Computing the earth gravity field with spherical harmonics

Abstract

The expensive evaluation of the spherical-harmonic series expansion of the earth gravity field is optimised by transition to 4-dimensional coordinates. That makes disappear square roots and trigonometric functions. The singularity at the poles inherent in spherical coordinates is removed by the increase of dimension. Instead of the associated Legendre functions we obtain a basis of hypergeometric Jacobi polynomials that reproduces under derivation. Thus, the calculation of their derivatives cancels in the Forsythe summation technique; for the Clenshaw summation, the recursions of function value and derivatives are decoupled. © 2008 Springer-Verlag Berlin Heidelberg.