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In this paper, we investigate on the direct and inverse scattering problem by an unbounded penetrable rough surface in a lossless medium. The cases that the transmission coefficient 1 and =1, which creates certain difficulties in the direct and inverse problem, respectively, are both considered. We first estalish the well-posedness of the direct problem using the integral equation method through an elaborate analysis. Then we carefully consider the singularity of the solutions to the problem with incident point source or hypersingular point source, where a simple and novel perspective is given for the derivation of the singularity. Finally, a global uniqueness result is proven for the inverse problem on the unique determination of the unbounded rough surface, the transmission coefficient and the wave number in the lower half plane from the measurements of the near field only on a line segment above the interface at a fixed frequency.
Wu et al. (Mon,) studied this question.