# Normal Mode Tomography

## Normal Mode Tomography

After major earthquakes, continuing oscillations can be observed even hours and sometimes days after the initial event. These oscillations are called free oscillations or normal modes. They are comparable to the oscillations of a church bell. If the bell is struck, then it starts oscillating at large. These oscillations are superpositions of oscillations with single frequencies. The whole package of vibrations is transferred to the surrounding air, and our ear recognizes this as a sound. Similarly, the Earth is struck by a major earthquakes and performs oscillations which are superpositions of many different waves and can be measured by special seismic stations. In the same manner as the sound of a church bell depends on its composition, the frequencies that can be observed for the Earth's normal modes depend on the structures inside the planet. Moreover, the measured frequencies depend on the locations of the epicentre and the seismic station which records the oscillations. This spatial dependence is due to the lateral heterogeneity of the Earth's structure. The aim of normal mode tomography is to determine the structures inside the Earth out of the frequency spectra of the normal modes. In a project, which is funded by the German Research Foundation (DFG), the Geomathematics Group in Siegen is developing new mathematical methods for solving this problem.

References:

- P. Berkel, D. Fischer, V. Michel:
*Spline multiresolution and numerical results for joint gravitation and normal mode inversion with an outlook on sparse regularisation*, International Journal on Geomathematics, 1 (2011), 167-204. - P. Berkel, V. Michel: On mathematical aspects of a combined inversion of gravity and normal mode variations by a spline method, Mathematical Geosciences, 42 (2010), 795-816.
- D. Fischer: Sparse Regularization of a Joint Inversion of Gravitational Data and Normal Mode Anomalies, PhD Thesis, Geomathematics Group Siegen, 2011.