3.2.4. Trend Analysis

The Mann–Kendall test and Theil–Sen median analysis were used to explore the variation trend in snow phenology from hydrological years 2001–2018. In the Mann– Kendall test, to calculate whether the trend of snow phenology was increasing or decreasing at the 0.05 confidence level, the *Z* value was divided into five types: significant decrease (*Z* < 1.96), slow decrease ( −1.96 ≤ *Z* < 0), nonsignificant change ( *Z* = 0), slow increase (0 > *Z* ≥ 1.96) and significant increase ( *Z* > 1.96). The formulas are as follows:

$$Z = \begin{cases} \frac{S - 1}{\sqrt{\text{var}(S)}}, S > 0\\ 0 \quad \text{,} S = 0\\ \frac{S + 1}{\sqrt{\text{var}(S)}}, S < 0 \end{cases} \tag{7}$$

where,

$$var(S) = \frac{n(n-1)(2n+5)}{18} \tag{8}$$

$$S = \sum\_{i=1}^{n-1} \sum\_{j=i+1}^{n} \text{sgn}\left(\mathcal{S}\_{j} - \mathcal{S}\_{i}\right) \tag{9}$$

$$\text{sgn}(S\_{\dot{j}} - S\_{\dot{i}}) = \begin{cases} \text{1}, S\_{\dot{j}} - S\_{\dot{i}} > 0 \\ \text{0}, S\_{\dot{j}} - S\_{\dot{i}} = 0 \\ -\text{1}, S\_{\dot{j}} - S\_{\dot{i}} < 0 \end{cases} \tag{10}$$

When *Z* > 0, the trend is upward, and when *Z* < 0, it is downward. *Si*/*Sj* represent the value in years *i/j*, n is the length of the time series. When | *Z*| > *Z*1−*<sup>α</sup>*/2 (α is the significance level), the trend is significant in the time series. In this paper, α = 0.05 was used.

In the Theil–Sen median analysis, to explore the details of the variations in snow phenology, *Ssnow* was divided into seven types: < −4 d/a, -< −4– −2 d/a, −2–0 d/a, 0 d/a, 0–2 d/a, 2–4 d/a and >4 d/a. The formula is as follows:

$$S\_{\text{suow}} = \operatorname{Median} \left( \frac{S\_j - S\_{\bar{i}}}{j - i} \right), \quad \forall j > i \tag{11}$$

where *Ssnow* > 0 and *Ssnow* < 0 represent upward and downward trends, respectively.

3.2.5. Relative Importance of Multiple Factors to Snow Phenology

The use of geodetector is a common statistical approach that can analyze spatial variations and reveal the driving factors behind them [36]. A geodetector contains four subdetectors: factor detector, risk detector, ecological detector and interaction detector [37]. In this research, we employed a factor detector to quantify the relative contributions of vegetation and geographical (altitude, slope, aspect, latitude and longitude) and meteorological (temperature and precipitation) factors to snow phenology variations, and the dominant driving factor was then the highest contributor to snow phenology variations.

The factor detector is calculated by the following *q*-statistic:

$$q = 1 - \frac{\sum\_{h=1}^{L} N\_h \sigma\_h^2}{N\sigma^2} = 1 - \frac{SSW}{SST} \tag{12}$$

$$SSW = \sum\_{h=1}^{L} N\_h \sigma\_{h'}^2 \text{ } SST = \text{ } N\sigma^2 \tag{13}$$

where 0 ≤ *q* ≤ 1, and the larger the value, the greater the influence of the factor. *h* is the number of strata for variables or factors, *N* represents the number of units in stratum *h*, and *σ*<sup>2</sup> and *σ*<sup>2</sup> *h* denote the variance in the entire study area and stratum h, respectively. *SW* and *SST* are the sum of squares within the data and the total sum of squares, respectively.
