## 3 Methods
### Product Algorithm  
The NI-LUE model is based on the LUE model considering temperature and water, and further integrates the effect of CO\({}_{2}\) fertilization, radiation components and canopy N concentrations. The schematic workflow of the NI-LUE model was shown in Figure 2 and GPP can be estimated as follows:  
\[ \begin{split} GPP=(p\times[NI]+q)\times f(T)\times f(W)\times f( CO_{2})\\ \times[f(PAR_{su})\times APAR_{su}+f(PAR_{sh})\times APAR_{sh}] \end{split}  \tag{2}\]  
where \(p\) and \(q\) are parameters optimized for different vegetation types; [\(NI\)] represents the vegetation index selected to characterize the canopy N concentrations. This study collected the vegetation indices used in previous studies to estimate canopy N (Table 1). These indices were used separately to drive the model and to obtain the optimal vegetation index that would result in the best model performance. Figure 1: Optimum temperature calculation based on a satellite SIF pixel. Figure 2: Datasets and workflow of NI-LUE model to calculate GPP. \(f\) (\(T\)) and \(f\) (\(W\)) are the scalars of temperature and water, respectively. \(f\) (\(T\)) was calculated according to the Terrestrial Ecosystem Model (TEM) as the following formula [26]:  
\[f(T)=\frac{(T_{a}-T_{min})(T_{a}-T_{max})}{(T_{a}-T_{min})(T_{a}-T_{max})-\big{(}T _{a}-T_{opt}\big{)}^{2}} \tag{3}\]  
where \(T_{a}\) represents the air temperature (\({}^{\circ}\)C). T\({}_{\rm opt}\), T\({}_{\rm min}\) and T\({}_{\rm max}\) are the optimum, minimum, and maximum air temperature for vegetation growth, respectively. In this model, T\({}_{\rm opt}\) was the global \(T_{opt}^{exo}\) calculated in Section 3.1. \(T_{min}\) and \(T_{max}\) were set as shown in Table 2. For \(f\) (\(W\)), we described it using the actual evapotranspiration (\(E\)) and the potential evapotranspiration (\(E_{p}\)) as follows:  
\[f(W)=0.5+0.5\bigg{(}\frac{E}{E_{p}}\bigg{)} \tag{4}\]  
where \(E\) was calculated based on the revised Penman-Monteith model. Since the Penman-Monteith model requires the canopy resistance of different vegetation types under sufficient water conditions [65; 66], we used the Priestley-Taylor equation for \(E_{p}\) calculations [67; 68; 69; 70; 71]. The detailed calculation method can be found in [PERSON] et al. [67]. The effect of CO\({}_{2}\) fertilization (\(f\) (CO\({}_{2}\))) was integrated into our model with reference to the revised EC-LUE model [26] as follows:  
\[f(CO_{2})=\frac{C_{i}-\varphi}{C_{i}+2\varphi} \tag{5}\]  
\[C_{i}=[CO_{2}]\times\chi \tag{6}\]  
where \(\varphi\) is the CO\({}_{2}\) compensation point in the absence of dark respiration, which was set for different vegetation types according to [PERSON] et al. [26] as shown in Table 2; \(C_{i}\) indicates the internal leaf CO\({}_{2}\) concentrations and was calculated using the product of the atmospheric CO\({}_{2}\) concentrations ([CO\({}_{2}\)]) and \(\chi\), which represents the ratio of the \(C_{i}\) and [CO\({}_{2}\)] which can be calculated as follows:  
\[\chi=\frac{\xi}{\xi+\sqrt{VPD}} \tag{7}\]  
\[VPD=SVP\times(1-RH) \tag{8}\]  
\[SVP=0.6112\times e^{\frac{VZ\xi\gamma_{L}}{1+2\xi\gamma_{L}}} \tag{9}\]  
Figure 2:0  
Figure 2:1  
where parameter \(\xi\) represents the sensitivity of \(VPD\) to \(\chi\); \(K\) is the Michaelis-Menten coefficient of Rubisco; \(\eta^{*}\) is the viscosity of water relative to its value at 25 \({}^{\circ}\)C.  
Figure 2:2  
Figure 2:3  
Figure 2:4  
where \(P_{o}\) is the partial pressure of O\({}_{2}\), approximated as 21,278.25 Pa; \(K_{c}\) and \(K_{o}\) are the Michaelis-Menten constants of CO\({}_{2}\) and O\({}_{2}\), respectively; \(T_{K}\) is the air temperature with unit K; and \(R\) is the molar gas constant and is set to 8.314 J/mol/K. We further considered the effect of radiation components. Since the differences between the LUE of sunlit and shaded leaves are mainly influenced by light intensity, their \(\varepsilon_{max}\) should be similar [3; 72].