| The focus of this research is the development of a rigorous theoretical and computational framework for the rapid, three-dimensional imaging of underground obstacles using seismic waves. To this end, an elastodynamic boundary integral equation (BIE) approach and a linear sampling method are proposed. In the first approach, solution to the inverse problem is reduced to a nonlinear minimization of the misfit between experimental observations and BIE predictions of the ground motion for an assumed obstacle location. With the aid of an adjoint field method, sensitivities of the featured misfit-type functional are obtained explicitly in terms of a boundary integral formula. An in-depth treatment of the radiation condition for semi-infinite solids, which is essential to both forward and inverse scattering problems, is elucidated.; Unfortunately, the success of any gradient-based minimization technique is strongly dependent on the choice of an initial “guess” for the unknown scatterer. Moreover, nonlinear minimization methods require the prior knowledge of boundary conditions satisfied by the field quantities on the surface of a hidden obstacle. These difficulties have led to the generalization of a recently developed technique for solving inverse scattering problems, called the linear sampling method. Originally introduced for far-field acoustics, this new technique has since been adapted to deal with the far-field electromagnetic and elastic scattering problems in infinite domains. So far, however, there have not been any attempts to apply this method to the interpretation of near-field elastic waveforms that are relevant to underground obstacle identification. Aimed at bridging this gap, the 3D inverse analysis of elastic waves scattered by an obstacle is formulated as a linear integral equation of the first kind whose solution becomes unbounded as the sampling point approaches the boundary of an unknown scatterer. This unboundedness property of the solution is used to determine the support of a hidden obstacle. Details of extending the validity of the linear sampling method to near-field elastodynamics and semi-infinite domains are illuminated. A set of numerical examples is included to illustrate the performance of both methods. |
