Bayesian approach to estimating margin of safety for total maximum daily load development

A Bayesian-updating approach is presented to the estimation of total uncertainty-based Margin of Safety (MOS) for Total Maximum Daily Load (TMDL) calculations. Probability distributions are presented to construct the likelihood function, the prior probability distribution, and the posterior (total uncertainty) probability distribution. The Bayesian-updating approach is demonstrated through a case study for the Lower Amite River, Louisiana. The posterior probability distribution-based on the Bayesian approach updates the standard deviation of summer dissolved oxygen in the Amite River from 1.88 mg/L to 2.10 mg/L when the total uncertainty is considered. Results from the Bayesian-updating approach are compared with two conventional methods. The dissolved oxygen reserve based on a conventional margin of safety of 20% is estimated to be 45,682.26 kg/Day. The second conventional method, where we consider the standard deviation of 1.88 mg/L, produces a dissolved oxygen reserve of 40,516.09 kg/Day. The Bayesian approach yields the dissolved oxygen reserve of 38,614.43 kg/Day with the first level (μ-σ) of MOS, producing a deficit of 5606.65 kg/Day in dissolved oxygen. The dissolved oxygen reserve deficit increases to 23,895.13 kg/Day when the second level (μ-2σ) of MOS is used, which escalated to 42,383.52 kg/Day when the highest level (μ-3σ) of MOS is used. While the total uncertainty-based Bayesian approach is demonstrated for a TMDL development on the Amite River, the overall approach could be applied in any river system with similar available data.