## 2 Materials and Methods
### Methods
#### 2.3.5 Sensitivity Analysis and Contribution Rate Calculation  
The sensitivity coefficient method is used to analyze the sensitivity of climate factors to \(ET_{0}\) quantitatively [38]. The commonly used sensitivity analysis method is to assume that other variables are fixed, and to evaluate the impact of a variable change on the model output. The sensitivity coefficient of the \(ET_{0}\) to a climate variable \(x\) is defined by [39; 40]:  
\[S_{x}=\underset{\Delta x\to 0}{lim}\left(\frac{\Delta ET_{0}/ET_{0}}{ \Delta x/x}\right)=\frac{\partial ET_{0}}{\partial x}\cdot\frac{x}{ET_{0}} \tag{8}\]  
A first-order Taylor series approximation was applied to calculate \(S_{x}\)[41; 42]:  
\[S_{x}=\frac{\Delta ET_{0}}{\Delta x}\cdot\frac{x}{ET_{0}} \tag{9}\]  
where \(S_{x}\) is the sensitivity coefficient of the climate variable \(x\), \(\Delta x\) is the relative change of climate factor \(x\), and \(\Delta ET_{0}\) is the relative change in \(ET_{0}\) induced by \(\Delta x\).  
In this study, \(ET_{0}\) was recalculated by making a \(\pm\)10% change to each meteorological factors (solar radiation (\(RS\)), relative humidity (\(RH\)), air temperature (\(T_{man}\)) and wind speed (\(U_{2}\))) while keeping the other factors constant; sensitivity coefficient \(S_{x}\) was calculated from Equation (9). A positive value of \(S_{x}\) indicates that \(ET_{0}\) increases as \(x\) increases, and vice versa. The greater the absolute value of the sensitivity coefficient, the greater the effect of the variable on \(ET_{0}\).  
The sensitivity coefficient reflects the influence of meteorological variables on \(ET_{0}\), but it is not equal to the \(ET_{0}\) change caused by a certain variable. Here, the contribution rate is introduced to analyze the change of \(ET_{0}\) caused by a single meteorological factor. It is the product of the sensitivity coefficient of a climate factor and its multi-year change rate [11]. A positive contribution means that the variety of a meteorological factor has a positive effect on the variety of \(ET_{0}\) and vice versa. The larger the absolute value is, the greater the impact on \(ET_{0}\) will be. The contribution rate was calculated using the following formulas [43,44]:  
\[CR_{x}=S_{x}\cdot RC_{x} \tag{10}\]  
\[RC_{x}=\frac{n\cdot a_{x}}{\overline{x}}\times 100\% \tag{11}\]  
where \(CR_{x}\) is the contribution rate climate variable \(x\) to \(ET_{0}\); \(RC_{x}\) is the multi-year change rate of \(x\); \(n\) is the number of years; \(a_{x}\) represents the trend rate of meteorological factor \(x\) during the corresponding study period; \(\overline{x}\) is the multi-year absolute average value of \(x\) during the evaluation period. In this paper, we calculated the sensitivity of \(ET_{0}\) to \(RS\), \(RH\), \(T_{men}\) and \(U_{2}\), and calculated the contribution rate of these four climatic factors to \(ET_{0}\). The evaluation period includes three periods: during the period from 1979 to 2018 and two periods before and after the mutation point.  
\[C_{-}(ET_{0})=CR_{RS}+CR_{RH}+CR_{Tmean}+CR_{U2}+\varepsilon \tag{12}\]  
where \(CR_{RS}\), \(CR_{RH}\), \(CR_{Tmean}\), \(CR_{U2}\) are the individual contributions to the long-term trend in \(ET_{0}\) as a result of the change in \(RS\), \(RH\), \(T_{man}\) and \(U_{2}\), respectively. \(C_{-}(ET_{0})\) is the sum of the contribution of variation in \(RS\), \(RH\), \(T_{man}\) and \(U_{2}\) to the change in \(ET_{0}\). \(\varepsilon\) is the error between \(C(ET_{0})\) and \(ET_{0}\) trend calculated by P-M model (\(LR_{-}(ET_{0})\)).