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).