Reconstruction of the shape and surface impedance from acoustic scattering data for an arbitrary cylinder

The inverse scattering for an obstacle D subset of R-2 with mixed boundary condition can be considered as a prototype for radar detection of complex obstacles with coated and noncoated parts of the boundary. We construct some indicator functions for this inverse problem using the far-field pattern directly, without the necessity of transforming the far field to the near field. Based on careful singularity analysis, these indicator functions enable us to reconstruct the shape of the obstacle and distinguish the coated from the noncoated part of the boundary. Moreover, an explicit representation formula for the surface impedance in the coated part of the boundary is also given. Our reconstruction scheme reveals that the coated part of the obstacle is less visible than the noncoated one, which corresponds to the physical fact that the coated boundary absorbs some part of the scattered wave. Numerics are presented for the reconstruction formulas, which show that both the boundary shape and the surface impedance in the coated part of the boundary can be reconstructed accurately. The theoretical reconstruction techniques proposed in this work can be applied in the practical 3-dimensional electromagnetic inverse scattering problems with promising numerical performance. Such problems are of great importance in the design of nondetectable obstacles.