Nonlinear empirical modeling to estimate phosphorus exports using continuous records of turbidity and discharge

We tested an empirical modeling approach using relatively low-cost continuous records of turbidity and discharge as proxies to estimate phosphorus (P) concentrations at a subhourly time step for estimating loads. The method takes into account nonlinearity and hysteresis effects during storm events, and hydrological conditions variability. High-frequency records of total P and reactive P originating from four contrasting European agricultural catchments in terms of P loads were used to test the method. The models were calibrated on weekly grab sampling data combined with 10 storms surveyed subhourly per year (weekly+ survey) and then used to reconstruct P concentrations during all storm events for computing annual loads. For total P, results showed that this modeling approach allowed the estimation of annual loads with limited uncertainties (approximate to -10%+/- 15%), more reliable than estimations based on simple linear regressions using turbidity, based on interpolated weekly+ data without storm event reconstruction, or on discharge weighted calculations from weekly series or monthly series. For reactive P, load uncertainties based on the nonlinear model were similar to uncertainties based on storm event reconstruction using simple linear regression (approximate to 20%+/- 30%), and remained lower than uncertainties obtained without storm reconstruction on weekly or monthly series, but larger than uncertainties based on interpolated weekly+ data (approximate to -15%+/- 20%). These empirical models showed we could estimate reliable P exports from noncontinuous P time series when using continuous proxies, and this could potentially be very useful for completing time-series data sets in high-frequency surveys, even over extended periods. Plain Language Abstract Phosphorus (P) loads transported by rivers and streams have to be estimated reliably, but this is a difficult task because P loads can be transported during very short period of time, like during storm events, and most P surveys are executed with low sampling frequencies. Because continuous surveys of P are costly, we tested a modeling approach using commonly used low-cost continuous records of turbidity and discharge as surrogate variables. This had to take into account nonlinear relationships and the fact that the relationship between P and turbidity or discharge is different during a rising phase or a descending phase. The model we developed estimates P concentration variations during storm events and provides continuous time series of P. From the model estimations, total annual loads of P could be predicted with low uncertainty ranges when using turbidity as a surrogate variable, showing its ability at estimating phosphorus exports values closer to the reality. The soluble reactive form was however less reliably predicted by discharge records, but our method could still be improved.