# Characterizing Micro-Displacements on Active Faults in the Gobi Desert with Time-Series InSAR
## 3 Methodology
### InSAR Technique  
Interferometric synthetic aperture radar (InSAR) is a technique used to quantify ground surface deformation using the differences in the phase of the electromagnetic waves returning to the satellite [33; 34; 35]. The deformation can be measured through the comparison of two radar images, but there are several mechanisms which contribute to a change in the measured phase. Algorithmic approaches have been developed to estimate these phase contributions based on statistical analysis of the data. The typical model for these phase terms is as follows [36; 37]:  
\[\Phi\ =\Phi\ dem+\Phi\ def+\Phi\ atm+\Phi\ orb+\Phi\ scat \tag{1}\]  
where \(\Phi\) is measured phase change, \(\Phi\ dem\) is topographic error, \(\Phi\ orb\) is satellite orbital errors generated by inaccurate orbit determination, \(\Phi\ atm\) is atmospheric phase delay caused by water vapor, \(\Phi\ scat\) is clutter and thermal noise and processing errors, and \(\Phi\ def\) is the deformation, which is a function of radar wavelength (\(\lambda\)) and radar line-of-sight (LoS) distortion (\(\Delta R\)):  
\[\Phi\ def=\frac{4\pi}{\lambda}\Delta R \tag{2}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Fault No.** & **Dipping Direction** & **Vertical Shearing Component** & **Lateral Shearing Component** & **Dip Angle** & **Reference** \\ \hline F\({}_{1}\) & South or North & Reverse & Dextral or sinistral & – & [23; 25] \\ F\({}_{6}\) & South or North & Reverse & Sinistral & 60\({}^{\circ}\)–75\({}^{\circ}\) & [27; 28] \\ F\({}_{14}\) & Northwest & Normal & Sinistral & 63\({}^{\circ}\) & [25] \\ F\({}_{15}\) & Northwest or Southeast & Normal & Sinistral & \(\approx\)90\({}^{\circ}\) & [25] \\ F\({}_{16+1}\) & Northwest & Normal & – & – & [25] \\ F\({}_{16+2}\) & Northwest & Normal & Sinistral & 75\({}^{\circ}\) & [25] \\ F\({}_{17}\) & Northwest & Normal & Sinistral & 80\({}^{\circ}\) & [25] \\ F\({}_{18}\) & Northwest or Southeast & Normal & – & 70\({}^{\circ}\)–80\({}^{\circ}\) & [23; 25] \\ F\({}_{19}\) & Northwest & Normal & – & \(\approx\)90\({}^{\circ}\) & [25] \\ F\({}_{20+1}\) & Northwest & Normal & Sinistral & – & [23; 25] \\ F\({}_{20+2}\) & Northwest & Normal & Sinistral & – & [23; 25] \\ F\({}_{20-3}\) & Northwest & Normal & Sinistral & – & [23; 25] \\ F\({}_{20-4}\) & Northwest & Normal & Sinistral & – & [23; 25] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The characteristics of Jiujing region faults in the literature. where a change in the distance between the sensor and scatterer will produce a shift in the measured phase, corresponding to 2\(\pi\) radians doubled due to the two-way travel path-per-wavelength of the signal. The deformation (\(\Phi\)_def_) of the targeted scatterer during the data acquisition period can be obtained after the removal of satellite orbital errors (\(\Phi\)_orb_), topography (\(\Phi\)_dem_), atmospheric phase delay (\(\Phi\)_atm_), and temporal and geometrical decorrelation noise (\(\Phi\)_noise_). This is achieved using a time-series InSAR analysis technique, such as persistent scatterer interferometry (PSI or PSInSAR(tm)). The SAR data were processed in ENVI SARScape (Harris Geospatial Solutions) using the PSI technique applied to a series of Sentinel-1 SAR datasets. PSI is a stack processing method that uses multiple images taken at regular intervals to achieve millimetric-scale measurements. Compared to the traditional DInSAR stacking method, (1) PSI has a strongly reduced atmospheric error while DInSAR has no reduction on it, (2) PSI usually has a high coherence threshold (0.7\(>\)) on a single pixel while DInSAR only has a coherence threshold about 0.3, (3) PSI has Digital elevation model (DEM) accuracy around 100 m while the DEM accuracy in DInSAR is baseline dependent, and (4) PSI allows the stacking of more than 30 SAR images and DInSAR only allows 2 [36]. These make PSI much more precise than the DInSAR, which only has centimetre level precision [38]. The PSI method was developed from the work of [PERSON] [36; 37] and the method can be summarised in four steps:  
1.