Estimating bootstrap and Bayesian prediction intervals for constituent load rating curves

Abstract

[1] 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.Citation: Vigiak, O., and U. Bende-Michl (2013), Estimating bootstrap and Bayesian prediction intervals for constituent load rating curves, Water Resour. Res., 49,[8565][8566][8567][8568][8569][8570][8571][8572][8573][8574][8575][8576][8577][8578]

Quantifying uncertainty on sediment loads using bootstrap confidence intervals

“…By leaving out step 1 (bootstrapping the Q−h pairs) and just using Q as predicted by the discharge rating curve from all observed data points, confidence intervals can be obtained that only take into account the uncertainty on the sediment rating curve. If the resulting confidence intervals closely resemble the confidence intervals calculated with the full approach, this would mean that the uncertainty in the sediment concentration is what drives the uncertainty in the loads, thus supporting the finding that the error in the discharge is negligible compared with other sources of uncertainty (e.g., Némery et al, 2013, Vigiak andBende-Michl, 2013).…”

“…To introduce this second source of error on the sediment rating curve, Rustomji and Wilkinson (2008) and Vigiak and Bende-Michl (2013) added an additional step to the bootstrap process: a randomly drawn residual from the original regression equation was added to the expected value of the constituent concentration, so that the predicted concentration included both the uncertainty of the parameters of the rating curve due to having a finite sample, and the uncertainty that arises from the fact that sediment concentrations simply cannot be perfectly predicted by any equation, regardless of how large the observed dataset would be. However, by randomly resampling from the residuals, it is assumed that these residuals are independent.…”

“…Improving upon the bootstrap percentile method, Efron and Tibshirani (1993) proposed bias-corrected and accelerated intervals, used by Vigiak and Bende-Michl (2013). Unfortunately, this approach requires an even larger number of bootstrap replicates than the percentile method to sufficiently reduce the Monte Carlo sampling error.…”

“…Walling and Webb (1988) suggested that seasonal differences of the relationship between discharge (Q) and constituent concentration, non-simultaneity of Q and concentration peaks during storms, hysteresis and exhaustion effects are the most important causes of inaccuracy in concentration predictions. Few studies to date have taken this error on the discharge rating curve into account explicitly when calculating uncertainty on load estimates (Vigiak and Bende-Michl, 2013;Rustomji and Wilkinson, 2008). Uncertainties of discharge, however, depend on several factors: the method chosen to estimate the discharge, site conditions and the time interval over which water levels are measured (Harmel et al, 2009;Hamilton and Moore, 2012;McMillan et al, 2012;Tomkins, 2014).…”

“…This approach is suitable for testing different methods and different temporal measurement resolutions of load estimation, but it does not assess the uncertainty of the "true" load, as it is assumed that, with sufficiently high sampling frequency (in their case, daily), the measured load is equivalent to the actual load. More recently, two new candidate approaches have emerged to calculate confidence intervals on loads that have the potential to be generally applicable, regardless of the method used to calculate the load and the distributional assumptions made: bootstrap methods (Mailhot et al, 2008;Rustomji and Wilkinson, 2008;Vigiak and Bende-Michl, 2013) and Bayesian methods that result in credibility intervals (Pagendam et al, 2014;Vigiak and Bende-Michl, 2013).…”