## 2 Methods
### Data Processing Method
#### 2.3.1 SAR Data Processing Method  
To acquire the deformation spatio-temporal evolution law of the slopes in the deep excavated expansive-soil canal section of the South-North Water-Diversion Middle-Line Project, we collected 11 TerraSAR images from ascending track 51. These images were collected between 10 June 2020 and 7 February 2021. Initially, we employed DInSAR technology to process all eligible interferometric pairs. Subsequently, the SBAS-InSAR method (as shown in Figure 5) was utilized to extract the timeseries deformation of the deep excavation expansive-soil channel slope. During the utilization of DInSAR technology, Gamma software [25] was employed, and a multilook ratio of 10:2 in the range and azimuth directions was set to effectively mitigate noise to some extent. To ensure relatively high co-registration accuracy, a combination of methods accounting for scene topography and spectral diversity, considering the interferometric phase of the burst-overlap region, was adopted [26]. Precise orbits provided by the agency and a 30 m shuttle radar topography mission digital elevation model were utilized to compensate for topography effects [27]. A power spectrum filter was applied to reduce the impact of phase noise [28]. Subsequently, the interferograms were unwrapped using the branch cut method [29], followed by the WGS84 geographic coordinates. Using the SBAS-InSAR technology, we used the open-source MintPy package to process the regional timeseries of the deformation by employing the small baseline method [30; 31; 32]. We selected the image taken on 10 June 2020 from all of the SAR images as the master image and registered the other images with the master image. Then, the M-amplitude interferograms that conformed to the spatiotemporal baseline threshold were selected using the freely combined differential interferometry pairs. The interference phase (\(\psi\)) of each interferogram is composed of a reference ellipsoid phase (\(\varphi_{\text{ref}}\)), terrain phase (\(\varphi_{\text{top}}\)), LOS surface deformation phase (\(\varphi_{\text{def}}\)), atmospheric delay phase (\(\varphi_{\text{atm}}\)), and random noise (\(\varphi_{\text{noi}}\)), as shown in Equation (1):  
\[\psi=\varphi_{\text{ref}}+\varphi_{\text{top}}+\varphi_{\text{def}}+\varphi_{ \text{atm}}+\varphi_{\text{noi}} \tag{1}\]  
Figure 5: SBAS-InSAR workflow. After the reference ellipsoid phase removal, topographic phase removal, and minimum-cost flow phase unwinding, the interferogram phase can be expressed as Equation (2):  
\[\Delta\varphi=A\varphi+\Delta\varphi_{z} \tag{2}\]  
where \(\Delta\varphi=\left[\Delta\varphi^{1},\ldots,\Delta\varphi^{M}\right]^{T}\) is the interferometric phase of each interferogram; \(\varphi=\left[\varphi^{2},\ldots,\varphi^{N+1}\right]^{T}\) is the temporal interferometric phase of other images relative to the reference image (assuming that the phase of the reference image is \(\varphi_{1}\)); \(\Delta\varphi_{z}=\left[\Delta\varphi_{z}{}^{1},\ldots,\Delta\varphi_{z}{}^{M }\right]^{T}\) is the residual interferometric phase error; and \(A\) is an \(M\times N\) design matrix, which represents the interferometric combination mode and consists of 1, 0, and \(-1\), where \(-1\) represents the master image, and \(1\) represents the slave image. The least-square norm method can be used to calculate the optimal estimation value of the timeseries of the interferometric phase:  
\[\hat{\varphi}=argmin\left\|W^{1/2}(\Delta\varphi-A\varphi)\right\|_{2}=\left(A ^{T}WA\right)^{-1}A^{T}W\Delta\varphi \tag{3}\]  
where \(\hat{\varphi}\) is the best estimation of the timeseries of the interferometric phase, \(W\) is an \(M\)\(\times\)\(M\) diagonal weight matrix, and the weight matrix method adopted in this paper was the Fisher information matrix (FIM). The interferometric phase of each CT that conformed to the SC threshold can be obtained with DInSAR processing.