A novel integral equation for scattering by locally rough surfaces and application to the inverse problem (Q2873877)

The direct and inverse acoustic or electromagnetic scattering problems by a locally perturbed, perfectly reflecting, infinite plane (locally rough surface) are studied in the paper. The discussion is restricted to the two-dimensional case, so that the total wave field \(u\) satisfies the Helmholtz equation \( \Delta u + k^2 u =0 \) in \(D_{+}\), where \(D_{+}\) represents a homogeneous medium above a locally rough surface. The direct scattering problem defined on a bounded curve with two corners is formulated as an integral equation. The integral equation can be solved efficiently by using the Nystrom method with the graded mesh introduced by \textit{R. Kress} [Numer. Math. 58, No. 2, 145--161 (1990; Zbl 0707.65078)] and is able to deal with large wave number cases. To cope with the inverse problem, the integral formulation is also used. Namely, to reconstruct the local perturbation of the plane from multiple-frequency far-field data a modification of the Newton iteration method is proposed. Numerical examples are presented to confirm the stability and accuracy of the reconstruction method even for the case of multiple-scale profiles.