## Results
### Relationships between GPP and Vegetation Indices in Combination with Environmental Stresses  
To further address how each VI in combination with environmental factors could explain the variations in GPP, we also examined the relationships of GPP with VI \(\times\) f\({}_{\rm{Topt}}\), VI \(\times\) f\({}_{\rm{Topt}}\)\(\times\) f\({}_{\rm{VPD}}\) and VI \(\times\) f\({}_{\rm{Topt}}\)\(\times\) f\({}_{\rm{VPD}}\)\(\times\) PAR at the monthly timescale for each site separately. On average, the VIs with temperature stress (VIs \(\times\) f\({}_{\rm{Topt}}\)) explained higher variance in GPP (R\({}^{2}\) = 0.64-0.72, p \(<\) 0.0001) than  
Figure 8: Daily VIs, temperature, precipitation, temperature stress (f\({}_{\rm{Topt}}\)) and water stress (f\({}_{\rm{VPD}}\)) at a grassland site (CH-Oe1) in a normal year (2002) and a drought year (2003). Figure 9: Daily VIs, temperature, precipitation, temperature stress (f\({}_{\rm{Topt}}\)) and water stress (f\({}_{\rm{VPD}}\)) at a deciduous broadleaf forest site (DK-Sor) in a normal year (2014) and a low temperature year (2007). VIs with temperature and water stresses (R\({}^{2}\) = 0.60-0.70, p \(<\) 0.0001) or VIs alone (R\({}^{2}\) = 0.48-0.70, p \(<\) 0.0001); VIs \(\times\) f\({}_{\text{Topt}}\)\(\times\) f\({}_{\text{VPD}}\)\(\times\) PAR explained the highest variance in GPP (R\({}^{2}\) = 0.72-0.76, p \(<\) 0.0001) at the monthly timescale for 121 sites. Figure 10 summarizes the R\({}^{2}\) values for the relationships between the GPP and the VIs combined with f\({}_{\text{Topt}}\), f\({}_{\text{VPD}}\), and PAR in each biome. In general, VIs \(\times\) f\({}_{\text{Topt}}\)\(\times\) f\({}_{\text{VPD}}\)\(\times\) PAR had the strongest relationship with GPP in ten biomes except EBF and SAV where VIs \(\times\) f\({}_{\text{Topt}}\)\(\times\) f\({}_{\text{VPD}}\) had the best performance. This indicates that GPP was mainly driven by VIs and also influenced by the environmental stresses and PAR. The seasonal cycles of the flux tower GPP, two environmental scalars (f\({}_{\text{Topt}}\) and f\({}_{\text{VPD}}\)), and PAR at the NL-Loo site are shown in Figure 11. f\({}_{\text{Topt}}\) exhibited clear seasonal cycles while f\({}_{\text{VPD}}\) was equal to 1.0 most of the time, indicating that temperature limited GPP and VPD was usually not a limiting factor at this site. We chose NIRV, BRDF to examine how VIs combined with environmental scalars and PAR could explain the variance in GPP because NIRV, BRDF had the best performance for predicting GPP (Figure 11). NIRV, BRDF, f\({}_{\text{Topt}}\), f\({}_{\text{VPD}}\), and PAR alone explained 73%, 79%, 23%, and 85% of the variance in GPP, respectively. NIRV, BRDF \(\times\) f\({}_{\text{Topt}}\) (87%) explained the same portion of the variance in GPP as NIRV, BRDF \(\times\) f\({}_{\text{Topt}}\)\(\times\) f\({}_{\text{VPD}}\) (86%). Meanwhile, NIRV, BRDF \(\times\) f\({}_{\text{Topt}}\)\(\times\) PAR explained the same variance in GPP (R\({}^{2}\) = 0.93, p \(<\) 0.0001) as NIRV, BRDF \(\times\) f\({}_{\text{Topt}}\)\(\times\) f\({}_{\text{VPD}}\)\(\times\) PAR, indicating that VPD was not a dominant controlling factor of photosynthesis and had negligible contribution at this forest ecosystem (Figure 11). Figure 10: The plot of the averaged R\({}^{2}\) values for the relationships of the GPP with VIs combined with environmental factors (f\({}_{\text{Topt}}\) and f\({}_{\text{VPD}}\) and PAR) for each biome. The letters on the x axis stand for: A - NDVI, B - EVI, C - EVI2, D - NDVI\({}_{\text{BRDF}}\), E - EVI\({}_{\text{BRDF}}\), F - EVI\({}_{\text{BRDF}}\), G - NIRV, and H - NIRV, BRDF. The biome types are as follows: evergreen needleleaf forest (ENF); evergreen broadleaf forest (EBF); deciduous needleleaf forest (DNF); mixed forest (MF); closed and open shrublands (COSH); woody savannas (WSA); savannas (SAV); grasslands (GRA); wetlands (WET); croplands (CRO).