## 2 Materials and Methods
### Conductance Method  
Though the linear SIF-GPP relationship (Equation (3)) looks simple, the parameter k1 is not a constant in the real environment [42]. Previous work has also reported on the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Site** & **Longitude** & **Latitude** & **Period** & **Land Cover** & **Reference** \\ \hline \multirow{2}{*}{Daman (DM)} & \multirow{2}{*}{100.37\({}^{\circ}\) E} & \multirow{2}{*}{38.85\({}^{\circ}\) N} & Jun 2017–Sep 2017; & \multirow{2}{*}{Maize (C\({}_{4}\))} & [28; 29; 30] \\  & & & Jun 2018–Sep 2018 & & \\ \hline \multirow{2}{*}{Huailai (HL)} & \multirow{2}{*}{115.78\({}^{\circ}\) E} & \multirow{2}{*}{40.33\({}^{\circ}\) N} & Jul 2017–Oct 2017; & \multirow{2}{*}{Maize (C\({}_{4}\))} & [30; 31; 32] \\  & & Jul 2018–Oct 2018 & & & \\ \hline Niwot Ridge (NR) & 105.55\({}^{\circ}\) W & 40.03\({}^{\circ}\) N & Jun 2017–Jul 2018 & Evergreen needle leaf forest (C\({}_{3}\)) & [19; 33; 34] \\ \hline Harvard Forest (HF) & 72.17\({}^{\circ}\) W & 42.54\({}^{\circ}\) N & Jun 2013–Nov 2013 & Mixed temperate forest (C\({}_{3}\)) & [20; 35] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the sites used to build the models. hyperbolic relationship between SIF and GPP [43; 44]. The link between SIF and GPP is due to the close relationship between SIF and J as above-mentioned and J can be derived by SIF in [12]:  
\[\mathrm{J}=\mathrm{a}\times\mathrm{q}_{\mathrm{L}}\times\mathrm{SIF}\times \frac{1}{\Omega_{\mathrm{c}}} \tag{7}\]  
where a is an empirical factor supposed to be a constant for a specific ecosystem; \(\mathrm{q}_{\mathrm{L}}\) is the fraction of open Photosystem II reaction centers, indicating the \"traffic jam\" in the electron transport pathway from Photosystem II to Photosystem I. Note that, Equation (7) is designed for broadband SIF from Photosystem II. Single NIR band SIF was used here instead by assuming a linear relationship between single-band SIF and full band SIF. \(\mathrm{q}_{\mathrm{L}}\) ranges from 0 to 1 and decreases with increased photosynthetically active radiation [12; 45]. Here, \(\mathrm{q}_{\mathrm{L}}\) is derived by \(\mathrm{q}_{\mathrm{L}}=\exp(-\beta\times\mathrm{PAR})\), where \(\beta\) is a parameter denoting the sensitivity of \(\mathrm{q}_{\mathrm{L}}\) to the illumination. Benefiting from the carbon-pump mechanism, the GPP of \(\mathrm{C}_{4}\) plants is linearly related to J. For \(\mathrm{C}_{4}\) plants, \(\mathrm{GPP}_{\mathrm{gs}}\) can be derived by [12]:  
\[\mathrm{GPP}_{\mathrm{gs}}=\mathrm{J}/4=\frac{\mathrm{a}\times\mathrm{q}_{ \mathrm{L}}\times\mathrm{SIF}}{4\Omega_{\mathrm{C}}} \tag{8}\]  
gs of \(\mathrm{C}_{4}\) plants is derived by inserting Equation (8) into the Ball-Berry model (Equation (9)) [46], which is consistent with the SCOPE model, then we have the conductance model of \(\mathrm{C}_{4}\) plants (Equation (10)):  
\[\mathrm{gs}=\mathrm{m}\frac{\mathrm{GPP}_{\mathrm{gs}}\times\mathrm{Rh}}{ \mathrm{C}_{\mathrm{a}}}+\mathrm{g}_{0} \tag{9}\]  
\[\mathrm{gs}=\mathrm{m}\frac{\mathrm{a}\times\mathrm{q}_{\mathrm{L}}\times \mathrm{SIF}}{4\Omega_{\mathrm{c}}}\mathrm{Rh}/\mathrm{C}_{\mathrm{a}}+ \mathrm{g}_{0} \tag{10}\]  
where m is an empirical slope parameter, which is often treated as a constant for a specific ecosystem [47]. \(\mathrm{C}_{\mathrm{a}}\) is the ambient carbon dioxide concentration and \(\mathrm{g}_{0}\) is the minimum conductance, which is set as 0. Lack of an efficient mechanism gathering \(\mathrm{CO}_{2}\) from the air, \(\mathrm{C}_{3}\) plants rely more on stomata to absorb \(\mathrm{CO}_{2}\) for the Calvin cycle. The relationship between SIF and GPP is also affected by the dark reactions for \(\mathrm{C}_{3}\) plants, which can be expressed by [12]:  
\[\mathrm{GPP}_{\mathrm{gs}}=\mathrm{a}\frac{\mathrm{C}_{\mathrm{i}}-\Gamma}{4 \mathrm{C}_{\mathrm{i}}+8\Gamma}\mathrm{q}_{\mathrm{L}}\times\mathrm{SIF} \times\frac{1}{\Omega_{\mathrm{c}}} \tag{11}\]  
\(\mathrm{C}_{\mathrm{i}}\) is the intercellular \(\mathrm{CO}_{2}\) concentration.