Ray theoretical investigations using matching pursuits

The three-dimensional structure of the Earth's interior shapes its geomagnetic and gravity fields, and can thus be constrained by observing these fields. 3-D Earth structure also causes seismological observables to deviate from those predicted for approximated, spherically symmetrical reference models. Travel time tomography is the inverse problem that uses these observed differences to constrain the 3-D structure of the interior.
On the planetary scale, i.e. in a spherical geometry, this linearized inverse problem has been parameterized with a variety of basis systems, either global (e.g. spherical harmonics) or local (e.g. finite elements). The Geomathematics Group Siegen has developed alternative approximation methods for certain applications from the geosciences: the Inverse Problem Matching Pursuits (IPMPs). These methods combine different basis systems by calculating an approximation in a so-called best basis, which is chosen iteratively from a so-called dictionary, an intentionally overcomplete set of diverse trial functions. In each iteration, the choice of the next best basis element reduces the Tikhonov functional. A particular numerical expertise has been gained for applications on spheres or balls. Hence, the methods were successfully applied to, for instance, the downward continuation of the gravitational potential as well as the MEG-/EEG-problem from medical imaging.
Our aim is to remodel the IPMPs for travel time tomography. This includes developing the data-dependent operator, deciding for specific trial functions and applying the operator to them. We also have to define termination criteria and develop the regularization in theory and practice. We introduce the IPMPs and show results from our remodelling.