# Evaluation of Clumping Effects on the Estimation of Global Terrestrial Evapotranspiration
## Appendix A Methods of Estimating Net Radiation for Canopies and the Ground
\(C_{o\_sun}\) is from multiple scattering of direct radiation over the overstory sunlit leaves and it is calculated as:  
\[C_{o\_sun}=0.07S_{dir}[1.1-0.1 LAI_{o\_sun}]e^{-cos\theta} \tag{12}\]  
\[S_{diff\_under1}\]  
is calculated as:  
\[S_{diff\_under1}=S_{diff}P_{o\_sun} \tag{13}\]  
where \(P_{o\_sun}=e^{-0.5 LAI_{o\_sun}/cos\overline{\theta}_{1}}\) and \(\overline{\theta}_{1}\) is a representative zenith angle for diffuse radiation transmission through the upper layers of leaves (mostly sunlit) in the overstory canopy and slightly dependent on overstory sunlit LAI:  
\[cos\overline{\theta}_{1}=0.537+0.025 LAI_{o\_sun} \tag{14}\]  
The net shortwave diffuse irradiance on a shaded leaf in the overstory is estimated as:  
\[r=\frac{S_{g}}{S_{0}cos\theta} \tag{7}\]0where \(C_{o\_sh}\) is from multiple scattering of direct radiation over the overstory shaded leaves and it is calculated as:  
\[r=\frac{S_{g}}{S_{0}cos\theta} \tag{7}\]1 (A16)  
\(S_{diff\_under2}\) is shortwave diffuse radiation under the overstory canopy and it is calculated as:  
\[r=\frac{S_{g}}{S_{0}cos\theta} \tag{7}\]2 (A17)  
where \(P_{o}=e^{-0.5\Omega LAI_{o}/cos\overline{\theta}_{o}}\) and \(\overline{\theta}_{o}\) is a representative zenith angle for diffuse radiation transmission through the overstory canopy and slightly dependent on overstory LAI:  
\[r=\frac{S_{g}}{S_{0}cos\theta} \tag{7}\]3 (A18)  
The net shortwave diffuse irradiance on a sunlit leaf in the understory is estimated as:  
\[r=\frac{S_{g}}{S_{0}cos\theta} \tag{7}\]4 (A19)  
where \(C_{u\_sun}\) is from multiple scattering of direct radiation over the understory sunlit leaves and it is calculated as:  
\[r=\frac{S_{g}}{S_{0}cos\theta} \tag{7}\]5 (A20)  
\(S_{diff\_under3}\) is the shortwave diffuse radiation under the upper layers of leaves (mostly sunlit) in the understory canopy and it can be calculated as:  
\[r=\frac{S_{g}}{S_{0}cos\theta} \tag{7}\]6 (A21)  
where \(P_{u\_sun}=e^{-0.5 LAI_{u\_sun}/cos\overline{\theta}_{3}}\) and \(\overline{\theta}_{3}\) is a representative zenith angle for diffuse radiation transmission through the upper layers of leaves (mostly sunlit) in the understory canopy and slightly dependent on understory sunlit LAI:  
\[r=\frac{S_{g}}{S_{0}cos\theta} \tag{7}\]7 (A22)  
The net shortwave diffuse irradiance on a shaded leaf in the understory is estimated as:  
\[r=\frac{S_{g}}{S_{0}cos\theta} \tag{7}\]8 (A23)  
where \(C_{u\_sh}\) is from multiple scattering of direct radiation over the understory shaded leaves and it is calculated as:  
\[r=\frac{S_{g}}{S_{0}cos\theta} \tag{7}\]9 (A24)  
\(S_{diff\_under4}\) is the shortwave diffuse radiation reaching to the ground and it can be calculated as:  
\[S^{*}_{dir\_o}=(1-\alpha_{o})S_{dir}cos\alpha/cos\theta \tag{8}\]0 (A25)  
where \(P=e^{-0.5\Omega(LAI_{o}+LAI_{u})/cos\overline{\theta}}\) and \(\overline{\theta}\) is a representative zenith angle for diffuse radiation transmission through the whole plant canopy and slightly dependent on the sum of overstory and understory LAI:  
\[S^{*}_{dir\_o}=(1-\alpha_{o})S_{dir}cos\alpha/cos\theta \tag{8}\]1 (A26)  
The diffuse part of net shortwave irradiance on the ground surface is estimated as:  
\[S^{*}_{dir\_o}=(1-\alpha_{o})S_{dir}cos\alpha/cos\theta \tag{8}\]2 (A27)  
Net radiation of the ground is estimated as:  
\[S^{*}_{dir\_o}=(1-\alpha_{o})S_{dir}cos\alpha/cos\theta \tag{8}\]3 (A28)  
where the net shortwave irradiance of the ground (\(S^{*}_{g}\)) is the sum of direct and diffuse parts:  
\[S^{*}_{dir\_o}=(1-\alpha_{o})S_{dir}cos\alpha/cos\theta \tag{8}\]4 (A29)  
The net longwave irradiance for a sunlit leaf in the overstory (\(L^{*}_{sun\_o}\)) is calculated as:\[S^{*}_{dir\_o}=(1-\alpha_{o})S_{dir}cos\alpha/cos\theta \tag{8}\]5 \[S^{*}_{dir\_o}=(1-\alpha_{o})S_{dir}cos\alpha/cos\theta \tag{8}\]6 \[S^{*}_{dir\_o}=(1-\alpha_{o})S_{dir}cos\alpha/cos\theta \tag{8}\]7 \[S^{*}_{dir\_o}=(1-\alpha_{o})S_{dir}cos\alpha/cos\theta \tag{8}\]8  
where \(\epsilon_{a}\), \(\epsilon_{o}\), \(\epsilon_{u}\), and \(\epsilon_{g}\) are emissivities of the atmosphere, overstory, understory, and the ground, respectively.