Seismic migration using the WKB approximation

Abstract

Summary. Acoustic wave propagation in a horizontally stratified medium is considered. A Fourier transform with respect to time and horizontal space variables results in two coupled first‐order differential equations with respect to the depth variable. These equations are transformed to give new equations which govern the interaction between the up‐ and downgoing wavefields. The reflected wavefield from a point source is recorded with an areal array at the surface. The zero‐order WKB approximation (consisting of neglecting all interaction between the down‐ and upgoing wavefields) is used to downward extrapolate both the downgoing wavefield from the source and the recorded upgoing wavefield. The ratio between the up‐ and the down‐ going wavefields is used to compute the depth‐migrated seismic trace. An extension of the zero‐order migration scheme is found by using the first‐order WKB approxitnation where the effect of the downgoing wavefield on the upgoing wavefield is considered. This‐results in an additional term in the migrated seismic traces which only depends on the generalized reflectivity function. It is shown that the Hankel transform may be used in the prestack migration of seismic line data. The zero‐order and first‐order WKB approximations result in similar algorithms as derived for areal seismic data. An exact equation is derived for zero‐offset seismic data, neglecting second‐order derivatives with respect to offset coordinates results in an inhomogeneous acoustic wave equation with half the velocity and twice the density of the real medium. Neglecting all transmission effects in this equation results in the classical frequency–wavenumber migration algorithm. The zero‐order WKB approximation gives a new migration algorithm which takes the transmission effects into account. The first‐order WKB approximation results in the same migrated section plus an additional term which takes into account the effect of the downgoing wavefield on the upgoing wavefield. Copyright © 1984, Wiley Blackwell. All rights reserved