## 2 Materials and Methods
### Framework for GPP Generation
#### 2.3.1 The Improved GPP Estimation Model  
CASA model represents GPP as the multiplication of PAR, the fraction of PAR being absorbed by the plant canopy (FPAR) and the efficiency \(\varepsilon\) with which this absorbed PAR can be converted into biomass [23]. The equation for the classical CASA model to estimate GPP can be expressed as follows:  
\[GPP(x,y,t)=APAR(x,y,t)\times\varepsilon(x,y,t) \tag{1}\]  
where \(APAR\) (\(x\), \(y\), \(t\)) represents the actual \(PAR\) absorbed by the vegetation, and \(\varepsilon\) (\(x\), \(y\), \(t\)) indicates the actual light use efficiency of the green plant, which is reflected by air temperature and precipitation. \(APAR\) (\(x\), \(y\), \(t\)) and \(\varepsilon\) (\(x\), \(y\), \(t\)) can be expressed in the following expression. \[APAR(x,y,t)=PAR(x,y,t)\times FPAR(x,y,t)\times 0.5 \tag{2}\]  
\[\varepsilon(x,y,t)=\varepsilon_{max}\times f(T_{\varepsilon 1}(x,y,t))\times f(T_{ \varepsilon 2}(x,y,t))\times f(W_{\varepsilon}(x,y,t)) \tag{3}\]  
where \(PAR\) (\(x\), \(y\), \(t\)) (MJ\(\cdot\)m\({}^{-2}\)) is the incident \(PAR\) per time period (e.g., day, month, or year), \(FPAR\) (\(x\), \(y\), \(t\)) is the fraction of \(PAR\) (\(x\), \(y\), \(t\)) absorbed by the vegetation canopy, \(\varepsilon_{max}\) (g C\(\cdot\)m\({}^{-2}\cdot\)MJ\({}^{-1}\)\(APAR\)) is the potential light use efficiency without environmental stresses. \(f(T_{\varepsilon 1}\) (\(x\), \(y\), \(t\))), \(f(T_{\varepsilon 2}\) (\(x\), \(y\), \(t\))) and \(f(W_{\varepsilon}\) (\(x\), \(y\), \(t\))) are the scalars quantifying the temperature stress and water stress, respectively. As the feedstock for green plant photosynthesis, CO\({}_{2}\) also has a limiting effect on photosynthetic rates. Canopy CO\({}_{2}\) penetrates through the stomata into the interior of the leaf and participates in the carbon assimilation process during the dark phase, converting the carbon in CO\({}_{2}\) into organic matter in the green plant. Therefore, this study integrates  
Figure 2: The framework of the improved GPP estimation model. atmospheric CO\({}_{2}\) concentration data into the traditional CASA model, and proposes an improved GPP estimation model which expresses the main improvements as follows:  
\[\varepsilon(x,y,t)=\varepsilon_{max}\times f(T_{\varepsilon 1}(x,y,t))\times f(T_{ \varepsilon 2}(x,y,t))\times f(W_{\varepsilon}(x,y,t))\times f(C_{\varepsilon}(x,y,t)) \tag{4}\]  
where \(f\) (\(C_{\varepsilon}\) (\(x\), \(y\), \(t\))) is the scalar quantifying the atmospheric CO\({}_{2}\) fertilization. In the C-fix model, atmospheric CO\({}_{2}\) fertilization was defined as the augmentation in carbon assimilation resulting from CO\({}_{2}\) levels surpassing the atmospheric background level (or reference level) [2; 24]. Thus, \(f(C_{\varepsilon}\) (\(x,y,t\))) can be calculated by:  
\[f(C_{\varepsilon}(x,y,t))=\frac{[C_{i}](x,y,t)-\frac{[O_{2}]}{2\tau}}{[C_{i}]^ {ref}(x,y,t)-\frac{[O_{2}]}{2\tau}}\times\frac{K_{m}\Big{(}1+\frac{[O_{2}]}{K_ {0}}\Big{)}+[C_{i}]^{ref}(x,y,t)}{K_{m}\Big{(}1+\frac{[O_{2}]}{K_{0}}\Big{)}+[ C_{i}](x,y,t)} \tag{5}\]  
where \([C_{i}]\) (\(x\), \(y\), \(t\)) is the leaf-internal CO\({}_{2}\) concentration at current level, \([C_{i}]^{ref}\) (\(x\), \(y\), \(t\)) is leaf-internal CO\({}_{2}\) concentration at the reference level, \([O_{2}\) ] is atmospheric O\({}_{2}\) concentration and is expressed in %, \(\tau\) is CO\({}_{2}\)/O\({}_{2}\) specificity ratio and is dimensionless, \(Km\) is the affinity constant for CO\({}_{2}\) of Rubisco and is expressed in ppmv, \(K_{0}\) is inhibition constant for O\({}_{2}\) and is expressed in %. \([O_{2}]\), \(\tau\), \(K_{m}\) and \(K_{0}\) were set to 20.9%, 2550, 948 ppmv and 30%, respectively [2; 24].