Evaluation of fast spherical transforms for geophysical applications

Abstract

Just as the FFT has revolutionized data processing and numerical solution of differential equations in Cartesian geometry, so also would a fast spherical harmonic transform revolutionize many geophysical problems in spherical geometry. Algorithms have recently been published with a theoretical asymptotic operation count of O(d(log2 d)2), where d → ∞ is the number of harmonics. We have developed and extended one such algorithm that uses recurrence relations for associated Legendre functions for both increasing and decreasing degree. The algorithm limits the ranges of spherical harmonic degree spanned by the recurrence relations automatically to produce a given accuracy. Tests on synthetic series and a Magsat lithospheric anomaly model show the new algorithm to be faster than conventional Gauss-Legendre quadrature for maximum degree L = 127, and three times faster for L = 511. However, numerical instabilities prevent the theoretical asymptotic speed from being reached, and further gains at higher degree are unlikely.