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A simple method for solving inverse scattering problems in the resonance region. (English) Zbl 0859.35133

Summary: This paper is concerned with the development of an inversion scheme for two-dimensional inverse scattering problems in the resonance region which does not use nonlinear optimization methods and is relatively independent of the geometry and physical properties of the scatterer. It is assumed that the far field pattern ${u}_{\infty }\left(\varphi ;\theta \right)$ corresponding to the observation angle $\varphi$ and plane waves incident at angle $\theta$ is known for all $\varphi$, $\theta \in \left[-\pi ,\pi \right]$. From this information, the support of the scattering obstacle is obtained by solving the integral equation

${\int }_{-\pi }^{\pi }{u}_{\infty }\left(\varphi ;\theta \right)g\left(\theta \right)d\theta ={e}^{-ik\rho cos\left(\varphi -\alpha \right)},\phantom{\rule{1.em}{0ex}}\varphi \in \left[-\pi ,\pi \right],$

where $k$ is the wave number and ${y}_{0}=\left(\rho cos\alpha ,\rho sin\alpha \right)$ is on a rectangular grid containing the scatterer. The support is found by noting that ${|g|}_{{L}^{2}\left(-\pi ,\pi \right)}$ is unbounded as ${y}_{0}$ approaches the boundary of the scattering object from inside the scatterer. Numerical examples are given showing the practicability of this method.

##### MSC:
 35R30 Inverse problems for PDE 35J05 Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation 78A45 Diffraction, scattering (optics)