Utskrift från Malmö universitets webbplats www.mah.se

Inverse resonance scattering problem for Rayleigh waves

Contact person: Samuele Sottile
Responsible: Alexei Lantchenko
Funding: Malmö University
Timeframe: 2018-01-01 -- 2021-12-31
Faculty/Department: School of Technology, Department of Materials Science and Applied Mathematics

Inverse problems is a branch of applied Mathematics, which studies indirect measurements. In Inverse problems we want to deduce uniquely (or in the most unique possible way) the cause of an event from its effects.

The most typical historical example is the inverse problem of the reconstruction of the shape of a drum from its eigenfrequencies ( from the paper “Can you hear the shape of a drum?”, Kac 1966, URL: http://www.jstor.org/stable/2313748).

In seismology, this leads to the reconstruction of the Lame’ of Earth’s inner layers from the measured eigenvalues and resonances of the Rayleigh operator. The Lame’ parameters tells us about the elastic properties of the medium, while the eigenvalues are the energies of the seismic waves measured (through a seismograph), and the resonance are the energy and decay time of the resonant waves.

In the past, those seismic inverse problems were studied only using the eigenvalues, and they brought to results, which are true only for certain wave frequencies or only under “strong” assumptions. The innovative character of our research is to use also resonances (which are measurable in most cases) as an additional information for the reconstruction of the Lame’parameters. In order to do so, we make use of various methods and concepts of Semiclassical Analysis, spectral theory, scattering theory and advanced complex analysis.

The result that we are obtaining in this framework, it will be a big step in the mathematical theory of Seismology and can be applied not only in the case of Earth (expecially interesting for oil exploration missions), but also for Mars, which has recently showed to be seismically active thanks to the new seismic station (Insight mission).

The improvement that we bring to theory of Inverse problems can be extended to many other areas of science, for example in Medicine (X-ray tomography), Cosmology (black holes as quasi-normal modes), and so on. In the future we plan to apply our results also in those areas of science, apart from Seismology.

 

Senast uppdaterad av Magnus Jando