## 3 Materials and Methods
### Methodology Framework
#### 3.2.1 Data Processing  
SAR Data Processing  
To quantify ground displacement across the study area from January 2015 to December 2022, we processed 432 C-band Sentinel-1 Synthetic Aperture Radar (SAR) images, comprising 234 ascending- and 198 descending-orbit acquisitions. The Sentinel-1 Single  
Figure 4: Technical roadmap of this study.  
Look Complex (SLC) products, provided by the European Space Agency (ESA), undergo pre-processing such as instrument processing facility (IPF) that includes mitigation of Radio Frequency Interference (RFI) to ensure data quality [61]. The dual-orbit strategy aimed to (i) mitigate SAR geometrical distortions (e.g., layover and shadowing) and (ii) cross-validate displacement estimates and land subsidence identifications through multi-geometry analysis. Employing the Small Baseline Subset (SBAS) technique via the LiCSBAS package [62], we generated interferometric pairs using maximum spatial baselines of 100 m (ascending) and 200 m (descending). Interferograms were multi-looked with a 4:1 (range: azimuth) ratio to enhance phase stability, achieving detection thresholds for slope movements as small as 100 m spatially. After filtering noisy unwrapped phases, displacement time series were reconstructed from high-coherence interferometric pairs, with each epoch linked to 3-4 temporally adjacent SAR scenes. The LiCSBAS workflow includes quality control measures, such as coherence-based selection and phase filtering, which further minimize the impact of any residual noise, including potential RFI effects. Temporal coherence degradation was observed during two intervals: (i) post-December 2021, following Sentinel-1B's orbital retirement, and (ii) mid-June to July 2019 due to ascending data gaps. The spatiotemporal baseline configuration of the SBAS network is illustrated in Figure 5a (ascending) and Figure 5b (descending).  
A common decomposition (assuming eastward and vertical motion dominate or ignoring the north component if geometry and data availability permit) is as follows:  
\[\mathrm{v_{asc}}=\mathrm{v_{x}\cdot sin}\theta_{\mathrm{asc}}\cdot\cos\alpha_{ \mathrm{asc}}+\mathrm{v_{y}\cdot sin}\theta_{\mathrm{asc}}\cdot sin\alpha_{ \mathrm{asc}}+\mathrm{v_{z}\cdot cos}\theta_{\mathrm{asc}} \tag{1}\]  
\[\mathrm{v_{desc}}=\mathrm{v_{x}\cdot sin}\theta_{\mathrm{desc}}\cdot\cos \alpha_{\mathrm{desc}}+\mathrm{v_{y}\cdot sin}\theta_{\mathrm{desc}}\cdot sin \alpha_{\mathrm{desc}}+\mathrm{v_{z}\cdot cos}\theta_{\mathrm{desc}} \tag{2}\]  
One way to solve explicitly for \(v_{z}\) and (for instance) \(v_{x}\) is by arranging these equations in matrix form or by applying known trigonometric simplifications [63; 64]. A frequently cited simplified pair (assuming motion is primarily east (\(v_{x}\)) and vertical (\(v_{z}\))) is as follows:  
\[\begin{bmatrix}\mathrm{v_{asc}}\\ \mathrm{v_{desc}}\end{bmatrix}=\begin{bmatrix}\sin\theta_{\mathrm{asc}}\cdot \cos\alpha_{\mathrm{asc}}&\cos\theta_{\mathrm{asc}}\\ \sin\theta_{\mathrm{desc}}\cdot\cos\alpha_{\mathrm{desc}}&\cos\theta_{\mathrm{ desc}}\end{bmatrix}\begin{bmatrix}\mathrm{v_{x}}\\ \mathrm{v_{z}}\end{bmatrix} \tag{3}\]  
One can invert this 2 \(\times\) 2 system (when the look angles differ sufficiently) to isolate \(v_{x}\) and \(v_{z}\). For more general 3D decompositions (east, north, and up), additional constraints or multi-orbit data (multiple incidence/azimuth angles) are typically required [65; 66]. This combination of velocities allows us to calculate vertical velocity for our analysis.  
Figure 5: Baseline plots of the interferograms utilized for the ascending (**a**) and descending (**b**) Sentinel-1 radar satellite.