Near-field imaging of small perturbed obstacles for elastic waves

Consider an elastically rigid obstacle which is buried in a homogeneous and isotropic elastic background medium. The obstacle is illuminated by an arbitrary time-harmonic elastic incident wave. This work is concerned with an inverse obstacle scattering problem for elastic waves, which is to reconstruct the obstacle's surface from the displacement field measured at a circle surrounding it. The surface of the obstacle is assumed to be a small and smooth perturbation of a disk. Based on the Helmholtz decomposition, the displacement field is split into its compressional and shear parts. A coupled system of boundary value problems is derived for the decomposed scalar potentials by using the transparent boundary conditions. Utilizing the transformed field expansion method, we reduce the coupled system into to a successive sequence of one-dimensional two-point boundary value problems which are solved in closed forms. The inverse problem is linearized by dropping high order terms in the power series expansion and an explicit reconstruction formula is obtained. Moreover, a simple nonlinear correction algorithm is proposed to improve the accuracy of the reconstructions. Ample numerical examples show that, using this method, it is possible to reconstruct the surface with subwavelength resolution.