On the Accuracy of the Numerical Detection of Complex Obstacles from Far Field Data Using the Probe Method

Abstract

We deal with the acoustic inverse scattering problem for detecting an obstacle with mixed boundary conditions from the far field map. We show how the geometrical properties and the material parameter distributed on the surface are involved in the obstacle reconstruction numerically. The main advance of this research is the numerical analysis and implementation of our recent theoretical work [J.J. Liu and M. Sini, How to make the reconstruction of obstacles more (or less) accurate from exterior measurements, RICAM preprint 2008-04, Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria, 2008] regarding the higher order asymptotic expansion of the probe indicator function and the introduction of complex-valued surface impedance. An efficient numerical implementation scheme is proposed based on the properties of minimum norm solutions. Precisely, using the relation between the surface impedance and the obstacle curvature contained in the higher order expansion of the indicators, we reveal how the obstacle curvature and the surface impedance on the coated part influence the blowing-up behavior. Using this theoretical result, we can specify the complex surface impedance in terms of the obstacle curvature to make the reconstruction more (or less) accurate. By establishing properties of the minimum norm solution for approximating the singular sources, efficient realizations for approximating the multipoles by the Herglotz wave function and therefore the implementations of the probe methods are developed with an error estimate. We finally show extensive numerical tests explaining how and to what extent the coupling relation between the curvature and the surface impedance changes the accuracy of the reconstruction of the obstacles.