Transmission inverse problem with partial information about the source

In commonly studied coefficient identification problems in wave propagation it is assumed that the wavefield data being measured arises in response to a known source function, often an idealization such as a delta function. However it may be the case that the source is itself difficult or impossible to know with any precision, thus some works have treated the source itself, subject to some constraints, as an unknown in the problem, together with the coefficient(s) of interest. In this paper we will discuss two problems of this type involving the determination of an unknown impedance in a one dimensional wave equation from corresponding transmission data. In the first version it is assumed that the phase of the Fourier transform of the source function is known, but not its amplitude. In the second version no specific knowledge of the source is assumed, but some restrictions are imposed on the locations of the complex zeros of its Fourier transform. The strategy in both cases is to first infer the data for a related inverse spectral problem.