## Appendix 3 Separating Pure Canopy PAR Reflection from the Total Reflected PAR  
We assume that the canopy is macro-homogeneous and unlimited large, and canopy reflectance and canopy transmittance are all isotropic. Any kind of gap is allowed in our simplified canopy model, including gaps between tree crowns, within tree crowns, between leaves, or even between cellular gaps. We attribute the canopy transmission of PAR to the effect of all these gaps, without considering canopy scattering. The possible error due to this [54] is regarded as minor for snow-free conditions. An incoming PAR beam onto the canopy undergoes multiple transmission and reflection between canopy and ground (Figure 10).  
The downward-looking sensor above the canopy measures:  
\[RPAR_{eco}=R_{0}+R_{1}+R_{2}+R_{3}+\cdots\]  
\[=PAR_{0}\cdot rPAR_{c}+PAR_{0}\cdot rPAR^{2}\cdot rPAR_{g}+PAR_{0}\cdot rPAR ^{2}\cdot rPAR_{g}\cdot\left(rPAR_{c}\cdot rPAR_{g}\right)\]  
\[+PAR_{0}\cdot rPAR^{2}\cdot rPAR_{g}\cdot\left(rPAR_{c}\cdot rPAR_{g}\right)^ {2}+\cdots\]  
\[=PAR_{0}\cdot[rPAR_{c}+\frac{\tau PAR^{2}\cdot rPAR_{g}}{1-rPAR_{c}\cdot rPAR _{g}}],\]  
where \(rPAR_{c}\) is pure canopy reflectance, \(\tau PAR\) is the \"true\" canopy transmittance, and \(rPAR_{g}\) is the \"true\" ground reflectance.  
Thus:  
\[rPAR_{eco}=rPAR_{c}+\frac{\tau PAR^{2}\cdot rPAR_{g}}{1-rPAR_{c}\cdot rPAR_{g}}. \tag{13}\]  
The upward-looking sensor below canopy measures:  
\[TPAR_{m}=T_{0}+T_{1}+T_{2}+\cdots\]\[=PAR_{0}\cdot\tau PAR+PAR_{0}\cdot\tau PAR\cdot(rPAR_{g}\cdot rPAR_{c})+PAR_{0} \cdot rPAR\cdot(rPAR_{g}\cdot rPAR_{c})^{2}+\cdots\]  
\[=PAR_{0}\cdot\frac{\tau PAR}{1-rPAR_{c}\cdot rPAR_{g}}.\]  
Thus, the measured transmittance of PAR has the following relation with the \"true\" PAR transmittance:  
\[\tau PAR_{m}=\frac{\tau PAR}{1-rPAR_{c}\cdot rPAR_{g}}.\] (A14)  
The downward-looking sensor below canopy measures ground reflection:  
\[RPAR_{g\_m}=Rg_{0}+Rg_{1}+Rg_{2}+\cdots\]  
\[=PAR_{0}\cdot\tau PAR\cdot rPAR_{g}+PAR_{0}\cdot\tau PAR\cdot(rPAR_{g}\cdot rPAR _{c})\cdot rPAR_{g}+PAR_{0}\cdot\tau PAR\]  
\[\cdot(rPAR_{g}\cdot rPAR_{c})^{2}\cdot rPAR_{g}+\cdots\]  
\[=PAR_{0}\cdot\frac{\tau PAR\cdot rPAR_{g}}{1-rPAR_{c}\cdot rPAR_{g}}.\]  
Thus, the measured ground reflectance of PAR has the following relation with the \"true\" ground reflectance of PAR:  
\[rPAR_{g\_m}=\frac{RPAR_{g\_m}}{TPAR_{m}}=rPAR_{g},\] (A15)  
and the measured ground reflectance is the same as the \"true\" reflectance. From Equations (A13-A15), we get the expressions of pure canopy reflectance \(rPARc\) and \"true\" canopy transmittance \(\tau PAR\):  
\[rPAR_{c}=\frac{rPAR_{eco}-\tau PAR_{m}^{2}\cdot rPAR_{g}}{1-\tau PAR_{m}^{2} \cdot rPAR_{g}^{2}},\] (A16)  
\[\tau PAR=\tau PAR_{m}\cdot\big{(}1-rPAR_{c}\cdot rPAR_{g}\big{)}.\] (A17)  
The canopy intercepted PAR is:  
\[IPAR=PAR_{0}-T_{0}+(Rg_{0}-R_{1})+(Rg_{1}-R_{2})+\cdots\]  
\[=PAR_{0}-T_{0}+(Rg_{0}+Rg_{1}+\cdots)-(R_{1}+R_{2}+\cdots)\]  
\[=PAR_{0}(1-\tau PAR)+PAR_{0}\cdot\frac{\tau PAR\cdot rPAR_{g}}{1-rPAR_{c} \cdot rPAR_{g}}-PAR_{0}\cdot\frac{\tau PAR^{2}\cdot rPAR_{g}}{1-rPAR_{c} \cdot rPAR_{g}}.\]  
Finally, we get the \(fIPAR\):  
\[fIPAR=(1-\tau PAR)\cdot\big{(}1+\tau PAR_{m}\cdot rPAR_{g}\big{)}.\] (A18)