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 equation, with form similar to the 15 equation plus an
additional correction term.
Although splitting their equations results in less azimuthal
anisotropy than with the standard 45 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 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.

7/5/1998