Conventional seismic data processing can be improved by modifying wide-offset data so that dipping events stack coherently. A procedure to achieve this improvement is proposed here, which is basically a 'partial' migration of common offset sections prior to stack. It has an advantage over full migration before stack in that, in the case of the latter, the final product is a migrated section. However, the prestack partial migration provides the interpreter with a high-quality common midpoint (CMP) stacked section which can be subsequently migrated.The theory of prestack partial migration is based on the double square-root equation, which describes seismic imaging with many shots and receivers. The double square-root operator in midpoint-offset space can be separated approximately into two terms, one involving only migration effects and the other involving only moveout correction. This separation provides an analysis of conventional processing. Estimation of errors in the separation yields the equation for prestack partial migration.Extension of the theory for separable approximation to incorporate lateral velocity variation yields a significant term proportional to the product of the first powers of offset, dip, and lateral velocity gradient. This term was used to obtain a rough estimate of lateral velocity variation from a field data set.