An event-based stochastic model of phosphorus loading into a lake

In most lakes, phosphorus (P) is the nutrient controlling the trophic state. Thus, for effective control of eutrophication, the uncertainty in P-loading should be encoded as a probability density function (pdf). Specifically, the pdf of P-loading Y from non-point agricultural sources is sought by means of an event-based stochastic model.P-loading events are triggered by precipitation events (X₁, X₂, T), in which X₁ is the rainfall amount, X₂ the duration, and T the interarrival time between events. (X₁, X₂) are dependent random variables, while T is assumed to be exponentially distributed. The precipitation event causes runoff, which carries dissolved P into the lake with a concentration C₁ and sediment yield, Z, which carries fixed or sorbed P into the lake in a fraction C₂ of Z. Seasonal loading of P is calculated by adding random numbers of random variables. The model accounts separately for dissolved P and sorbed P. Explicit expressions are given for the mean and variance of each type of P-loadings. The case study of a sub-watershed of Lake Balaton, Hungary, is used to illustrate the methodology. Precipitation data, empirical rainfall-runoff-sediment yield relationships and a small number of observations of events are used to calibrate the model and estimate the means and variances of loading per event and per season. Then a simulation method is used to estimate complete pdf of these random variables. Use of the model for alternative methods of controlling P-loading is briefly discussed, as well as the economics of control.