Risk analysis using stochastic reliability methods applied to two cases of deterministic water quality models

For the computation of extreme event statistics with respect to pollutant loads and environmental effects, the uncertainty in model parameters of deterministic models and the inherent stochastic variability in input variables have to be taken into account. Two stochastic reliability methods—combinations of the First Order Reliability Method (FORM) with Directional Simulation with Importance Sampling (FORM/DIS) or with Latin Hypercube Sampling (FORM/LHS)—were used for risk analysis related to acute environmental effects. The range of probabilities of exceedence at which these combined methods are more efficient than crude Monte Carlo simulation (CMC) or Latin Hypercube Sampling (LHS) only, was investigated. Their efficiency in relation to the required number of model simulations to accurately estimate the small exceedence probabilities of extremes compared to that of Monte Carlo simulation was demonstrated with two case studies: (1) to estimate exceedence probabilities of limit pollutant concentrations in a lake after a loading event using a simple numerical model. At exceedence probabilities smaller than 0.1 both FORM/DIS and FORM/LHS gave more accurate results than crude Monte Carlo simulation, and this improvement increased rapidly with decreasing probabilities. (2) to estimate the frequencies of occurrence of extremely low dissolved oxygen (DO) concentrations in a stream receiving multiple COD loads from combined sewer overflows and storm water outlets using a issolved xygen tream del (DOSMO). In this example the relationships between stochastic parameters and the model output are strongly non-linear and the number of loading events that determine the in-stream DO concentration is variable. Here, FORM/LHS was more efficient than than FORM/DIS in estimating very small probabilities. A general limit value for the probability of exceedence below which FORM/LHS or FORM/DIS were more efficient than Crude Monte Carlo simulation and Latin Hypercube Sampling could not be given, but they were lower for the complex stream DO model than for the simple lake model. Based on these very different case studies it is concluded that the methods can generally be applied to a large variety of numerical models within the field of environmental engineering