Our goal is to study two inverse problems: using seismic data to invert for earthquake parameters and using tide gauge data to invert for earthquake parameters. We focus on the feasibility of using a combination of these inverse problems to improve tsunami runup prediction.

A considerable part of the thesis is devoted to studying the seismic forward operator and its modeling using immersed interface methods. We develop an immersed interface method for solving the variable coefficient advection equation in one dimension with a propagating singularity and prove a convergence result for this method. We also prove a convergence result for the one-dimensional acoustic system of partial differential equations solved using immersed interface methods with internal boundary conditions. Such systems form the building blocks of the numerical model for the earthquake.

For a simple earthquake-tsunami model, we observe a variety of possibilities in the recovery of the earthquake parameters and tsunami runup prediction. In some cases the data are insufficient either to invert for the earthquake parameters or to predict the runup. When more data are added, we are able to resolve the earthquake parameters with enough accuracy to predict the runup. We expect that this variety will be true in a real world three dimensional geometry as well.