Introduction

Within the exploration industry, geophysicists are realizing the inherent limitations of Kirchhoff methods when it comes to accurately modeling the effects of finite-frequency wave propagation. This is fueling interest in ``wave-equation'' migration algorithms, such as those based on wavefield extrapolation, that do accurately model finite-frequency effects.

As with all migration algorithms, there is a tradeoff amongst extrapolators: cost versus accuracy. For wavefield extrapolators, however, the tradeoff goes three ways: accuracy at steep dips versus the ability to accurately handle lateral velocity variations versus cost again. Fourier finite-difference migration Ristow and Ruhl (1994) strikes an effective balance between the accuracy priorities, combining the steep dip accuracy of phase-shift migration in media with weak lateral velocity contrasts, and the ability to handle lateral variations with finite-difference.

Unfortunately, as with other implicit finite-difference, the cost does not scale well for three-dimensional problems without additional approximations that often expensive and may compromise accuracy. In an earlier paper, Rickett et al. (1998) solved the costly matrix inversion for implicit extrapolation with the 45equation with an approximate LU decomposition based on the helical transform Claerbout (1998b). In this paper, the same approach allows me to extrapolate with a more accurate operator.