Spherical Harmonic Transforms Using Quadratures and Least Squares

Spherical Harmonic Transforms (SHTs) which are essentially Fourier transforms on the sphere are critical in global geopotential and related applications. For analysis purposes, discrete SHTs are difficult to formulate for an optimal discretization of the sphere, especially for applications with requirements in terms of near-isometric grids and special considerations in the polar regions. With the enormous global datasets becoming available from satellite systems, very high degrees and orders are required and the implied computational efforts are very challenging. Among the best known strategies for discrete SHTs are quadratures and least squares. The computational aspects of SHTs and their inverses using both quadrature and least-squares estimation methods are discussed with special emphasis on information conservation and numerical stability. Parallel and grid computations are imperative for a number of geodetic, geophysical and related applications, and these are currently under investigation.