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Natural resource managers require information concerning the frequency, duration, and long-term probability of occurrence of water-quality indicator (WQI) violations of defined thresholds. The timing of these threshold crossings often is hidden from the observer, who is restricted to relatively infrequent observations. Here, a model for the hidden process is linked with a model for the observations, and the parameters describing duration, return period, and long-term probability of occurrence are estimated using Bayesian methods. A simulation experiment is performed to evaluate the approach under scenarios based on the equivalent of a total monitoring period of 5-30 years and an observation frequency of 1-50 observations per year. Given constant threshold crossing rate, accuracy and precision of parameter estimates increased with longer total monitoring period and more-frequent observations. Given fixed monitoring period and observation frequency, accuracy and precision of parameter estimates increased with longer times between threshold crossings. For most cases where the long-term probability of being in violation is greater than 0.10, it was determined that at least 600 observations are needed to achieve precise estimates. An application of the approach is presented using 22 years of quasi-weekly observations of acid-neutralizing capacity from Deep Run, a stream in Shenandoah National Park, Virginia. The time series also was sub-sampled to simulate monthly and semi-monthly sampling protocols. Estimates of the long-term probability of violation were unbiased despite sampling frequency; however, the expected duration and return period were over-estimated using the sub-sampled time series with respect to the full quasi-weekly time series.

Evaluation of Bayesian Estimation of a Hidden Continuous-Time Markov Chain Model with Application to Threshold Violation in Water-Quality Indicators