## 3 Method
### Inversely Solving V\({}_{\text{m25}}\) by Coupling the BESS and LRC
#### 3.1.2 Photosynthesis Rates of Sunlit and Shaded Leaves Estimated by LRC  
Light response curve is a commonly utilized tool to depict the correlation between the photosynthetic rate and photosynthetically active radiation (PAR). The rectangular hyperbola is employed to characterize the shape of the light response curve [78]:  
\[\text{A}=\frac{\alpha\text{IP}}{\alpha\text{I}+\text{P}} \tag{5}\]  
where A is the gross photosynthesis rate; \(\alpha\) is the maximum light use efficiency (LUE) obtained by the initial slope of the curve; I is the absorbed photosynthetically active radiation (APAR) of the leaf per unit; P is the gross photosynthetic rate under saturated radiation.  
The light response curves are employed to estimate \(\text{A}_{\text{sun}}\) and \(\text{A}_{\text{shade}}\), respectively [31; 33]. Then, the total canopy GPP of Equation (2) can be expressed as:  
\[\text{GPP}=\text{LAI}_{\text{sun}}\frac{\alpha\text{I}_{\text{sun}}\text{P}_ {\text{sun}}}{\alpha\text{I}_{\text{sun}}+\text{P}_{\text{sun}}}+\text{LAI}_{ \text{shade}}\frac{\alpha\text{I}_{\text{shade}}\text{P}_{\text{shade}}}{ \alpha\text{I}_{\text{shade}}+\text{P}_{\text{shade}}} \tag{6}\]  
where \(\text{I}_{\text{sun}}\) and \(\text{I}_{\text{shade}}\) are PAR absorbed by sunlit leaves and shade leaves (APAR\({}_{\text{Sun}}\) and APAR\({}_{\text{Sh}}\)), respectively, and are calculated using the two-leaf canopy radiative transfer model in BESS (Appendix B); and \(\text{P}_{\text{sun}}\) and \(\text{P}_{\text{shade}}\) are given by:  
\[\text{P}_{\text{sun}}=\frac{\text{P}_{0}\text{k}\left(1-\text{e}^{-\text{kLAI }-\text{k}_{\text{kLAI}}}\right)}{\left(\text{k}+\text{k}_{\text{n}}\right) \times\left(1-\text{e}^{\text{kLAI}}\right)} \tag{7}\]  
\[\text{P}_{\text{shade}}=\frac{\text{P}_{0}\big{(}\big{(}2\text{cos}\theta\text{ k}_{\text{n}}\text{k}-\left(\text{k}_{\text{n}}+\text{k}\right)\text{e}^{\text{kLAI}} \big{)}\text{e}^{-\text{kLAI}-\left(\left(2\text{cos}\theta\text{k}_{\text{n}} -1\right)\text{k}-\text{k}_{\text{n}}\right)}\big{)}}{\left(\text{k}_{\text{n}} ^{2}+\text{k}_{\text{n}}\text{k}\right)\left(\text{LAI}-2\text{cos}\theta\text{ }\big{(}1-\text{e}^{-\text{kLAI}}\big{)}\right)} \tag{8}\]where \(P_{0}\) is \(P\) on the top of the canopy; \(k_{n}\) describes the rate at which leaf nitrogen content decreases with increasing depth into the canopy, and is taken as 0.3 following the previous study [21].  
\[k=\frac{0.5\Omega}{\cos\theta} \tag{9}\]  
Here, \(\alpha\) and \(P_{o}\) represent the two unknown parameters, both of which are contingent on the biological conditions of the leaves. During periods of low incoming radiation, such as in the morning or afternoon, the photosynthetic rate is presumed to be primarily influenced by radiation, leading to the maximum LUE at these times. To determine \(\alpha\) for the day, GPP data are selected with incoming PAR less than 350 \(\mu\)mol m\({}^{-2}\)s\({}^{-1}\) for regression analysis against APAR. The resulting slope of this regression is considered as \(\alpha\) for the given day. The daily value of \(P_{0}\) is then obtained through the optimization of eddy covariance GPP data after determining \(\alpha\).