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.

- High-order plane-wave destructors
- Slope estimation
- Examples of data regularization
- Plane-wave destruction and B-splines
- Plane-wave destruction in 3-D

12/28/2000