Error analysis of maximum likelihood estimates of physical parameters from one or more dispersive waves

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 (ML) estimation and Cramer-Rao (CR) bounds theory describes the measurement under the assumption of correct values for 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 should be included in the estimate to improve the total error on the desired parameters