Optimizing Irrigation Depth Using a Plant Growth Model and Weather Forecast

Numerical models of crop response to irrigation and weather forecasts with internet access should be fully utilized in modern irrigation management. In this respect, we developed a new numerical scheme to optimize irrigation depth that maximizes net income. Net income was calculated as a function of cumulative transpiration over irrigation interval which depends on irrigation depth. To evaluate this scheme, we carried out a field experiment for groundnut (Arachis hypogaea L.) grown in a sandy field of the Arid Land Research Center, Tottori University, Japan. Two treatments were established to compare the net income of the proposed scheme with that of an automated irrigation system. Results showed that although the proposed scheme gave a larger amount of seasonal irrigation water 28%, it achieved 2.18 times of net income owing to 51% higher yield compared to results of the automated irrigation system. This suggests that the proposed scheme would be more economical tool than automated irrigation systems to optimize irrigation depths.


Introduction
Irrigation is a vital factor for agriculture in both arid and semi-arid regions.Even in the humid and sub-humid regions, it is essential for rain-fed crops during drought periods when rainfall fails to provide sufficient moisture for stabilized crop production (Debaeke & Aboudrare, 2004).Approximately, 70% of global water resources are used for irrigation (WWAP, 2012).By 2050, the global population is forecasted to reach 9 billion (United Nations, 2007); therefore, the world needs to produce at least 50% more food (World Bank 2016).This gives the irrigation a great challenge in the coming decades to satisfy world's requirements from food, particularly in countries with limited water resources.
To manage irrigation more efficiently, both frequency and amount of watering must be determined carefully.With adopting computer and electronic technologies in agriculture, farmers may schedule irrigation water more efficiently.Mbabazi et al. (2017) used an average of the previous 5-day crop evapotranspiration to develop an irrigation scheduling for Avocado using technology of mobile irrigation applications.Yet irrigation scheduling is more efficient if methodologies of soil water sensors are used (Irrigation Association, 2011).Consequently, automated irrigation systems with sensors are widely used to meet crop water needs more precisely (Cancela et al., 2015;Osroosh et al., 2015).Liang et al. (2016) used data of soil water tension from wireless soil moisture sensors and the van Genuchten model (van Genuchten, 1980) to schedule irrigation water.Stirzaker et al. (2017) used electronic detectors for wetting front of infiltrated irrigation water through the soil profile to close a solenoid valve at a certain value to manage irrigation water.Those technologies, however, require high initial investment; therefore, the foundation of cheap technologies will encourage farmers to save irrigation water.For example, numerical simulation of water flow and crop growth can be utilized as a substitute for sensing drought stress.reference evapotranspiration.Delgoda et al. (2016) used weather forecasts and AquaCrop model (Steduto, Hsiao, Raes, & Fereres, 2009) to validate their framework that based on model predictive control to minimize both root zone soil moisture deficit and irrigation depth under water scarcity conditions.Combination of a multi-objective function and weather forecasts were used to give users a choice of optimal yield-irrigation combinations (Linker & Sylaios, 2016).This optimization was based on the end of seasonal yield and irrigation.Wang and Cai (2009) used a genetic algorithm (GA) to schedule irrigation water assuming perfect weather forecasts for either non-overlapping two weeks or the entire growing season.
Irrigation scheduling is generally targeted to improve water use efficiency; however, it is worth to consider net income as well.Concerning the economic benefits in relation to irrigation water, Yang et al. (2017) developed a flexible irrigation scheduling decision support system using fuzzy programming and interval optimization approaches.They used four multiple objective functions with different purposes (1) to maximize the gross economic profit; (2) to maximize the net economic profit; (3) to maximize the economic benefits per unit acreage of cultivated land; and (4) to maximize the economic benefits per unit cubic meter of irrigation water supply.Note that those functions were based on uncertain data of crop evapotranspiration; that would be a major constraint of that model.Moreover, Wang and Cai (2009) developed an optimization framework combined the SWAP model (Van Dam et al., 1997) and the GA to search for both irrigation dates and depths that maximize profits.They calculated net income for the entire season based on seasonal yield and seasonal fixed irrigation cost.
Water scarcity threatens the future of world food; therefore, governments typically set a price on water to motivate farmers to save irrigation water.Bozorg-Haddad et al. ( 2016) estimated farmer's response to the price of agricultural water.No effect on water use was found under low prices compared to non-priced water.Fujimaki et al. (2015) developed an optimization scheme to determine irrigation depths that maximize net income at fixed irrigation interval.That scheme was incorporated into a two-dimensional model of water, solute, and heat movement in soils (WASH_2D).To test that method, they carried out two preliminary field experiments in two different locations, soils and crops.The first experiment was carried out at the Institute des Régions Arides (IRA), Medenine, Tunisia, during 2011-2012.The measured crop was barley (Hordeum vulgar L. cv.Ardhaui) grown in loamy sand soil.The second experiment was carried out at the Arid Land Research Center, Tottori University, Japan, in 2013; the measured crop was sweet corn (Zea mays, cv.Amaenbou86) grown in sandy soil.Results of those experiments, however, are not satisfied to validate that scheme; it still needs more field experiments under different combinations of climate, soil, and crop to give users more confidence in its effectiveness.The objective of this paper, therefore, was to evaluate the optimization scheme to determine irrigation depth that maximizes net income using a major crop, groundnut.The specific goal was to replace capital-intensive automated irrigation methods with a low-cost scheme based on freely available weather data and numerical simulation.

