  This paper is concerned with the inverse problem of scattering of
time-harmonic acoustic waves by an inhomogeneous penetrable obstacle in a
piecewise homogeneous medium. The well-posedness of the direct problem is first
established by using the integral equation method. We then proceed to establish
two tools that play an important role for the inverse problem: one is a mixed
reciprocity relation and the other is a priori estimates of the solution on
some part of the interfaces between the layered media. For the inverse problem,
we prove in this paper that both the penetrable interfaces and the possible
inside inhomogeneity can be uniquely determined from a knowledge of the far
field pattern for incident plane waves.
