The theory of plane-wave prediction in three dimensions is described
by Claerbout (1993, 1997). Predicting a
local plane wave with *T*-*X* filters amounts to finding a pair of
two-dimensional filters for two orthogonal planes in the 3-D space.
Each of the filters predicts locally straight lines in the
corresponding plane. The system of two 2-D filters is sufficient for
predicting all but purely vertical plane waves, in which case a third
2-D filter for the remaining orthogonal plane is needed.
Schwab (1998) discusses this approach in more detail.

Using two prediction filters implies dealing with two filtering output volumes for each input volume. In many applications, this situation leads to a lot of inconvenience. It happens, for example, when one uses the prediction output as a measure of coherency in the input volume Claerbout (1993); Schwab et al. (1996). Two outputs are obviously more difficult to interpret than one, and there is no natural way of combining them into one image. Another difficulty arises when plane-wave predictors are used for regularizing linear inverse problems Clapp et al. (1997). In this case, we cannot apply an efficient recursive preconditioning Claerbout (1998a) unless the regularization operator is square, or, in other words, only one prediction filter is involved.

Helical filtering Claerbout (1998b) brings us new tools for addressing this problem. In this paper, I show how to combine orthogonal 2-D plane predictors into a single three-dimensional filter with similar spectral properties. The 3-D filter can then work for coherency measurements or for preconditioning 3-D inverse problems. The construction employs the Wilson-Burg method of spectral factorization, adapted for multidimensional filtering with the help of the helix transform Sava et al. (1998).

I use simple synthetic examples to demonstrate the applicability of plane-wave prediction to 3-D problems.

10/25/1999