## 2 Materials and Methods
### Methods
#### 2.3.2 GPP Estimation Model  
The original EC-LUE (Eddy Covariance Light Use Efficiency) model, belonging to the LUE model, was developed by [PERSON] et al. in 2007 [28] and is driven by inputting NDVI, PAR, air temperature, and the Bowen ratio. However, in this study, we used the revised EC-LUE model [53] derived by integrating the CO\({}_{2}\) fertilization effect on GPP and adding the limit of VPD to GPP. Compared with other LUE models, the LUE calculated from \(\varepsilon_{\mathrm{max}}\) (the maximum LUE) is only limited by the most limiting factor between air temperature and the atmospheric water demand according to [PERSON]'s law. The model estimates terrestrial ecosystem GPP as follows:  
\[\mathrm{GPP}=\mathrm{PAR}\times\mathrm{fPAR}\times\varepsilon_{\mathrm{max}} \times\mathrm{C_{s}}\times\mathrm{min}(\mathrm{T_{s}},\mathrm{W_{s}}) \tag{3}\]  
where PAR is the incident photosynthetically active radiation (MJ m\({}^{-2}\) day\({}^{-1}\)); fPAR is the fraction of PAR absorbed by plant canopy; \(\varepsilon_{\mathrm{max}}\) is the maximum LUE; and \(\mathrm{C_{s}}\), \(\mathrm{T_{s}}\), and \(\mathrm{W_{s}}\) are the regulation scalars for the CO\({}_{2}\) fertilization effect, the limitations from air temperature, and the atmospheric water demand on LUE, respectively.  
In EC-LUE, fPAR is calculated by a linear equation that describes the relationship with NDVI established by a set of empirical constants [28]:  
\[\mathrm{fPAR}=1.24\times\mathrm{NDVI}-0.168 \tag{4}\]  
For \(\mathrm{C_{s}}\),  
\[\mathrm{C_{s}}=\frac{\mathrm{C_{i}}-\Gamma^{*}}{\mathrm{C_{i}}+2\Gamma^{*}} \tag{5}\]  
where \(\mathrm{C_{i}}\) is the intercellular concentration of CO\({}_{2}\) (ppm); \(\Gamma^{*}\) is the CO\({}_{2}\) compensation point in the absence of dark respiration (ppm) depending on temperature. [PERSON] et al. (2001) estimated \(\Gamma^{*}\) through a biochemical rate parameter as follows:  
\[\Gamma^{*}=4.22\times e^{\frac{3780\times(\mathrm{T_{a}}-298.15)}{298138\mathrm{ H_{a}}}} \tag{6}\]  
where R is the molar gas constant (8.314 J mol\({}^{-1}\) K\({}^{-1}\)), \(\mathrm{T_{a}}\) is the air temperature (K), and 4.22 Pa is the photorespiratory point at 25 \({}^{\circ}\)C.  
\(\mathrm{C_{i}}\) is calculated through the ratio of leaf internal to ambient CO\({}_{2}\) (\(\chi\)) as follows:  
\[\mathrm{C_{i}}=\mathrm{C_{a}}\times\chi \tag{7}\]  
where \(\chi\) depends on \(\mathrm{T_{a}}\) and VPD as follows:  
\[\chi=\frac{\gamma}{\gamma+\sqrt{\mathrm{VPD}}},\ \mathrm{where}\ \gamma=\sqrt{\frac{356.51K}{1.6 \eta^{*}}} \tag{8}\]where \(\eta^{*}\) is the viscosity of water as a function of T\({}_{\rm a}\); \(K\) is the Michaelis-Menten coefficient of Rubisco, calculated as follows:  
\[K=K_{c}\left(1+\frac{P_{o}}{K_{o}}\right) \tag{9}\]  
\[K_{c}=39.97\times e^{\frac{79.43\times(\rm T_{a}-298.15)}{298.18\rm R_{a}}} \tag{10}\]  
\[K_{o}=27480\times e^{\frac{36.38\times(\rm T_{a}-298.15)}{298.15\rm R_{a}}} \tag{11}\]  
where \(K_{c}\) and \(K_{o}\) are the Michaelis-Menten coefficient of Rubisco for carboxylation and oxygenation, respectively, expressed in partial pressure units; \(P_{o}\) is the partial pressure of O\({}_{2}\) (ppm); and e is the base of the natural logarithm.  
The temperature and water dependency factor are defined by [53] as follows:  
\[\rm T_{s}=\frac{(T_{a}-T_{min})\times(T_{a}-T_{max})}{(T_{a}-T_{min})\times(T_{ a}-T_{max})-\left(T_{a}-T_{opt}\right)^{2}} \tag{12}\]  
\[\rm W_{s}=\frac{VPD_{0}}{VPD+VPD_{0}} \tag{13}\]  
where T\({}_{min}\) (0 \({}^{\circ}\)C), T\({}_{max}\) (40 \({}^{\circ}\)C), and T\({}_{opt}\) (20.33 \({}^{\circ}\)C) are the minimum, maximum, and optimum ambient temperature, respectively, for photosynthetic activity. VPD\({}_{0}\) is the empirical coefficient of the water demand constraint equation.