Reverse‐time wave‐field extrapolation, imaging, and inversion

Abstract

The scattered wave field propagated backward in time into an arbitrary background medium is related via a volume integral to perturbations in velocity about the background, which are expressed as a scattering potential. In general, there is no closed‐form expression for the kernel of this integral representation, although it can be expressed asymptotically as a superposition of plane waves backpropagated from the receiver array. When the receiver array completely surrounds the scatterer, the kernel reduces to the imaginary part of the Green’s function for the background medium. This integral representation is used to relate the images obtained by imaging algorithms to the actual scattering potential. Two such relations are given: (1) for the migrated image, obtained by deconvolving the extrapolated field with the incident field; and (2) for the reconstructed image, obtained by applying a one‐way wave operator to the extrapolated field and then deconvolving by the incident field. The migrated image highlights rapid changes in the scattering potential (interfaces), whereas the reconstructed image can, under ideal conditions, be a perfect reconstruction of the scattering potential. “Ideal” conditions correspond to (1) weak scattering about a smoothly varying background medium, (2) a receiver array with full angular aperture, and (3) data of infinite bandwidth. Images obtained from a multioffset vertical seismic profile (VSP) illustrate some of the practical differences between the two imaging algorithms. The reconstructed image shows a much clearer picture of the target (a reef structure), in part because the one‐way imaging operator eliminates artifacts caused by the limited aperture of the receiver array.