Claerbout 1997a describes a helical coordinate to cast multi-dimensional filtering as one dimensional, enabling the use of some well-developed signal processing theory in applications including missing data interpolation Fomel et al. (1997) and low-cut filtering Claerbout (1998). Missing data interpolation with PEFs is typically done in small patches where dips are approximately stationary Abma and Claerbout (1995), to account for nonstationarity in the data. Claerbout 1997b describes a method for estimating smoothly time-varying PEFs without patching. We use the helix to extend the idea of smooth time-variable PEF estimation to smooth time- and space-variable PEF estimation. The new PEFs can perform better at interpolating missing data than PEFs estimated in patches. More applications will come later.
Clapp et al. 1997 show how to use space-variable inverse flag (or ``steering") filters to smooth in adjustable directions, and they show how to solve empty-bin problems filling in missing data along the directions of the flags. We use space-variable flag filters to control the direction of smoothness between PEFs. We orient the flag filters radially in a CMP gather to encourage PEFs to have the same dip information along lines of constant x/t, where data tends to have constant dip spectra. In this paper we review the theory for estimating nonstationary PEFs, and show examples of their application to missing data interpolation. Finally, we describe an improvement to filter estimation for CMP gathers using ``radial-flag filters.''