Mathematical problems and solutions in sprinkler irrigation

Pressurized irrigation is quickly replacing surface irrigation systems in Spain due to the impulse of irrigation modernization programs. The change in irrigation system is resulting in reduced labour and increased irrigation efficiency and crop yield. The main constraint for sprinkler irrigation is the wind, which severely reduces irrigation uniformity and increases evaporation water losses. Many areas of Spain are characterized by strong winds, and therefore require specific design and management techniques. Such is the case of certain areas of the Ebro Valley depression, where yearly wind averages can exceed 3 m/s. Mathematical simulation models of sprinkler irrigation are required to predict irrigation performance under different hardware, operation and environmental conditions. Such models are based on ballistic theory, and require the numerical solution of the equations of movement applied to a drop moving in the air from the sprinkler nozzle to the soil surface or the crop canopy. A fourth-order Runge-Kutta method has often been used to solve the governing equations. While ballistic models of sprinkler irrigation were proposed decades ago, the evaluation of their adequacy, their calibration, the optimization of their execution time, and their application to environment-wise irrigation scheduling remain active fields of research. In this paper, recent advances on these issues are presented. The characterization of sprinkler drops is first presented through a photographic method. In a second step, an algorithm is presented to improve the quality of drop measurements produced with an optical spectropluviometer. In a third phase, a computer model is presented for the ballistic simulation of sprinkler irrigation performance. Next, the model is optimized for computational speed using the technique of Runge-Kutta pairs. Finally, the resulting model is applied to the problem of collective sprinkler irrigation scheduling. The use of mathematical models of sprinkler irrigation in combination with real time meteorological information and remote control of collective irrigation networks will result in relevant water conservation, increased water productivity and the generation of high technology jobs in the agricultural sector. All these benefits are required to ensure the sustainability of irrigated agriculture.