  We propose a class of spherical wavelet bases for the analysis of geophysical
models and forthe tomographic inversion of global seismic data. Its
multiresolution character allows for modeling with an effective spatial
resolution that varies with position within the Earth. Our procedure is
numerically efficient and can be implemented with parallel computing. We
discuss two possible types of discrete wavelet transforms in the angular
dimension of the cubed sphere. We discuss benefits and drawbacks of these
constructions and apply them to analyze the information present in two
published seismic wavespeed models of the mantle, for the statistics and power
of wavelet coefficients across scales. The localization and sparsity properties
of wavelet bases allow finding a sparse solution to inverse problems by
iterative minimization of a combination of the $\ell_2$ norm of data fit and
the $\ell_1$ norm on the wavelet coefficients. By validation with realistic
synthetic experiments we illustrate the likely gains of our new approach in
future inversions of finite-frequency seismic data and show its readiness for
global seismic tomography.
