Accuracy bounds for normal-incidence acoustic structure estimation

Abstract

Determnation of the structure of a medium from normal-incidence acoustic reflection data is a basic problem in fields as diverse as medical technology and the earth sciences; this research examines the accuracy with which quantitative structure estimates can be made from noise-corrupted measurements of reflected energy. Two classes of simple physical models, which exclude geometrical spreading and attenuation, are developed: one in which the properties of the medium change continuously with depth, and one in which they change discretely. Given these reasonable models, estimation accuracy is studied by computing a statistical lower bound on estimator performance, the Cramer-Rao bound, for three cases of interest. (1) The bound is computed for the estimation of unknown, nonrandom reflection coefficients in a medium containing only discrete reflectors; special attention is given to the one- and two-reflector situations. The bound's ability to predict estimator performance is demonstrated, as is the inadequacy of a particular ad-hoc estimdtion method based on the Wiener- Levinson algorithm of stochastic filtering theory. (2) The bound is developed for estimation in a continuous medium whose structure (acoustic impedance, for exaiple) parametrized by a set of unknown, non-random coefficients, and for which the reflection response may be computed in closed form. The problem of estimating the parameters of a single, isogradient velocity layer of known depth is studied in detail. It is demonstrated that one can identify the parameters of such a layer from normal-incidence measurements given an appropriate source and experimenc geometry. (3) A unique extension of some known results in random process estimation is used to derive a pointwise bound for estimation in a continuous medium whose structure (reflection coefficient density) is a random process. Again we give special consideration to the problem of identifying a single isolated layer structure. We demonstrate that for a weakly scattering structure, estimation accuracy is independent of the mean or nominal structure.