# Climate, CO2, and Anthropogenic Drivers of Accelerated Vegetation Greening in the Haihe River Basin
## 2 Materials and Methods
### Methods  
To estimate the effects of CO\({}_{2}\) concentration, climate change, and human activities on NDVI variation, an analytical framework of attribution analysis [49; 50] was adopted. The multivariate function describing the relationship between NDVI and driven factors can be expressed as follows:  
\[NDVI=f(Tem,Pre,Rad,AAF,NTL,CO_{2}) \tag{1}\]  
where Tem is the air temperature (\({}^{\circ}\)C), Pre is the precipitation (mm/yr), Rad is the solar radiation (W/m\({}^{2}\)), AAF is the accumulative area of afforestation (hm\({}^{2}\)), NTL is nighttime light (dimensionless), and CO\({}_{2}\) is atmospheric CO\({}_{2}\) concentration (ppm). We assumed that there is a linear formulation between NDVI and the driving variables, and the first-order partial derivative equation is deduced as follows:  
\[dNDVI= \left(\frac{\partial NDVI}{\partial tem}dTem+\frac{\partial NDVI}{ \partial pre}dPre+\frac{\partial NDVI}{\partial Rad}dRad\right)+\left(\frac{ \partial NDVI}{\partial AAAF}dAAF+\frac{\partial NDVI}{\partial NTL}dNTL\right) \tag{2}\] \[+\left(\frac{\partial NDVI}{\partial tc_{2}}dCO_{2}\right)+\epsilon\] \[\frac{dNDVI}{dt}= \left(\frac{\partial NDVI}{\partial dm}\frac{dTem}{dt}+\frac{ \partial NDVI}{\partial pre}\frac{dpre}{dt}+\frac{\partial NDVI}{\partial Rad }d\frac{dRad}{dt}\right)+\left(\frac{\partial NDVI}{\partial AAAF}\frac{dAAF}{ dt}+\frac{\partial NDVI}{\partial NTL}\frac{dNTL}{dt}\right)\] (3) \[+\left(\frac{\partial NDVI}{\partial tc_{2}}\frac{dCO_{2}}{dt} \right)+\epsilon\]  
where \(\frac{dNDVI}{dt}\), \(\frac{dTem}{dt}\), \(\frac{dPre}{dt}\), \(\frac{dRad}{dt}\), \(\frac{dAAF}{dt}\), \(\frac{dNTL}{dt}\), and \(\frac{dCO_{2}}{dt}\) are the slopes of the annual Tem, Pre, Rad, AAF, NTL, and CO\({}_{2}\) from 2000 to 2018. \(\frac{\partial NDVI}{\partial tem}\), \(\frac{\partial NDVI}{\partial pre}\), \(\frac{\partial NDVI}{\partial had}\), \(\frac{\partial NDVI}{\partial AAF}\), \(\frac{\partial NDVI}{\partial NTL}\), and \(\frac{\partial NDVI}{\partial SC_{2}}\) are the ridge regression coefficients of the annual Tem, Pre, Rad, AAF, NTL, and CO\({}_{2}\) with NDVI, respectively [1]. The parameter value of ridge regression with the lowest GCV (cross validation criteria) was chosen as the optimal value. It was set to 2 in this study. As given in Equation (3), the absolute change in NDVI is a linear sum of the absolute changes in Tem, Pre, Rad, AAF, NTL, and CO\({}_{2}\).  
We used Tem, Pre, and Rad to describe the elements of climate change, whereas AAF and NTL were employed to convey the composition of human or social activities. Thus, the contribution of climate change (CLI), human activities (HUM), and CO\({}_{2}\) to the variation in NDVI can be expressed as follows:  
\[Slope\ of\ NDVI=CLI+HUM+CO_{2} \tag{4}\]  
\[RC=100*\frac{CLI}{|CLI|+|HUM|+|CO_{2}|} \tag{5}\]  
\[RH=100*\frac{HUM}{|CLI|+|HUM|+|CO_{2}|} \tag{6}\]  
\[RCO_{2}=100*\frac{CO_{2}}{|CLI|+|HUM|+|CO_{2}|} \tag{7}\]  
where RC, RH, and RCO\({}_{2}\) (%) represent the relative contribution rates of climate change, human activities, and elevated atmospheric CO\({}_{2}\) concentration to NDVI variation, respectively.