High resolution spherical and ellipsoidal harmonic expansions by Fast Fourier Transform

Abstract

High resolution transformations between regular geophysical data and harmonic model coefficients can be most efficiently computed by Fast Fourier Transform (FFT). However, a prerequisite is that the data grids are given in the appropriate geometrical domain. For example, if the data are situated on the ellipsoid at equi-angular reduced latitudes, spherical harmonic analysis can be employed and the coefficients subsequently converted by Jekeli’s transformation. This results in the spherical harmonic spectrum in the domain of geocentric latitudes. However, the data are most likely given at geodetic (ellipsoidal) latitudes which means that the FFT base needs to be shifted by latitude dependent phase lags in order to obtain the correct spherical harmonic spectrum. This requires appropriate sample rate conversion about the shifted latitudes by means of Fourier summation and cannot be treated efficiently by an FFT algorithm. In this article another solution is discussed instead. Since the variable heights between the spherical and ellipsoidal surfaces can be accurately approximated by a series of Tschebyshev polynomials, they can be convolved into the spherical basis. It will be shown how this new type of transformation to and from the ellipsoid in combination with Jekeli’s conversion of the spectra between the two surfaces allows eventually the sample rate conversion to shifted latitudes. This avoids the inexpedient Fourier summation mentioned previously. In this paper three applications for FFT in the domain of spherical and ellipsoidal surfaces, and using geocentric, reduced and geodetic latitudes are discussed. The Earth gravitational model EGM2008 of 5 arcminutes resolution has been used to demonstrate numerical results and computational advantages.