Reconstruction of the Shape and Surface Impedance from Acoustic Scattering Data for an Arbitrary Cylinder

Abstract

Abstract. The inverse scattering for an obstacle D ⊂ R 2 with mixed boundary condition can be considered as a prototype for radar detection of complex obstacles with coated and non-coated parts of the boundary. We construct some indicator functions for this inverse problem using the far-field pattern directly, without the necessity to transform the far-field to the near field. Based on the careful singularity analysis, these indicator functions enable us to reconstruct the shape of the obstacle and distinguish the coated from the non-coated 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 non-coated 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 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 hopeful numerical performances, which are of great importance in the design of non-detectable obstacles.

Recovering the Dirichlet-to-Neumann map in inverse scattering problems using integral equation methods

“…More explicitly, in the original version of the probe method, the indicator function is constructed by firstly computing the D-to-N map (1.5) for unknown D from the far-field data {w ∞ (θ, d) : θ, d ∈ S}, with suitable choices of and f related to some detection point z outside D. We then get the profile of the scatterer by capturing the blow-up behavior of the indicator function as z → ∂ D. If the scatterer is an obstacle with the boundary of impedance type, then we can also recover the surface impedance from the Dto-N map by moment method as suggested in [1]. Hence, the reconstruction of the D-to-N map from the far-field measurements is very interesting and important in inverse scattering problems, although the indicator function can be constructed directly from the far-field data as done in [12][13][14]. Using Green's representation theorem and properties of layer potentials, this problem can be reduced to solving an integral equation of the second kind, where the kernels are singular and involve the derivatives of the scattered waves for point sources.…”

On the reconstruction of Dirichlet-to-Neumann map in inverse scattering problems with stability estimates

“…These numerical implementations did not involve the process of reconstructing Dto-N map from the far-field data, which is the first step in the initial version of probe method using the physical measurement data. Although the probe method can use an indicator constructed from the far-field data directly as done in [15][16][17], the original version based on the D-to-N map is still of great importance for the thorough understanding on the probe method, i.e., the probe method essentially extracts the information about the Green function of the boundary value problem from the far field data to reconstruct the obstacle boundary. Moreover, to the authors' knowledge, the numerical implementations of D-to-N map from the far-field pattern have not yet been realized efficiently up to now.…”

“…In the original version of this scheme, the indicator is constructed by firstly computing the D-to-N map (5) using the far-field data for suitably chosen Ω and boundary value f . Here, we would like to point out that the indicator in the probe method can also be constructed directly from the far-field pattern in one step [15], which is firstly noticed in [17] and then is developed in [16]. For other reconstruction schemes for the obstacle with impedance boundary, we refer to [3] and the references therein.…”

Obstacle and Boundary Determination from Scattering Data

“…For the 2 dimensional case, we need to use more singular sources to capture the values of the surface impedance. This has been analyzed in [13] and the corresponding formulas have been justified theoretically and tested numerically. We refer to that paper for more details on how the formulas are used numerically.…”