Introduction

Prestack data are usually transformed to zero offset by the application of normal moveout (NMO) followed by dip moveout (DMO). Different algorithms have been proposed to perform DMO assuming a constant velocity medium (Deregowski and Rocca, 1981; Hale, 1983; Jakubowitz, 1984; Biondi and Ronen, 1987). These conventional DMO operators are kinematically correct when velocity is constant and they are approximately correct when velocity linearly increases with depth (Hale, 1983), but they cannot be applied when velocity varies rapidly. In this case the prestack partial migration (PSPM) process cannot even be split in NMO followed by DMO, but must be applied as a single process.

A PSPM algorithm that transforms the prestack data to zero offset, even when velocity is a general 2-D function, is an attractive alternative to full prestack migration because it requires less computer resources. In 2-D the spatial aperture of the PSPM operator in constant offset sections is limited by the offset, while the corresponding full prestack migration operator has unlimited aperture. In 3-D the gain in efficiency is even more important; the impulse response of full prestack migration is a a 3-D ellipsoid, while the impulse response of the PSPM operator is a simple 2-D ellipse lying in a vertical plane passing through the source and the receiver. Another application where PSPM has potential advantages over full prestack migration is velocity estimation. The velocity function assumed for applying PSPM influences the alignment in time of the reflections over offset but not their absolute position. On the contrary, the choice of the migration velocity influences the absolute position of the reflections (Fowler, 1988).

The determination of the correct amplitudes is important for many applications. In particular when DMO is used for Amplitude Versus Offset (AVO) analysis of the prestack data (Gardner and Forel, 1988; Black and Egan, 1988; Beasley and Mobley, 1988).

We propose in this paper an algorithm for computing the impulse response of the PSPM operator for any v(z) velocity model. The method can be easily adapted for v(x,z). We apply the proposed algorithm for computing the kinematics of the impulse response when velocity is constant and for different velocity functions of depth. We show that the impulse response in variable velocity media not only differs from the constant velocity one by a time shift but in certain cases triplications appear in the impulse response. The proposed algorithm can be applied to partially migrate the prestack data using the impulse response as an integral operator.