For each given frequency, an F-X prediction-error filter (PEF) can be thought of as a Z-transform polynomial. The roots of the polynomial correspond precisely to predicted plane waves Canales (1984). Therefore, F-X PEFs simply represent a spectral (frequency-domain) approach to plane-wave destruction. This powerful and efficient approach is, however, not theoretically adequate, when the plane wave slopes or the boundary conditions vary both spatially and temporally.
Multidimensional T-X prediction-error filters Claerbout (1992a, 1999) share the same purpose of predicting local plane waves. They work well with spatially aliased data and allow for both temporal and spatial variability of the slopes. In practice, however, T-X filters appear as very mysterious creatures, because their construction involves many non-intuitive parameters. The user needs to choose such parameters as the number of filter coefficients, the gap and the exact shape of the filter, the size, number, and shape of local patches for filter estimation, the number of iterations and the amount of regularization. Recently developed techniques for handling non-stationary PEFs Clapp et al. (1999); Crawley et al. (1998, 1999); Crawley (1999), demonstrated an excellent performance in a variety of applications Brown et al. (1999); Clapp and Brown (2000); Crawley (2000), but the large number of adjustable parameters still requires a significant human interaction and remains the down side of the method.
Clapp et al. (1997) have recently revived the original plane-wave destructors for preconditioning tomographic problems with predefined dip field Clapp et al. (1998); Clapp and Biondi (1998, 2000). The filters were named steering filters because of their ability to steer the solution in the direction of the local dips.
In this paper, I revisit Claerbout's original technique of finite-difference plane-wave destruction. First, I develop an approach for increasing the accuracy and dip bandwidth of the method. Applying the improved filter design to several data interpolation and noise attenuation problems, I discover that the finite-difference filters often perform as well as or even better than T-X PEFs. At the same time, the number of adjustable parameters is kept at minimum, and the only estimated quantity has a clear physical meaning of the local plane-wave slope.
The encouraging results of this paper suggest further experiments with plane-wave destructors. One can apply similar approaches to wave fields, characterized by more complicated differential equations, such as the offset continuation equation Fomel (2000c).