Bayesian parameter estimation in a mixed-order model of BOD decay

We describe a generalized version of the BOD decay model in which the reaction is allowed to assume an order other than one. This is accomplished by making the exponent on BOD concentration a free parameter to be determined by the data. This “mixed-order” model may be a more appropriate representation of the aggregation of underlying processes that contribute to overall oxygen consumption in organic wastes and therefore has the potential to result in improved model fit. In order to directly compare the relative plausibility of alternative choices for a reaction order, we adopt a Bayesian approach to parameter estimation. This approach uses Bayes’ theorem to develop a joint probability distribution for all parameter values conditional on the observed data. From this joint distribution, we employ a numerical integration method to derive marginal parameter distributions that can be used to directly compare the relative plausibility of competing parameter values. For the data sets we examine, reaction orders other than one are generally much better supported by the data, and the often-proposed second-order model does not appear to be an adequate alternative. For practical use, the mixed-order model formulation results in a better fit to observations and yields more realistic predictions of ultimate BOD than the first-order expression. In addition, the probabilistic nature of the Bayesian model we describe facilitates explicit consideration of uncertainty in subsequent water quality management and decision-making.