Near-field imaging of biperiodic surfaces for elastic waves

This paper is concerned with the direct and inverse scattering of elastic waves by biperiodic surfaces in three dimensions. The surface is assumed to be a small and smooth perturbation of a rigid plane. Given a time-harmonic plane incident wave, the direct problem is to determine the displacement field of the elastic wave for a given surface; the inverse problem is to reconstruct the surface from the measured displacement field. The direct problem is shown to have a unique weak solution by studying its variational formulation. Moreover, an analytic solution is deduced by using the transformed field expansion method and the convergence is established for the power series solution. A local uniqueness is proved for the inverse problem. An explicit reconstruction formula is obtained and implemented by using the fast Fourier transform. The error estimate is derived for the reconstructed surface function, and it provides an insight on the trade-off among resolution, accuracy, and stability of the solution for the inverse problem. Numerical results show that the method is effective to reconstruct biperiodic scattering surfaces with subwavelength resolution.