## 2 Background
### Irrigation Demand  
The daily irrigation demand is determined using a modification of the daily periods used by [PERSON] et al. (2016) to the Jensen relative yield equation ([PERSON], 2016):  
\[Y_{r}=\frac{Y_{a}}{Y_{max}}=\prod_{i=1}^{N}\prod_{j=1}^{M_{i}}\left[1-K_{y}(i) \cdot\left(1-\frac{ET_{a}(j)}{ET_{p}(j)}\right)\right]^{1/M_{i}} \tag{6}\]  
where \(Y_{a}\) is the actual yield, \(Y_{max}\) is the maximum yield, \(N\) is the number of growth stages, \(K_{y}(i)\) is a response factor in the absence of water for each of the stages of the crop, and \(M_{i}\) is the duration of each growth stage in days. To obtain \(Y_{r}=1\) where \(Y_{a}=Y_{max}\), the actual evapotranspiration of the crop must be equal to the potential, i.e., \(ET_{a}(j)=ET_{p}(j)\). Evaptranspiration calculations are obtained from the reference evapotranspiration \(ET_{o}\) as measured in mm/day according to the Penman--Monteith equation ([PERSON], 1965):  
\[ET_{o}(k)=\frac{0.408\Delta(k)\cdot\left(R_{n}(k)-G(k)\right)+\gamma\frac{900} {T(k)+273}u_{2}(k)\cdot\left(e_{s}(k)-e_{a}(k)\right)}{\Delta(k)\cdot+\gamma(1 +0.34u_{2}(k))} \tag{7}\]  
where \(G\) is the thermal flux density in MJ/m\({}^{2}\) day, \(R_{n}\) is the net radiation in MJ/m\({}^{2}\) day, \(T\) is the average daily temperature in \({}^{\circ}\)C, \(u_{2}\) is the average wind speed at a height of 2 m in m/s, \(e_{s}\) is the saturation vapor pressure in kPa, \(e_{a}\) is the actual vapor pressure in kPa, \(\Delta\) is the slope of the vapor saturation pressure with temperature in kPa \({}^{\circ}\)C, and \(\gamma=c_{p}P/0.622\Delta\) is the psychometric constant in kPa \({}^{\circ}\)C, where \(c_{p}\) is the specific heat at a constant pressure in kJ/kg \({}^{\circ}\)C, \(\lambda\) is the latent heat of vaporization in MJ/kg, and \(P\) is the atmospheric pressure in kPa. From \(ET_{o}\), the potential evapotranspiration (\(ET_{p}\)) in which there are no irrigation restrictions on the crop is obtained as ([PERSON], 2016):  
\[ET_{p}(k)=K_{c}(k)\cdot ET_{o}(k), \tag{8}\]  
where \(K_{c}\) is a dimensionless coefficient that depends on the crop and its growth stage.  
If there are no water limitations for irrigation, the actual evapotranspiration is equal to the potential (\(ET_{a}(k)=ET_{p}(k)\)). Otherwise, the lowest available water content in the land implies that \(ET_{a}(k)<ET_{p}(k)\). The generated stress is quantified with the coefficient \(K_{s}(k)\), which relates the supplied irrigation water with the crop yield for day \(k\)([PERSON], 2016):\[ET_{a}(k)= K_{s}(k)\cdot ET_{p}(k)=K_{s}(k)\cdot K_{c}(k)\cdot ET_{o}(k), \tag{9a}\] \[K_{s}(k)= \begin{cases}\frac{TAW(k)-D_{r}(k)}{TAW(k)-RAW(k)}&\text{if }RAW(k)<D_{r}(k),\\ 1&\text{otherwise},\end{cases}\] (9b) \[TAW(k)= 1.000\cdot(\theta_{FC}-\theta_{WP})\cdot Z_{r}(k),\] (9c) \[D_{r}(k)= D_{r}(k-1)-P(k)+RO(k)-I(k)-CR(k)+DP(k)+ET_{p}(k),\] (9d) \[D_{r}(0)= 1.000\cdot(\theta_{FC}-\theta_{0})\cdot Z_{r}(0),\] (9e) \[I(k)= 1,000\cdot\frac{\tilde{V}_{r,c}(k)}{A_{c}},\] (9f) \[0\leq D_{r}(k)\leq TAW(k), \tag{9g}\]  
where \(TAW\) is the total available water from the soil in mm, \(D_{r}\) is the root zone depletion in mm, \(RAW\) is the fraction of \(TAW\) for crops that can be extracted from the soil without suffering stress (fulfilling \(RAW=p\cdot TAW\)), \(\theta_{FC}\) is the water content that the soil naturally retains in m\({}^{3}\cdot\) m\({}^{-3}\), \(\theta_{WP}\) is the permanent wilting point in m\({}^{3}\cdot\) m\({}^{-3}\), \(Z_{r}\) is the depth of roots in m, \(P\) is the rainfall in mm, \(RO\) is the runoff loss in mm, \(I\) is the irrigation in mm, \(CR\) is the absorption of water by capillarity in mm, and \(DP\) is the water lost by filtration. As the water system shown in Figure 1 uses the yield irrigation requirements, the irrigation \(I\) is obtained from the required irrigation volume \(\tilde{V}_{r,c}\) per crop in m\({}^{3}\) and its acreage \(A_{c}\) in m\({}^{2}\).