Discrete spherical harmonic transforms: Numerical preconditioning and optimization

Abstract

Spherical Harmonic Transforms (SHTs) which are essentially Fourier transforms on the sphere are critical in global geopotential and related applications. Among the best known strategies for discrete SHTs are Chebychev quadratures and least squares. The numerical evaluation of the Legendre functions are especially challenging for very high degrees and orders which are required for advanced geocomputations. The computational aspects of SHTs and their inverses using both quadrature and least-squares estimation methods are discussed with special emphasis on numerical preconditioning that guarantees reliable results for degrees and orders up to 3800 in REAL*8 or double precision arithmetic. These numerical results of spherical harmonic synthesis and analysis using simulated spectral coefficients are new and especially important for a number of geodetic, geophysical and related applications with ground resolutions approaching 5 km. © 2008 Springer-Verlag Berlin Heidelberg.