On a transmission inverse problem

In crosswell seismic experiments, seismic sources are fired in one well, and the wave-fields generated are measured in another well. The goal of the crosswell seismology is to find physical parameters, especially the velocities, of the rocks between the wells from these measurements. This amounts to the mathematical problem of solving a coefficient inverse problem of the multidimensional acoustic wave equation. We consider two inversion methods in this thesis: traveltime inversion via traveltime tomography and waveform inversion via differential semblance optimization. The main results are obtained for traveltime tomography and differential semblance optimization under the non-caustic assumption. The main result for traveltime tomography is that the objective function is smooth, and vanishing gradient implies a global minimizer if the data is noise free. The main result for DSO approach is that the objective function is smooth and a critical point is kinematically close to the true velocity model if the noise level is low enough, the source wavelet is oscillatory enough, and the DSO parameter is small enough. We also discuss to some extent the two methods in the presence of caustics