Probabilistic programming for nitrate pollution control: Comparing different probabilistic constraint approximations

Agricultural nitrate emissions within a river catchment are, due to rainfall and other sources of natural variation, uncertain. A regulator aiming to reduce nitrate emissions into surface and groundwater faces a trade-off between reliability in achieving emission standards and the cost of compliance to agriculture. This paper explores this trade-off by comparing different assumptions about the probability distribution of nitrate emissions and thus the probabilistic constraint included in the catchment model. Three categories of probabilistic constraints are considered: (1) non-parametric, (2) normal and (3) lognormal. The results indicate that the restrictiveness of the non-parametric assumption could lead to a significant reduction in profit relative to the normal and lognormal. The lognormal assumption, although it is theoretically correct, cannot be generalised to the case of correlated emissions. However, ignoring the dependence between different sources of nitrate emissions introduces more bias than mis-specifying their distribution. Therefore a probabilistic constraint based on a correlated normal distribution of emissions gives the best approximation for nitrate emissions in this study.