Andrew D. Gronewold

Modeling the relationship between most probable number (MPN) and colony-forming unit (CFU) estimates of fecal coliform concentration.

Most probable number (MPN) and colony-forming-unit (CFU) estimates of fecal coliform bacteria concentration are common measures of water quality in coastal shellfish harvesting and recreational waters. Estimating procedures for MPN and CFU have intrinsic variability and are subject to additional uncertainty arising from minor variations in experimental protocol. It has been observed empirically that the standard multiple-tube fermentation (MTF) decimal dilution analysis MPN procedure is more variable than the membrane filtration CFU procedure, and that MTF-derived MPN estimates are somewhat higher on average than CFU estimates, on split samples from the same water bodies. We construct a probabilistic model that provides a clear theoretical explanation for the variability in, and discrepancy between, MPN and CFU measurements. We then compare our model to water quality samples analyzed using both MPN and CFU procedures, and find that the (often large) observed differences between MPN and CFU values for the same water body are well within the ranges predicted by our probabilistic model. Our results indicate that MPN and CFU intra-sample variability does not stem from human error or laboratory procedure variability, but is instead a simple consequence of the probabilistic basis for calculating the MPN. These results demonstrate how probabilistic models can be used to compare samples from different analytical procedures, and to determine whether transitions from one procedure to another are likely to cause a change in quality-based management decisions.

Calibrating and validating bacterial water quality models: A Bayesian approach

Water resource management decisions often depend on mechanistic or empirical models to predict water quality conditions under future pollutant loading scenarios. These decisions, such as whether or not to restrict public access to a water resource area, may therefore vary depending on how models reflect process, observation, and analytical uncertainty and variability. Nonetheless, few probabilistic modeling tools have been developed which explicitly propagate fecal indicator bacteria (FIB) analysis uncertainty into predictive bacterial water quality model parameters and response variables. Here, we compare three approaches to modeling variability in two different FIB water quality models. We first calibrate a well-known first-order bacterial decay model using approaches ranging from ordinary least squares (OLS) linear regression to Bayesian Markov chain Monte Carlo (MCMC) procedures. We then calibrate a less frequently used empirical bacterial die-off model using the same range of procedures (and the same data). Finally, we propose an innovative approach to evaluating the predictive performance of each calibrated model using a leave-one-out cross-validation procedure and assessing the probability distributions of the resulting Bayesian posterior predictive p-values. Our results suggest that different approaches to acknowledging uncertainty can lead to discrepancies between parameter mean and variance estimates and predictive performance for the same FIB water quality model. Our results also suggest that models without a bacterial kinetics parameter related to the rate of decay may more appropriately reflect FIB fate and transport processes, regardless of how variability and uncertainty are acknowledged.

Short communication: A software tool for translating deterministic model results into probabilistic assessments of water quality standard compliance

Most assessments of whether a water body will comply with pollutant standards after modification of land use, loading, or climate change are based on the results of deterministic simulation models. These models, including those used to support the United States Environmental Protection Agency (USEPA) total maximum daily load (TMDL) program, typically do not account for common sources of assessment uncertainty. Instead, model results are typically represented by a time series of predicted pollutant concentration values or the parameters of a frequency-based distribution of these values over a specified time period. The rate of exceedance of relevant pollutant limits is then assessed directly from this time series or distribution to determine standard compliance. In this way, sampling and analysis-based variability and model uncertainty are typically ignored, although they may substantially influence the probability of non-compliance. To help address this problem, we introduce ProVAsT (Probabilistic Water Quality Standard Violation Assessment Tool), a software tool encoded in the graphical model-based package Analytica^(R). Here, we present a version of ProVAsT which translates model-predicted in situ fecal indicator bacteria (FIB) pollutant concentrations into the expected frequency of water quality standard violations and provides a Bayesian measure of the degree of confidence in this assessment. We call this version ProVAsT-FIB. Along with inputting their own simulation model results, users can specify the particular water quality analysis methods employed (e.g. the analytic procedure used and the number and volume of sample aliquots) as well as the numeric limits pertaining to local water quality standards. It is our hope that ProVAsT will encourage the rational consideration of uncertainty and variability in water quality assessments by reducing the burden of complex statistical calculations.