An Uncertainty-Based Approach to the Total Maximum Daily Load Development

Abstract

A Bayesian-updating approach is developed to the estimation of the total uncertainty-based Margin of Safety (MOS) for Total Maximum Daily Load calculations using the watershed modeling tool HSPF. In order to determine the prior, likelihood, and posterior distributions of uncertainties involved in Bayesian approach, various probability density functions are derived. The temperature measurement induced uncertainty in dissolved oxygen simulation is found to be normally distributed. The temporal scale uncertainty in weather data decreases with decreasing temporal resolution. The temporal-scale uncertainties in the rainfall and temperature data follow a Weibull and general extreme value distributions, respectively. The spatial-resolution uncertainty in simulated dissolved oxygen follows a general extreme value distribution. Duration curves are developed to examine the output computation-induced uncertainty. Duration curves for dissolved oxygen and nitrate-nitrogen exhibit high variability in the load estimated using daily data as compared to those based on bi-weekly and monthly data. It is found that, the temporal scale-induced uncertainty in model outputs is linearly and inversely correlated with the logarithm of the time scale. Regression equations are presented to extrapolate near real time flow and water quality data, greatly simplifying flow and water quality monitoring and reducing the cost involved in flow and water quality monitoring. The temporal scale-induced uncertainties in simulated dissolved oxygen and nitrate-nitrogen follow a general extreme value and gamma distributions while the temporal scale uncertainty in flow is normally distributed. The new Bayesian updating approach is demonstrated through a case study for the Amite River, Louisiana. The posterior probability distribution-based on the above distributions updates standard deviation of summer dissolved oxygen from 1.88 mg/L to 2.10 mg/L for the Amite River. The Bayesian method yields the dissolved oxygen reserve of 38,614.43 Kg/Day with first level MOS, producing a deficit of 5,606.65 Kg/Day in dissolved oxygen. The dissolved oxygen reserve deficit increases to 23,895.13 Kg/Day when the second level MOS is used, which escalates to 42,383.52 Kg/Day when highest level of MOS is used. The total uncertainty-based Bayesian approach developed in this study provides a useful tool for the adaptive and risk based TMDL implementation and watershed restoration.