## 2 Materials and Methods
### Methods
#### 2.2.1 LUE Estimation  
We established 3 approaches to compute LUE. Two of them were parameterization approaches in which we first determined the maximum LUE for each type of vegetation by an optimization algorithm developed at the University of Arizona (SCE-UA), and then we adjusted the maximum LUE with temperature and water stress factors, finally estimating the actual LUE. The difference between the two parameterization approaches was whether we considered the effect of solar radiation by adopting the CI in the determination of the maximum LUE. The third approach was the Cubist regression approach, which considers the contributions of temperature, LAI, EF, CI and vegetation type to the LUE. * **Parameterization approach without CI (approach V1)** LUE is estimated by Formula (2) in the parameterization approach without CI (approach V1): \[\text{LUE}=\text{LUE}_{max}\text{*f}(\text{T})\text{*f}(\text{W})\] (2) where LUE is the actual LUE and can be calculated by Formula (1) at FLUXNET sites, and \(\text{LUE}_{max}\) is the maximum LUE without stress in each type of vegetation. \(\text{f}(\text{T})\) represents the temperature stress and can be described as [65]: \[\text{f}(\text{T})=\frac{1.1814}{\left[1+e^{0.2(Topt-10-T)}\right]\times \left[1+e^{0.3(-Topt-10+T)}\right]}\] (3) where T represents the average temperature and comes from the FLUXNET measurements, and Topt is the monthly mean temperature when the vegetation reaches the maximum LAI for each vegetation type. \(\text{f}(\text{W})\) describes the water stress and is related to EF by [65]: \[\text{f}(\text{W})=0.5+0.5\times EF\] (4)  
At the site scale, EF was calculated by Formula (5), where LE and H were collected from the FLUXNET data. For global LUE and GPP estimation, EF was derived by Formula (6), where ET and PET represent the actual and potential evapotranspiration, respectively. \[EF=\frac{LE}{LE+H} \tag{5}\]  
\[EF=\frac{ET}{PET} \tag{6}\]  
Previous studies suggest that the Penman-Monteith (P-M) formula is a biophysically sound and robust framework for estimating daily evapotranspiration at regional and global scales with remotely sensed data [66]. In this study, we used a modified P-M approach with biome-specific canopy conductance to estimate daily actual evapotranspiration, which can be partitioned into soil evaporation and canopy transpiration [67; 68]. Potential evapotranspiration is calculated using the Priestley and Taylor (P-T) formula [69]. More details can be found in [PERSON], et al. [70]. The maximum LUE values for different vegetation types in approach V1 were determined by the SCE-UA optimization algorithm. The SCE-UA optimization algorithm, developed and described by [PERSON], et al. [71], is both a global and a probabilistic optimization algorithm. This approach is structured by four basic ideas: (1) the combination of random and deterministic approaches; (2) the concept of clustering; (3) the concept of a systematic evolution towards global improvement; and (4) the concept of competitive evolution [72]. The details of the SCE-UA method refer to the literature [71; 73]. Due to its high efficiency of solving global optimal solutions under nonlinear constraints and never depending on the initial value of the mode, SCE-UA has been widely used for the optimization of parameters and data assimilation [74; 75; 76; 77]. In this study, we set a valid and reasonable range for \(\text{LUE}_{max}\). The costfunction equals the root-mean-square error (RMSE) between the actual LUE and the estimated value. Then, we minimized the cost function to obtain \(\text{LUE}_{max}\) in each vegetation type (Table 1).