## 3 Materials and Methods
### Validation of the PT-JPL-Thermal ET  
RS \(ET\) estimations derived from (i) the 2-sources PT-JPL-thermal model and (ii) the global 2-sources \(ET\) product MOD16 were compared against WATEN \(ET\) calibrated data in the mixed-irrigation BXII from 2003 to 2012 where WATEN outputs were available. Monthly \(ET\) estimations as well as the average seasonal (monthly) and inter-annual \(ET\) dynamics were evaluated and compared. In this work, \(ET\) has been expressed in mm/day because of the importance in this region of irrigation in which \(ET\) is usually given per mm/day. This value is calculated by dividing monthly \(ET\) by the number of days in the month.  
Table 2 illustrates the inputs and parameters required to derive \(ET\) from the three methods. Potential \(ETp\) data series from PT-JPL-thermal and MOD16 \(ET\) were further compared against \(ETp\) at the agro-climatic station at (BXII), Lebrija I.  
The performance of the PT-JPL-thermal and MOD16 \(ET\) models was evaluated through the Pearson correlation coefficient \(\rho\) (Equation (6)), \(p\) probability value, the Mean Absolute Error (\(MAE\)) (Equation (7)), the bias (Equation (8)), and the Root Mean Square error (\(RMSE\)) (Equation (9)).  
\[\rho_{x,y} = \frac{cov(x,y)}{\sigma_{y}\cdot\sigma_{x}} \tag{6}\]  
\[MAE = \frac{1}{n}\sum_{i=1}^{n}|y_{i}-x_{i}| \tag{7}\]  
\begin{table}
\begin{tabular}{c c c c} \hline  & WATEN & \(PT\)-JPL-Thermal & **MOD16** \\ \hline \multicolumn{2}{c}{**Inputs**} & & \\ \hline \multicolumn{2}{c}{**\(\bullet\)**RS Data**} & & \(LAI,f_{APAR}\) (MOD15A2) & \(LAI,f_{APAR}\) (MOD15A2) \\ \multicolumn{2}{c}{} & & Broadband \(\alpha\) (MCD43B3) & Broadband \(\alpha\) (MOD43C1) \\ \multicolumn{2}{c}{} & & \(NDVI\) (MOD13A2) & \(EVI\) (MOD13A2) \\ \multicolumn{2}{c}{} & & Day \(LST,\epsilon_{S},T_{APAR  
\[bias = \frac{1}{n}\sum_{i=1}^{n}\left(y_{i}-x_{i}\right) \tag{8}\] \[RMSE = \sqrt{\frac{1}{n}.\sum_{i=1}^{n}\left(y_{i}-x_{i}\right)^{2}} \tag{9}\]  
where cov (\(x\),\(y\)) is the covariance between the observed \(ET\) values \(x_{i}\) and the estimated RS \(ET\) values, \(y_{i}\); \(\sigma_{x}\) and \(\sigma_{y}\) are the standard deviation of the variables, and (\(y_{i}~{}-~{}x_{i}\)) the estimated RS model error.  
The coefficient of efficiency (\(e_{2}\)) [67] was used to determine the relative magnitude of the residual variance compared to the measured data variance [68] (Equation (10)).  
\[e_{2} = 1-\frac{\sum_{i=1}^{n}\left(y_{i}-x_{i}\right)^{2}}{\sum_{i=1}^{n} \left(x_{i}-\overline{x}\right)^{2}} \tag{10}\]  
In addition, [PERSON]'s inequality decomposition was useful for breaking down the error in monthly \(ET\) rates into three different characteristic sources: bias (\(u_{m}\)) in which large values indicate a systematic error, variance (\(u_{s}\)) in which large values indicate large difference in the fluctuation of the series, and covariance (\(u_{c}\)) in which low values indicate unsystematic errors due to randomness [69; 70] (Equation (11)).  
\[1 = u_{m}+u_{s}+u_{c} \tag{11}\] \[= \frac{n.\left(\overline{y}-\overline{x}\right)^{2}}{\sum_{i=1}^{ n}\left(y_{i}-x_{i}\right)^{2}}+\frac{\left(n-1\right).\left(\sigma_{y}-\sigma_{x} \right)^{2}}{\sum_{i=1}^{n}\left(y_{i}-x_{i}\right)^{2}}\] \[+\frac{2\left(n-1\right).\left(1-\rho_{x}y\right).\sigma_{y}. \sigma_{x}}{\sum_{i=1}^{n}\left(y_{i}-x_{i}\right)^{2}}\]  
According to [PERSON] and [PERSON] [69], [PERSON]'s inequality components distribution is close to the optimal value when the sum of all three components follows the ideal distribution \(u_{m}\cong u_{s}\cong 0\) and \(u_{c}\cong 1\). Otherwise, undesirable occurrences would point to a revision of the evaluated model.