Applications of operator separation in reflection seismology

Abstract

The principle of operator separation, a generalization of operator splitting, is applied to some problems in reflection seismology. In particular, the examples of wave‐equation migration of seismic data in a three‐dimensional medium and accurate depth migration in a laterally varying medium are considered in light of this theory. For the case of a stratified three‐dimensional medium, the standard dimensional splitting technique used in the downward‐continuation step of the migration process can be replaced with full dimensional separation. The computational implications of this result are that the wave field array need only be transposed once during the downward continuation rather than 2n times, where n is the number of finite‐difference steps taken in the calculation. For the example of downward continuation in a laterally varying medium, the ideas of operator separation can be used to split the downward continuation operator into two parts, one that looks like the conventional downward‐continuation operator for a stratified medium, and a second that represents the correction for the effects of lateral variation.