Discrete Spherical Harmonic Analysis: Neumann’s Approach

Abstract

When making the transition from continuous to discrete spherical harmonic analysis, it appears that orthogonality of the base functions in latitude direction is lost, whereas orthogonality in longitudinal direction is preserved. The latter holds for equiangular gridding along each parallel. In order to overcome the latitudinal orthogonality problem, Franz Neumann devised-in the previous century-two exact quadrature methods (Neumann,1838), which have only sparsely been used in geodesy. One method resembles Gaussian quadrature, where the evaluation points are fixed, but not uniformly distributed. The other method does not restrict the choice of evaluation points, i.e. one may freely choose the position of the parallels. Both methods will be presented in this paper as a weighted least squares problem.