INTRODUCTION. Residual migration has proved to be a useful tool in imaging and in velocity analysis.

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1 Stanford Exploration Project, Report, June 3, 999, pages 5 59 Short Note On Stolt prestack residual igration Paul Sava keywords: Stolt, residual igration INTRODUCTION Residual igration has proved to be a useful tool in iaging and in velocity analysis. Rothan (983) shows that post-stack residual igration can be successfully used to iprove the focusing of the igrated sections. He also showed that igration with a given velocity is equivalent to igration with a reference velocity followed by residual igration with a velocity r that can be expressed as a function of and. Residual igration has also been used as a tool in velocity analysis. Al-Yahya (987) discusses a residual igration operator in the prestack doain, and shows that it can be posed as a function of a nondiensional paraeter that is the ratio of the correct velocity and the reference velocity used for the initial igration. Etgen (988, 989) defines a kineatic residual igration operator as a cascade of NMO and DMO, and shows that it, again, is only a function of the nondiensional paraeter defined by Al-Yahya. Finally, Stolt (996) defines a prestack residual igration operator in the f k doain, and shows that it depends on the reference ( ) and the correct ( ) igration velocities. In this short note, I review the prestack residual Stolt igration, and show that it also can be forulated as a function of a nondiensional paraeter that is the ratio of the reference ( ) and correct ( ) velocities. Consequently, we can use Stolt residual igration in the prestack doain to obtain a better focused iage without aking any assuption about the velocity. This approach has a direct application to igration velocity analysis, for instance in cases when we repeatedly do residual igration on data that have been depth-igrated with an arbitrary velocity function that cannot be approxiated by a constant velocity (Biondi and Sava, 999). eail: paul@sep.stanford.edu 5

2 5 Sava SEP STOLT MIGRATION Prestack Stolt igration (PrSM) was suarized as (Claerbout, 985) p t y h P k y k h P k z k y k h p z y h An iportant coponent of PrSM is the reapping fro the k y k h doain to the k z k y k h doain, where, k z represent, respectively, the frequency and the vertical wavenuber, and k y k h represent the idpoint and offset wavenubers. If we consider the alternative representation of the input data in shot-geophone coordinates, the apping takes the for k z k g k s () where k g and k s stand for, respectively, the geophone and the source wavenubers. It is desirable to ipleent the reapping as a pull operator, to avoid nuerical probles in the inverse Fourier transfor. A detailed discussion on the advantages and disadvantages of the different appings is done by Levin (994). We can, therefore, express as a function of k z fro Equation () as: 6k z 4k z 4k z () or k z k z k h k z k y (3) -D RESIDUAL STOLT MIGRATION In general, residual igration represents a ethod of iproving the quality of the iage without having to reigrate the original data, but rather only applying a transforation to the current igration iage. In residual prestack Stolt igration (RPrSM), we attept to correct the effects of igrating with an inaccurate reference velocity by applying a transforation to the data that have been transfored to the Fourier doain (Figure ). Supposing that the initial igration was done with the velocity, and that the correct velocity is, we can then write k z k g k g k s k s (4)

3 SEP Residual igration 53 Kh Ky Kh Ky V Kz Figure : A sketch of Stolt residual igration paul-stolt [NR] Vo Vr Kh Ky Kzo The goal of RPrSM is to obtain fro k z. If we use the first equation of (4) to substitute in the second equation of (4), we obtain or 4k z 4k z 6k z k g 4k z 4k z 6k z k s (5) kz k h kz k y k kz y k h k z k h k z k y k z k y k h (6) Equation (6) represents the RPrSM equation in two diensions. For post-stack data, the sae equation takes the failiar for k z k y k y (7) 3-D RESIDUAL STOLT MIGRATION The RPrSM equations in three diensions can be written siilarly to those in two diensions if we consider the relationship between source-geophone and idpoint-offset coordinates as follows: k g k y k h k s k y k h The equivalent prestack residual igration equation in source-geophone coordinates thus becoes 4k z 4k z 6k z k g 4k z 4k z 6k z k s (8)

