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# Introduction

There has been a great deal of debate over explicit versus implicit methods for seismic wavefield extrapolation. Stability has traditionally been one of the most compelling advantages of implicit methods. Conventional 45-degree finite-difference migration, for example, is based on an implicit wavefield extrapolation that is guaranteed to be stable (Claerbout, 1985). In contrast, the derivation of stable explicit downward continuation operators is more difficult; their derivation, and proof of stability, assumes the velocity to be locally constant (Blanquiere, 1989; Hale, 1990)

Despite the stability problem, explicit methods are attractive because they are efficient on either vector or parallel computers. In addition to efficiency, another advantage shared by explicit methods for depth migration of seismic wavefields is the ease with which they can be extended for use in 3-D depth migration. On the contrary, the solution of the linear system of equations required by implicit methods is particularly expensive in 3-D.

Recently, a new method for deriving unconditionally stable explicit time extrapolators was presented by Richardson et al. (1991). Their method is based on the decomposition of the banded matrix derived by a finite-difference approximation of the wave equation. This banded matrix is decomposed in block matrices, that can be analytically exponentiated. We apply this new method to derive a stable downward continuation operator from the one-way wave equation. The banded matrix in our case comes from a Taylor approximation of the square root of a matrix. When the Taylor series is truncated after the second term, we get the conventional 15-degree approximation. Adding more terms to the Taylor series improves the accuracy for steep dips. However, the downward continuation operator becomes longer, and its derivation becomes fairly complicated.

Next: ONE-WAY WAVE EQUATION AND Up: Ji and Biondi: Explicit Previous: Ji and Biondi: Explicit
Stanford Exploration Project
12/18/1997