Parameter Uncertainty in Estimation of Spatial Functions: Bayesian Analysis

Linear estimation has found many applications in the inference of spatial functions in surface and subsurface hydrology. The effect of parameter uncertainty is examined in a Bayesian framework with emphasis on the derivation of the Bayesian distribution (and its first two moments) of unknown quantities given some measurements. This distribution accounts not only for natural variability but also for parameter uncertainty. For known covariance parameters the Bayesian distribution is Gaussian (for Gaussian processes) with the mean being a given linear function of the data. This linear estimator is equivalent to the conventional Gaussian conditional mean estimator for a priori known drift coefficients and is the same with kriging for diffuse prior distribution of the drift coefficients; however, the developed procedure is more general. When both drift and covariance function parameters are uncertain, the Bayesian distribution is generally not Gaussian, and the Bayesian conditional mean is a nonlinear estimator. The case of diffuse priors is examined in some detail; it is shown that the posterior distribution of the covariance function parameters is given by the restricted likelihood function, i.e., the likelihood function of generalized increments. The results provide insight into the applicability of maximum likelihood versus restricted maximum likelihood parameter estimation, and conventional linear versus kriging estimation. A more general procedure which includes these methods as special cases is presented.