## Appendix A Calculation of Auxiliary Variables for the TSEB Algorithm  
The \(\mathrm{r_{ah}}\) term is calculated using Equation (A1) as follows (as in [69]):  
\[\mathrm{r_{ah}}=\frac{\ln\left(\frac{\mathrm{Z_{u}-d}}{\mathrm{Z_{eh}}}\right) -\mathrm{\psi_{h}}}{\mathrm{u_{*}k}}\] (A1)  
where \(\mathrm{z_{u}}\) is the height of wind speed measurement (m); \(\mathrm{d}\) is the zero-plane displacement height (m); \(\mathrm{z_{oh}}\) is the roughness length for heat transfer (m); \(\mathrm{\psi_{h}}\) is the atmospheric stability correction function for heat transfer (dimensionless); \(\mathrm{k}\) is the von [PERSON] constant and is set to 0.41 [70; 71].  
The shear or friction velocity (Equation (A2)) is calculated as follows (as in [69]):  
\[\mathrm{u_{*}}=\frac{\mathrm{Uk}}{\ln\left(\frac{\mathrm{z_{u}-d}}{\mathrm{Z_ {om}}}\right)-\mathrm{\psi_{m}}}\] (A2)  
where \(\mathrm{u_{*}}\) is the mean shear velocity (m/s), \(\mathrm{\psi_{m}}\) is the atmospheric stability correction function for momentum transfer (dimensionless); \(\mathrm{z_{om}}\) is the roughness length for momentum transfer (m); and \(\mathrm{U}\) is the mean horizontal wind speed (m/s).  
The \(\mathrm{\psi_{h}}\) and \(\mathrm{\psi_{m}}\) are equal to zero for quasi-neutral atmospheric conditions. When thermal stratification exists, the Monin-Obukhov stability length (\(\mathrm{L_{MO}}\)) and theory is considered to correct the estimations of momentum and heat transfer [72]. For unstable (L\({}_{\rm MO}\) \(<\) 0) and stable (L\({}_{\rm MO}\) \(>\) 0) atmospheric conditions, Equations (A3) and (A4) present the models for the atmospheric stability corrections for heat transfer [73; 74; 75]:  
\[\psi_{h}=\left\{\begin{aligned} 2\times ln\left(\frac{1+x_{1}^{2}}{ 2}\right)-2\times\ln\left(\frac{1+x_{1}^{2}}{2}\right),&\ L_{ \rm MO}<0\\ -5\times\left(\frac{z_{u}-d}{L_{\rm MO}}\right),&\ L_{ \rm MO}>0\end{aligned}\right.\] (A3)  
\[x_{1}=\left[1-16\left(\frac{z_{u}-d}{L_{\rm MO}}\right)\right]^{0.25}\] (A4)  
where L\({}_{\rm MO}\) is the Monin-Obukhov stability length (m) and is calculated as indicated by Equation (A5) below:  
\[L_{\rm MO}=-\frac{u_{*}^{3}\ T_{a}\ p_{a}\ C_{pa}}{g\ k\ H}\] (A5)  
where g is the gravitational acceleration (\(\approx\)9.81 m/s\({}^{2}\)).  
For unstable and stable atmospheric conditions, Equation (A6) indicates the models for the atmospheric stability corrections for momentum transfer [73; 74; 75]:  
\[\psi_{m}=\left\{\begin{aligned} 2\ ln\left(\frac{1+x_{1}}{ 2}\right)+ln\left(\frac{1+x_{1}^{2}}{2}\right)-2\arctan(x_{1})+\frac{\pi}{2},&\ L_{\rm MO}<0\\ -5\times\left(\frac{z_{u}-d}{L_{\rm MO}}\right),&\ L_{ \rm MO}>0\end{aligned}\right.\] (A6)  
The roughness elements d, z\({}_{\rm om}\), and z\({}_{\rm oh}\) are calculated as indicated by Equations (A7)-(A9), respectively [76]:  
\[z_{om}=\left\{\begin{aligned} z_{o}^{\prime}+0.28\ h_{c}\ \sqrt{J},&\ 0\leq J\leq 0.20\\ 0.3\ h_{c}\Big{(}1-\frac{d}{h_{c}}\Big{)},&\ 0.20<J\leq 2 \end{aligned}\right.\] (A7)  
\[d=h_{c}\ \left[ln\Big{(}1+J^{\frac{1}{6}}\Big{)}+0.03\ ln\Big{(}1+J^{6} \Big{)}\right]\] (A8)  
\[z_{oh}=0.10\times z_{om}\] (A9)  
where z\({}_{0}^{\prime}\) is the roughness length of the soil surface (z\({}_{\rm o}^{\prime}\approx\) 0.01 m), and J is equal to 20% of LAI (m\({}^{2}\)/m\({}^{2}\)).