## 2 Materials and Methods
### Sentinel-1 SAR Interferometry  
#### 2.3.1 Basic Concept  
SAR is an active imaging system where the sensor emits microwave radiation to illuminate the Earth's surface and receives the reflected signal, called backscatterer. It works any time of the day and in any cloud-cover condition towards the surface. The backscattered signals are expressed as an image of which each pixel contains both amplitude (\(A\)) and phase (\(\varphi\)) information, recorded as complex numbers (\(c\)) [46]:  
\[c=Ae^{i\varphi} \tag{1}\]  
where \(i\) is the imaginary number. By pairing two SAR images obtained at distant periods with disparate look angles over the same location, one image is set as master (\(m\)) and the other as slave (\(s\)), an interferogram (\(I\)) is generated. An interferogram is computed by multiplying the amplitude of both images and calculating the difference in phase (Equation (2)). The InSAR technique exploits the phase difference between \(\varphi_{m}\) and \(\varphi_{s}\), named interferometric phase (\(\varphi_{ip}\)), expressed in Equation (3). \[I=c_{m}c_{s}=A_{m}A_{s}e^{i(\varphi_{m}-\varphi_{s})} \tag{2}\]  
\[\varphi_{m}-\varphi_{s}=\varphi_{ip}=\varphi_{elev}+\varphi_{defo}+\varphi_{flat} +\varphi_{atm}+\varphi_{noise} \tag{3}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Application** & **Test Site** & **Method** & **Image Pair** & **BTemp (Days)** & **BPerp (m)** \\ \hline DEM & A & InSAR & 20180321\(-\)0402 & 12 & 170 \\ Earthquake & B & DInSAR & 20171023\(-\)1104 & 12 & 33 \\ Tunneling & C & A-DInSAR & 20171104\(-\)1116 & 12 & 67 \\ Landside & D & A-DInSAR & Figure 4a & 12\(-\)576 & 1\(-\)117 \\ \multicolumn{1}{c}{Landslide} & D & A-DInSAR & Figure 4b & 12\(-\)372 & 1\(-\)125 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: SAR dataset used. Figure 4: Master-slave configuration for (**a**) tunneling-induced ground deformation in Site C and (**b**) rainfall-induced landslide in Site D. The time interval between successive SAR acquisition is equal to or less than 24 days. All perpendicular baselines for each interferometric pair were less than 150 m. Each point in the graphs represents a SAR image and each line represents an interferometric pair. Horizontal and vertical axes define the temporal (BTemp) and perpendicular (BPerp) baselines, respectively. where \(\varphi_{elev}\) is the topographic phase, \(\varphi_{d\varphi o}\) is the phase contribution due to surface deformation, \(\varphi_{flat}\) is the flat-earth phase, \(\varphi_{atm}\) is the atmospheric phase screening, and \(\varphi_{noise}\) is the phase noise related to some processing-induced factors. Through interferometry, the desired phase contributor, which is typically the elevation or the surface deformation, can be isolated from the other phase components. However, the final quantitative measurements of elevation or surface deformation from interferometry will greatly depend of the quality of the generated interferogram. While the SAR technology creates satellite images that cover almost the entire planet, not every resolution cell contains valuable information. The main condition for the InSAR application is the degree of coherence. When ground targets or objects change between acquisition periods, the coherent sum of backscattered signals varies resulting from coherence degradation due to decorrelation. To quantify the impact of decorrelation for our InSAR application, the coherence (\(\gamma\)) is calculated:  
\[\gamma=\frac{\left|\sum_{n=1}^{N}c_{m}c_{s}e^{i(\varphi_{m}-\varphi_{n})}\right| }{\sqrt{\sum_{n=1}^{N}}|c_{m}|^{2}\Sigma_{n=1}^{N}|c_{s}|^{2}} \tag{4}\]  
where \(N\) is the coherence window size. The calculated coherence indicates how identical each pixel is between master and slave scenes from a scale of 0 (completely out of phase) to 1 (completely in phase). Any ground deformation associated with natural hazard generates different radar signal paths for the same ground target between two observation periods.