The thesis investigates the stability/sensitivity of the inverse problem of recovering velocity fields in a bounded domain from the boundary measurements. The problem has important applications in geophysics where people are interested in finding the inner structure (the velocity field in the elastic wave models) of earth from measurements on the surface. Two types of measurements are considered. One is the boundary dynamic Dirichlet-to-Neumann map (DDtN) for the wave equation. The other is the restricted Hamiltonian flow induced by the corresponding velocity field at a sufficiently large time and with domain the cosphere bundle of the boundary, or its equivalent form the scattering relation. Relations between these two type of data are explored. Three main results on the stability/sensitivity of the associated inverse problems are obtained: (1). The sensitivity of recovering scattering relations from their associated DDtN maps. (2). The sensitivity of recovering velocity fields from their induced boundary DDtN maps. (3). The stability of recovering velocity fields from their induced Hamiltonian flows.In addition, a stability estimate for the X-ray transform in the resence of caustics is established. The X-ray transform is introduced by linearizing the operator which maps a velocity field to its corresponding Hamiltonian flow. Micro-local analysis are used to study the X-ray transform and conditions on the background velocity field are found to ensure the stability of the inverse transform. The main results suggest that the DDtN map is very insensitive to small perturbations of the velocity field, namely, small perturbations of velocity field can result changes to the DDtN map at the same level of large perturbations. This differs from existing H$\ddot{o}$lder type stability results for the inverse problem in the case when the velocity fields are simple. It gives hint that the methodology of velocity field inversion by DDtN map is inefficient in some sense. On the other hands, the main results recommend the methodology of inversion by Hamiltonian flow (or its equivalence the scattering relation), where the associated inverse problem has Lipschitz type stability.
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