Wave equation migration with the phase‐shift method

Abstract

Accurate methods for the solution of the migration of zero‐offset seismic records have been developed. The numerical operations are defined in the frequency domain. The source and recorder positions are lowered by means of a phase shift, or a rotation of the phase angle of the Fourier coefficients. For applications with laterally invariant velocities, the equations governing the migration process are solved very accurately by the phase‐shift method. The partial differential equations considered include the 15 degree equation, as well as higher order approximations to the exact migration process. The most accurate migration is accomplished by using the asymptotic equation, whose dispersion relation is the same as that of the full wave equation for downward propagating waves. These equations, however, do not account for the reflection and transmission effects, multiples, or evanescent waves. For comparable accuracy, the present approach to migration is expected to be computationally more efficient than finite‐difference methods in general.