Addressing equifinality and uncertainty in eutrophication models

[1] Large simulation models of eutrophication processes are commonly used to aid scientific understanding and to guide management decisions. Confidence in models for these purposes depends on uncertainty in model equations (structural uncertainty) and on effects of input uncertainties (model parameters, initial conditions, and forcing functions) on model outputs. Our objective herein is to illustrate two strategies, a generalized likelihood uncertainty estimation (GLUE) approach combined with a simple Monte Carlo sampling scheme and a Bayesian methodological framework along with Markov Chain Monte Carlo (MCMC) simulations, for elucidating the propagation of uncertainty in the high-dimensional parameter spaces of mechanistic eutrophication models. We examine the ability of the two approaches to offer insights into the degree of information about model inputs that the data contain, to quantify the correlation structure among parameter estimates, and to obtain predictions along with uncertainty bounds for modeled output variables. Our analysis is based on a four-state-variable (phosphate-detritus-phytoplankton-zooplankton) model and the mesotrophic Lake Washington (Washington State, United States) as a case study. Scientific knowledge, expert judgment, and observational data were used to formulate prior probability distributions and characterize the uncertainty pertaining to 14 model parameters. Despite the conceptual differences for addressing model equifinality, that is, wide ranges of parameter values subject to complex multivariate relationships that result in plausible observed behaviors and produce equivalently accurate predictions, we found that the two strategies provided fairly consistent estimates of the posterior parameter correlation structure and output uncertainty. Nonetheless, our analysis also shows that MCMC can more efficiently quantify the joint probability distribution of model parameters and make inference about this distribution. The latter finding can be explained by the basic idea underlying the MCMC methodology, that is, the configuration of a Markov process whose stationary distribution approximates the joint posterior distribution of all the stochastic model nodes; as a result, Monte Carlo samples are not drawn from the prior parameter space, and problems of wide or highly correlated prior distributions can be overcome. Finally, our study stresses the lack of perfect simulators of natural system dynamics and introduces two statistical formulations that can explicitly account for the discrepancy between mathematical models and environmental systems.