On Spherical Multiresolution Analysis and Synthesis

Global array computations are greatly facilitated by Spherical Harmonic Transforms (SHTs) just as planar array computations benefit from discrete Fast Fourier Transforms (FFTs). Multiresolution analysis and synthesis involve linear filtering or convolutions, decimation and dilation. Spherical convolutions in the continuous sense differ from planar ones because of the noncommutativity of spherical rotations and hence are restricted to isotropic filtering. However in discrete computations, the situation is different which can be exploited in practical applications. Different spherical strategies for multiresolution analysis based on regular grids (e.g. first generation wavelet analysis), Reuter grids (spherical wavelet analysis) and tessellations are presented with numerical simulations to exhibit their respective advantages and disadvantages. Concluding remarks with some recommendations for various geoscience applications are also included.