Estimating bootstrap and Bayesian prediction intervals for constituent load rating curves

Assessment of constituent loads in rivers is essential to evaluate water quality of streams and estuaries; however, uncertainty in load estimation may be large and must be considered and communicated together with estimates. In this comparative study, the usefulness of two existing methods (bootstrap and Bayesian inference) to assess uncertainty in constituent loads estimated with an improved eight-parameter rating curve is demonstrated. Bootstrap prediction intervals and Bayesian credible intervals were estimated for daily and monthly loads obtained with a rating curve applied to routine monitoring sampling data sets of nitrate (NO3-N), reactive phosphorus (RP), and total phosphorus (TP) of the Duck River, in Tasmania (Australia). Predicted loads and prediction intervals were compared to benchmark loads obtained by an independent, high frequency monitoring program. The eight-parameter rating curve resulted in better prediction of NO3-N and TP than RP loads. Both inference methods successfully generated prediction intervals. The bracketing frequency (i.e., the fraction of prediction intervals that comprised benchmark loads) of bootstrap prediction intervals was 50-65% of daily or monthly benchmark loads. Bracketing frequency of Bayesian credible intervals was consistently higher, and included 74-85% of benchmark daily loads and 80% or more of benchmark monthly loads. Both methods proved to be robust to the presence of an artificial outlier. Prediction intervals were affected by the distribution of the regression error, hence they reflected uncertainty in the regression data set and limitations in the rating curve formulation. They did not account for other sources of uncertainty, i.e., they were still conservative predictions of load uncertainty.