The Process Model
A two-dimensional physically based model, WASH_2D was used.It can simulate water, solute, and heat movement in soils with the finite difference method.It includes a module for simulating root water uptake and crop growth.This software is freely distributed with source code under a general public license from the website of the Arid Land Research Center, Tottori University (http://www.alrc.tottori-u.ac.jp/fujimaki/download/WASH_2D).A detailed description of the model was informed by Fujimaki et al. (2015).

Maximization of Net Income
Net income, I n ($ ha -1 ) was calculated at each irrigation interval in proportion to the increment in dry matter attained during the irrigation interval (Fujimaki et al., 2015): where, P c is the producer's price of crop ($ kg -1 DM), ε is transpiration productivity of the crop ((produced dry matter (kg ha -1 ) divided by cumulative transpiration (kg ha -1 )), τ i is cumulative transpiration during the interval (kg ha -1 ), k i is the income correction factor, P w is the price of water ($ kg -1 ), W is the irrigation depth (1 cm = 100 000 kg ha -1 ), and C ot is other costs ($ ha -1 ).
In Equation (1), the income correction factor was considered to avoid possible underestimation for the contribution of initial transpiration to the entire quantum of growth; because transpiration in the initial growth stage is smaller than that in later stages.Estimation of k i was suggested in Fujimaki et al. (2015).
The transpiration rate, (cm s -1 ), was calculated by integrating the water uptake rate, S, over the root zone: where, L x and L z are width and depth of root zone.A macroscopic root water uptake model (Feddes & Raats, 2004) was used to predict the water uptake rate, S (cm s -1 ): where, T p , α w and β are potential transpiration (cm s -1 ), reduction coefficient of root water uptake and normalized root density distribution, respectively.The T p was calculated by multiplying reference evapotranspiration by basal crop coefficient, K c , as follows: where, E p is reference evapotranspiration (cm s -1 ), calculated by the Penman-Monteith equation (Allen, Pereira, Raes, & Smith, 1998).Since the crop coefficient is largely affected by growth stage, it was expressed as a function of transpiration as follows: where, a kc , b kc and c kc are fitting parameters.Estimated value of those parameters depends on each growth stage of the plant.Fujimaki et al. (2015) suggested their values by measuring cumulative transpiration rate via a weighing lysimeter.
The reduction of the water uptake rate, α is a function of drought and osmotic stresses; WASH_2D model uses so-called additive function as follows: where, ψ and ψ o are the matric and osmotic heads, respectively, and ψ 50 , ψ o50 , and p are fitting parameters (van Genuchten, 1987).
In this paper, the equation that describes the normalized root activity, β, is modified as follows: where, b rt is a fitting parameter; d rt and g rt are the depth and width of the root zone (cm), respectively; x is the horizontal distance; z is the soil depth; and z r0 is the depth below which roots exist (cm).In general, the roots of cultivated plants start from about 2.5 cm below the soil surface, therefore, we have added as a new parameter to make the model more realistic.The d rt was also expressed as a function of transpiration as follows: where, a drt , b drt and c drt are fitting parameters.By expressing both K c and d rt as functions of cumulative transpiration as independent variables instead of days after sowing, WASH_2D may express plant growth more dynamically responding to drought or salinity stresses.

Optimization of Irrigation Depth
To minimize repetition of numerical prediction in non-linear optimization, we used the following scheme proposed by Fujimaki et al. (2015).
First, it is assumed that cumulative transpiration rate at each irrigation interval may be empirically described as: where, a τ and b τ are fitting parameters and τ 0 is τ at W = 0. Note that even when W = 0, the plant can still uptake remained available water from the soil.
Second, maximum I n is achieved when the derivative of Equation (1) for W becomes zero: The values of a τ and b τ must be known; therefore, two additional points of transpiration at maximum (W max , τ max ) and intermediate (W mid , τ mid ) irrigation depths should be assessed:

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