A new method for computing the ellipsoidal correction for Stokes's formula

Abstract

This paper generalizes the Stokes formula from the spherical boundary surface to the ellipsoidal boundary surface. The resulting solution (ellipsoidal geoidal height), consisting of two parts, i.e. the spherical geoidal height N 0 evaluated from Stokes's formula and the ellipsoidal correction N 1, makes the relative geoidal height error decrease from O(e 2) to O(e 4), which can be neglected for most practical purposes. The ellipsoidal correction N 1 is expressed as a sum of an integral about the spherical geoidal height N 0 and a simple analytical function of N 0 and the first three geopotential coefficients. The kernel function in the integral has the same degree of singularity at the origin as the original Stokes function. A brief comparison among this and other solutions shows that this solution is more effective than the solutions of Molodensky et al. and Moritz and, when the evaluation of the ellipsoidal correction N 1 is done in an area where the spherical geoidal height N 0 has already been evaluated, it is also more effective than the solution of Martinec and Grafarend.