Reachability of water quality targets while avoiding adverse impacts on agricultural productivity

Efforts to reduce riverine nutrient loads have not been as fruitful as expected or hoped. Load reduction goals have been delayed in numerous U.S. states and watersheds, yet the question remains – are goals achievable by the new target dates? Moreover, what actions would be required in order to meet water quality targets, and how would the required actions affect agricultural productivity or other objectives? Here, we frame the nutrient loss reduction problem as a “reach-while-avoid” problem, employing probabilistic reachability methods from control theory. We provide a mathematical formalism for the problem of reaching a water quality target set at a future date while avoiding (during the intermediate time) an unsafe set describing agricultural nutrient deficiency. In the present study, we specifically consider phosphorus management and construct a low-dimensional, stochastic model of watershed scale phosphorus dynamics for the analysis. The framework naturally emphasizes uncertainty in initial conditions (i.e., built up nutrient concentrations) and incorporates uncertainty in future rainfall. We resolve the problem for a variety of admissible control sets representing access to different portfolios of agricultural and wastewater technology interventions. This work will produce a novel characterization of plausible future nutrient load trajectories under alternative management strategies. Broadly, this work also introduces the water resources planning and management community to the extensive developments in reachability analysis within the control theory and safety verification communities.