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

Abstract

Abstract. 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. It is common in hydrogeologic practice to utilize nonlinear regression methods for parameter estimation. There are currently several private and public domain software packages that are used for this purpose. In order to achieve unique solutions and avoid unstable behavior in the results, it is necessary to parameterize or zone the problem in such a way as to define a smaller (usually much smaller) number of parameters than data points. As stated above, aquifers are highly heterogeneous, and hydraulic properties can vary significantly over very short distances thus making conceptual models based on large zones questionable in our opinion. Another difficulty lies 2081

Incorporating subjective and stochastic uncertainty in an interactive multi-objective groundwater calibration framework

“…The other major framework for addressing model uncertainty is the formal Bayesian approach, which has been used by Draper (1995), Kass and Raftery (1995), Hoeting et al (1999), Woodbury and Ulrych (2000), Neuman (2003), Neuman and Wierenga (2003), and Ye et al (2004), among others. This approach, called Bayesian model averaging-or more specifically maximum likelihood Bayesian model averaging (MLBMA), in the case of Neuman (2003)-uses a more statistically consistent methodology to assess the Bayesian posterior probabilities for a given conceptual model.…”

Stochastic rainfall interpolation and downscaling

“…Later, Kitanidis (1986) introduced a Bayesian approach to Kriging, more specifically to a special formulation of Kriging based upon the generalised covariance functions (Kitanidis, 1983), and appropriately discussed the problem of uncertainty induced by parameter estimation. Additional Bayesian approaches have been developed, among others, by Omre and Halvorsen (1989), Le and Zidek (1992) and Woodbury and Ulrych (2000). Bayesian approaches have the advantage of assessing and reducing the effects of uncertainty on model parameters, although at the expense of extensive numerical integration, generally based upon Monte Carlo or Markov chain, Monte Carlo techniques.…”

Evaluation of Groundwater Leakage into a Drainage Tunnel in Jinping-I Arch Dam Foundation in Southwestern China: A Case Study

“…The inverse modeling approaches become an alternative effective tool for reassessment of the rock permeability using the field observations obtained by the groundwater monitoring system (Garzonio et al 2014;Perello et al 2014). In hydrology or hydrogeology, numerous inverse models (e.g., Neuman 1973;Woodbury and Ulrych 2000;Valstar et al 2004) have been proposed using various optimization techniques (e.g., Karpouzos et al 2001;Rao et al 2003;Gill et al 2006) for back calculations of the aquifer parameters with an acceptable computational cost. Most of the models, however, relied only on the hydraulic head observations for construction of the objective function (e.g., Shigidi and Garcia 2003;Garcia and Shigidi 2006;Karahan and Ayvaz 2008;Virbulis et al 2013), making the inverse solution plagued with the non-uniqueness problem (Mao et al 2013).…”