Migration deconvolution vs. least squares migration

Both migration deconvolution (MD) and least squares migration (LSM) are capable of improving the resolution and suppress acquisition footprints in migrated images. In this paper, we investigate the relative performance of these two technologies in enhancing migration image quality, suppressing artifacts and computational efficiency. Both MD and LSM were tested on SEG/EAGE overthrust models. The results indicate that both MD and LSM improve the energy focusing and illumination of the migration images. LSM sharpens the reflection events with increasing iterations at the cost of much more increase in CPU time. In comparison, MD produces significant improvements in the spatial resolution and migration noise suppression of migration images but at a low computation cost. Introduction Least squares migration has the potential to reconstruct more accurate reflectivity distributions of buried structures. With increasing iterations, LSM is able to significantly suppress the migration artifacts arising from incomplete data and enhance the illumination of migration images (Duquet et al, 2000; Nemeth et al., 1999; Schuster 1997). A similar method, migration deconvolution, has also been shown to be an effective method in improving the spatial resolution, attenuating migration noise, and correcting for some migration amplitude distortion in 2D and 3-D P-P data and PS wave data (Hu et al., 2001; Yu, 2002). The purpose of this abstract is to evaluate the performance of the MD and LSM methods in improving the spatial resolution of the migration image and suppressing migration noise. Here, MD and LSM were tested on synthetic data generated from the point scatterer model and SEG/EAGE overthrust model. Experimental evidence suggests that MD is both more efficient and effective in suppressing migration noise and enhancing spatial resolution. Its computation is more efficient than LSM does, at least for the SEG/EAGE overthrust data set tested. In the following, we first present the brief description of of MD and LSM and then demonstrate the application of MD and LSM with two examples. Least squares migration The reflectivity distribution r of the earth can be inverted from the observed seismic data d by minimizing the l2 seismogram misfit function as = 1 2 ||Lr− d|| + 1 2 ||Cr||, (1) where L denotes a forward modeling operator that approximates the actual wave propagation in the earth; the d and r represent the data and model vectors, respectively; and || || represents the l2 norm. The ||Cr|| term is the regularization functional and C is the constraint matrix. The reflectivity distribution that minimizes equation (1) is obtained by solving the following normal equation [LL + CTC]−1r = Ld, (2) where L is the transpose to the forward modeling operator L. Dropping the regularization term, equation (2) reduces to r = [LTL]−1LTd. (3) Equation (2) or (3) can be solved using the conjugate gradient algorithm. Migration deconvolution method The basic idea of MD approach can be understood by relating the true reflectivity distribution m to the seismic data d via Lm = d, (4) where L is the forward modeling operator and d is the observed seismic data. The migrated image can be constructed by applying the adjoint operator L to the data, i.e., Ld = LLm = m′, (5) where m’ is the migrated image. Here we see that the migration image is a blurred vision of the true reflectivity m, where LL is a blurring operator. To deblur m′, we apply the deblurring operator [LTL]−1 to m′ and get [LTL]−1LTLm = [LTL]−1m′. (6) or m = [LTL]−1m′, (7) MD vs. LSM where [LTL]−1 is the deblurring operator that can be obtained by calculating migration Green’s function associated with acquisition geometry and velocity model. The detailed description of the MD implementation is presented in Hu et al (2001). Comparing equations 3 and 7, we see that both expressions yield the true reflectivity. However their implementations are different: one is applied to data and the other to the migration image. Numerical results The MD and LSM methods were tested on synthetic data associated with a point scatterer model and the SEG/EAGE overthrust model. Point scatterer model In this example, the source position is located at the surface, and the model is represented by a 100 × 600 grid of reflectivity values. The time sample interval is 2 ms, and there are ten impulsive scatterers buried around the central part of the subsurface model (see the left panel in Figure 1). Applying standard Kirchhoff migration to the data results in the image shown in the middle frame of Figure 1. The right frame in Figure 1 is the migration result after application of MD. It is clear that MD reduces the migration noise and improves the spatial resolution. The LSM results are depicted in Figure 2. The left, middle and right frames show the LSM images, respectively, after 10, 15 and 19 iterations. Compared with Figure 1, LSM also improves the quality of migration image, attenuates the migration noise, and produces the ideal image. It is seen that the LSM image has better spatial resolution than the MD image after a sufficient number of iterations. This is not surprising because MD assumes a very wide aperture of seismic traces. SEG/EAGE overthrust model Both the MD and LSM methods were also applied to a complex data set associated with the SEG/EAGE overthrust model. The portion of velocity model is depicted in Figure 3, which is gridded into a 311× 187 image with grid point dimensions of 25 × 25 m. The time sample interval is 8 ms, and the source and receiver depths are at -25 m. Before migrating the overthrust seismic data, some pre-processing steps, such as data muting, were used because of the existence of multiples (Schuster, 1997). The Kirchhoff migration result is shown in the top panel of Figure 4. The middle panel of Figure 4 corresponds to the LSM image after 15 iterations, which slightly sharpens the reflection events and enhances the amplitude of some events. The reflectivity boundaries in the LSM image become clearer after 15 iterations compared with the Kirchhoff migration result. The resulting image after applying MD is shown in the bottom panel of Figure 4. Compared with Kirchhoff migration, MD produces a high resolution image with less migration noise. Furthermore, it is observed that the MD image is better than that of LSM in attenuating migration noise in the complex part of the overthrust structure. The closeups of the migrated sections for Kirchhoff migration, LSM and MD are presented in Figure 5. The top frame in Figure 5 represents the Kirchhoff migration result. The middle and bottom frames in Figures 5 are the zoom views of the LSM result with iteration numbers 15 and MD image, respectively. Comparing these closeups, it is noted that both MD and LSM generated the higher resolution images than the Kirchhoff migration result. But MD is superior to LSM in attenuating migration noise. Additionally, MD costs about one migration compared to more than ten migrations for LSM. Table 1 gives the comparison of the performance of MD and LSM in enhancing image resolution, attenuating migration noise, and computational cost. Conclusions The MD and LSM methods are applied to synthetic data from the SEG/EAGE overthrust models. Both MD and LSM are capable of enhancing the resolution of migration images. The results from the SEG/EAEG overthrust model data show MD is superior to LSM in suppressing migration noise in this test. However, MD is an order of magnitude cheaper in computation efficiency compared with LSM. Acknowledgments We are grateful to the sponsors of the University of Utah Tomography and Model/Migration (UTAM) Consortium for their financial support of this project. References Duquet B., Marfurt K., and Dellinger J., 2000, Kirchhoff modeling, inversion for reflectivity, and subsurface illumination: Geophysics, Vol. 65 (4), pp.1195-1209. Gonzales, R., and Woods, R., 1992, Digital Image Processing: Addison-Wesley Publ. Co., Menlo Park, Calif.. Hu, J., Schuster, G., and Valasek, P., 2001, Poststack migration deconvolution: Geophysics, Vol.66(3), pp. 939-952. Hu J., 1997, Migration deconvolution, UTAM Annual Report, 37-71.