Best basis construction and comparison of trial functions for ill-posed inverse problems in Earth sciences - studied at the examples of global-scale seismic tomography and gravitational field modelling

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The choice of basis functions can essentially influence the result of an inverse problem. In view of today's demands on the accuracy of models, we are, consequently, confronted with the question how the obtained results can be confirmed or improved, respectively, by verifying or correcting (if necessary) the used basis functions. The arising questions are: can there be artefacts due to the used numerical method in individual structures in the mapping of e.g. a seismic velocity field or of the gravitational field? To what extent are data sensitive to single regional changes in the solution? Vice versa, how can local variations in the Earth or at the Earth's surface or, alternatively, known local errors in a current model be considered in a model, without deteriorating the model elsewhere? The latter appears to be better to achieve with local basis functions (such as radial basis functions) than with global basis functions (such as spherical harmonics).In the recent years, the principal investigator and his research group have developed several algorithms (the Regularized Functional Matching Pursuit, RFMP, and its enhancements) which are able to iteratively construct a kind of a best basis for an inverse problem. These methods were particularly elaborated for scenarios on the sphere or the ball. The applicability to several problems has already been demonstrated. However, the methods still have some limitations. E.g. large data sets, as they are common for the gravitational field, cannot be handled up to now, and the traveltime tomography does not allow efficient formulae for the forward calculations.Within this project, these algorithms shall be further enhanced, in order to make them better applicable to realistic problems in Earth sciences. We particularly consider global-scale seismic tomography and high-dimensional modelling of the gravitational potential. We expect especially new insights into these two practical problems. In the former case, the above mentioned question arises concerning possible artefacts in velocity models. RFMP and its variants yield the possibility to automatize the multi-scale adaptation of grid structures, which has previously been done manually. We anticipate an improvement of the accuracy of the calculated models. Moreover, the set of trial functions (which is called a dictionary) may be varied, in order to test how stable some aspects of a model are. This way, artefacts due to the choice of the basis functions can be better identified. In the case of gravitational field modelling, efficient ways shall be found which enable us to approximate local anomalies as locally concentrated and as accurately as possible, also in high-resolution models, by the choice of optimal additional basis functions (splines, wavelets, Slepians). For this purpose, new innovative ways of improving existing algorithms need to be found.