Plane-wave destructor filters, introduced by Claerbout (1992), serve the purpose of characterizing seismic images by a superposition of local plane waves. They are constructed as finite-difference stencils for the plane-wave differential equation. In many cases, a local plane-wave model is a very convenient representation of seismic data. Unfortunately, early experiences with applying plane-wave destructors for interpolating spatially aliased data showed that they performed poorly in comparison with that of industry-standard F-X prediction-error filters Spitz (1991).
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 (1992, 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 objects, because their construction involves many non-intuitive parameters. The user needs to choose a raft of parameters, such 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), have 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 level of human interaction and remains the drawback of the method.
Clapp et al. (1997) have recently revived the original plane-wave destructors for preconditioning tomographic problems with a 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 section, 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 regularization problems, I discover that the finite-difference filters often perform as well as or even better than T-X PEFs. At the same time, they keep the number of adjustable parameters to a minimum, and the only quantity we estimate has a clear physical meaning of the local plane-wave slope.