This thesis has been directed toward the goal of identifying objects in a finite-depth, constant or variable index of refraction ocean. The mathematical procedure is based on the physical idea of scattering one or more "plane waves" off the unidentified object and then trying to identify the object from its far-field pattern. In this thesis, we have developed a mathematical theory, numerical algorithm and some examples of computer implementation for a class of direct and inverse scattering problems for time harmonic acoustic waves in shallow oceans.;We consider the finite depth ocean to occupy a slab ;In order to solve the inverse problem we must first solve the direct (propagation) problem i.e., for given incoming plane waves ;The underwater acoustic inverse problem is doubly ill-posed. First, as only a finite number of modes propagate, there is incomplete information in the far-field. Furthermore, even in space ;Based on the theory developed as above, we have reconstructed objects in finite depth oceans, with a constant refraction index. We have also constructed the propagating modes for the variable index case.;Some other problems in the shallow ocean such as the reconstruction of refraction index and optimal control problem are also considered in the thesis.
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