Stable explicit depth extrapolation of seismic wavefields

Stability has traditionally been one of the most compelling advantages of implicit methods for seismic wavefield extrapolation. The common 4S-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 tilters. Because a very long extrapolator is required to extrapolate waves propagating at angles approaching 90 degrees, stable explicit extrapolators muy be less elticient than implicit extrapolators for high propagation angles. For more modest propagation angles of 50 degrees of less, stable explicit e_xtrapolmrs are !ikely !r.? be more &Gent than current implicit extrapolators. Furthermore, unlike implicit extrapolators, stable explicit extrapolators naturally attenuate waves propagating at high angles for which the extrapolators are inaccurate. INTRODUCTION coupled equations for the filtered output samples. Partly because it is simpler, explicit filtering is likely to be impleImplicit filtering methods are widely used to extrapolate mented more efficiently on various computer architectures seismic wavefields in depth. For example, the well-known (vector, parallel. super-scalar. etc.) than is implicit filtering. 45degree, finite-difference method for depth migration is In addition to simplicity, another advantage of explicit based on a recursive application of implicit filtering (e.g.. methods for depth extrapolation of seismic wavefields is the Claerbout, 1985). Implicit methods are most attractive beease with which explicit methods can be extended for use in cause they are guaranteed to be stable. Specifically, implicit 3-D depth migration. The solution of the linear system of methods for depth extrapolation will not permit the ampliequations required by implicit methods is particularly awktude of the extrapolated wavefield to grow with depth. In ward in this application. For example, an accurate extension contrast, the most straightforward explicit extrapolation of the implicit 45-degree, finite-difference method to 3-D methods are unstable, tending to amplify wavefield amplidepth migration is difficult and may be computationally tudes exponentially with depth. impractical (Claerbout, 1985, p. 101: Yilmaz, 1987, p. 405). Notwithstanding stability, explicit filtering is attractive Explicit depth extrapolation methods, in contrast, are easily because it resembles convolution, for which each filtered extended to 3-D depth migration, as demonstrated by Blacoutput sample can be computed independently, perhaps in quiere et al. (1989). parallel with other output samples. Implicit filtering, in These advantages of computational simplicity, efficiency, contrast. is accomplished by solving a linear system of and extendability motivate the development of a method for Manuscript received by the Editor August 16. 1990;. revised manuscript received February 20, 1991. *Department of Geophysics. Colorado School of Mines. Golden, CO 80401.