Optimization of Spherical Harmonic Transforms and Applications

Spherical harmonics have traditionally been used to represent potential fields of the Earth such as gravity and magnetic fields. A Spherical Harmonic Transform (SHT) is simply a non-Abelian Fourier transform on the sphere and its discretization can be achieved using quadratures and related strategies. Given a spherical band-limited function to degree N, i.e. with its corresponding spectral coefficients zero for all degrees ≥ N, most SHTs using quadratures on Gaussian and equiangular grids are not one-to-one or generally invertible. This research has investigated the invertibility and related aspects of SHTs for various quadrature strategies using Gaussian and equiangular grids. Such considerations are very important for numerous data processing applications of global satellite and related observations. For dense global datasets, very high degree and order spherical harmonic expansions are required. Among the computational challenges are the stability of numerical computations and the large computational efforts required. For kilometer scale resolution, this reseach has explored both challenges for analysis and synthesis, and experimented with recent advances in 64-bit arithmetic as well as parallel and grid computations. The developed algorithms are computationally efficient and numerically stable for all kinds of applications.