Inverse Surface Scattering Problems for Elastic Waves

The elastic wave scattering problems have received ever increasing attention in both the engineering and mathematical communities for their significant applications in many scientific areas such as geophysics and seismology. In this thesis, we develop a novel method for solving the inverse elastic surface scattering problem which arises from near-field imaging applications in two dimensions and three dimensions. The method utilizes the transformed field expansion along with the Fourier series expansion to deduce an analytic solution for the direct problem. Implemented via the fast Fourier transform, an explicit reconstruction formula is obtained to solve the linearized inverse problem. Numerical examples show that the method is efficient and effective to reconstruct scattering surfaces with subwavelength resolution. We have also investigated the convergence analysis of the proposed method. The well-posedness is established for the solution of the direct problem. The convergence of the power series solution is examined. A local uniqueness result is proved for the inverse problem where a single incident field with a fixed frequency is needed. The error estimate is derived for the reconstruction formula. It provides a deep insight on the trade-off among resolution, accuracy, and stability of the solution for the inverse problem. At last, we study the direct obstacle scattering problems for elastic waves in two dimensions. We develop an exact transparent boundary condition and show that the direct problem has a unique weak solution