Bayesian Framework for Water Quality Model Uncertainty Estimation and Risk Management

A formal Bayesian methodology is presented for integrated model calibration and risk-based water quality management using Bayesian Monte Carlo simulation and maximum likelihood estimation (BMCML). The primary focus is on lucid integration of model calibration with risk-based water quality management and total maximum daily load (TMDL) estimation under conditions of uncertainty. The sources of uncertainty considered in the analysis are modeling errors, observational data errors and fuzziness of the water quality standard. The difference between observed data or transformation thereof and corresponding model response is assumed to follow first-order Markov process, a specific case of which is statistically independent Gaussian errors. The BMCML method starts with sampling parameter sets from prior probability distributions of the model parameters and uses Bayes theorem and the maximum likelihood technique to estimate the triplicate (variance of residual errors, bias and autocorrelation coefficient of total errors) for each parameter set and the corresponding likelihood value. By approximating integration over the entire parameter space discretely, analytical expressions are derived for the cumulative probability distributions of model outputs and probability of violating water quality standards. The solution of the TMDL problem and related margin of safety (MOS) is then framed in the context of the developed Bayesian framework. Three example applications of varying complexities are utilized to demonstrate the versatility of the Bayesian methodology for water quality management. The BMCML methodology is validated using a hypothetical lake-phosphorus model and familiar statistical benchmarks. It is shown that the risk-based framework can estimate the reliability of an arbitrarily selected MOS as demonstrated in the Fork Creek bacteria and Shunganunga Creek dissolved oxygen TMDL case-studies. It is also shown that neglecting covariation among model parameters (i.e., by sampling parameter values from their posterior marginal distributions) influences the estimation of probability of exceedance and could potentially lead to the overestimation of the MOS at low risk levels.