Accuracy bounds for normal-incidence acoustic structure estimation
Theriault, Kenneth B.
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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.
Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution August, 1977
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