## 2 Materials and Methods
### Research Methodology  
Trends in the data were analyzed to assess the spatio-temporal changes. For long-time-scale raster grids, the trends in the unit image elements over time can be calculated by using the one-dimensional linear regression method [28]. In this study, we calculated the slope of each image element in the raster grids for the snow cover and NDVI parameters in the study area (i.e., the inter-annual rate of variation in each parameter) using MATLAB v2019a code. The spatial distribution of the inter-annual variation in each parameter in the study area was obtained from 1981 to 2020 using the following equation:  
\[Slope=\frac{m\times\sum_{i=1}^{m}i\times X_{i}-\sum_{i=1}^{m}i\times\sum_{i=1} ^{m}X_{i}}{m\times\sum_{i=1}^{m}i^{2}-(\sum_{i=1}^{m}X_{i})^{2}} \tag{1}\]  
where \(i\) is the year number, \(m\) is the length of the time series, and \(X_{i}\) is the value of each parameter in the \(i\)-th year. \(Slope>0\) indicates an increasing trend, and \(Slope<0\) indicates a decreasing trend. The significance of changes in slope was tested using the F-test, where we considered whether the trend was significant at the 95% confidence level, i.e., the trend was not significant when \(p\geq 0.05\), whereas the trend was significant when \(p<0.05\). Pearson's correlation coefficients were calculated for correlation analysis between the snow cover phenology and climatic and vegetation factors [14]. Pearson's correlation coefficient measures the closeness of the correlation between two series by the product of their deviations from the respective means. It has the advantages of a simple and intuitive  
Figure 2: Experimental protocol. calculation method, and the correlation coefficient is a standardized dimensionless value that is comparable between different studies. Suppose that we have series \(X\) and series \(Y\):  
\[ \begin{cases}X=\left\{X(k)|k=1,2,\ldots,n\right\}\\ Y=\left\{Y(k)|k=1,2,\ldots,n\right\}\end{cases}  \tag{2}\]  
where \(k\) denotes the sampling points in sequence \(X\) and sequence \(Y\), and \(n\) is the total number of sampling points in sequences \(X\) and \(Y\), then the Pearson's correlation coefficient \(r\) between sequence \(X\) and sequence \(Y\) is calculated as:  
\[r=\frac{\sum_{i=1}^{m}\left(X_{i}-\overline{X}\right)\left(Y_{i}-\overline{Y} \right)}{\sqrt{\sum_{i=1}^{m}\left(X_{i}-\overline{X}\right)^{2}}\cdot\sqrt{ \sum_{i=1}^{m}\left(Y_{i}-\overline{Y}\right)^{2}}} \tag{3}\]  
where \(r\) is the correlation coefficient between \(X\) and \(Y\), \(X_{i}\) is the snow depth parameter in the year \(i\); \(Y_{i}\) is the annual precipitation, annual average temperature, annual average solar radiation, or NDVI in year \(i\); \(\overline{X}\) is the average multi-year snow depth; \(\overline{Y}\) is the average precipitation, annual average temperature, annual average solar radiation, or NDVI; and \(m\) is the number of years in the monitoring period. \(r>0\) denotes a positive correlation between the two variables, \(r<0\) denotes a negative correlation, \(|r|\leq 0.2\) denotes an insignificant correlation, and \(|r|>0.2\) denotes a significant correlation between the two variables. The response of vegetation to climate does not always occur in real time, and thus a time lag effect may be evident in the response of vegetation to changes in climate [29]. In this study, based on [PERSON]'s correlation test and the actual situation on the QTP, we designed a test to clarify the lagged effect of snow on vegetation. The effects of the snow depths were calculated in winter (December-February in the following year), spring (March-May), and the plateau cold season (September-March in the following year) on the spring maximum NDVI and summer maximum NDVI (the warm season maximum NDVI was equal to the summer maximum NDVI).