Open AccessEvaluation of fast spherical transforms for geophysical applications
Author(s) -
Lesur V.,
Gubbins David
Publication year - 1999
Publication title -
geophysical journal international
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0956-540X
DOI - 10.1046/j.1365-246x.1999.00957.x
Subject(s) - spherical harmonics , legendre polynomials , degree (music) , legendre function , cartesian coordinate system , quadrature (astronomy) , mathematics , fast fourier transform , harmonics , gaussian quadrature , gauss , mathematical analysis , geometry , algorithm , physics , integral equation , optics , nyström method , quantum mechanics , voltage , acoustics
Summary 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 (log 2 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.