Aarhus University Seal / Aarhus Universitets segl

No. 376: Empirical lake models for associations between inlet and sea concentrations of phosphorus and nitrogen

Søndergaard, M., Nielsen, A., Levi, E.E., Johansson, L.S., Sørensen, P.B. & Trolle, D. 2020. Empiriske sømodeller for sammenhænge mellem indløbs- og søkoncentrationer af fosfor og kvælstof. Aarhus Universitet, DCE – Nationalt Center for Miljø og Energi, 32 s. - Videnskabelig rapport nr. 376. 


The level of nutrients – especially phosphorus and nitrogen – sets the overall framework for the water quality of lakes and their ecological status. In order to be able to assess the extent to which efforts are needed to obtain at least good ecological status, it is therefore important to have a good understanding of the nutrient supply to the lakes and the expected resulting lake concentrations.

The purpose of this report is to use a larger and up-to-date data set, as well as to carry out re-calculations of the nutrient contribution from the unmonitored catchment, in order to assess whether the existing empirical models for the relations between the inputs of nitrogen and phosphorus to lakes and the resulting lake concentrations need to be revised. The hydraulic retention time is included to create simple empirical models for the relations between the inputs and lake concentrations of phosphorus and nitrogen.

The models are founded on up-to-date and longer time series than earlier measurements of nutrient inputs and lake concentrations. The basis of the models is that of Vollenweider, which was also used for the preparation of the first water plans (2009-2015), and the modified OECD model used in the current river basin management plans (2015-2021). The dataset used includes a gross list of 50 lakes whose number was reduced in the final analysis to include data from 20 lakes (189 lake years). This was done to exclude lakes that are not considered to be in equilibrium with the external loading and which therefore do not meet the criteria for these empirical analyses. Compared with previous analyses, where data from 2000 and onwards were used, separate analyses have been conducted including only data from 2005 and onwards. Finally, analyses have been undertaken where data from lakes and years are only used if at least half of the calculated phosphorus or nitrogen input comes from monitored sub-catchments. In this way data where unmonitored catchments contribute the major part of the loading – which probably entails more uncertain calculations – are omitted from the analyses.

The modest number of 20 lakes makes the analysis results uncertain, not least if a subdivision into specific lake types is requested. Uncertainty also increases due to the fact that it is not possible to ascertain whether the lakes used in the analyses are in complete equilibrium with the external nutrient input. In addition, the available dataset consists almost exclusively of nutrient-rich lakes, which means that the models are best at elucidating relations between nutrient inputs and lake concentrations in turbid lakes, which expectedly have moderate, poor or bad ecological status. At the same time, however, these are the lakes for which a reduction of nutrient inputs and application of empirical models are particularly important.

For phosphorus, the new analyses show that it is possible to establish empirical relations where the explanatory value (R2) increases compared with the currently applied models. Thus, by using the modified OECD model with three coefficients, the explained variation of the data is 59% if all years from 2000-2018 are used and if only lakes where at least half of the phosphorus input is derived from measurements are included. It is recommended to apply these empirical models in future calculations of the relationship between phosphorus inputs and lake concentrations. In the previously used models (data from 2000-2011), the explanatory value was 45%. The explanatory values and MSE values (Mean Squared Error) of the new models do not change significantly whether data from 2000-2018 or from 2005-2018 are used, indicating that the possible lack of equilibrium conditions in the period 2000-2005 had no major influence on the empirical relations. On the other hand, exclusion of data from lakes where unmonitored catchments constitute the majority of the total phosphorus contribution gives an increased explanatory value. This suggests that the actual calculation of the phosphorus input from an unmonitored-catchment constitutes a significant factor of uncertainty for the empirical relations between the input and the in-lake concentration of phosphorus. A comparison of the empirically modelled phosphorus concentrations based on the ‘old’ and the ‘new’ model expression is primarily distributed around the 1:1 line. For quite a few of the lakes (lake years), there are significant differences and for lakes and years the difference between the 'old' and 'new' modelled TP varies more than a factor two.

For nitrogen, it is possible to establish somewhat stronger empirical relations between the external nitrogen input and the measured lake concentration than for phosphorus. The explanatory value is approx. 75%. Neither the explanatory value nor the dispersion (represented by MSE) are significantly different from those of the models applied in the current river basin management plans. In contrast to phosphorus, the explanatory value does not increase when excluding measurements in which the nitrogen input constitutes more than half of the total nitrogen input.