Least Squares, Galerkin and BVPs Applied to the Determination of Global Gravity Field Models

Abstract

The determination of global gravity potential models is a central issue in Geodesy. In principle the existence of solutions, i.e. of harmonic potentials, given certain boundary conditions, depends on the solvability of certain boundary value problems (BVP) for the Laplace equation, an item that has recently received quite an impulse in terms of abstract theorems. Yet, the real problem is that of approximating such solutions. In general, least squares (l.s.) in Hilbert spaces can give an easy and converging solution to such a problem. Yet, l.s. is very demanding from the numerical point of view because of the notable variability of the boundary, be it the actual surface of the earth or the telluroid. Another, somewhat simpler, technique to approximate a BVP is the so called Galerkin method. The relation between the l.s. and Galerkin is analyzed and clarified. Finally, one has to recognize that in reality the true method used in geodesy to construct approximations is the use of the downward continuation followed by orthogonality relations. The paper shows that this geodetic approach can be seen as an approximation of the solution of Galerkin’s system and, at the same time, as an accelerator of the so called change of boundary method.