Data interpolation can be formulated as an inverse problem. If the data and model are in the same space, the data fitting goal is simply a mask operator, which guarantees that the model fits the data at the known points. The model is unconstrained where there are unknown data, leaving a family of possible models with a large nullspace. Unfortunately, there is an ambiguity in how to constrain the solution so that a reasonable result is produced. Claerbout 1999 suggests that the covariance of the known data can be used as the covariance of the model. The data covariance can be characterized by a prediction-error filter, which is estimated from the known data. In the second stage of this method, the PEF is then used to regularize the inversion and constrain the nullspace.
In some cases, the sparse data may be so sparsely sampled as to make conventional PEF estimation impossible. For the special case of interpolating between regularly sampled traces, Crawley 2000 spaces the coefficients of the PEF at multiple scales, and successfully interpolates aliased events. Fomel 2001 uses a nonlinear method to estimate dips within data. The method works well, but is computationally expensive. A test case for sparse data interpolation has been developed by Brown, et al. 2000, which consists of a single plane wave that is sparsely and irregularly sampled.
We develop a method that correctly interpolates a more difficult test case, and provides an overall strategy to interpolate sparse, irregular data when existing methods fail. To do this, we develop a PEF estimation scheme where a single PEF is estimated with multiple scales of regridded data, by simultaneously autoregressing for a common filter. We show that this method is more robust than estimating a PEF on a single scale of data, and provides more equations than estimating the PEF with multiple scales of filters. Once we estimate the PEF, we use the interpolated data as a starting guess for a nonlinear iterative method, which gives promising results. This method is then compared favorably to starting guesses based upon Laplacian interpolation and PEFs estimated from a single scale of data.