# Climate Change and Inverse Problems - A Novel Algorithm for the Analysis of Geosatellite Data

Volker Michel^{*}

Information about the Earth's interior can be obtained from satellite data of the gravitational field and from different kinds of seismic data. Moreover, climate-based mass transports can also be identified in gravitational field data. Such geophysical tomography problems are ill-posed for at least two reasons: The solution is instable and non-unique. The former in combination with the enormous size of the data sets is the reason why sophisticated regularization techniques are necessary to obtain reliable results. The latter is the reason why the focus of future research must be on the realization of a joint inversion. Several approaches have been developed for geophysical tomographic problems so far. Among them are tools which use global basis systems (such as a truncated singular value decomposition based on orthogonal polynomials) and methods based on localized trial functions (such as spline and wavelet methods). Each approach has its intrinsic advantages and disadvantages. Motivated by techniques for Euclidean domains, a greedy algorithm for tomographic problems on the 3D-ball is presented in this talk: The Regularized Functional Matching Pursuit (RFMP) uses a dictionary of heterogeneous systems of trial functions (e.g., global and localized functions) to combine the different features of the basis systems. Iteratively, a regularized solution is computed via a best-basis algorithm. In the talk, the method and its theoretical properties are presented. Moreover, numerical results for the local inversion of gravitational data and the identification of climate-based mass transports are presented. $\\$ The novel method has several advantages: In comparison to previous methods, the same (or a better) accuracy of the result can be achieved with essentially less trial functions (the solution is, in this respect, sparse). Larger data sets can be handled than with, e.g., a spline method. Moreover, the obtained solution is stable with respect to irregularities in the data grid. Furthermore, the choice of the localized basis functions by the algorithm is correlated with the density of the detail structures in the solution. The presented results are from joint works with D.$\ $Fischer and R.$\ $Telschow.

Mathematics Subject Classification: 65R32 65D07 65T60 86A22

Keywords: inverse problem; tomography; Mathematics of Planet Earth 2013; integral equation; wavelet; spline; greedy algorithm; numerical analysis; gravitation

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