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On the sphere, global Fourier transforms are non Abelian and usually called Spherical Harmonic Transforms (SHTs). Discrete SHTs are defined for various grids of data but most applications have requirements in terms of preferred grids and polar considerations. Chebychev quadrature has proven most appropriate in discrete analysis and synthesis to very high degrees and orders. Multiresolution analysis and synthesis that involve convolutions, dilations and decimations are efficiently carried out using SHTs. The high-resolution global datasets becoming available from satellite systems require very high degree and order SHTs for proper representation of the fields. The implied computational efforts in terms of efficiency and reliability are very challenging. The efforts made to compute SHTs and their inverses to degrees and orders 3600 and higher are discussed with special emphasis on numerical stability and information preservation. Parallel and grid computations are imperative for a number of geodetic, geophysical and related applications where near kilometre resolution is required. Parallel computations have been investigated and preliminary results confirm the expectations in terms of efficiency. Further work is continuing on optimizing the computations.

Abstract. Spherical Harmonic Transforms (SHTs) which are essentially Fourier transforms on the sphere are critical in global geopotential and related applications. Discrete SHTs are more complex to optimize computationally than Fourier transforms in the sense of the well-known Fast Fourier Transforms (FFTs). Furthermore, 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. The computational aspects of SHTs and their inverses to very high degrees and orders (over 3600) 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.

Most boundary value problems of the geopotential field have integral and series solutions in terms of Green's convolution kernels. These solutions are advantageously evaluated using fast Spherical Harmonic Transforms (SHTs) for regular arrays of simulated or observed global data. However, the computational complexity and numerical conditioning of SHTs for relatively dense data are quite challenging and recent algorithmic developments warrant further investigations for geodetic and geophysical applications. Global multiresolution applications for scalar, vector and tensor fields on the Earth and its neighborhood require spherical harmonic analysis and synthesis using convolution filters with data decimation and dilation. For global spherical grid applications, efficient and reliable SHTs are needed just as Fast Fourier Transforms (FFTs) are used in regional planar applications. With the availability of enormous quantities of space, surface and subsurface data, extensive data structuring and management are unavoidable for most array computations. Different methodologies imply very different strategies and conflicting claims often appear in the literature. Discussions of the implicit and other assumptions with simulated results would undoubtedly help to clarify the situation and help decide on appropriate data structuring strategies for different computational applications. For every linear boundary value problem of the Earth's geopotential, it is possible to define a source or a Green's function. If this Green's function can be formulated explicitly, then the boundary value problem (BVP) is solved formally in terms of integral or series forms. In general, Green's function for a linear partial differential operator BVP is the solution for a Dirac delta impulse and homogeneous boundary conditions. Corresponding to the nonhomogeneous Dirichlet, Newmann and Robin BVPs of the Laplace operator, Green's functions for the sphere are well known with the solutions in integral and series forms readily available. Explicitly, the BVP solutions are expressed as convolutions of Green's functions and normal derivatives thereof with discrete measurements on or near the surface of the Earth. Convolution operations are fundamental in linear filtering, solving BVPs using Green's functions and numerous other applications. In planar contexts, Fourier transform techniques are well established with FFTs, as the computational complexity