Georgia Institute of Technology College of Sciences

We give a brief analysis of the oscillations of the earth and then extract the system of equations describing acousto-elastic, seismic waves. Processes in Earth's interior are encoded in the coefficients of this system, which also parametrize its structure and material properties. We introduce the seismic inverse problem with its different aspects including a dual time-frequency point of view. Central in the analysis is the formulation as an inverse boundary value problem with the Dirichlet-to-Neumann map or Neumann-to-Dirichlet map as the data. We discuss various conditional Lipschitz stability estimates for this problem for coefficients containing discontinuities, and with partial boundary data, which involves the introduction of an unstructured tetrahedral mesh. Quantitative estimates of the stability constants play acritical role in analyzing convergence for iterative reconstruction schemes, making use of Hausdorff warping and leading to a multilevel approach requiring hierarchical, multi-scale compression. We present computational experiments on the regional and geophysical exploration scales. We conclude with some results pertaining to the high-frequency inverse boundary value or geometric inverse problems, again, in the presence of discontinuities.