## 2 Materials and Methods
### Retrieval of EOS  
Numerous methods have been used to fit the NDVI changes from seasonal vegetation cycles. After comparing the fitting results of HANTS [38], Polyfit [39], and Double logistic [40] in the nine subregions, we found that the RMES of HANTS (1.26 \(\pm\) 0.24 \(\times\) 10\({}^{-5}\)) and Polyfit (1.28 \(\pm\) 0.24 \(\times\) 10\({}^{-5}\)) were similar and smaller than the Double logistic results (1.93 \(\pm\) 0.43 \(\times\) 10\({}^{-5}\)) (Figure A1). HANTS and Polyfit, were therefore selected as the two most simple and effective methods to fit the NDVI change curves. Dynamic thresholds were adopted to determine the EOS. Further details of these two fitting methods are described below.  
The HANTS method involves the harmonic analysis of a time series, is adapted from the fast Fourier transform, and eliminates cloud noise using the least square method [38; 40]. The HANTS methods can quickly smooth the data, remove outliers, and fill gaps of missing data. The following Equation (1) was used to fit the NDVI seasonal fluctuation curve:  
\[NDVI(t)=a_{0}+\sum_{i=1}^{n}a_{i}cos(2\pi t-\varphi_{i}) \tag{1}\]  
where \(t\) is the Julian date, \(a_{0}\) is the average of all \(NDVI\) observations, and \(\varphi_{i}\) and \(a_{i}\) are the phase and amplitude of the curve, respectively.  
\begin{table}
\begin{tabular}{c c c c c} \hline ID & **Subregion Names** & **AGDD\({}_{0}\) Means (\({}^{\circ}\)C)** & **MI Means** & **Main Provines** \\ \hline I & Alpine temperate steppe of the Qinghai Lake Basin & 1311.45 & 0.62 & Qinghai, Gansu \\ \hline II & Alpine meadow steppe on the Zoige Plateau & 981.29 & 1.01 & Qinghai, Sichuan \\ \hline III & Alpine meadow steppe on the Yushu-Naqua Plateau & 670.04 & 0.91 & Qinghai, Tibet \\ \hline IV & Alpine meadow steppe on the sources of the Yangtze and Yellow rivers & 496.14 & 0.57 & Qinghai \\ \hline V & Alpine and cold grassland on the Southern Chang Tang Plateau & 824.56 & 0.45 & Tibet \\ \hline VI & Alpine temperate grassland of the Brahnapputra River Basin & 917.33 & 0.59 & Tibet \\ \hline VII & Alpine and cold grassland on the Northern Chang Tang Plateau & 618.61 & 0.38 & Tibet \\ \hline VIII & Alpine and cold grassland on the Upper Indus River Basin & 827.01 & 0.24 & Tibet \\ \hline IX & Alpine and cold desert grassland of the Kumbian Mountains & 571.07 & 0.35 & Tibet, Xinjiang \\ \hline X & Alpine desert in the Qaidam Basin & 1699.63 & 0.18 & Qinghai \\ \hline XI & Alpine forestland in the Hengluan Mountain & 2043.25 & 1.14 & Sichuan, Yunnan \\ \hline XII & Subtropical forestland in the southern Tibet & 3941.97 & 1.86 & Tibet \\ \hline \end{tabular}
\end{table}
Table 1: Description of the Qinghai-Tibetan Plateau subregions. \(\text{AGDD}_{0}\): Annual accumulated temperature above 0 \({}^{\circ}\)C. MI: Moisture index calculated by the ratio of the mean annual precipitation to the annual equilibrium evapotranspiration.  
The Polyfit method adopts a polynomial function to fit the _NDVI_ records [39]. The following sixth order Equation (2) is used to describe the _NDVI_ curve:  
\[NDVI(t)=a_{0}+a_{1}t+a_{2}t^{2}+\ldots+a_{6}t^{6} \tag{2}\]  
where \(a_{0}\)-\(a_{6}\) are regression coefficients determined using the Levenberg-Marquardt method.  
The EOS was determined by the day when the smoothed curve of the 34-year mean passed a designated threshold. We first fitted the _NDVI_ changes with HANTS and Polyfit methods and then calculated the NDVI\({}_{\text{ratio}}\) (described in Equation (3)) for 365 days with multi-year mean _NDVI_ values, next detected the time \(t\) with the minimum \(NDVI_{\text{ratio}}\) and used the corresponding \(NDVI(t+1)\) at time \((t+1)\) as the _NDVI_ threshold for the EOS. Finally, we obtained the EOS for 34 years using the threshold:  
\[NDVI_{\text{ratio}}(t)=\frac{NDVI(t+1)-NDVI(t)}{NDVI(t)} \tag{3}\]