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Abstract:
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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. |
