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).

9/5/2000