A wide class of interpolation methods, including thin-plate and tension splines, kriging, sinc functions, equivalent-source, and radial basis functions, can be encompassed in a common mathematical framework involving continuous global surfaces (CGSs). The difficulty in applying these techniques to geophysical data sets has been the computational and memory requirements involved in solving the large, dense matrix equations that arise. We outline a three-step process for reducing the computational requirements: (1) replace the direct inversion techniques with iterative methods such as conjugate gradients; (2) use preconditioning to cluster the eigenvalues of the interpolation matrix and hence speed convergence; and (3) compute the matrix–vector product required at each iteration with a fast multipole or fast moment method.
We apply the new methodology to a regional gravity compilation with a highly heterogeneous sampling density. The industry standard minimum-curvature algorithms and several scale-dependent CGS methods are unable to adapt to the varying data density without introducing spurious artifacts. In contrast, the thin-plate spline is scale independent and produces an excellent fit. When applied to an aeromagnetic data set with relatively uniform sampling, the thin-plate spline does not significantly improve results over a standard minimum-curvature algorithm.