## 2 Material and Methods
### Statistical Methods
#### 2.3.3 GPP Inversion of Total Primary Productivity  
In this study, the GPP inversion of total primary productivity is obtained by using the MODIS GPP model. The precise quantification of GPP at the landscape level is however challenging as there is no direct measurement technique beyond the leaf level. Forest productivity is primarily estimated using the data gathered through temporal forest inventories and established empirical allometric equations in terms of biomass. The MODIS GPP model is established based on the linear relationship between GPP and the photosynthetically active radiation absorbed by vegetation. The calculation process is as follows:  
\[\text{GPP}=\text{APAR}\times\epsilon_{\text{max}}\times f(\text{$a$min}) \times f(\text{VPD}) \tag{4}\]  
\[\text{APAR}=\text{SWRad}\times 0.45\times(1-e^{k\times\text{LAI}}) \tag{5}\]  
where GPP is the total primary productivity, the unit is g c-m\({}^{-2}\cdot\)s\({}^{-1}\); \(\epsilon_{\text{Max}}\) is the maximum light energy utilization rate, the unit is kg C/MJ. The light energy utilization rate (E\({}_{\text{max}}\)) is the ratio of the energy stored to the energy of light absorbed. The amount of energy stored can only be estimated because many products are formed, and they vary with the plant species and environmental conditions. APAR (Absorbed Photosynthetic Active Radiation) is the photosynthetically active radiation absorbed by vegetation, and the unit is MJm\({}^{-2}\cdot\)s\({}^{-1}\), thus, representing the product with a 45% incident short-wave radiation (SWRad) and a photosynthetically active radiation ratio which is absorbed by the vegetation canopy. The SWRad is radiation at wavelengths shorter than 4 microns. Usually, radiation is in the visible and near-infrared wavelengths. This radiation ratio is calculated using LAI (Leaf Area Index) through simple Beer's law; this study used the GlobMap LAI data from 2000 to 2017. GlobMapLAI is a global long-term series LAI product generated based on AVHRR/MODIS data. \(k\) is the canopy extinction coefficient--generally, its value is 0.5. \(f\) (VPD) and \(f\) (\(a_{\text{min}}\)) are the correction factors of the vapor pressure difference and air temperature at a 2 m height--they are calculated using the following equation:  
\[f(\text{VPD})=\frac{\text{VPD}_{\text{max}}-\text{VPD}}{\text{VPD}_{\text{max }}-\text{VPD}_{\text{min}}} \tag{6}\]  
\[f(T_{\text{min}})=\frac{T_{\text{min}}-T_{\text{min,min}}}{T_{\text{min,max}}- T_{\text{min,min}}} \tag{7}\]  
where VPD\({}_{\text{max}}\) and \(T_{\text{min,max}}\) are the maximum daily vapor pressure difference (Pa) and maximum daily air temperature (\({}^{\circ}\)C) at the time of maximum photosynthetic efficiency, respectively; VPD\({}_{\text{min}}\) and \(T_{\text{min,min}}\) are the minimum vapor pressure difference (Pa) and daily minimum temperature (\({}^{\circ}\)C) when photosynthesis is 0, respectively. These parameters are default parameters in the BPLUT table [42; 43; 44]. In this study, the 500 m MODIS reflectance data (MOD09 GA) were used for spectral unmixing. The MOD09 GA product was provided along with daily 500 m spatial resolution surface reflectance data, which were generated from Terra-MODIS bands 1 to 7 (620 nm-2155 nm). The MODIS data collection and resolution methods used in this study are available and well explained in the website ([[https://modis-images.gsfc.nasa.gov/_docs/CMUSERSGUIDE.pdf](https://modis-images.gsfc.nasa.gov/_docs/CMUSERSGUIDE.pdf)]([https://modis-images.gsfc.nasa.gov/_docs/CMUSERSGUIDE.pdf](https://modis-images.gsfc.nasa.gov/_docs/CMUSERSGUIDE.pdf)), 12 October 2018), which is proposed by [PERSON] in 2012 [45].