Prestack phase-shift migration of separate offsets in laterally inhomogeneous media

Using the stationary-phase method, prestack phase-shift migration is implemented one offset at a time. This separate-offset implementation allows for a Fourier (reasonably fast) wave-equation-type migration on data with irregular offset sampling. However, the separate-offset phase-shift migration, like its zero-offset counterpart, handles only vertically inhomogeneous media. Using the combination of the split-step and phase-shift-plus-interpolation (PSPI) approaches, the separate-offset phase-shift migration is extended to handle laterally inhomogeneous media. The cost of the separate-offset implementation is practically equivalent to that of the conventional zero-offset version. However, due to the lack of exact source and receiver ray-trajectory information in the separate-offset implementation, the combined split-step and PSPI handles only smooth lateral inhomogeneity. Specifically, it produces images equivalent to those resulting from smoothing the velocity model laterally over a window equal to the half offset. Thus, for zero-offset or laterally homogeneous media, the separate-offset migration is equivalent to any wave-equation-based migration. Errors might occur for finite-offset data in laterally inhomogeneous media. Such errors depend primarily on the strength of lateral inhomogeneity. Using this separate-offset phase-shift migration, accurate images of synthetic data of a model with large reflector dips and good images from real data from offshore Trinidad are obtained.