## 2 Methods
### Numerical Indexing of Related Spatial Factors
#### 2.2.1 Terrain Category  
* Elevation, slope, and aspect  
On the basis of the framework of the slope unit, the highest elevation in each unit was extracted and represented as the elevation factor, as shown in Equation (1). According to the height change caused by the horizontal movement distance, the slope factor is expressed by a tangent function on average, as indicated in Equation (2). The aspect factor refers to the direction of the maximum elevation change in the slope unit. It is calculated by the angle with the true north direction, as shown in Equation (3), where the true north direction is 0\({}^{\circ}\), and the angle increases to 360\({}^{\circ}\) in the clockwise direction. \[I_{elevation}=\max(Z_{i}) \tag{1}\]  
\[I_{slope}=\tan\overline{\theta_{s}}=\frac{\Delta\overline{Z}}{\Delta \overline{L}} \tag{2}\]  
\[I_{aspect}=\frac{180^{\circ}}{\pi}\tan^{-1}[\max(\theta_{s})] \tag{3}\]  
where \(Z_{i}\) is elevation, \(\Delta\overline{Z}\) is the mean elevation difference, \(\Delta\overline{L}\) is the mean horizontal distance, and \(\theta_{s}\) is the main slope angle. * Terrain roughness  
Terrain roughness represents the degree of height change. When the undulating terrain faces the effect of large gravity, the smaller resistance force makes the slope have a higher possibility of landslide. The elevation standard deviation \(\sigma\) is used to describe the degree of elevation change in the slope unit (Equation (4)). \[\sigma=\sqrt{\frac{\sum_{i}(Z_{i}-\overline{Z})^{2}}{n_{s}-1}} \tag{4}\]  
where \(\overline{Z}\) is the average elevation in a slope unit, and \(n_{s}\) is the number of grids in the slope unit. * Profile curvature  
The profile curvature is expressed as the slope steepness. This study used the spatial analysis module of the software ArcMap to calculate the profile curvature of each slope unit on the basis of a 3 \(\times\) 3 moving grid, which is the default grid size in ArcMap. A negative (positive) value of the curvature represents a convex (concave) slope. * Vegetation index  
Plants can effectively stabilize the rock and soil on slopes, but the exposed soil area may suffer from repeated landslide and displacement problems. Hence, the vegetation index is defined as the proportion of vegetation area in the slope unit, as shown in (Equation (5)). \[I_{\text{veg.}}=\frac{A_{\text{veg.}}}{A_{s}} \tag{5}\]  
where \(A_{\text{veg. }}\) is the area of the vegetation and \(A_{s}\) is the area of the slope unit. * Annual displacement velocity gradient of InSAR  
InSAR technology calculates the phase difference to estimate the displacement of the ground through more than two periods of SAR observations. The InSAR-derived ground displacement can be regarded as a direct observation of ground stability and was thus proposed as an essential index for landslide susceptibility analysis in this study. However, the original displacements from InSAR observations suffer from various influencing factors, such as vegetation changes and orbital variations of SAR satellites. In order to reduce the periodical or systematic noises due to those uncontrollable factors and to extract ameaningful index for evaluating the ground stability, the annual velocity gradients derived from InSAR displacement fields were used in this study. First, the annual displacement information of InSAR is placed in the range from \(-1\) to \(1\) by mean normalization, which is shown in Equation (6), to unify the scale and reduce the systematic error of InSAR data. \[Z_{\overline{\varsigma_{i}}}=\frac{Z_{\overline{\varsigma_{i}}}-\mu}{\max(Z_{ \overline{\varsigma_{i}}})-\min(Z_{\overline{\varsigma_{i}}})} \tag{6}\]  
where \(Z_{\overline{\varsigma_{i}}}\) is the normalized InSAR displacement value, \(Z_{\overline{\varsigma_{i}}}\) is the annual displacement of InSAR, and \(\mu\) is the average annual displacement. The annual displacement velocity of InSAR is obtained as the slope value in first-order linear fitting (Equation (7)).