Reliable estimates of the nutrient fluxes carried by rivers from land-based sources to the sea are needed for efficient abatement of marine eutrophication. Although nutrient concentrations in rivers generally display large temporal variation, sampling and analysis for nutrients, unlike flow measurements, are rarely performed on a daily basis. The infrequent data calls for ways to reliably estimate the nutrient concentrations of the missing days. Here, we use the Gaussian state space models with daily water flow as a predictor variable to predict missing nutrient concentrations for four agriculturally impacted Finnish rivers. Via simulation of Gaussian state space models, we are able to estimate aggregated yearly phosphorus and nitrogen fluxes, and their confidence intervals.

The effect of model uncertainty is evaluated through a Monte Carlo experiment, where randomly selected sets of nutrient measurements are removed and then predicted by the remaining values together with re-estimated parameters. Results show that our model performs well for rivers with long-term records of flow. Finally, despite the drastic decreases in nutrient loads on the agricultural catchments of the rivers over the last 25 years, we observe no corresponding trends in riverine nutrient fluxes.