The probability of breaching water quality standards – A probabilistic model of river water nitrate concentrations

Fred Worrall*, Brandon Kerns, Nicholas J.K. Howden, Tim P. Burt, Helen P. Jarvie

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

2 Citations (Scopus)
22 Downloads (Pure)


In this study we propose an approach to predicting the probability that river waters will exceed a water quality standard. The study used a two-part generalised linear modelling approach within a Bayesian framework. Binomial regression was used to model the probability that a water quality standard would be exceeded and included two factors – the difference between sampling sites and difference between years of sampling. Using a Bayesian approach meant that information could be drawn from all observations from all sites, across all years, and that all results would come with a measure of uncertainty. Furthermore, although some known factors could not be included in the binomial regression, they could be included using Bayes’ rule to enhance and inform the results. This approach was applied to assessing the probability of nitrate concentrations in English river waters exceeding the current nitrate water quality standard of 11.3 mg N/l. The study showed that the Bayesian approach decreased the measures of uncertainty in the predicted outcomes was reduced by an average of 60% and increased the effective sample size by 64%.The best-fit model had a root mean square error (RMSE) of 7.9% which equated to an error of ±1 sample above the water quality standard for the median site. When interaction of factors could be included, then RMSE decreased to 3.8%. It was not possible to include a diurnal cycle, owing to a paucity of sub-daily sampling, but there was a significant seasonal cycle and so outputs could be adjusted by means of Bayes’ rule to predict water quality standard exceedance each month. Comparison with the current method of classification shows no significant difference between five out of the six lowest classifications with only the highest classification being correlated with the estimated exceedence rate. With respect to nitrate in English river waters, the average exceedance rate was 8.3% but was declining at a statistically-significant rate of 0.09%/yr2.

Original languageEnglish
Article number124562
JournalJournal of Hydrology
Early online date18 Jan 2020
Publication statusPublished - Apr 2020


  • Bayesian analysis
  • Binomial regression
  • Nitrate
  • Water quality standards


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