Implicit finite-difference wavefield extrapolation played an
exceptionally important role in the early development of seismic
migration methods. Using limited-degree approximations to the one-way
wave equation, implicit schemes have provided efficient and
unconditionally stable numerical wave extrapolation operators
Claerbout (1985); Godfrey et al. (1979). Unfortunately, the
advantages of *implicit* methods were lost with the development
of three-dimensional seismic exploration. While the cost of 2-D
implicit extrapolation is linearly proportional to the mesh size, the
same approach, applied in the 3-D case, leads to a nonlinear
computational complexity. Primarily for this reason, implicit
extrapolators were replaced in practice by *explicit* ones,
capable of maintaining linear complexity in all dimensions. A number
of computational tricks Hale (1991b) allow the commonly
used explicit schemes to behave stably in practical cases. However,
their stability is not unconditional and may break in unusual
situations Etgen (1994).

In this paper, we present an approach to three-dimensional extrapolation, based on the helix transform of multidimensional filters to one dimension Claerbout (1997b). The traditional approach involves an inversion of a banded matrix (tridiagonal in the 2-D case and blocked-tridiagonal in the 3-D case). With the help of the helix transform, we can recast this problem in terms of inverse recursive filtering. The coefficients of two-dimensional filters on a helix are obtained by one-dimensional spectral factorization methods. As a result, the complexity of three-dimensional implicit extrapolation is reduced to a linear function of the computational mesh size. This approach doesn't provide an exact solution in the presence of lateral velocity variations. Nevertheless, it can be used for preconditioning iterative methods, such as those described by Nichols (1991). In this paper, we demonstrate the feasibility of 3-D implicit extrapolation on the example of laterally invariant velocity continuation and, in the final part, discuss possible strategies for solving the problem of lateral variations.

The main application of finite-difference wave extrapolation is
*post-stack* depth migration. An application of similar methods
for *prestack* common-shot migration is constrained by the
limited aperture of commonly used seismic acquisition patterns.
Recently developed acquisition methods, such as the vertical cable
technique Krail (1993), open up new possibilities for 3-D wave
extrapolation applications. An alternative approach is common-azimuth
migration Biondi and Palacharla (1994); Biondi (1996). Other interesting
applications include finite-difference data extrapolation in offset
Fomel (1995), migration velocity Fomel (1996a),
and anisotropy Alkhalifah and Fomel (1997).

10/9/1997