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

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