## 2 Materials and Methods
### VPM Model and Simulation  
#### 2.5.1 VPM Model  
The VPM model estimates the daily GPP by multiplying the amount of PAR absorbed by chlorophyll in the crop canopy (APAR\({}_{\text{chl}}\)) and the light use efficiency [37; 38]. The model equations are the following:  
\[GPP=\varepsilon_{g}\times APAR_{chl}, \tag{4}\]  
\[APAR_{chl}=FPAR_{chl}\times PAR. \tag{5}\]  
The fraction of PAR absorbed by chlorophyll in the canopy (FPAR\({}_{\text{chl}}\)) is estimated as a linear function of EVI [55]:  
\[FPAR_{chl}=1.25\times(EVI-0.1). \tag{6}\]  
The effects of air temperature and water on the maximum light use efficiency (\(\varepsilon_{0}\)) are calculated as:  
\[\varepsilon_{g}=\varepsilon_{0}\times T_{solar}\times W_{solar}. \tag{7}\]  
The parameters \(\varepsilon_{g}\) and \(\varepsilon_{0}\) refer to the light use efficiency and maximum light use efficiency (umol CO\({}_{2}\) umol\({}^{-1}\) photosynthetic photon flux density (PPFD) or g C mol\({}^{-1}\) PPFD), respectively. The \(W_{solar}\) and \(T_{solar}\) scalars represent the effects of air temperature and water on the light use efficiency of vegetation, respectively. The maximum light use efficiency (LUE) parameter values are obtained from a literature survey and/or analysis of NEE, GPP, and PAR data from CO\({}_{2}\) EC flux tower sites [62]. In this study, we used 0.075 mol of CO\({}_{2}\) mol\({}^{-1}\) PPFD (0.9 g C mol\({}^{-1}\) PPFD) as the parameter \(\varepsilon_{0}\) for sugarcane (C4 plant), based on the findings of an early study of the CO\({}_{2}\) uptake and quantum yield of photosynthesis in different sugarcane clones at leaf scale [63]. In that study, the authors investigated how leaf nitrogen content affects the quantum yield at the leaf scale for a variety of sugarcane plants, and they found that quantum yield is a linear function of leaf nitrogen content and reached 0.075 mol of CO\({}_{2}\) mol\({}^{-1}\) PPFD when the leaf nitrogen content is the largest value in the experiment. In this study, we assume that this quantum yield value at the leaf scale is the maximum LUE parameter at the ecosystem scale.  
We calculated the effect of the air temperature on photosynthesis (\(T_{solar}\)) using Equation (6):  
\[T_{solar}=\frac{(T-T_{min})(T-T_{max})}{\left[(T-T_{min})(T-T_{max})\right]- \left(T-T_{opt}\right)^{2}}. \tag{8}\]  
In Equation (8), \(T_{min}\), \(T_{opt}\), and \(T_{max}\) are the minimum, optimal, and maximum air temperature of photosynthesis, respectively. The \(T_{min}\) and \(T_{max}\) parameters for these two sugarcane sites were set to be \(-1\)\({}^{\circ}\)C and 48 \({}^{\circ}\)C, using the same parameter values of the cropland biome in the publication [55]. The site-specific optimum air temperature (\(T_{opt}\)) parameter was estimated by the relationship between the GPPEc or vegetation index (NDVI, EVI) and the daily mean air temperature and daytime mean air temperature. This procedure was already used to estimate the optimum air temperature parameter in studying the carbon fluxes of paddy rice fields [64] and grasslands [65]. According to the results in Section 3.2, the T\({}_{\text{opt}}\) parameter value is approximately \(\sim\)25 \({}^{\circ}\)C (both GPP-based and VI-based) at the Brazil sites and \(\sim\)28 \({}^{\circ}\)C (GPP-based) or \(\sim\)25 \({}^{\circ}\)C (VI-based) at the USA site.  
We calculated the effect of water on photosynthesis (\(W_{scalar}\)) and this was estimated by the LSWI, and the maximum LSWI during the plant growing season is assumed to be the \(LSWI_{\text{max}}\):  
\[W_{scalar}=\frac{1+LSWI}{1+LSWI_{max}}. \tag{9}\]