  We consider an inverse acoustic scattering problem in simultaneously
recovering an embedded obstacle and its surrounding inhomogeneous medium by
formally determined far-field data. It is shown that the knowledge of the
scattering amplitude with a fixed incident direction and all observation angles
along with frequencies from an open interval can be used to uniquely identify
the embedded obstacle, sound-soft or sound-hard disregarding the surrounding
medium. Furthermore, if the surrounding inhomogeneous medium is from an
admissible class (still general), then the medium can be recovered as well. Our
argument is based on deriving certain integral identities involving the
unknowns and then inverting them by certain harmonic analysis techniques.
Finally, based on our theoretical study, a fast and robust sampling method is
proposed to reconstruct the shape and location of the buried targets and the
support of the surrounding inhomogeneities.
