Shape detection and localization of a scatterer embedded in a halfplane
The inverse problem of detecting the shape and location of a rigid scatterer fully embedded in a halfplane based solely on surficial measurements of the scatterer's response to illumination by plane waves, is solved numerically in the frequency domain, using integral equations within the general framework of PDE-constrained optimization. Two different, but closely related, physical problems are considered: first, scatterers embedded in the soil where SH waves are used for detection, and secondly, scatterers embedded in an acoustic fluid, where pressure waves are used for detection. The elastic case of SH waves gives rise to a traction-free surface and an associated Neumann condition, whereas the acoustic case gives rise to a pressure-free surface and a Dirichlet condition, respectively. The measurement stations are sparsely located on the free surface and depending on the physical problem, either displacements are measured (SH case), or fluid velocities (or pressure gradients) are recorded (acoustic case). Localizing and detecting the shape of the scatterer entails matching the observed response to the response resulting from the scatterer's assumed location and shape. There arises a misfit minimization problem that is tackled using a PDE-constrained optimization approach, which, in turn, results in state, adjoint, and control problems, necessary for the satisfaction of the first- order optimality conditions. Boundary integral equations are used throughout, whereas operations over moving interfaces that arise naturally during the iterative search process, are treated using the apparatus of total differentiation.To alleviate inherent difficulties with solution multiplicity, amplitude-based misfits and continuation schemes are used. Numerical results, attesting to the efficacy of the methodology in detecting shapes and localizing scatterers, are discussed.