Smooth fitting of geophysical data using continuous global surfaces

Continuous global surfaces (CGS) are a general framework for interpolation and smoothing of geophysical data. The first of two smoothing techniques we consider in this paper is generalized cross validation (GCV), which is a bootstrap measure of the predictive error of a surface that requires no prior knowledge of noise levels. The second smoothing technique is to define the CGS surface with fewer centers than data points, and compute the fit by least squares (LSQR); the noise levels are implicitly estimated by the number and placement of the centers relative to the data points. We show that both smoothing methods can be implemented using extensions to the existing fast framework for interpolation, so that it is now possible to construct realistic smooth fits to the very large data sets typically collected in geophysics. Thin-plate spline and kriging surfaces with GCV smoothing appear to produce realistic fits to noisy radiometric data. The resulting surfaces are similar, yet the thin-plate spline required less parameterization. Given the simplicity and parsimony of GCV, this makes a combination of the two methods a reasonable default choice for the smoothing problem. LSQR smooth fitting with sine functions defined on a regular grid of centers, effectively low-pass filters the data and produces a reasonable surface, although one not as visually appealing as for splines and kriging.