Spline Representations of Functions on a Sphere for Geopotential Modeling

Abstract

Three types of spherical splines are presented as developed in the recent literature on constructive approximation, with a particular view towards global (and local) geopotential modeling. These are the tensor-product splines formed from polynomial and trigonometric B-splines, the spherical splines constructed from radial basis functions, and the spherical splines based on homogeneous Bernstein-Bézier (BB) polynomials. The spline representation, in general, may be considered as a suitable alternative to the usual spherical harmonic model, where the essential benefit is the local support of the spline basis functions, as opposed to the global support of the spherical harmonics. Within this group of splines each has distinguishing characteristics that affect their utility for modeling the Earth’s gravitational field. Tensor-product splines are most straightforwardly constructed, but require data on a grid of latitude and longitude coordinate lines. The radial-basis splines resemble the collocation solution in physical geodesy and are most easily extended to three-dimensional space according to potential theory. The BB polynomial splines apply more generally to any sphere-like surface (e.g., the geoid or the Earth’s surface) and have a strong theoretical legacy in the field of spline approximations. This report provides a review of these three types of splines, their application to the geodetic boundary-value problem, and formal expressions for determining the model coefficients using data with observational errors.