On the factorization method for a far field inverse scattering problem in the time domain

We develop a factorization method to obtain an explicit characterization of a (possibly nonconvex) Dirichlet scattering object from measurements of time-dependent causal scattered waves in the far field regime. In particular, we prove that far fields of solutions to the wave equation due to particularly modified incident waves characterize the obstacle by a range criterion involving the square root of the time derivative of the corresponding far field operator. Our analysis makes essential use of a coercivity property of the solution of the Dirichlet initial boundary value problem for the wave equation in the Laplace domain. This forces us to consider this particular modification of the far field operator. The latter in fact can be chosen arbitrarily close to the true far field operator given in terms of physical measurements.