Introduction

Implicit 2-D finite-difference wavefield extrapolation has proved itself as a robust, accurate migration method Claerbout (1985). It naturally and efficiently deals with lateral variations in velocity without the need for asymptotic approximations, such as ray-tracing. The implicit formulation also ensures unconditional stability. Unfortunately, despite the rapid growth of 3-D seismology applications, implicit 3-D wavefield extrapolation has yet to find wide-spread application. Whereas 2-D extrapolation requires the inversion of a tridiagonal system, the simple extension from 2-D to 3-D leads to a blocked tridiagonal system, which is prohibitively expensive to solve.

Typically, the matrix inversion problem is avoided by an explicit finite-difference approach Holberg (1988). Explicit extrapolation has proved itself effective for practical 3-D problems; since stable explicit filters can be designed Hale (1990b), and McClellan filters provide an efficient implementation Hale (1990a). However, unlike implicit methods, stability can never be guaranteed if there are lateral variations in velocity Etgen (1994). Additionally, accuracy at steep dips requires long explicit filters, which cannot handle rapid lateral velocity variations, and can be expensive to apply.

The problem can also be avoided by splitting the operator to act sequentially along the x and y axes. Unfortunately this leads to azimuthal operator anisotropy, and requires an additional phase correction operator Graves and Clayton (1990); Li (1991). Zhou and McMechan (1997) have presented an alternative to the traditional 45$^\circ$ equation, with form similar to the 15$^\circ$ equation plus an additional correction term. Although splitting their equations results in less azimuthal anisotropy than with the standard 45$^\circ$ equation, the splitting approximation is still needed to solve the equations.

We apply helical boundary conditions Claerbout (1997), to simplify the structure of the matrix, reducing the 2-D convolution to an equivalent problem in one dimension. The 1-D convolution matrix can be factored into a pair of causal and anti-causal filters, thereby providing an LU decomposition. The factorization is based on Kolmogoroff's spectral method, but with an extension to handle cross-spectra Claerbout (1998). The filters are then inverted efficiently by recursive polynomial division. We also allow for laterally variable velocity by factoring spatially varying filters, followed by non-stationary deconvolution.

Very accurate implicit methods have been developed for 2-D migrations (e.g. Jenner et al., 1997) without obvious extensibility to 3-D. Although we only solve the 45$^\circ$ wave equation in this paper, the helical boundary conditions provide a practical way to apply implicit migrations of higher accuracy in 3-D. In addition, helical boundary conditions and the common-azimuth formulation Biondi and Palacharla (1996) may enable wave-equation based 3-D prestack depth migration with finite-differences.