Recovering source location, polarization, and shape of obstacle from elastic scattering data

Abstract

We consider an inverse elastic scattering problem of simultaneously reconstructing a rigid obstacle and the excitation sources using near-field measurements. Specifically, we are concerned with the inverse elastic scattering of a bounded obstacle embedded in an unbounded, homogeneous, and isotropic medium. The obstacle is illuminated by the time-harmonic incident field generated by each point source with a single given frequency, and the time-harmonic measurements are the total field along an entire circular curve enclosing the obstacle. A two-phase numerical method is proposed to achieve the co-inversion of multiple targets. In the first phase, we develop several indicator functionals to determine the source locations and the polarizations from the total field data, and then we manage to obtain the approximate scattered field. In this phase, only the inner products of the total field with the fundamental solutions are involved in the computation, and thus it is direct and computationally efficient. In the second phase, we propose an iteration method of Newton's type to reconstruct the shape of the obstacle from the approximate scattered field. Using the layer potential representations on an auxiliary curve inside the obstacle, the scattered field together with its derivative on each iteration surface can be easily derived. Theoretically, we establish the uniqueness of the co-inversion problem and analyze the indicating behavior of the sampling-type scheme. An explicit derivative is provided for the Newton-type method. Numerical results are presented to corroborate the effectiveness and efficiency of the proposed method.