Reconstruction of a penetrable cavity and the external obstacle

The inverse acoustic scattering of point sources by a penetrable cavity and the external obstacle is considered. Making use of the internal measurements of scattered field on a closed curve inside the cavity, we first derive a factorization method which provides a rigorous characterization of the support of the cavity without knowing the external object. Then under the condition of the cavity is known in advance, we show that the linear sampling method can be applied to recover the outside obstacle with the help of some Green function.