To recover the density of the Earth we invert Newton's gravitational potential. It is a well-known fact that this problem is ill-posed. Thus, we need to develop a regularization method to solve it appropriately.
We apply the idea of a Matching Pursuit (see Mallat and Zhang 1993) to recover a solution stepwise. At every step, the next expansion function and the corresponding weight are selected to best match the data structure.
One big advantage of this method is that all kinds of different functions may be taken into account to improve the solution stepwise and, thus, the sparsity of the solution may be controlled directly. Moreover, this new approach generates models with a resolution that is adapted to the data density as well as the detail density of the solution.
In the numerical part of this work, we reconstruct the density distribution of the Earth.
For the area of South America, we perform an extensive case study to investigate the performance and behavior of the new algorithm. Furthermore, we research the mass transport in the area of the Amazon where the proposed method shows great potential for further ecological studies, i.e. to reconstruct the mass loss of Greenland or Antarctica.
However, from gravitational data alone it is only possible to recover the harmonic part of the density. To get information about the anharmonic part as well, we need to be able to include other data types, e.g. seismic data in the form of normal mode anomalies. In this work, we will perform such an inversion and present a new model of the density distribution of the whole Earth.