## 2 Materials and Methods
### Two-Leaf Lue-Gpp Model  
[PERSON] et al. [9] considered the transmission of scattered light in the canopy and the difference in \(LUE_{\rm max}\) (\(LUE\) without environmental constraints) between shaded and sunlit leaves and proposed a two-leaf model to calculate separately the \(GPP\) values of shaded and sunlit leaves. [PERSON] et al. [40] found that the two-leaf LUE-GPP model performed better than the MODIS model based on data from 98 EC sites globally. The expression of the two-leaf LUE-GPP model is as follows:  
\[GPP=(LUE_{\rm su-max}APAR_{\rm su}+LUE_{\rm sh-max}APAR_{\rm sh})\times T_{\rm s }\times W_{\rm s} \tag{20}\]  
\[APAR_{\rm su}=(1-\alpha)\bigg{[}PAR_{\rm dir}\times\frac{\cos(\beta)}{\cos( \theta)}+\frac{PAR_{\rm dif}-PAR_{\rm dif,u}}{LAI}+C\bigg{]}\times LAI_{\rm su} \tag{21}\]  
\[APAR_{\rm sh}=(1-\alpha)(\frac{PAR_{\rm dif}-PAR_{\rm dif,u}}{LAI}+C)\times LAI _{\rm sh} \tag{22}\]  
where \(LUE_{\rm su-max}\) and \(LUE_{\rm sh-max}\) represent the \(LUE_{\rm max}\) values of sunlit and shaded leaves, respectively; \(\alpha\) represents the leaf albedo, set to 0.23 for crop [41]; \(APAR_{\rm su}\) and \(APAR_{\rm sh}\) represent the absorbed PAR values of sunlit and shaded leaves, respectively; \(LAI_{\rm su}\) and \(LAI_{\rm sh}\) represent the estimated canopy LAI values of sunlit and shaded, respectively; \((PAR_{\rm dif}-PAR_{\rm dif,u})/LAI\) represents the diffuse PAR on the unit leaf area of the canopy [42]; C represents the parameter for quantifying PAR scattering inside the canopy [43]; \(\theta\) represents the solar zenith angle; \(\beta\) represents the mean leaf-sun angle set to \(60^{\circ}\) for a canopy with spherical leaf-angle distribution [9]; \(PAR_{\rm dif}\) and \(PAR_{\rm dif}\) represent the diffuse PAR and direct PAR, respectively; and \(PAR_{\rm dif,u}\) represents the \(PAR_{\rm dif}\) below the canopy. \(T_{\rm s}\) and \(W_{\rm s}\) were calculated according to the MODIS model.  
The \(LAI_{\rm su}\) and \(LAI_{\rm sh}\) values were calculated as follows [9; 42; 43]:  
\[LAI_{\rm su}=2\times cos(\theta)\times(1-exp(\frac{-0.5\Omega LAI}{cos\theta})) \tag{23}\]  
\[LAI_{\rm sh}=LAI-LAI_{\rm su} \tag{24}\]  
\[PAR_{\rm dif}=PAR\times D_{\rm f} \tag{25}\]  
\[D_{\rm f}=0.7527+3.8453K_{\rm t}-16.31K_{\rm t}^{2}+18.962K_{\rm t}^{3}-7.0802 K_{\rm t}^{4} \tag{26}\]  
\[C=0.07\Omega PAR_{\rm dif}(1.1-0.1 LAI)exp(-\cos\theta) \tag{27}\]  
where \(D_{\rm f}\) is the proportion of diffuse \(PAR\) to total \(PAR\) determined by \(K_{\rm t}\); \(K_{\rm t}\) is the clearness index used to quantify cloud thickness [44]; and \(\Omega\) is the aggregation index set to 0.75 based on [PERSON] et al. [45].  
\[K_{\rm t}=\frac{R_{\rm g}}{R_{\rm a}} \tag{28}\]  
\[R_{\rm a}=\frac{24\times 60}{\pi}G_{\rm sc}d_{\rm r}(\omega_{\rm s}sinsin\delta+ cos\varrho cos\delta cosw_{\rm s}) \tag{29}\]  
\[d_{\rm r}=1+0.033 cos(\frac{2\pi I}{365}) \tag{30}\]  
\[\delta=0.409 sin(\frac{2\pi I}{365}-1.39) \tag{31}\]  
\[w_{\rm s}=arccos(-tan\theta\delta) \tag{32}\]where \(R_{\rm a}\) is the solar radiation at the top of the atmosphere (MJ m\({}^{-2}\) d\({}^{-1}\)), \(G_{\rm sc}\) is the solar constant (0.082 MJ m\({}^{-2}\) min\({}^{-1}\)), \(d_{\rm r}\) is the inverse relative distance between the sun and earth, \(w_{\rm s}\) is sunset angle (rad), \(\varphi\) is the latitude (rad), \(\delta\) is the solar declination (rad), and \(J\) is the number of days.  
\[PAR_{\rm dif,u}=PAR_{\rm dif}\times exp(\frac{-0.5\Omega LAI}{\cos\overline{ \theta}}) \tag{33}\]  
where \(\overline{\theta}\) is a representative zenith angle for the transmission of diffuse radiation and is slightly related to \(LAI\)[42]. It is calculated as below:  
\[cos\overline{\theta}=0.537+0.025 LAI \tag{34}\]