Claerbout (1999) presents a two-stage methodology for missing data interpolation. In the first stage of the so-called GEE approach, a prediction error filter (PEF) is estimated from known data. In the second stage, the PEF is used in a least squares interpolation scheme to regularize the undetermined (missing) model points. Crawley (2000) extends the two-stage procedure to use nonstationary PEF's.
The first stage (PEF estimation) of the two-stage procedure consists of convolving the unknown filter coefficients with the known data, and adjusting the coefficients such that the residual is minimized. Conceptually, in the process of convolution, a filter template is slid past every point in the data domain. The GEE approach adheres to the following convention: unless every point in the filter template overlies known data, the regression equation for that output point is ignored, and will not contribute to the PEF estimation.
Unfortunately, when the known data is very sparsely distributed, all the regression equations may depend on missing data, making PEF estimation impossible. The motivation of this paper is not to present a new methodology for estimating a PEF from sparse data, but instead to create a very simple test case which fulfills the following criteria: