Prediction-error filters (PEFs) can be used to interpolate data Claerbout (1999); Crawley (2000); Spitz (1991). In order to generate a PEF, a least-squares problem is solved where regularly-sampled data is convolved with an unknown PEF. When estimating a PEF with missing data, the equations that contain missing data can be eliminated from the inversion. When dealing with sparse data, however, all equations may contain missing data, so a PEF cannot be estimated.
This problem has been addressed by regridding the data to multiple different scales, and then introducing those coarser copies of the data to the problem to gain more fitting equations Curry and Brown (2001); Curry (2002). When creating coarser copies of the data, there is freedom in choosing the method used to regrid the data, as well as the choice of the coarser grid itself.
Existing methods use multiple different grid sizes in order to extract more information from the sparse data. More fitting equations can be generated by not only varying the size of the grid, but also by varying the positioning of that grid, i.e., the location of the origin of the grid. As the grid becomes coarser, the number of possible grid positions increases, and as the dimensionality of the problem increases, that number increases further.
By varying the grid location, a PEF can be more accurately determined on sparse data. 2D and 3D examples are shown for non-stationary PEF estimation, with a noticeable improvement in the interpolated result.