r Analysis of aximum Likelihood stimates of Physical Parameters ispersive Waves
Christopher V. Kimball, Pawel Lewicki, N.I. Wijeyesekera · 1995
Measurement of physical parameters from one or more dispersive waves propagating in a band of frequencies across a linear array of receivers is considered. The waves propagate according to a known physical model having real-valued parameters and have spatially uncorrelated white Gaussian noise added to them. The objective of the measurement is to estimate a specified subset of the physical parameters, the remaining physical parameters are externally provided and have known error statistics. Existing maximum likelihood (MI,) estimation and Cram&-Rao (CR) bounds theory describes the measurement under the assumption of correct values €or the externally provided parameters. A review of this theory is presented for completeness. Then a bounding estimate on the total error on the estimated parameters is derived which takes into account both the CR bounds and errors in the externally provided parameters. Suggestions for the application of the theory to practical problems are provided. A criterion is derived for determining whether additional, unwanted (nuisance) parameters shouid be included in the estimate to improve the total ermr on the desired parameters. I. XNTRODUCTION ROPAGATING waves provide measurements of the phys- ical parameters of the medium in which they propagate. Such measurements require a model which describes the propagation in terms of the physical parameters. The model allows calculation of the sensitivity of the complex phase slowness of each wave to a particular physical parameter, but such individual sensitivities don't assure that a parameter can be practically estimated from the waves. This paper provides a general method of evaluating physical parameter estimates in the framework of maximum likelihood (ML) and least mean-squared error (LMSE) estimation. Consider the estimation of physical parameters fiom measurements of q 2 1 propagating waves. At each radian frequency w these waves have complex phase slowness, ak(w),k = l,...,q which are known functions of a z > 0 element vector of real-valued, physical parameters