## 3 Methodology
### The LUE-RE GPP Model  
Firstly, the LUE GPP values in each unit are divided into forest GPP, grassland GPP, crop GPP, and shrub GPP based on the vegetation type. For example, the LUE GPP is separated into different types of GPP according to the land use datasets. \[GPP=GPP_{1}+GPP_{2}+GPP_{3}+GPP_{4} \tag{2}\]  
where \(GPP_{1}\), \(GPP_{2}\), \(GPP_{3}\), and \(GPP_{4}\) represent forest GPP, grassland GPP, crop GPP, and shrub GPP, respectively. On the pixel scale, a multivariate linear regression model between \(GPP_{s}\) and each factor of a vegetation is set up as follows:  
\[GPP_{s}=\beta_{0s}+\beta_{1s}TEMP+\beta_{2s}PREC+\beta_{3s}SRAD+\beta_{4s}eCO_ {2}+\varepsilon_{s} \tag{3}\]  
where \(\beta_{0s},\beta_{1s},\beta_{2s},\beta_{3s},\beta_{4s}\) are parameters for the response of the gross primary productivity of the \(s\) vegetation type to temperature, precipitation, solar radiation, vapor pressure deficit, and elevated CO\({}_{2}\), respectively. \(\varepsilon_{s}\) is an unobservable random variable with a mean of zero and a variance of \(\sigma^{2}>0\), which is called the Error term and is usually assumed \(\varepsilon\sim N(0,\sigma^{2})\). If a total of \(n\) years of time series data have been obtained in the existing years of history, there are:  
\[\left\{ \begin{array}{l}GPP_{s1}=\beta_{0s}+\beta_{1s}TEMP_{1}+\beta_{2s}PREC_ {1}+\beta_{3s}SRAD_{1}+\beta_{4s}eCO_{21}+\varepsilon_{s1}\\ GPP_{s2}=\beta_{0s}+\beta_{1s}TEMP_{2}+\beta_{2s}PREC_{2}+\beta_{3s}SRAD_{2}+ \beta_{4s}eCO_{22}+\varepsilon_{s2}\\ \vdots\\ GPP_{sn}=\beta_{0s}+\beta_{1s}TEMP_{n}+\beta_{2s}PREC_{n}+\beta_{3s}SRAD_{n}+ \beta_{4s}eCO_{2n}+\varepsilon_{sn}\end{array} \right. \tag{4}\]  
\[Y_{s}=\left[ \begin{array}{l}GPP_{s1}\\ GP_{s2}\\ \vdots\\ GP_{sn}\end{array} \right],\;\beta_{s}=\left[ \begin{array}{l}\beta_{0s}\\ \beta_{1s}\\ \vdots\\ \beta_{4s}\end{array} \right],\;\varepsilon_{s}=\left[ \begin{array}{l}\varepsilon _{s1}\\ \varepsilon_{s2}\\ \vdots\\ \varepsilon_{sn}\end{array} \right] \tag{5}\]  
\[X=\left[ \begin{array}{cccc}1&TEMP_{1}&PREC_{1}&SRAD_{1}&eCO_{21}\\ 1&TEMP_{2}&PREC_{2}&SRAD_{2}&eCO_{22}\\ 1&\vdots&\vdots&\vdots&\vdots\\ 1&TEMP_{n}&PREC_{n}&SRAD_{n}&eCO_{2n}\end{array} \right] \tag{6}\]  
The above formula is expressed as matrix representation:  
\[\left\{ \begin{array}{l}Y_{s}=X\beta_{s}+\varepsilon_{s}\\ \varepsilon_{s}\sim N(1,\sigma^{2}In)\end{array} \right. \tag{7}\]  
Using the least squares method to calculate the estimated values of regression parameters, by minimizing the sum of squares of errors and seeking the best functional match of the data, the optimal estimation value of the regression parameters is obtained to minimize the sum of squares of deviations:  
\[Q(\widehat{\beta_{s}})=\sum_{i=1}^{n}{(GPP_{s1}-\widehat{\beta_{0s}}- \widehat{\beta_{1s}}TEMP_{i}-\widehat{\beta_{2s}}PREC_{i}-\widehat{\beta_{3s}} SRAD_{i}-\widehat{\beta_{4s}}eCO_{2i})^{2}} \tag{8}\]  
According to the principle of extreme value in calculus, \(\widehat{\beta}\) should meet the following equations:\[\left\{ \begin{array}{l}\left.\begin{array}{l}\frac{\partial Q}{\partial\beta_{0}} \Big{|}_{\beta_{\text{bs}}=\widehat{\beta_{0}}}=-2\sum\limits_{i=1}^{n}\left( GPP_{si}-\widehat{ GPP_{si}}\right)=0\\ \frac{\partial Q}{\partial\beta_{1}}\Big{|}_{\beta_{1s}=\widehat{\beta_{1s}}}=- 2\sum\limits_{i=1}^{n}\left(GPP_{si}-\widehat{GPP_{si}}\right)TEMP_{i}=0\\ \frac{\partial Q}{\partial\beta_{2}}\Big{|}_{\beta_{2s}=\widehat{\beta_{2s}}}= -2\sum\limits_{i=1}^{n}\left(GPP_{si}-\widehat{GPP_{si}}\right)PREC_{i}=0\\ \frac{\partial Q}{\partial\beta_{3}}\Big{|}_{\beta_{is}=\widehat{\beta_{sa}}}= -2\sum\limits_{i=1}^{n}\left(GPP_{si}-\widehat{GPP_{si}}\right)SRAD_{i}=0\\ \frac{\partial Q}{\partial\beta_{4}}\Big{|}_{\beta_{4s}=\widehat{\beta_{4s}}}= -2\sum\limits_{i=1}^{n}\left(GPP_{si}-\widehat{GPP_{si}}\right)eCO_{2i}=0 \end{array}\right.\end{array} \right.