## 2 Materials and Methods
### Methods
#### 2.2.4 Data Analysis Methods  
(1) Model performance evaluation during the calibration process  
We conducted Monte Carlo simulations to calibrate parameter g1 and V\({}_{\rm cmax}\) for each biome. The optimal values of g1 for different biomes were set as driving parameters for the FGM, while the optimal values of V\({}_{\rm cmax}\) for different biomes were only used for the quantification of the V\({}_{\rm cmax}\) change effect on GPP. The Monte Carlo simulations identified \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Simulation**} & **All** & **PC1** & **PC2** & **PC3** \\  & **(LCC-Derived V\({}_{\rm cmax}\))** & **(LCC-Derived V\({}_{\rm cmax}\))** & **(PFT-Specific V\({}_{\rm cmax}\))** & **(PFT-Specific V\({}_{\rm cmax}\))** \\ \hline LULC & \(\triangle\) & \(\triangle\) & \(\triangle\) & \(\triangle\) & \(\triangle\) & \(\triangle\) \\ LAI & \(\triangle\) & \(\triangle\) & \(\triangle\) & \(\triangle\) & \(\triangle\) \\ \(\Omega\) & \(\triangle\) & \(\triangle\) & \(\triangle\) & \(\triangle\) & \(\triangle\) \\  & & & \(\blacklozenge\) & \(\blacklozenge\) & \(\blacklozenge\) \\ V\({}_{\rm cmax}\) & (8-day, 2001 to 2016) & (8-day, 2001) & & \(\blacklozenge\) & \(\blacklozenge\) & \(\blacklozenge\) \\  & & & & & \\  & & & & & \\ DSR & \(\triangle\) & \(\triangle\) & \(\triangle\) & \(\triangle\) & \(\triangle\) \\ T\({}_{\rm a}\) & \(\triangle\) & \(\triangle\) & \(\triangle\) & \(\triangle\) & \(\triangle\) \\ RH & \(\triangle\) & \(\triangle\) & \(\triangle\) & \(\triangle\) & \(\triangle\) \\ CO\({}_{2}\) & \(\triangle\) & \(\triangle\) & \(\triangle\) & \(\triangle\) & \(\triangle\) \\ \hline \hline \end{tabular} \({}^{1}\) See [PERSON] et al. (2009).  
\end{table} 
Table 2: Scenario designs used to quantify the effects of changes in photosynthesis capacity (PC) on GPP based on the FGM. The symbol '\(\triangle\)’ indicates that the input variable changes over time, while the symbol ‘\(\blacklozenge\)’ indicates that the seasonality of V\({}_{\rm cmax}\) on a large scale is included. The symbol ‘\(\blacklozenge\)’ indicates that the input variable is fixed at a biome-specific constant. optimal parameters when the simulated data reached the strongest agreement with the observed data using a weighted R\({}^{2}\) (wR\({}^{2}\)) as a performance indicator [30; 100]:  
\[\text{wR}^{2}=\left\{ \begin{array}{l}|\text{b}|\text{R}^{2},\text{ b}<1\\ |\text{b}|^{-1}\text{R}^{2},\text{ b}\geq 1\end{array} \right. \tag{11}\]  
where b and R\({}^{2}\) are the slope and coefficient of determination for the regression of the modeled GPP and GPP estimates derived from eddy covariance data (EC-GPP) when the intercept is forced to zero, respectively [101]. A coefficient of determination (R\({}^{2}\)) equal to one and b equal to one (wR\({}^{2}\) = 1) indicate a perfect model performance. The range of R\({}^{2}\) is 0 to 1, which describes the proportion of the observed dispersion that is explained by the prediction. Quantification of the accuracy of the FGM GPP  
To quantify the accuracy of the FGM GPP directly, first, we compared the simulated GPP with the EC-GPP collected during the daytime at the 19 validation sites. The R\({}^{2}\) and root mean square error (RMSE) were estimated using the regression lines between the modeled GPP and EC-GPP to evaluate the FGM model performance. Second, we quantified the accuracy of the spatial pattern of European GPP predicted by the FGM. The multi-year mean of the European vegetation photosynthesis rate was calculated based on different GPP products and the remotely sensed sun-induced chlorophyll fluorescence (SIF) data for the same study area. The results of the multi-year means of the annual European GPP and SIF data with coarse spatial resolutions were resampled to the target spatial resolution of 500 m by nearest neighbor interpolation. We calculated the spatial correlation matrix between the different GPP products and the SIF data.