On Detection and Visibility of a Complex Obstacle Using Multipoles Sources from Far Field Data

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 geometric properties and the material parameter distributed on the surface are involved in the obstacle reconstruction numerically. The main advance of this research on our recently work (SIAM J. Applied Math, 67(4), 2007) is the higher-order asymptotic expansion of the indicator function and the introduction of complex valued surface impedance, which makes the reconstruction more (or less) accurate depending on how we choose this surface impedance. 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 obstacle more (or less) visible. By establishing the property of the minimum norm solution for approximating the singular sources, efficient realizations for approximating the multipoles by Herglotz wave function and therefore the implementations of the probing methods are developed with an error estimate. Such an estimate gives a thorough explanation on the reasons why these probing methods can reach only finite accuracy of reconstruction. We finally show extensive numerical tests explaining how and to what extent the coupling relation between the curvature and the surface impedance changes the visibility of the obstacles.