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 stea dy state groundwater flow. The approach combines both Bayesian and maximum entropy viewpoints of probability. In the first phase, log transmissivity m easurements are incorporated into Bayes' theorem, and the prior probability density function is updated, yielding posterior estimates of the mean valu e of the log transmissivity field and covariance. The two central moments a re generated assuming that the prior mean, variance, and integral scales ar e "hyperparameters"; that is, they are treated as random variables in thems elves which is contrary to classical statistical approaches. The probabilit y density functions (pdfs) of these hyperparameters are, in turn, determine d 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 alte rnative to kriging for spatial interpolation. The final step consists of up dating 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 pertin ent hyperparameters are removed by marginalization. Finally, what is produc ed 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 methodol ogy yields good estimates of transmissivities, even when hydraulic head mea surements are noisy and little or no information is specified on mean value s of In(T), variance of In(T), and integral scales.