Gravitation, Magnetics, and Seismology from a Geo-Mathematician's Point of View

 We have a look at the mathematical foundations of the modelling and inversion of geoscientific observables such as gravitational, magnetic or seismic data. For instance, the understanding of the governing equations requires very similar mathematical objects and techniques such as the Laplace/Poisson/Helmholtz equations, spherical harmonics, radial basis functions as well as the Helmholtz and the Mie decomposition. With the understanding of these principles, we observe that some modelling techniques appear as natural choices for the practical and numerical application. Moreover, synergies can be created by transferring knowledge from one application to the other, also outside Earth sciences (such as medical imaging).
 Moreover, numerous basis systems have been developed for problems in Earth sciences - each with their pros and cons. Therefore, the construction of so-called best basis algorithms has become more and more interesting. An example of such a class of algorithms has been developed by the speaker and his geomathematics group. These methods will also be presented in the talk together with some numerical results for a selection of applications.