Abstract

Stability has traditionally been one of the most compelling advantages of implicit methods for seismic wavefield extrapolation. The common 45-degree, finite-difference migration algorithm, for example, is based on an implicit wavefield extrapolation that is guaranteed to be stable. Specifically, wavefield energy will not grow exponentially with depth as the wavefield is extrapolated downwards into the subsurface. Explicit methods, in contrast, tend to be unstable. Without special care in their implementation, explicit extrapolation methods cause wavefield energy to grow exponentially with depth, contrary to physical expectations.The Taylor series method may be used to design finite-length, explicit, extrapolation filters. In the usual Taylor series method, N coefficients of a finite-length filter are chosen to match N terms in a truncated Taylor series approximation of the desired filter's Fourier transform. Unfortunately, this method yields unstable extrapolation filters. However, a simple modification of the Taylor series method yields extrapolators that are stable.The accuracy of stable explicit extrapolators is determined by their length--longer extrapolators yield accurate extrapolation for a wider range of propagation angles than do shorter filters. Because a very long extrapolator is required to extrapolate waves propagating at angles approaching 90 degrees, stable explicit extrapolators may be less efficient than implicit extrapolators for high propagation angles. For more modest propagation angles of 50 degrees of less, stable explicit extrapolators are likely to be more efficient than current implicit extrapolators. Furthermore, unlike implicit extrapolators, stable explicit extrapolators naturally attenuate waves propagating at high angles for which the extrapolators are inaccurate.

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