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Many flexible parameterizations exist to represent data on the sphere. Inaddition to the venerable spherical harmonics, we have the Slepian basis,harmonic splines, wavelets and wavelet-like Slepian frames. In this paper wefocus on the latter two: spherical wavelets developed for geophysicalapplications on the cubed sphere, and the Slepian "tree", a new constructionthat combines a quadratic concentration measure with wavelet-likemultiresolution. We discuss the basic features of these mathematical tools, andillustrate their applicability in parameterizing large-scale global geophysical(inverse) problems.

Wavelets and wavelet-like transforms on the sphere and their application to geophysical data inversion