# USING BAYESIAN STATISTICS TO ASSESS PARAMETER AND MODEL UNCERTAINTY IN GROUNDNWATER MODELING

• Dr. Bruno Mendes – Department of Applied Mathematics and Statistics – University of California, Santa Cruz – USA
• FCUL (DEIO) – Campo Grande – Bloco C6 – Piso 4 – Sala 6.4.30 – 12h
• Sexta-feira, 23 de Junho de 2006

The issue of uncertainty of models in groundwater contamination has always been present in the minds of modelers. Everyone is well aware of the difficulties in characterizing geophysical systems, and how different models can fit observable data equally well and yet offer quite different extrapolations (predictions) away from the data. We begin with an uncertainty assessment framework we developed earlier [www.ams.ucsc.edu/~draper/writings.html items 43 and 55], in which all forms of uncertainty in groundwater contamination modeling can be broken down hierarchically into four types: * scenario (there may be uncertainty about relevant inputs to the physical process under study), * structure (conditional on scenario, the precise mathematical form of the best model to capture all relevant physical processes (advection, diffusion, …) may not be known, * parametric (conditional on scenario and structure, the model will typically have one or more unknown physical constants that need to be estimated from data), and * predictive (conditional on scenario, structure, and parameters, the model predictions may still not agree perfectly with the observed data). Two of the most fundamental goals of scientific work are accurate prediction of future data and accurate (well-calibrated) uncertainty assessments for the predictions. If scenario and structural uncertainty are present, it will not be enough in creating well-calibrated uncertainty assessments merely to qualitatively compare the results of different scenario and structural choices; it is necessary to combine uncertainty across (or between) these different choices in addition to quantifying uncertainty within (conditional on) them. A Bayesian approach to solving this problem appears most natural to us. To show the applicability of our approach we chose a data set available from a reputable source [Gonzalez et al., Groundwater Hydraulics, 1984], and well established analytical/ numerical modeling methods [Javandel et al., “Groundwater Transport: Handbook of Mathematical Models, 1984]. In the work on which I will report in this talk, we increase the physical realism of the modeling by making a direct comparison of one-, two-, and three-dimensional advection-diffusion models on the same data set. In each case we use Markov Chain Monte Carlo methods to account fully for parameter uncertainty given the specific model structure; we compare the models with a log-scoring criterion [www.ams.ucsc.edu/~draper/ writings.html, item 71] which rewards models that make accurate and well-calibrated predictions of the observable data; we create composite predictive distributions that capture not only within-model parametric and predictive uncertainty but also between-model uncertainty; and we demonstrate that these composite predictive distributions are better calibrated than those obtained by ignoring model uncertainty or treating it in a qualitative way. Our approach can be applied quite generally, e.g., not only in modeling research but also in risk analysis for accidental contaminations.