On Simultaneous Identification of the Shape and Generalized Impedance Boundary Condition in Obstacle Scattering

We consider the inverse obstacle scattering problem of determining both the shape and the “equivalent impedance” from far field measurements at a fixed frequency. In this work, the surface impedance is represented by a second order surface differential operator (referred to as generalized impedance boundary condition) as opposed to a scalar function. The generalized impedance boundary condition can be seen as a more accurate model for effective impedances and is widely used in the scattering problem for thin coatings. Our approach is based on a least square optimization technique. A major part of our analysis is to characterize the derivative of the cost function with respect to the boundary and this complex surface impedance configuration. In particular, we provide an extension of the notion of shape derivative to the case where the involved impedance parameters do not need to be surface traces of given functions, which leads (in general) to a nonvanishing tangential boundary perturbation. The efficiency of considering this type of derivative is illustrated by several two-dimensional numerical experiments based on a (classical) steepest descent method. The feasibility of retrieving both the shape and the impedance parameters is also discussed in our numerical experiments.