A revaluation of lake-phosphorus loading models using a Bayesian hierarchical framework

We revisit the phosphorus-retention and nutrient-loading models in limnology using a Bayesian hierarchical framework. This methodological tool relaxes a basic assumption of regression models fitted to data sets consisting of observations from multiple systems, i.e., the systems are assumed to be identical in behavior, and therefore the models have a single common set of parameters for all systems. Under the hierarchical structure, the models are dissected into levels (hierarchies) that explicitly account for the role of significant sources of variability (e.g., morphometry, mixing regime, geographical location, land-use patterns, trophic status), thereby allowing for intersystem parameter differences. Thus, the proposed approach is a compromise between site-specific (where limited local data is a problem) and globally common (where heterogeneous systems in wide geographical areas are assumed to be identical) parameter estimates. In this study, we used critical values of the mean lake depth $$ \left( {\bar{z} = 10.3\,{\text{m}}} \right) $$ and the hydraulic residence time (τw = 2.6 years) to specify the hierarchical levels of the models. Our analysis demonstrates that the hierarchical configuration led to an improvement of the performance of six out of the seven hypothesized relationships used to predict lake-phosphorus concentrations. We also highlight the differences in the posterior moments of the group-specific parameter distributions, although the inference regarding the importance of different predictors (e.g., inflow-weighted total phosphorus input concentration, and hydraulic retention time) of lake phosphorus or the relative predictability of the models examined are not markedly different from an earlier study by Brett and Benjamin. The best fit to the observed data was obtained by the model that considers the first-order rate coefficient for total phosphorus loss from the lake as an inverse function of the lake hydraulic retention time. Finally, our analysis also demonstrates how the Bayesian hierarchical framework can be used for assessing the exceedance frequency and confidence of compliance of water-quality standards. We conclude that the proposed methodological framework will be very useful in the policy-making process and can optimize environmental management actions in space and time.