  We study an inverse scattering problem for a generic hyperbolic system of
equations with an unknown coefficient called the reflectivity. The solution of
the system models waves (sound, electromagnetic or elastic), and the
reflectivity models unknown scatterers embedded in a smooth and known medium.
The inverse problem is to determine the reflectivity from the time resolved
scattering matrix (the data) measured by an array of sensors. We introduce a
novel inversion method, based on a reduced order model (ROM) of an operator
called wave propagator, because it maps the wave from one time instant to the
next, at interval corresponding to the discrete time sampling of the data. The
wave propagator is unknown in the inverse problem, but the ROM can be computed
directly from the data. By construction, the ROM inherits key properties of the
wave propagator, which facilitate the estimation of the reflectivity. The ROM
was introduced previously and was used for two purposes: (1) to map the
scattering matrix to that corresponding to the single scattering (Born)
approximation and (2) to image i.e., obtain a qualitative estimate of the
support of the reflectivity. Here we study further the ROM and show that it
corresponds to a Galerkin projection of the wave propagator. The Galerkin
framework is useful for proving properties of the ROM that are used in the new
inversion method which seeks a quantitative estimate of the reflectivity.
