  This paper is concerned with analysis of electromagnetic wave scattering by
an obstacle which is embedded in a two-layered lossy medium separated by an
unbounded rough surface. Given a dipole point source, the direct problem is to
determine the electromagnetic wave field for the given obstacle and unbounded
rough surface; the inverse problem is to reconstruct simultaneously the
obstacle and unbounded rough surface from the electromagnetic field measured on
a plane surface above the obstacle. For the direct problem, a new boundary
integral equation is proposed and its well-posedness is established. The
analysis is based on the exponential decay of the dyadic Green function for
Maxwell's equations in a lossy medium. For the inverse problem, the global
uniqueness is proved and a local stability is discussed. A crucial step in the
proof of the stability is to obtain the existence and characterization of the
domain derivative of the electric field with respect to the shape of the
obstacle and unbounded rough surface.

Analysis of PDEs