Interpolation using a generalized Green's function for a spherical surface spline in tension

A variety of methods exist for interpolating Cartesian or spherical surface data onto an equidistant lattice in a procedure known as gridding. Methods based on Green's functions are particularly simple to implement. In such methods, the Green's function for the gridding operator is determined and the resulting gridding solution is composed of the superposition of contributions from each data constraint, weighted by the Green's function evaluated for all output-input point separations. The Green's function method allows for considerable flexibility, such as complete freedom in specifying where the solution will be evaluated (it does not have to be on a lattice) and the ability to include both surface heights and surface gradients as data constraints. Green's function solutions for Cartesian data in 1-, 2- and 3-D spaces are well known, as is the dilogarithm solution for minimum curvature spline on a spherical surface. Here, the spherical surface case is extended to include tension and the new generalized Green's function is derived. It is shown that the new function reduces to the dilogarithm solution in the limit of zero tension. Properties of the new function are examined and the new gridding method is implemented in Matlab (R) and demonstrated on three geophysical data sets.