A Simple Method for Deriving the Transformation Formula of Spherical Harmonic Expansion Coefficients and its Application

Abstract

Suppose an arbitrary function expressed in a surface spherical harmonic expansion series when the pole is translocated at an arbitrary point on the surface of a sphere. The coefficients of the expansion series are functions of the latitude and longitude of the translocated pole. GANEKO [6] has derived the general transformation formula of spherical harmonic expansion coefficients in a form of the modified Jacobi polynomials. It can be applied to theoretical problems of geodesy and geophysics including the translocation of the polar axis. This paper shows a simple method for deriving GANEKO's transformation formula and its inverse one, with their applications to the global geoidal height.