4 54 Sava SEP In idpoint-offset coordinates, the sae equation becoes 4k z k y k h k y k h 4k z k y k h k y k h 6k z k y k h 4k z k y k h k y k h 4k z k y k h k y k h 6k z k y k h (9) which, for the post-stack case, takes the for k z k y k y () EXAMPLES In this section, I present two -D post-stack synthetic exaples, shown in Figures and 3, to prove the applicability of the equations derived in the preceding sections. In the first exaple, the input is a set of three spikes. Initially, I do forward odeling with a velocity of 3 k/s. Next, I do Stolt igration with a velocity of 3 6 k/s, and residual Stolt igration with a ratio (Figure ). I then take the sae input and perfor Stolt igration with a velocity of 4 k/s, followed by residual Stolt igration with a ratio 8 (Figure 3). In both cases, the data are correctly collapsed at the location of the original spikes. Residual igration can be done without knowing the absolute values of the velocity. In the next exaple, shown in Figure 4, I apply the sae ethodology to a real dataset (Ecker, 998). All three panels are the result of residual igration as described in the preceding theory sections. The corresponding ratios are 96 for the top panel, 98 for iddle panel, and for the botto panel. Residual igration with ratio= is equivalent to no residual igration at all. It is apparent that different iages are focused better in one region or another, although the iage corresponding to the ratio 98 sees to have the highest overall energy. We can use such an observation to obtain an iage that has the best focusing in all the regions. This can be achieved by doing residual igration for a range of velocity ratios, and then interpolating the iage that is best focused everywhere. One possible application is in wave-equation igration velocity analysis, where we can iprove the focusing of a depth-igrated dataset by residual igration, without aking any assuption about the original velocity distribution (Biondi and Sava, 999). CONCLUSIONS Equations (5) and (6) show that residual prestack Stolt igration can be done without actual knowledge of the reference and correct velocities, but only by knowing or assuing their ratio. This understanding has iportant practical consequences, for exaple, in applications

5 SEP Residual igration 55 Figure : Top left: input data. Top right: Stolt forward odeling with the correct velocity 3 k/s. Botto left: Stolt igration with an incorrect velocity of 3 6 k/s. Botto right: Stolt residual igration with the ratio. paul-posp [ER]

6 56 Sava SEP Figure 3: Top left: input data. Top right: Stolt forward odeling with the correct velocity 3 k/s. Botto left: Stolt igration with an incorrect velocity of 4 k/s. Botto right: Stolt residual igration with the ratio 8. paul-pos [ER]

7 SEP Residual igration 57 Figure 4: Iages obtained after residual igration. Ratio = eans no residual igration. Different iages are focused better in different regions. The iage corresponding to ratio = 98 sees to have the best focusing. paul-hydrates [CR]

8 58 Sava SEP that use RPrSM as a ethod of iproving the focusing of an iage that has been depthigrated with an arbitrary velocity function. ACKNOWLEDGMENTS I would like to thank Sergey Foel for allowing e to use his interpolation code in the Stolt progras. REFERENCES Al-Yahya, K., 987, Velocity analysis by iterative profile igration: Ph.D. thesis, Stanford University. Biondi, B., and Sava, P., 999, Wave-equation igration velocity analysis: SEP, 34. Claerbout, J. F., 985, Iaging the Earth s Interior: Blackwell Scientific Publications. Ecker, C., 998, Seisic characterization of gas hydrates structures: Ph.D. thesis, Stanford University. Etgen, J., 988, Velocity analysis by prestack depth igration: Linear theory: SEP 57, Etgen, J., 989, Kineatic residual prestack igration: SEP 6, 79. Levin, S. A., 994, Stolt without artifacts? dropping the Jacobian: SEP 8, Rothan, D. H., Levin, S. A., and Rocca, F., 983, Residual igration: SEP 35, Stolt, R. H., 996, Short note a prestack residual tie igration operator: Geophysics, 6, no.,

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