# Performance of Satellite-Based Evapotranspiration
## 3 Methods
### Surface Energy Balance Models and Eapotranspiration Modelling  
The SEBS, SEBAL, METRIC and SSEBop models are all based on the surface energy balance (Equation (5)):  
\[R_{\rm n}=H+\lambda E+G_{0} \tag{6}\]  
where R\({}_{\rm n}\) is net radiation [W m\({}^{-2}\)], H is the sensible heat flux [W m\({}^{-2}\)], \(\lambda E\) to the latent heat flux [W m\({}^{-2}\)] and G\({}_{0}\) is the soil heat flux [W m\({}^{-2}\)], which is negligible for daily values [55]. The SEBS model uses meteorological and remote sensing data in order to estimate ET\({}_{\rm a}\) and evaporative fraction (\(\Lambda\)), which is adjusted to the sensible heat flux by the Monin-Obukhov similarity [56] and energy balance under dry and wet limits. \[\Lambda_{\rm r}=1-\frac{H-H_{\rm wet}}{H_{\rm dry}-H_{\rm wet}} \tag{7}\]  
where \(\Lambda_{\rm r}\) is the relative evaporative fraction, H\({}_{\rm wet}\) is sensible heat flux at the wet limit through a combination equation similar to the Penman-Monteith equation, and H\({}_{\rm dry}\) is the sensible heat flux at the dry limit condition (H\({}_{\rm dry}=R_{\rm n}-G_{0}\)). In this way, the latent heat flux was estimated as:  
\[\lambda E=\Lambda_{\rm r}\lambda E_{\rm wet} \tag{8}\]  
Daily ET\({}_{\rm a}\) [mm day\({}^{-1}\)] was estimated as:  
\[\rm ET_{a}=\frac{\lambda E}{\lambda}86400C_{di} \tag{9}\]  
where \(\lambda\) is the latent heat of vaporization and C\({}_{\rm di}\) is the ratio of daily and instantaneous net radiation [57]. SEBAL is a physical model which estimates H by the parametrization of the vertical difference between the aerodynamic temperature and the air temperature close to the surface (dT), assuming a linear relationship between T\({}_{\rm s}\) and dT. \[H=\rho C_{\rm p}\frac{dT}{\tau_{\rm ah}} \tag{10}\]  
To calibrate dT, it is necessary to select anchor pixels for each image under extreme conditions (dry and wet). The pixel of the dry limit is generally a bare agricultural field where H is maximum (\(\rm H_{hot}=R_{n}-G_{0}\)). The cold pixel can be found in water bodies or well-irrigated fields where \(\rm H_{cold}=0\). Therefore, \(\rm ET_{a}\) is estimated daily through Equation (8). METRIC is a variant of the SEBAL model. The main difference lies in the \"self-calibration\" done in each image, since METRIC applies on the 'cold-pixel', the \(\rm\lambda E\) estimation based on a reference ET over a well-watered alfalfa field (\(\rm ET_{r}\)) from hourly weather information. In this way just like SEBAL, \(\rm H\) is calibrated by correcting the buoyancy and parameterizing dT based on the anchor pixels for extreme conditions. Finally, SSEBop is a simplified surface energy balance model. It estimates \(\rm ET_{a}\) (Equation (10)) from \(\rm ET_{0}\), \(\rm\Lambda\) and a coefficient \(\rm k\) that scales the reference grass vegetation experiencing maximum ET by an aerodynamically rougher crop. \[\rm ET_{a}=\rm\Lambda kET_{0} \tag{11}\]  
where \(\rm k=1\), assuming a reference grass cover. The evaporative fraction was computed (Equation (11)) through \(\rm T_{s}\) and reference land surface temperature under dry and wet conditions (\(\rm T_{hot}\), \(\rm T_{cold}\)) based on \(\rm T_{s}\) and \(\rm T_{x}\). \[\rm\Lambda=\frac{T_{hot}-T_{s}}{dT}=\frac{T_{hot}-T_{s}}{T_{hot}-T_{cold}} \tag{12}\]  
dT was estimated using a reference value of \(\rm r_{ah}\) and maximum \(\rm H\) for bare soil surface, where \(\rm r_{ah}=110\) [s m\({}^{-1}\)] [58] and \(\rm H=R_{n}-G_{0}\). \[\rm dT=\frac{110R_{n}}{\rho_{a}C_{p}} \tag{13}\]  
The SEBS, SEBAL, METRIC and SSEBop models were evaluated by comparing the estimated pixels over the Oromo calibration site with in situ pasture evapotranspiration (\(\rm ET_{c}\)). \[\rm ET_{c}=k_{c}ET_{0} \tag{14}\]  
where \(\rm k_{c}\) is the crop coefficient, obtained from a linear function from Normalized Difference Vegetation Index (NDVI) from Landsat data over Oromo site.