Abstract
SUMMARY We propose a class of spherical wavelet bases for the analysis of geophysical models and for the tomographic inversion of global seismic data. Its multiresolution character allows for modelling 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 describe benefits and drawbacks of these constructions and apply them to analyse the information in two published seismic wave speed models of the mantle, using the statistics 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 � 2 norm of the data residuals and the � 1 norm of the model wavelet coefficients. By validation with realistic synthetic experiments we illustrate the likely gains from our new approach in future inversions of finite-frequency seismic data.
Highlights
While we have focused on the angular part of the cubed ‘sphere’ we have generalized our construction to the case of the ‘ball’ and provide an outlook for further research in global seismic tomography in the concluding Section 8
In preparing for the study of the suitability for solution of such massive inverse problems of the wavelet transforms that we introduced in the previous section, we take a detour by addressing the question: is the Earth sparse in a wavelet basis? we will never be able to answer this question with any degree of certainty, but we can investigate, at the very least, whether earth models are sparse in such bases
As noted by Loris et al (2007, 2010) there are more algorithms available to us than the one described in eq (17), and as we have argued in this paper there is a wealth of wavelet constructions that can be brought to bear on the inverse problem of global seismic tomography
Summary
As long as tomographic earth models remain the solutions to mixeddetermined (Menke 1989) inverse problems (Nolet 1987, 2008) there will be disagreement over the precise location, shape and amplitude of lateral and radial anomalies in seismic wave speed that exist within the Earth; there will be attempts to derive the bestfitting mean structure (e.g. Becker & Boschi 2002), and the needed efforts to validate them (e.g. Capdeville et al 2005; Qin et al 2009; Bozdag & Trampert 2010; Lekic & Romanowicz 2011). GJI Seismology continued interest and clear and present progress in the field (e.g. Foufoula-Georgiou & Kumar 1994; Klees & Haagmans 2000; Freeden & Michel 2004b; Oliver 2009), the use of wavelets is still no matter of routine in the geosciences, beyond applications in one and two Cartesian dimensions This despite, or perhaps because, there being a wealth of available constructions relevant for global geophysics, in other words: on the sphere In seismology, Chiao & Kuo (2001) were, to our knowledge, the first to develop a ‘biorthogonal-Haar’ wavelet lifting scheme (Schroder & Sweldens 1995) for a triangular surface tesselation of the sphere suitable for multiscale global tomography Later, these same authors formed a (biorthogonal) spline basis for a Cartesian cube useful in exploration geophysics (Chiao & Liang 2003) and for regional studies (Hung et al 2010). C 2011 The Authors, GJI, 187, 969–988 Geophysical Journal International C 2011 RAS
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