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Claerbout 1998b 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 (1998a).
To account for nonstationarity in the data,
missing data interpolation with PEFs is typically done in patches or gates where dips are assumed to be approximately stationary (); Spitz (1991).
Each patch constitutes an independent problem, though they may overlap.
The smaller the patch, the more stationary the data is likely to be within
the patch; however, there is a lower limit on the patch size, because a patch
must contain enough data to provide fitting equations for all the filter
coefficients.
Claerbout 1997 describes a method for estimating smoothly time-varying PEFs without patching.
The helix extends the idea of smooth time-variable PEF estimation to smooth time- and space-variable PEF estimation.
The smoothly-variable PEFs can perform better at interpolating missing data than PEFs estimated in independent patches.
Clapp et al. 1998 show
how to use space-variable inverse 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 steering.
I use space-variable steering filters to control the direction of smoothness between PEFs. I orient the steering 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 I review the theory for estimating smoothly varying PEFs, and show examples of their application to missing data interpolation. I describe an improvement to filter estimation for CMP gathers using ``radial-steering filters.'' Finally, I add the notion of signal and noise separation for interpolating
noisy data.

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Stanford Exploration Project

4/20/1999