## 2 Materials and Methods
### Methods  
#### 2.2.1 A Process-Based Farquhar GPP Model (FGM)  
We recently developed a large-scale Farquhar GPP model (FGM) based on eddy covariance data and remote sensing data [35]. The FGM model was initially developed from a stand-level GPP model based on [PERSON] et al. (2009). Derived from the [PERSON] et al. (2009) model, the FGM estimates GPP by integrating the Farquhar leaf-level biochemical photosynthesis model [40] with a two-leaf radiation interception simulation method. In the [PERSON] et al. (2009) model, GPP is solved based on three complex equations: [PERSON]'s law, the Farquhar photosynthesis model, and a model for stomatal conductance. This approach is computationally expensive. To reduce the computational need for large-scale GPP estimation at a high spatiotemporal resolution, we introduced the optimal stomatal conductance theory to compute GPP more efficiently with the FGM. The FGM simulates carbon assimilation using the Farquhar, von Caemmerer, and Berry (i.e., FvCB) [40] enzyme kinetic model, which couples electron transport and the Calvin-Benson cycle. The function of the leaf photosynthetic rate takes the minimum of the V\({}_{\text{cmax}}\)-limited photosynthesis rate (i.e., A\({}_{\text{v}}\)) and light-limited photosynthesis rate (i.e., A\({}_{\text{j}}\)). Some fundamental equations for GPP estimation with the FGM are described here:  
\[\text{A}_{\text{n}}\text{= min}\left\{ \begin{matrix}\text{A}_{\text{v}}\\ \text{A}_{\text{j}}\end{matrix} \right\} \tag{1}\]  
\[\text{A}_{\text{v}}=\frac{\text{V}_{\text{cmax}}(\text{C}_{\text{i}}-\Gamma^{ *})}{\text{C}_{\text{i}}\text{+}\text{K}_{\text{C}}(1+\text{O}/\text{K}_{ \text{O}})}-\text{R}_{\text{d}} \tag{2}\]  
\[\text{A}_{\text{j}}=\frac{\text{J}(\text{C}_{\text{i}}-\Gamma^{*})}{4.5\text{ C}_{\text{i}}\text{+}10.5\Gamma^{*}}-\text{R}_{\text{d}} \tag{3}\]  
where A\({}_{\text{n}}\) is the net photosynthesis rate; A\({}_{\text{v}}\) is the minimum of the V\({}_{\text{cmax}}\)-limited photosynthesis rate; A\({}_{\text{j}}\) is the light-limited photosynthesis rate; R\({}_{\text{d}}\) is the dark respiration rate; \(\Gamma^{*}\) is  
Figure 1: Spatial distribution of 40 EuroFLUX sites and land cover classification from MCD12Q1 in 2016 in Europe. Abbreviations: croplands (CRO), closed shrublands (CSH), deciduous broadleaf forest (DBF), deciduous needleleaf forest (DNF), evergreen broadleaf forest (EBF), evergreen needleleaf forest (ENF), grasslands (GRA), mixed forest (MF), open shrublands (OSH), and wetland (WET). the CO\({}_{2}\) compensation point; K\({}_{\rm C}\) and K\({}_{\rm O}\) are the Michaelis-Menten constants of Rubisco for CO\({}_{2}\) and O\({}_{2}\), respectively; O is the intercellular oxygen partial pressure in the leaves; and J is the rate of electron transport. According to the optimal stomatal conductance theory, plants adjust their stomata to minimize the combined unit costs of transpiration and carbon assimilation [61]. Assuming the residual conductance parameter g\({}_{0}\) equals zero, the ratio of the intercellular CO\({}_{2}\) concentration (C\({}_{\rm i}\)) to ambient CO\({}_{2}\) concentration (C\({}_{\rm a}\)) is regulated by the atmospheric vapor pressure deficit based on optimal stomatal theory [62; 63]:  
\[\frac{\rm C_{i}}{\rm C_{a}}\ \approx\ \frac{\rm g_{1}}{\rm g_{1}+\sqrt{D}} \tag{4}\]  
where D is the vapor pressure deficit in kPa, and g\({}_{1}\) is an empirical parameter in kPa\({}^{0.5}\). According to its theoretical interpretation, the parameter g\({}_{1}\) increases with the marginal water cost of carbon \(\lambda\) and the CO\({}_{2}\) compensation point \(\Gamma^{*}\). Thus, species with a high g\({}_{1}\) will have a low instantaneous water use efficiency, i.e., a lower ratio of photosynthesis to the transpiration rate.