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INTRODUCTION

Data interpolation is a long standing and persistent problem in exploration geophysics. Methods range from those based on the known behavior of the kinematics of seismic data Chemingui (1999); Fomel (2001); Vlad and Biondi (2001), to those that are based on transformations of the data into another domain, such as the Fourier Schonewille (2000) or Radon Trad et al. (2002) domains. Other methods use prediction-error filters (PEFs) in either the f-x Spitz (1991) or t-x Claerbout (1992, 1999) domains. Claerbout's method is cast as inverse problem, where the known data remains constant, and the empty bins are regularized by a PEF to constrain the null space. This is done in two stages, where a PEF is first estimated on known data, and is then used to constrain the unknown data by minimizing the output of convolution of the model with the PEF.

When the data are not stationary, a non-stationary filter may be used to fill the unknown data Crawley (2000), and in the case of coarsely sampled data, the filter can be stretched over various scales to fit the data. A PEF may be estimated on irregularly sampled data by scaling the data to various grid sizes and simultaneously estimating a single PEF on the various scales of data Curry and Brown (2001). This method of PEF estimation has also been shown to work for non-stationary filters Curry (2002).

One of the main drawbacks associated with using PEFs is the large number of non-intuitive parameters needed to create them. These include parameters relating to the structure of the PEF, it's size, shape, and the degree of non-stationarity in the filter. There are also many parameters needed in the PEF estimation process: the number of conjugate-direction iterations used to estimate the filter; and in the case of non-stationary PEFs, the type and amount of regularization used in the estimation.

In addition, the multi-scale method Curry and Brown (2001) requires another set of parameters to determine the scales of data to be used. In the case of non-stationary PEF estimation, the possible scales of data are limited by the size of the micropatches.

Currently, the only way to estimate many of these parameters is by trial and error. The spatial distribution of the data is known, but is not incorporated in a meaningful way into the estimation of these parameters. Also, when dealing with a non-stationary scenario, the importance of having all regions well-constrained by fitting equations is also an unresolved issue.

In this paper, I explore a general strategy to estimate certain parameters, where the number of valid fitting equations is counted at each position for every possible combination of parameters. A weighted sum of these equation maps may then be calculated, and from those results, optimal parameters can be obtained.


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Stanford Exploration Project
7/8/2003