## 3 Materials and Methods
### Hydrological Model WATEN  
Developed in [PERSON] et al. [65], WATEN is a conceptual model that approaches water balances where streamflow data are not available. The model is calibrated with the energy data to pump the drainage discharge from an irrigation district to the river and was applied to the mixed-irrigation  
\begin{table}
\begin{tabular}{l l l l} \hline \hline \multicolumn{1}{c}{**Variable Description**} & \multicolumn{2}{c}{**PT-JPL-Thermal Equations**} & \multicolumn{1}{c}{**Reference**} \\ \hline Evaportranspiration & \(ET\ =\ ET_{c}+ET_{s}\) & [41] \\ \hline
**Canopy Transpiration** & \(ET_{c}\ =\ ET_{p}\cdot f_{g}f_{w}\cdot f_{l}\) & [41] \\ \hline
**Potential Canopy Transpiration** & \(ETp_{c}\ =\ a_{PT}\frac{A}{A+\gamma^{\top}}\left(R_{ac}-G\right)\) & [41] \\ \(\bullet\) Net Canopy Radiation & \(R_{u}\ =\ R_{a}-R_{a_{c}}\) & [41] \\ \(\bullet\) Net Soil Radiation & \(R_{u}\ =\ R_{a}\epsilon^{(-k_{max}\cdot LA)}\) & [59] \\ \(\bullet\) Net Radiation & \(R_{a}\ =\ R_{a}^{\downarrow}-\ R_{1}^{\downarrow}+\ R_{1}^{\downarrow}-R_{1}^{ \downarrow}\ =\ R_{1}+\ R_{S}\) & [41] \\
**Instant. In. Longwave Radiation** & \(R_{Lou}^{\downarrow}\ =\ \sigma^{\prime}\left(T_{ap_{\text{success}- \text{max}}}+273.15\right)^{\star}\left[1-\left(\epsilon c^{(-d\ (T_{\text{success}- \text{max}}))}\right)\right]\) & [61] \\ _Air Temperature at MODIS pass-time_ & \(T_{ap_{\text{success}-\text{max}}}\ =\ \left(T_{\text{max}}-T_{\text{min}}\right) \cdot sin\left(\frac{\pi\cdot\pi}{N+2}\right)+T_{\text{min}}\) & [62] \\ _Number of Hours from T\({}_{\text{min}}\) until Sunset_ & \(m\ =\ MODIS_{Time}-\left(12-\left(\frac{N}{2}\right)+c\right)\) & [62] \\
**Instant Out. Longwave Radiation** & \(R_{Lou}^{\uparrow}\ =\ -\epsilon_{S}\sigma\cdot LST^{\star}\) & [52] \\ \hline Daily Shortwave Radiation & \(R_{S_{avg}}\ =\ R_{S_{avg}}^{\downarrow}\left(1-\alpha\right)\) & [52] \\ _Albedo_ & \(\alpha\ =\ 0.8\cdot n_{\text{SA}}+0.2\cdot n_{\text{WSA}}\) & [63] \\ _Instant. Shortwave Radiation_ & \(R_{S_{avg}}\ =\ \frac{k_{avg}}{f}\frac{2\alpha}{N}\) & [51] \\ _Conversion Factor Day-inst_ & \(I\ =\ \frac{2}{sin\left(\frac{\pi}{\pi}\right)}\) & [51] \\ _Instantaneous Net Radiation_ & \(R_{R_{max}}\ =\ R_{Lou}+R_{S_{max}}\) & [53] \\ \hline
**Canopy Transpiration Constraints** & \(\bullet\) Green Canopy Fraction & \(f_{g}\ =\ f_{APAR}/f_{IPAR}\) & [41] \\ \(\bullet\)**Plant Moisture Constraint** & \(f_{m}\ =\ \frac{f_{data}}{f_{data}}\) & [41] \\ \(\bullet\)**Plant Temperature Constraint** & \(f_{t}\ =\ 1.1814/\left[\left(1+\theta^{2.2\left(T_{pr}-10\cdot T_{m}\right)}\right) \right]/\left[\left(1+\theta^{2.3\left(-T_{pr}-10\cdot T_{m}\right)}\right) \right]\) & [52] \\ \hline
**Soil Evaporation** & \(ET_{S}\ =\ ET_{p}\cdot f_{gm}\) & [41] \\ \hline
**Potential Soil Evaporation** & \(ETp_{t}\ =\ a_{PT}\frac{A}{A+\gamma^{\top}}\left(R_{u_{t}}-G\right)\) & [41] \\ _Soil Evaporation Constraints_ & \(\bullet\) Soil Moisture Constraint & \(f_{gm}\ =\ \frac{AT-AT_{min}}{AT_{max}-AT_{max}}\) & [42] \\ _Apparent Thermal Inertia_ & \(AT\ =\ C\cdot\frac{1-4}{\left(\frac{1-4}{\left(\frac{1-4}{\left(\frac{1-4}{ \left(\frac{1-4}{\left(\frac{1-4}{\left(\frac{1-4}{\left(\frac{1-4}{ \left(\frac{1-4}{\left(\frac{1-4}{\left(\frac{1-4 BXII from 2002 to 2012. The accuracy of WATEN fitted the ideal Theil's proportion distribution for good model performance and reached a correlation coefficient up to \(\rho\) = 0.93 and a coefficient of efficiency \(e_{2}\) = 0.90. WATEN requires climatic, crop-soil, and farm management data and relies on the water balance (Equation (2)) to solve \(ET\) and drainage (\(D\)) based on changes in soil moisture (\(S\)) in mm/month. Considering \(\Delta t\) = 1 month, we obtain the equation below. \[P_{i}+I_{i} = ET_{i}+D_{i}+S_{i}-S_{i-1} \tag{2}\]  
where \(P_{i}\) is the precipitation in month \(i\), \(I_{i}\) is irrigation, \(D_{i}\) is drainage, and (\(S_{i}-S_{i-1}\)) is soil moisture variation between one month and the preceding.