## 2 Method  
### Model Description  
In this study, _GPP_ (g C m\({}^{-2}\) per time unit) is calculated as the product of _APAR_ and actual LUE. The actual LUE is the potential LUE (\(\varepsilon_{max}\)) constrained by environmental variables including air temperature, VPD, and water availability in a multiplicative manner. _GPP_ is then expressed as  
\[GPP=APAR\times\varepsilon_{max}\times F_{T}\times F_{VPD}\times F_{W} \tag{1}\]  
where _APAR_ is the absorbed photosynthetically active radiation (MJ m\({}^{-2}\) per time unit, e.g., hourly, daily, monthly, yearly) and can be estimated as the fraction of photosynthetically active radiation (_PAR_) (MJ m\({}^{-2}\) per time unit, e.g., hourly, daily, monthly, yearly):  
\[APAR=FAPAR\times PAR \tag{2}\]  
where _PAR_ is generally considered to account for a fraction (e.g., 0.44\(-\)0.5) of total solar shortwave irradiance [44; 45], 0.48 is taken in this study [46]. _FAPAR_ is the fraction of the absorbed photosynthetically active radiation that can be retrieved from satellite observations of multi-spectral reflectance or derived from NDVI or LAI from satellite observations using empirical relationship. In Equation (1), \(\varepsilon_{max}\) (g C MJ\({}^{-1}\)) is the potential LUE without environmental stresses (also known as maximum LUE), which defines the canopy photosynthetic capacity and varies across vegetation types and environment conditions [47; 48]. In this paper, the \(\varepsilon_{max}\) values are optimized using ground site data and interpolated according to climate zones for GPP estimate at global scale (see the optimization procedure in Section 2.2). The environmental constraint factors \(F_{T}\), \(F_{VPD}\) and \(F_{W}\) in Equation (1) are dimensionless factors in the range of \(0\leq F_{x}\leq 1\) with x denoting for T, VPD, or W (water availability), characterizing the constraints of ambient temperature, atmospheric water vapor pressure deficit and soil water availability on the potential LUE. The \(F_{x}\) values closer to \"0\" indicate stronger constraint by the environmental factor \"x\", resulting in the reduction of potential LUE and thus lower GPP. On the contrary, \(F_{x}\) values closer to \"1\" indicate the lower constraint by the corresponding environmental factor \"x\". \(F_{x}=1\) means that the constraint is not considered. The air temperature constraint factor (\(F_{T}\)) is usually defined as a ramp function or an asymmetric curve. Using the asymmetric formula, photosynthesis responds more smoothly to air temperature and reverses after the optimal temperature (\(T_{opt}\)),  
\[F_{T}=\frac{(T-T_{min})\times(T-T_{max})}{(T-T_{min})\times(T-T_{max})-\left(T- T_{opt}\right)^{2}} \tag{3}\]  
where \(T\) is the ambient temperature (\({}^{\circ}\)C); \(T_{min}\), \(T_{max}\), and \(T_{opt}\) are the minimum, maximum, and optimum air temperatures (\({}^{\circ}\)C) for vegetation photosynthesis, respectively. \(F_{T}\) is set to 0 when ambient temperature \(T\) is lower than \(T_{min}\) or higher than \(T_{max}\), resulting in \"0\" LUE. In this study, \(T_{min}\) and \(T_{max}\) are set as 0 \({}^{\circ}\)C and 40 \({}^{\circ}\)C, respectively. \(T_{opt}\) is optimized using GPP that is derived from CO\({}_{2}\) flux measurements collected at cropland flux tower sites. The atmospheric water Vapor Pressure Deficit (\(VPD\)) is the difference between the saturated water vapor pressure and the actual water vapor pressure at a certain air temperature, indicating the atmospheric water demand. As \(VPD\) increases, crops tend to close stomata to reduce the water loss thereafter absorb less CO\({}_{2}\). The constraint factor in response to \(VPD\) (\(F_{VPD}\)) is defined as concave functions by  
\[F_{VPD}=\frac{VPD_{0}}{VPD_{0}+VPD} \tag{4}\]  
where \(VPD_{0}\) is the half-saturation coefficient (kPa) and will be obtained by the optimization procedure described in Section 2.2.