Department of Mathematics
"Earth as an Unstructured Mesh: The Seismic Inverse Problem, Stability and Reconstruction via Hierarchical Compression"
The seismic inverse problem asks that we determine the coefficients - physically, elasticities and density of rock as function of position in the earth - of partial differential equations governing seismic wave motion from parts of their solutions (in reality, samples of seismic waves recorded by suitable sensors). An optimization approach to this problem (often called "full waveform inversion") was introduced in exploration and global seismology in the 1980s and has gained significant interest in recent years, primarily due to the increase in available computation power, allowing the problem to be posed in 3D. Since all data is noisy, a critical question is whether the problem is well-posed: do small changes in data result in small changes in the estimated earth structure (coefficients of the PDEs)? We have shown that the answer to this question is "yes", under certain circumstances. When the coefficients may be represented as piecewise constant (or piecewise wavelet, more generally) on a tetrahedral, unstructured mesh, the dependence of the full waveform inversion solution on the problem data is Lipshitz continuous, in suitable metrics. We show how use of a multi-level (multi-scale, multi-frequency) iterative method can mitigate the growth of the stability constants with the number of elements in the mesh.
These results are relevant to relatively low-frequency seismic wave data, the domain of most contemporary industrial full-waveform inversion. We end with some insights into similar stability results for higher-frequency data, which should lead to improved spatial resolution of earth structure.
This presentation summarizes joint research with E. Beretta, E. Francini, L. Qiu, O. Scherzer, G. Uhlmann, S. Vessella and J. Zhai.