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 problem corresponds to the reconstruction of the inner Earth’s structure from the close to surface seismic measurements. In isotropic case, the elastic properties of the medium are given by the Lamé parameters. The seismic measurements are performed by the seismometers placed on or near the surface of the Earth and the data measured are the frequencies of the surface and body waves from a seismic event (i.e. earthquake).

Mathematically, these waves can be associated with the discreet and essential spectrum of the Rayleigh operator: second-order matrix-valued differential operator in depth variable subject to the free surface Neumann boundary condition with coefficients expressed in Lamé parameters. Usually, in geophysical exploration, only eigenvalues are used for recovery of the parameters and information coming from the essential spectrum is neglected. However, as it is well known in Spectral theory, information from both discrete and essential spectrum is necessary for complete reconstruction of the coefficients.

The innovative character of our research is to use also essential spectrum in the reconstruction procedure. As essential spectrum itself is difficult to measure in practice, we use equivalent data – discrete set of complex scattering resonances. The scattering resonances are associated to the rates of oscillations and decay of the radiating solutions to the Rayleigh problem and can be observed on the surface.

In the present project we study direct and inverse problems for the scattering resonances. As Rayleigh operator is not of Sturm-Liouville type, new approach for solving the problems is needed. The anticipated results of the project will provide new mathematical theory and methods in Eart’h seismic waves exploration.

Senast uppdaterad av Magnus Jando