Accuracy bounds for normal-incidence acoustic structure estimation

Determnation of the structure of a medium from normal-incidenceacoustic reflection data is a basic problem in fieldsas diverse as medical technology and the earth sciences; thisresearch examines the accuracy with which quantitative structureestimates can be made from noise-corrupted measurementsof reflected energy. Two classes of simple physical models,which exclude geometrical spreading and attenuation, aredeveloped: one in which the properties of the medium changecontinuously with depth, and one in which they change discretely.Given these reasonable models, estimation accuracyis studied by computing a statistical lower bound on estimatorperformance, the Cramer-Rao bound, for three cases of interest.(1) The bound is computed for the estimation of unknown, nonrandomreflection coefficients in a medium containing onlydiscrete reflectors; special attention is given to the one- andtwo-reflector situations. The bound's ability to predictestimator performance is demonstrated, as is the inadequacy ofa particular ad-hoc estimdtion method based on the Wiener-Levinson algorithm of stochastic filtering theory. (2) Thebound is developed for estimation in a continuous medium whosestructure (acoustic impedance, for exaiple) parametrized by aset of unknown, non-random coefficients, and for which thereflection response may be computed in closed form. Theproblem of estimating the parameters of a single, isogradientvelocity layer of known depth is studied in detail. It isdemonstrated that one can identify the parameters of such alayer from normal-incidence measurements given an appropriate source and experimenc geometry. (3) A unique extension ofsome known results in random process estimation is used toderive a pointwise bound for estimation in a continuous mediumwhose structure (reflection coefficient density) is a randomprocess. Again we give special consideration to the problemof identifying a single isolated layer structure. We demonstratethat for a weakly scattering structure, estimationaccuracy is independent of the mean or nominal structure.