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Short Note
Equivalence of source-receiver migration and shot-profile

Biondo Biondi

At first glance, shot profile migration and source-receiver (survey-sinking) migration seem to be substantially different algorithms. The basic principles used by the two schemes are different. Shot profile migration is performed by independently propagating the source wavefield and the receiver wavefield. The image is obtained by cross-correlating (possibly normalized by the amplitude of the source wavefield) the two wavefields. Source-receiver migration is based on the concept of survey sinking, by which we recursively synthesize equivalent data sets at increasing depth. At each depth step imaging is performed by extracting the wavefield at zero time.

The issue of the relation between shot-profile migration and source-receiver migration has become more relevant since the recent introduction of methods for computing angle-domain common image gathers for source-receiver migration Prucha et al. (1999); Sava et al. (2001). Rickett and Sava (2002) extended one of these methods [Sava et al. (2001)] to downward-continuation shot-profile migration, and Biondi and Shan (2002) extended it to reverse-time shot-profile migration. Their extensions depend on the ``equivalence'' of the offset-domain common image gathers computed by shot-profile migration and source-receiver migration.

In this short note I demonstrate that the two migration methods produce exactly the same image cube; that is, the images are the same not only at zero subsurface offset, but also at non-zero subsurface offset. Wapenaar and Berkhout (1987) had already demonstrated the same result. Their focus, however, was on the stacked image, not on the whole image cube.

For the identity of the two methods to hold, the shot profile migration needs to satisfy three specific requirements: 1) the source function is an impulse at zero time and it has no spatial width, 2) the imaging condition is the cross-correlation of the source wavefield by the receiver wavefield, 3) the source and receiver wavefields are propagated by downward continuation. Another obvious assumption is that the same numerical algorithm is employed to downward continue the wavefields for both migration methods.

The demonstration of the equivalence of the two migration methods becomes fairly simple when we consider the migration of a single shot record by source-receiver downward continuation. In this case, the downward continuation of the sources is equivalent to the multiplication of the downward-continued receiver wavefield with many copies of the complex conjugate (time reversed) source wavefield appropriately shifted along the receiver axis.

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