# Estimating Global Gross Primary Production Using an Improved MODIS Leaf Area Index Dataset
## Appendix A The Structure of the Model
### Photosynthesis  
We divided the vegetation into C3 and C4 plants and calculated the photosynthesis of each pixel based on the ratio of C3 to C4 of each pixel. We used the biochemical photosynthesis model for C3 plants based on [PERSON]'s model [51] and for C4 plants based on the [PERSON]'s model [52].  
The net leaf photosynthesis (\(A_{n}\)) could be modeled as the minimum of three limiting rates after accounting for dark respiration (\(R_{d}\)).  
\[A_{n}=min(A_{c},A_{j},A_{p})-R_{d} \tag{104}\]  
where \(A_{c}\) is the rate of leaf photosynthesis when the Rubisco carboxylase is limited, which is expressed as follows:  
\[A_{c}=\left\{\begin{array}{ll}V_{cmax}\times\frac{C_{i}-\Gamma^{*}}{C_{i}+K _{c}(1+O_{i}/K_{o})}&,\text{for C3}\\ V_{cmax}&,\text{for C4}\end{array}\right. \tag{105}\]  
where \(V_{cmax}\) is the maximum rate of leaf carboxylation; \(C_{i}\) and \(O_{i}\) are the intercellular and atmospheric \(CO_{2}\) concentrations, respectively; \(\Gamma^{*}\) is the \(CO_{2}\) compensation point in the absence of dark respiration; and \(K_{c}\) and \(K_{o}\) are the Michaelis-Menten constants for Rubisco carboxylation and oxygenation, respectively.  
\(A_{j}\) is the rate of leaf photosynthesis when light is limited [112], which is expressed as follows:  
\[A_{j}=\left\{\begin{array}{ll}J\times\frac{C_{j}-\Gamma^{*}}{4C_{i}+8\Gamma^ {*}}&,\text{for C3}\\ 0.067\times Q&,\text{for C4}\end{array}\right. \tag{106}\]  
where \(J\) is the rate of electron transport and \(Q\) is the photosynthetically active photon flux density.  
\(A_{p}\) is the rate of photosynthesis when the product is limited for C3 plants and when the PEP carboxylase is limited for C4 plants, which is expressed as follows:  
\[A_{p}=\left\{\begin{array}{ll}0.5\times V_{cmax}&,\text{for C3}\\ k_{p}\times\frac{C_{i}}{P_{a}Tm}&,\text{for C4}\end{array}\right. \tag{107}\]  
where \(k_{p}\) is the initial slope of the C4 \(CO_{2}\) response curve and \(P_{a}tm\) is the atmospheric pressure.  
\(A_{c}\) and \(A_{j}\) are also affected by the diffusion constraints, which is expressed as follows:  
\[A_{c\text{ or }j}=(C_{a}-C_{i})\times G_{CO_{2}} \tag{108}\]  
where \(G_{CO_{2}}\) is the velocity of \(CO_{2}\) diffusion from the atmosphere into leaves.  
\(J\) depends on the photosynthetically active radiation absorbed by the leaf and expressed as follows [112]:  
\[\theta J^{2}-(\alpha Q+J_{max})\times J+\alpha QJ_{max}=0 \tag{109}\]  
where \(J_{max}\) is the maximum potential rate of electron transport, \(\theta\) is the curvature parameter of the light response curve, and \(\alpha\) is the quantum yield of electron transport.  
The calculation of \(K_{c}\) and \(K_{o}\) varies depending on the temperature threshold of 15 \({}^{\circ}\)C, which is expressed as follows:  
\[K_{o}=K_{o25}\times Q_{10,K_{o}}^{\frac{T-25}{100}} \tag{100}\]  
\[K_{c}=\left\{\begin{array}{ll}K_{o25}\times Q_{10,K_{o}}^{\frac{T-25}{100}}&, \text{for }T>15^{\circ}\text{C}\\ \frac{K_{o25}\times(Q_{10,K_{c}}\times 1.8)\frac{T-15}{10}}{Q_{10,K_{c}}}&, \text{for }T\leq 15^{\circ}\text{C}\end{array}\right. \tag{101}\]  
where \(T\) is the air temperature.  
We further calculated the vapor pressure deficit (VPD) as one of our input variables for the model by using dewpoint temperature and air temperature. The VPD, which is the difference between the saturated vapor pressure and the actual vapor pressure [113], is commonly used to predict leaf stomatal conductance and photosynthesis [50].  
\[es=0.61078\times e^{\frac{17.27\times T}{T+20.3}} \tag{102}\] \[ea=0.61078\times e^{\frac{17.27\times T}{T+20.3}}\] (103) \[VPD=es-ea \tag{104}\]  
where \(T_{d}\) is the dewpoint temperature.  
We used a temperature function and a high-temperature stress function to scale and describe the temperature dependences of \(V_{cmax}\), \(J_{max}\), \(\Gamma^{*}\), and \(R_{d}\)[112; 114; 115; 116].