## 3 Methodology
### Theil-Sen Median Trend Analysis and Mann-Kendall Test  
In this paper, Theil-Sen median trend analysis and the Mann-Kendall method were employed to examine the properties of the geographical distribution, temporal variation characteristics and variation trend of the TVMPDI in southeastern Tibet. These two methods were combined by many studies to examine the trend of lengthy regression analysis [48; 49; 50].  
The Theil-Sen median trend assay is an adjustment that could abate the access of abstract outliers [51]. The Theil-Sen median trend calculates the average abruptness of \(n(n-1)/2\) abstract portfolios (\(n\) is the arrangement length), and its blueprint is as follows:  
\[S_{TVMPDI}=Median\left(\frac{TVMPDI_{j}-TVMPDI_{i}}{j-i}\right),\;2003\leq i<j \leq 2018 \tag{7}\]  
\(S_{TVMPDI}>0\) indicates that the TVMPDI presents a growing trend; otherwise, the TVMPDI presents a degenerate trend.  
The Mann-Kendall assay is a non-parametric statistical analysis adjustment that is acclimated to adjudicate the acceptation of trend. It is not all-important for samples to chase an assertive distribution, and it is not diminished by a few outliers [52]. The calculation formula is as follows:  
\[Z=\left\{\begin{array}{l}\frac{S-1}{\sqrt{s}},S>0\\ 0,S=0\\ \frac{S-1}{\sqrt{s}},S<0\end{array}\right. \tag{8}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Drought Classification** & **TVMPDI** \\ \hline No dry & 0.50 \textless{} TVMPDI \\ Light dry & 0.40 \textless{} TVMPDI \textless{} 0.50 \\ Moderate dry & 0.30 \textless{} TVMPDI \textless{} 0.40 \\ Sever dry & 0.20 \textless{} TVMPDI \textless{} 0.30 \\ Extreme dry & TVMPDI \textless{} 0.20 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Drought classification.  
\[\mathrm{S}=\sum_{\mathrm{j=1}}^{\mathrm{n-1}}\sum_{\mathrm{i=j+1}}^{\mathrm{n}} \mathrm{sgn}(\mathrm{TVMDPI_{j}-TVMPDI_{i}}) \tag{9}\]  
\[\mathrm{sgn}(\mathrm{TVMDPI_{j}-TVMPDI_{i}})=\left\{\begin{array}{l}1,\mathrm{ TVMDPI_{j}-TVMPDI_{i}>0\\ 0,\mathrm{TVMDPI_{j}-TVMPDI_{i}=0\\ -1,\mathrm{TVMDPI_{j}-TVMPDI_{i}<0\end{array}}}\\ \mathrm{s}=\mathrm{n}\frac{(2\mathrm{n}+5)(\mathrm{n}-1)}{18}\end{array}\right. \tag{10}\]  
where \(\mathrm{TVMPDI_{i}}\) and \(\mathrm{TVMPDI_{j}}\) represent the TVMPDI value of pixel years i and j; n is the duration; and sgn is the symbol operation. The statistic Z ranges from \(-\infty\) to \(+\infty\).  
The mean TVMPDI of the five cities in the study area from 2003 to 2018 was statistically analyzed. When adding more hours to the TVMDI in the study area, it shows that drought intensifies; so, when the linear regression trend for timing (TVMPDI over time will increase), the drought trend becomes gradual over time, and when there is a yuan linear regression trend for negative, the TVMDI gradually decreases with time, with the drought increasing gradually over time.  
These two methods were accumulated to reflect the spatial administration characteristics of the TVMPDI trends in southeastern Tibet from 2003 to 2018. Since there is basically no range, \(\mathrm{ST_{VMPDI}}\) is equal to 0, with values between \(-\)0.0005 and 0.0005 classified as stable and values greater than or equal to 0.0005 classified as the advance area. \(\mathrm{ST_{VMPDI}}\) values lower than \(-\)0.0005 are classified according to the absolute values of \(\mathrm{ST_{TVMPDI}}\). The acceptation assay after-effects of the Mann-Kendall assay at 0.05 were divided into cogent change (Z > 1.96 or Z < \(-\)1.96) and bush change (\(-\)1.96 \(\leq\) Z \(\leq\) 1.96). Finally, the allocation after-effects of the Theil-Sen median trend assay and the allocation after-effects of the Mann-Kendall assay were superimposed to access the TMVPDI trend abstracts at the pixel scale, and the after-effects were divided into bristle change types (Table 4).