Maximum likelihood estimation with side information of 1-D layered media from noisy impulse reflection responses

We consider the problem of computing the maximum likelihood estimates of the reflection coefficients of a discrete 1-D layered medium from noisy observations of its impulse reflection response. We have side information in that a known subset of the reflection coefficients are known to be zero; this knowledge could come from either a, priori knowledge of a homogeneous subregion inside the scattering medium, or from a thresholding operation in which noisy reconstructed reflection coefficients with absolute values below a threshold are known to be zero. Our procedure is simple, noniterative, and requires only solutions of systems of linear equations. Numerical examples are provided which demonstrate not only the operation of the algorithm, but also that the side information improves the reconstruction of unconstrained reflection coefficients as well as constrained ones, due to the nonlinearity of the problem.