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Measurements made on the surface of the Earth are often sparse and unevenly distributed. For example, GPS displacement measurements are limited by the availability of ground stations and airborne geophysical measurements are highly sampled along flight lines but there is often a large gap between lines. Many data processing methods require data distributed on a uniform regular grid, particularly methods involving the Fourier transform or the computation of directional derivatives. Hence, the interpolation of sparse measurements onto a regular grid (known as gridding) is a prominent problem in the Earth Sciences.

Popular gridding methods include kriging, minimum curvature with tension (W. Smith & Wessel, 1990), and bi-harmonic splines (DT Sandwell, 1987). The latter belongs to a group of methods often called radial basis functions and is similar to the thin-plate spline (Franke, 1982). In these methods, the data are assumed to be represented by a linear combination of Green’s functions,