-squares migration of incomplete reflection data: Geophysics, 64, no. 1, 208 221. Prucha, M., Biondi, B., and Symes, W., 1999, Angle-domain common image gathers by waveequation migration: 69th Ann. Internat. Mtg, Soc. Expl. Geophys., Expanded Abstracts, 824 827. Rickett, J., Guitton, A., and Gratwick, D., 2001, Adaptive Multiple Subtraction with Non- Stationary Helical Shaping Filters: 63rd Mtg., Eur. Assn. Geosci. Eng., Expanded Abstracts, Session: P167. Rickett, J., 2001, Model-space vs. data-space normalization for finite-frequency depth migration: 71st Ann. Internat. Mtg, Soc. Expl. Geophys., Expanded Abstracts, 2081 2084.
SEP 113 Approximating the Hessian 359 Figure 6: The cmp location is 5550 m. (a) Migration result. (b) Inversion result. (c) Filtering result (ˆb m 1 ). antoine3-comp.ang.300 [CR,M] Figure 7: Sample of the estimated matched filters for Figure 5a. Each filter is 10 by 10. The shape of these filters prove that they tend to smooth energy along the main reflectors. antoine3-filter-minve0 [CR]
360 Guitton SEP 113 Figure 8: (a) Stack across angle of the migrated image m 1. (b) Stack across angle of the migrated image m 2. antoine3-comp.stack.ang1 [CR,M]
SEP 113 Approximating the Hessian 361 Figure 9: (a) Stack across angle of the inverted image. (b) Stack across angle of the filtered image ˆb m 1. antoine3-comp.stack.ang2 [CR,M]
362 Guitton SEP 113 Robinson, E. A., and Treitel, S., 1980, Geophysical signal analysis: Prentice-Hall, Inc. Sava, P., and Fomel, S., 2000, Angle-gathers by Fourier Transform: SEP 103, 119 130. Sava, P., Biondi, B., and Fomel, S., 2001, Amplitude-preserved common image gathers by wave-equation migration: 71st Ann. Internat. Mtg, Soc. Expl. Geophys., Expanded Abstracts, 296 299. Stolt, R. H., and Weglein, A. B., 1985, Migration and inversion of seismic data: Geophysics, 50, no. 12, 2458 2472. Thierry, P., Operto, S., and Lambare, G., 1999, Fast 2-D ray+born migration/inversion in complex media: Geophysics, 64, no. 1, 162 181. Weglein, A. B., and Stolt, R. H., 1999, Migration-inversion revisited (1999): The Leading Edge, 18, no. 8, 950 952.