## 3 Methods  
### Par-LUE  
The common structure of the LUE model can be formulated as follows:  
\[GPP=PAR\times FPAR\times\mathcal{E}_{max}\times f(\text{Ts})\times f(\text{Ws}) \tag{4}\]  
where PAR is photosynthetically active radiation, FPAR is the fraction of absorbed PAR, \(\mathcal{E}_{\text{max}}\) is maximal light use efficiency, and \(f\) (Ts) and \(f\) (Ws) are the scaled environmental stress indices of temperature and water on LUE, respectively. Referring to the existing research results [10; 21], FPAR, f (Ts), and f (Ws) were calculated as follows:  
\[FPAR=(EVI\ -\ 0.1)\ \times\ 1.25 \tag{5}\]  
Figure 1: Spatial distribution of selected FLUXNET 2015 sites. (Background is the map of the MCD12Q1 in 2014; CRO: cropland (19 sites), CSH: closed shrub (2 sites), DBF: deciduous broadleaf forest (22 sites), DNF: deciduous needleleaf forests (0 site), EBF: evergreen broad-leaf forest (11 sites), ENF: evergreen needleleaf forest (44 sites), MF: mixed forest (8 sites), GRA: grassland (30 sites), OSH: open shrub (11 sites), SAV: savanna (7 sites), WSA: woody savanna (6 sites), WET: wetland (11 sites)). \[T_{S}=\frac{(T\ -\ T_{min})(T\ -\ T_{max})}{(T\ -\ T_{min})(T\ -\ T_{max})\ -\ (T\ -\ T_{opt})^{2}} \tag{6}\]  
\[W_{S}=\frac{1+LSWI}{1+LSWI_{max}} \tag{7}\]  
where \(T\) is the daily temperature; \(T_{min}\), \(T_{max}\), and \(T_{opt}\) are 0 \({}^{\circ}\)C, 40 \({}^{\circ}\)C, and 20 \({}^{\circ}\)C, respectively; and \(\mathrm{LSWI_{max}}\) is the maximal LSWI in the growing season. Here, the growing season is defined according to the date of 75 days before and after the date of maximal EVI (i.e., [date_EVImax-75, date_EVImax +75]). In the PAR-LUE model, we proposed a PAR-based method to calculate dynamic \(\mathcal{E}_{\mathrm{max}}\) (i.e., _PAR-\(\mathcal{E}_{\mathrm{max}}\)_, and _PAR-\(\mathcal{E}_{\mathrm{max}}\)_ =_f_ (PAR)). Considering the nonlinear response of vegetation photosynthesis to PAR, we developed the PAR-LUE model based on two hypothesises. First, under the ideal condition that vegetation was only related to the PAR and unconstrained by other biotic and abiotic conditions (i.e., FPAR, f (_Ts_), and f (_Ws_) are equal to 1), GPP\({}_{i}\) can be represented as the product of PAR and \(\mathcal{E}_{\mathrm{max}}\) (8). Under the optimal temperature and water conditions (i.e., T = T\({}_{\mathrm{opt}}\), \(\mathrm{LSWI}\) = \(\mathrm{LSWI_{max}}\)), the f (_Ts_) and f (_Ws_) are equal to 1. For the FPAR, the maximum value of measured [40] and remotely sensed FPAR (e.g., for the formula (5), FPAR = 1 when EVI \(\geq\) 0.9) are close or equal to 1. Second, the maximum GPP (GPP\({}_{\mathrm{max}}\)) under different levels of PAR meets the ideal conditions (i.e., GPP\({}_{i}\) = GPP\({}_{\mathrm{max}}\)). Using the quality-screened daily GPP and PAR data from the FLUXNET 2015 dataset, the maximum value of GPP corresponding to PAR (PAR was sampled with a step of 1 MJ m\({}^{-2}\)) was sampled within the subrange of 1 \(\pm\) 0.25 MJ m\({}^{-2}\). Then, the sampled PAR and GPP\({}_{\mathrm{max}}\) were fitted using cubic polynomial (9). Finally, combining Formulas (8) and (9), _PAR-\(\mathcal{E}_{\mathrm{max}}\)_ was calculated with Formula (10), and the corresponding model was named PAR-LUE. \[GPP_{i}=PAR\times\mathcal{E}_{\mathrm{max}}=PAR\times f\ (PAR) \tag{8}\]  
\[GPP_{max}=a\times PAR^{3}+b\times PAR^{2}+c\times PAR \tag{9}\]  
\[PAR\text{-}\mathcal{E}_{\mathrm{max}}\text{=}a\times PAR^{2}+b\times PAR+c \tag{10}\]  
where GPP\({}_{i}\) is the GPP under ideal conditions, and \(a\), \(b\), and \(c\) are fitting parameters of the cubic polynomial. As shown in Figure 2, the orange fit curve indicated the vegetation GPP\({}_{\mathrm{max}}\) under different PAR levels, and the points below the fit curve indicated the true GPP constrained by vegetation physiology and environmental factors. It is important to note that PAR-\(\mathcal{E}_{\mathrm{max}}\) is the \(\mathcal{E}_{\mathrm{max}}\) of all vegetation in the ideal condition.