Predicting the Frequency of Water Quality Standard Violations Using Bayesian Calibration of Eutrophication Models

The water quality standard setting process usually relies on mathematical models with strong mechanistic basis, as this provides assurance that the model will more realistically project the effects of alternative management schemes. From an operational standpoint, the interpretation of model results should be coupled with rigorous error analysis and explicit consideration of the predictive uncertainty and natural variability. In this study, our main objective is to attain effective model calibration and rigorous uncertainty assessment by integrating environmental mathematical modeling with Bayesian analysis. We use a complex aquatic biogeochemical model that simulates multiple elemental cycles (org. C, N, P, Si, O), multiple functional phytoplankton (diatoms, green algae and cyanobacteria) and zooplankton (copepods and cladocerans) groups. The Bayesian calibration framework is illustrated using three synthetic datasets that represent oligo-, meso- and eutrophic lake conditions. Scientific knowledge, expert judgment, and observational data were used to formulate prior probability distributions and characterize the uncertainty pertaining to a subset of the model parameters, i.e., a vector comprising the 35 most influential parameters based on an earlier sensitivity analysis of the model. Our study also underscores the lack of perfect simulators of natural system dynamics using a statistical formulation that explicitly accounts for the discrepancy between mathematical models and environmental systems. The model reproduces the key epilimnetic temporal patterns and provides realistic estimates of predictive uncertainty for water quality variables of environmental management interest. Our analysis also demonstrates how the Bayesian parameter estimation can be used for assessing the exceedance frequency and confidence of compliance of different water quality criteria. The proposed methodological framework can be very useful in the policy-making process and can facilitate environmental management decisions in the Laurentian Great Lakes region.