Method of Successive Approximations in Solving Geodetic Boundary Value Problems: Analysis and Numerical Experiments

Abstract

After an introductory note reviewing the role and the treatment of boundary problems in physical geodesy, the explanation rests on the concept of the weak solution. The focus is on the linear gravimetric boundary value problem. In this case, however, an oblique derivative in the boundary condition and the need for a numerical integration over the whole and complicated surface of the Earth make the numerical implementation of the concept rather demanding. The intention is to reduce the complexity by means of successive approximations and step by step to take into account effects caused by the obliqueness of the derivative and by the departure of the boundary from a more regular surface. The possibility to use a sphere or an ellipsoid of revolution as an approximation surface is discussed with the aim to simplify the bilinear form that defines the problem under consideration and to justify the use of an approximation of Galerkin’s matrix. The discussion is added of extensive numerical simulations and tests using the ETOPO5 boundary for the surface of the Earth and gravity data derived from the EGM96 model of the Earth’s gravity field.