The theory of plane-wave prediction in three dimensions is described by Claerbout (1993, 1999). 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 the latter 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. This situation becomes inconvenient when plane-wave destructors are used for regularizing linear inverse problems. We cannot apply the efficient recursive preconditioning introduced in Chapter unless the regularization operator is square, or, in other words, only one plane-wave destructor is involved.
Helical filtering Claerbout (1998a) brings us new tools for addressing this problem. In this subsection, 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 preconditioning 3-D inverse problems, such as data regularization. The construction employs again the Wilson-Burg method of spectral factorization, adapted for multidimensional filtering with the help of the helix transform.
I use simple synthetic examples to demonstrate the applicability of plane-wave prediction to 3-D problems.