Inverse wave scattering by unbounded obstacles: uniqueness for the two-dimensional Helmholtz equation

Abstract

In this article we present some uniqueness results on inverse wave scattering by unbounded obstacles for the two-dimensional Helmholtz equation. We prove that an impenetrable one-dimensional rough surface can be uniquely determined by the values of the scattered field taken on a line segment above the surface that correspond to the incident waves generated by a countable number of point sources. For penetrable rough layers in a piecewise constant medium, the refractive indices together with the rough interfaces (on which the TM transmission conditions are imposed) can be uniquely identified using the same measurements and the same incident point source waves. Moreover, a Dirichlet polygonal rough surface can be uniquely determined by a single incident point source wave provided a certain condition is imposed on it.

Keywords:

• inverse scattering
• uniqueness
• rough surface
• Helmholtz equation
• point sources

AMS Subject Classifications:

• 35R30
• 78A46

Acknowledgements

The author thanks Dr J. Elschner for reading through the manuscript carefully and many helpful discussions which improved this article. This work was supported by the German Research Foundation (DFG) under Grant No. EL 584/1-2.