Goal: Allow hydrologists to cope quantitatively with conceptual model uncertainty via a well-founded and well-researched methodology, incorporating maximum likelihood parameter estimation in a multimodel Bayesian updating framework, which is feasible to implement in practice. Objectives: (1) To firm up the theoretical basis of a Maximum Likelihood Bayesian Model Averaging (MLBMA) method recently proposed by the PI (Neuman, 2002, 2003) for the rendering of optimum hydrologic predictions by means of several competing models and the assessment of their joint predictive uncertainty. (2) To implement, explore and demonstrate MLBMA on hydrogeologic data, distributed in three-dimensional space and time, collected earlier in unsaturated fractured rock at the Apache Leap Research Site (ALRS) in central Arizona. The Problem: Hydrologic analyses typically rely on a single conceptual-mathematical model of geologic or watershed makeup and corresponding hydrologic processes. Yet hydrologic environments are open and complex, rendering them prone to multiple interpretations and mathematical descriptions. This is true regardless of the quantity and quality of available data. Predictions and analyses of uncertainty based on a single hydrologic concept are prone to statistical bias (by committing a Type II error through reliance on an inadequate model) and underestimation of uncertainty (by committing a Type I error through under sampling of the relevant model space). The bias and uncertainty that result from reliance on an inadequate conceptual-mathematical model are often much larger than those introduced through an inadequate choice of model parameter values. Yet most hydrologic uncertainty analyses ignore the former and focus exclusively on the latter. This often leads to overconfidence in the predictive capabilities of the model, which the available hydrologic data seldom justify. Indeed, critiques of hydrologic analyses and scientific/regulatory/legal challenges to them typically focus on the validity of the underlying conceptual (and by implication mathematical) model. Existing method of dealing with the problem, most notably the Generalized Likelihood Uncertainty Estimation (GLUE) approach of Beven and Binley (1992; see also Beven and Freer, 2001) are, in the PIfs view, useful but not necessarily optimal for the purpose. There is a need for an innovative approach to the problem that rests on rigorous theory and is feasible to implement in practice. Approach: Under Objective 1 we propose to firm up the theoretical basis of MLBMA by exploring theoretically and through synthetic numerical studies (a) the accuracy and computational feasibility of MLBMA in comparison to Bayesian Model Averaging (BMA) implemented via Markov Chain Monte Carlo simulation (Hoeting et al., 1999); (b) the impact that availability or lack of prior hydrologic parameter measurements have on this comparison; (c) the difference between computing posterior model probabilities using Kashyap fs (1982) Bayesian information criterion KIC (Neuman, 2002, 2003) versus the asymptotic Bayesian criterion BIC (proposed by Raftery, 1993) or non-Bayesian information theoretic criteria such as Akaike fs (1974) AIC (proposed by Burnham and Anderson, 2002); (d) the unresolved issue of how to assign prior probabilities to various models; and (e) the rate at which the sensitivity of posterior parameter estimates and model probabilities to the choice of prior parameter and model probabilities diminishes with the information content (quantity and quality) of hydrologic data. Some of these same issues will also be addressed vis-a-vis real data under Objective 2. Objective 2 is to implement, explore and demonstrate the predictive capabilities of MLBMA on hydrogeologic data collected earlier at the ALRS. These include air permeability and air-filled porosity data from pneumatic injection tests in 1-m-length intervals along six vertical and inclined boreholes at the site, and transient pressure data from cross-hole pneumatic injection tests in these and ten additional boreholes. The data represent largely a continuum of interconnected fractures. Their analysis will be conducted in three stages. At Stage 1 we propose to consider alternative geological-geostatistical models of how the 1-m-scale log permeability ()10 log k and log porosity ()10 log data vary in space, based solely on 1-m-scale measurements. At Stage 2, we will examine the extent to which a subset of these models, as well as BMA and MLBMA, are capable of predicting air flow between boreholes during cross-hole tests at the site based solely on prior information about lo , 10 g k 10 log and alternative representations of forcing terms. The prior information will consist of measurements, statistics, geological-geostatistical models, projections and projection covariances of these quantities across a domain containing all boreholes, established at Stage 1. At Stage 3 we propose to calibrate (via ML) airflow models having alternative parameter structures against pressure data observed during one cross-hole test at the ALRS and examine their ability, as well as that of MLBMA, to predict pressures observed during other such (validation) tests. The cross-hole tests will be selected so that injection takes place into a different borehole in each of them. Intellectual Merit and Broad Impacts: A solid theory and a practical methodology of rendering optimum hydrologic predictions and an assessment of predictive uncertainty that account jointly for uncertainties in model structure (conceptual-mathematical frameworks) and parameters. The approach applies to a broad range of models representing natural processes in ubiquitously open and complex earth and environmental systems. Results will bedisseminated broadly to researchers and practitioners through various means.
