Maximum likelihood Bayesian averaging of spatial variability models in unsaturated fractured tuff

Abstract

[1] Hydrologic analyses typically rely on a single conceptual-mathematical model. Yet hydrologic environments are open and complex, rendering them prone to multiple interpretations and mathematical descriptions. Adopting only one of these may lead to statistical bias and underestimation of uncertainty. Bayesian model averaging (BMA) [Hoeting et al., 1999] provides an optimal way to combine the predictions of several competing models and to assess their joint predictive uncertainty. However, it tends to be computationally demanding and relies heavily on prior information about model parameters. Neuman [2002, 2003] proposed a maximum likelihood version (MLBMA) of BMA to render it computationally feasible and to allow dealing with cases where reliable prior information is lacking. We apply MLBMA to seven alternative variogram models of log air permeability data from single-hole pneumatic injection tests in six boreholes at the Apache Leap Research Site (ALRS) in central Arizona. Unbiased ML estimates of variogram and drift parameters are obtained using adjoint state maximum likelihood cross validation [Samper and Neuman, 1989a] in conjunction with universal kriging and generalized least squares. Standard information criteria provide an ambiguous ranking of the models, which does not justify selecting one of them and discarding all others as is commonly done in practice. Instead, we eliminate some of the models based on their negligibly small posterior probabilities and use the rest to project the measured log permeabilities by kriging onto a rock volume containing the six boreholes. We then average these four projections and associated kriging variances, using the posterior probability of each model as weight. Finally, we cross validate the results by eliminating from consideration all data from one borehole at a time, repeating the above process and comparing the predictive capability of MLBMA with that of each individual model. We find that MLBMA is superior to any individual geostatistical model of log permeability among those we consider at the ALRS.

Modeling of Groundwater Flow and Radionuclide Transport at the Climax Mine sub-CAU, Nevada Test Site

“…Ye et al (2004) applied MLBMA to seven alternative geostatistical models of log air permeability data from single-hole pneumatic injection tests in six boreholes at the Apache Leap Research Site (ALRS) in central Arizona. Predictive performance of MLBMA was evaluated through cross-validation by eliminating from consideration all data from one borehole at a time and comparing the predictive capability of MLBMA with that of each individual model.…”

“…can be calculated using likelihood functions (Beven [2006] and its reference for his method of Generalized Likelihood Uncertainty Estimation, GLUE) in chi-squared sense, the information criterion of AIC (Akaike, 1974) or AICc (Hurvich and Tsai, 1989) in KullbackLeibler sense (Poeter and Anderson, 2005), or the information criterion of BIC (Schwartz, 1978) or KIC (Kashyap, 1982) in Bayesian sense (Hoeting et al, 1999;Neuman, 2003;Ye et al, 2004). There is no consensus as to which method is superior.…”

Quantitative comparison of canopy conductance models using a Bayesian approach

“…Consequently, different models provide different estimates of system response, which may lead to different predictions and inferences regarding system functions [e.g., Pan et al, 1998;Cramer et al, 1999;Luckai and Larocque, 2002;Adams et al, 2004]. The inability to identify a unique model structure (i.e., structural uncertainty) out of the various possibilities is often taken into account in model prediction and the estimate of prediction uncertainty by using multiple models in an ensemble in methodologies such as Bayesian Model Averaging (BMA) [e.g., Neuman, 2003;Ye et al, 2004;Raftery et al, 2005;Ajami et al, 2007;Vrugt and Robinson, 2007]. However, an important objective of using semiempirical models in the analysis of complex Earth systems is to be able to make inferences about system processes.…”

Complexity in microbial metabolic processes in soil nitrogen modeling: a case for model averaging

“…The BMA weights sum up to unity. BMA methods were used in various forecasting applications such as surface water hydrological forecasting (Duan et al 2007;Ajami et al 2007;Vrugt and Robinson 2007;Wöhling and Vrugt 2008;Hsu et al 2009), ground water modeling (Neuman 2003;Neuman and Wierenga 2003;Ye et al 2004;Poeter and Anderson 2005;Refsgaard et al 2007;Ye et al 2008;Rojas et al 2009) and weather forecasting (Raftery et al 2003(Raftery et al , 2005 and have shown promising results in dealing with model predictive uncertainty.…”