Introduction

Prediction error filters (PEFs) are commonly used to interpolate data. This is typically carried out in a two step process. First the PEFs are estimated on areas where all of the filter coefficients land on known data. Then the PEFs are used to fill in the missing data. However, sometimes the data is so sparse that PEFs estimated on entirely known data do not adequately capture the nature of the data. In Curry (2003), a multi-scale approach estimates a non-stationary PEF on sparse data. This requires that the gaps in the data range continuously in size. If the data has only a few scales of gaps then this multi-scale approach fails.

I present an approach to estimate a stationary PEF on sparse data using non-linear conjugate gradients Claerbout (1999) where both the filter and the missing data are estimated simultaneously. I add a weight to use only fitting equations where a prescribed minimum number of filter coefficients are on known data. Then, I reduce the minimum number of filter coefficients and solve again. I repeat this bootstrapping process until all of the data is used. A variant of this method was first suggested by Jon Claerbout and described in Lomask (2002). In this paper, this method proves capable of estimating a 2D PEF on datasets where the known data occurs systematically as in the Madagascar satellite data Curry (2004b); Ecker and Berlioux (1995); Lomask (1998, 2002) or possibly the data used in Curry (2004a). If the data is not too radically non-stationary, this stationary PEF can, in principle, be used as a starting solution for tackling the non-stationary problem.

In the following paper, I first present the methodology for estimating the stationary PEF on sparse data. Then I illustrate its effectiveness on the Madagascar satellite data.