A full-Bayesian approach to the groundwater inverse problem for steady state flow

A full-Bayesian approach to the estimation of transmissivity from hydraulic head and transmissivity measurements is developed for two-dimensional steady state groundwater flow. The approach combines both Bayesian and maximum entropy viewpoints of probability. In the first phase, log transmissivity measurements are incorporated into Bayes' theorem, and the prior probability density function is updated, yielding posterior estimates of the mean value of the log transmissivity field and covariance. The two central moments are generated assuming that the prior mean, variance, and integral scales are hyperparameters; that is, they are treated as random variables in themselves which is contrary to classical statistical approaches. The probability density functions (pdfs) of these hyperparameters are, in turn, determined from maximum entropy considerations. In other words, pdfs are chosen for each of the hyperparameters that are maximally uncommitted with respect to unknown information. This methodology is quite general and provides an alternative to kriging for spatial interpolation. The final step consists of updating the conditioned natural logarithm transmissivity (In(T)) field with hydraulic head measurements, utilizing a linearized aquifer equation. It is assumed that the statistical properties of the noise in the hydraulic head measurements are also uncertain. At each step, uncertainties in all pertinent hyperparameters are removed by marginalization. Finally, what is produced is a In(T) field conditioned on measurements of both hydraulic heads and log transmissivity and covariances of the In(T) field. In addition, we can also produce resolution matrices, confidence (credibility) limits, and the like for the In(T) field. It is shown that the application of the methodology yields good estimates of transmissivities, even when hydraulic head measurements are noisy and little or no information is specified on mean values of In(T), variance of In(T), and integral scales.