This paper addresses the scattering of acoustic and electromagnetic wa
ves from a perturbed dissipative half-space. For simplicity, the pertu
rbation is assumed to have compact support. Section 1 discusses the ap
plication that motivated this work and explains how the scalar model u
sed here is related to Maxwell's equations. Section 2 introduces three
formulations for direct and inverse problems for the half-space geome
try. Two of these formulations relate to scattering problems, and the
third to a boundary value problem. Section 3 shows how the scattering
problems can be related to the boundary value problem. This shows that
the three inverse problems are equivalent in a certain sense. In sect
ion 4, the boundary value problem is used to outline a simple way to f
ormulate a multi-dimensional layer stripping procedure. This procedure
is unstable and does not constitute a practical algorithm for solving
the inverse problem. The paper concludes with three appendices, the f
irst two of which carry out a careful construction of solutions of the
direct problems and the third of which contains a discussion of some
properties of the scattering operator.