# Modeling Actual Evapotranspirationiration with MSI-Sentinel Images and Machine Learning Algorithms
## 2 Materials and Methods
### Response Variable  
The response variable, evapotranspiration fraction (ET\({}_{\rm r}\)F), was obtained for each pixel, with a 30 m \(\times\) 30 m spatial resolution, from the METRIC model. ET\({}_{\rm r}\)F is reckoned by dividing instantaneous evapotranspiration (ET\({}_{\rm inst}\)) in each pixel by the hourly reference evapotranspiration (ET\({}_{\rm r}\)) given by the meteorological station. [PERSON] (1977) standardized ET\({}_{\rm r}\) for alfalfa at 0.5 m high. According to the authors, when using this condition, the ET\({}_{\rm r}\)F can be considered equivalent to the crop coefficient (K\({}_{\rm c}\)). It also enables the extrapolation of the crop's actual evapotranspiration when the satellite switches to the daily 24 h level. Thus, the ET\({}_{\rm r}\)F is determined by Equation (1). \[\rm{ETrF}=\frac{ET_{\rm inst}}{ET_{\rm r}} \tag{1}\]  
where ET\({}_{\rm inst}\) is the instantaneous evapotranspiration (mm\(\cdot\)h\({}^{-1}\)) and ET\({}_{\rm r}\) is the reference evapotranspiration (mm\(\cdot\)h\({}^{-1}\)) standardized to alfalfa at 0.5 m height at the moment the satellite is passing. ET\({}_{\rm inst}\) is calculated from the latent energy consumed in the evapotranspiration (ET) process and the latent heat of vaporization Equation (2). \[\rm{ET}_{\rm inst}=3600\frac{LE}{\lambda\rho_{w}} \tag{2}\]  
where 3600 is the conversion from seconds to hours of the duration of the satellite pass; LE is the latent energy consumed in ET (W m\({}^{-2}\)); \(\rho_{w}\) is the density of water (\(\sim\)1000 kg m\({}^{-3}\)); \(\lambda\) is the latent heat of vaporization (J kg \({}^{-1}\)). \(\lambda\) represents the heat absorbed when one kg of water is evaporated, and it is determined by Equation (3). \[\lambda=[2.50-0.00236(T_{\rm s}-273.15)]\times 10^{6} \tag{3}\]  
where T\({}_{\rm s}\) is the surface temperature (\({}^{\circ}\)K) determined by band 10 of the TIRS sensor (Table 1). LE is calculated from the Earth's surface energy balance, Equation (4), which involves net radiation (R\({}_{\rm n}\)), a sensible flux of heat transferred to the ground (G), and a sensible flux of heat convected to air (H). These three components, responsible for determining LE, are usually expressed in W m\({}^{-2}\). \[\rm{LE}=R_{\rm n}-G-H \tag{4}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline Date & \multicolumn{3}{c}{**Landsat-8**} & \multicolumn{2}{c}{**Sentinel-2**} \\ \cline{2-5} (mm/dd/aaaaa) & **Time (hh:mm:sss)** & **Path/Row** & **Time (hh:mm:sss)** & **Tile Number** \\ \hline \multicolumn{5}{c}{Training and Test} \\ \hline
07/06/2018 & 09:55:17.157 & 218/71 & 10:12:41.024 & T23 LPD \\
05/22/2019 & 09:58:49.714 & 218/71 & 10:12:51.024 & T23 LPD \\
10/04/2019 & 10:02:19.874 & 219/70 & 10:12:49.024 & T23 LPD \\
10/29/2019 & 09:56:34.656 & 218/71 & 10:12:49.024 & T23 LPD \\
01/17/2020 & 09:56:20.969 & 218/71 & 10:12:41.024 & T23 LPD \\
05/31/2020 & 10:01:27.427 & 219/70 & 10:12:49.024 & T23 LPD \\
08/19/2020 & 10:02:00.995 & 219/70 & 10:12:49.024 & T23 LPD \\ \hline \multicolumn{5}{c}{Residual analysis} \\ \hline
12/02/2020 & 09:56:32.117 & 218/71 & 10:12:41.024 & T23 LPD \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Characteristics of the eight images selected to develop the models. R\({}_{\rm n}\) is the radiant energy of the surface that is partitioned into H, G, and LE and is determined by Equation (5). \[{\rm R_{n}}{\rm=R_{s\downarrow}-\alpha R_{s\downarrow}+R_{L\downarrow}-R_{L \uparrow}-(1-\epsilon_{0})R_{L\downarrow}} \tag{5}\]  
where R\({}_{s\downarrow}\) is the input of shortwave radiation (W m\({}^{-2}\)); \(\alpha\) is the surface albedo (adim.) determined by bands 2, 3, 4, 5, 6, and 7 of the Landsat-8 (Table 1); R\({}_{L\downarrow}\) and R\({}_{L\uparrow}\) are the input and output of long waves (W m\({}^{-2}\)), respectively; and \(\epsilon_{0}\) is the surface thermal emissivity. G is the heat storage flux in the ground due to heat conduction. When vegetation is present, G tends to have lower values, which means that the rate of heat storage in the soil is lower.