Accurate Multiresolution Modeling of the Earth's Gravitational Field

Abstract

Abstract : The thrust of this project has been two-fold, namely, (1) to improve the performance of existing models of Earth's gravitational field, mainly with respect to speed of evaluation; and (2) to develop new multi resolution estimation techniques to produce new gravitational models. We have constructed two local models of the gravitational field, both of which map the surface of a sphere to the surface of a cube. These models differ in the choice of basis functions. The first uses multi-wavelets to represent the gravitational field at a fixed distance from Earth, and the second model uses B-splines. Both models use polynomial interpolation to compute the variation in height of the gravity field. Significant progress has been made towards implementing a procedure for estimating gravity models directly from physical measurements. We have developed a new estimation algorithm, a multi-resolution rank-revealing QR decomposition. This algorithm produces the 'minimum detail' solution instead of the usual minimum norm solution of the ill-conditioned least squares problem. A basis for bandlimited functions has been constructed using a new method for computing the generalized Gaussian quadratures for exponentials. These bases are closely related to the prolate spheroidal wave functions, and we plan to create the next generation of models for evaluation and estimation of the gravity field using such bases. These bases are nearly optimal in terms of the number of coefficients necessary to represent a bandlimited function.