# Ground Displacement in East Azerbaijan Province, Iran, Revealed by L-band and C-band InSAR Analyses
## 3 Methods
### SBAS Analysis  
Several SBAS InSAR time series studies based on different X-band and C-band datasets have been done for NW Iran [45; 46; 47; 48; 51]. A small baseline selection is based on SBAS interferogram calculation. We assume that 15 PALSAR-2 images make up a set of single-look complex (SLC) images (\(N\)) over the same area, with the same imaging geometry from the same orbit acquired at ordered times (\(t_{1}\), \(t_{2}\), \(t_{3}\),..., \(t_{n}\)). The quality of interferogram correlation between 2 selected images is a product of the time period (\(T\)), the normal baseline of the two acquisitions (\(B\)), the Doppler centroid of the 2 images (\(D\)), and the thermal noise imposed by the sensor [52]. Thus, the conventional form of SAR interferometry is as follows:  
\[\rho=\rho_{temporal}\cdot\rho_{spatial}\cdot\rho_{Doppler}\approx(1-f(T_{\perp}-T_{\perp}^{c})).\ (1-f(B_{\perp}-B_{\perp}^{c})).\ (1-f(D_{\perp}-D_{\perp}^{c})) \tag{1}\]  
where \(\rho\) is the correlation, \(T_{\perp}\), \(B_{\perp}\), and \(D_{\perp}\) are temporal, normal interferometric, and Doppler baseline respectively, and \(T_{\perp}^{c}\), \(B_{\perp}^{c}\), and \(D_{\perp}^{c}\) are critical temporal, critical normal, and Doppler centroid baseline, respectively. If the correlations defined by Equation (1) meet the SBAS criteria, the corresponding SAR pair can be selected for interferogram generation of small baseline analysis. Instead of following [PERSON] et al. [52], here, for the SBAS analysis, we followed [PERSON] et al. [53]. For this analysis, the minimum number of pairs is 1 for 2 images and each interferogram consists of 2 images. Thus, the amount of phase value can be estimated considering \(N\) as the odd number of images and \(M\) as the number of interferograms:  
\[\frac{N+1}{2}\leq M\leq N\bigg{(}\frac{N+1}{2}\bigg{)}. \tag{2}\]  
Since the pixel values of SAR images are in radar coordinates, not geodetic/geographic coordinates, the pixel value of unwrapped interferogram \(j\) at times \(t_{B}\) and \(t_{A}\) in the azimuth and range direction will be as follows [53]:  
\[ \begin{split}\varphi_{j}(x,\ r)=\varphi(t_{B},\ x,\ r)=\varphi(t _{A},\ x,\ r)\approx\frac{4\pi}{\Lambda}\Big{[}d^{LP}(t_{B},\ x,\ r)-d^{LP}(t _{A},\ x,\ r)\Big{]}+\\ \Delta\varphi_{j}^{atm}(t_{B},\ t_{A},\ x,\ r)+\Delta\varphi_{j} ^{topo}(x,\ r).\end{split}  \tag{3}\]  
Figure 3: Histogram of PALSAR-2 images used in this study. where \(j\) is an integer number from 1 and \(M\), \(\varphi(t_{B},\ x,\ r)\) and \(\varphi(t_{A},\ x,\ r)\) are phases of 2 SAR images looked at multiple times at times \(t_{B}\) and \(t_{A}\) in the range and azimuth direction, \(d^{LP}(t_{B},\ x,\ r)\) and \(d^{LP}(t_{A},\ x,\ r)\) are displacement values of low-pass component for a time span from \(t_{A}\) to \(t_{B}\) in the LOS of the satellite, \(\Delta\varphi_{j}^{atm}(t_{B},\ t_{A},\ x,\ r)\) is the associated phase component of atmosphere from \(t_{A}\) to \(t_{B}\), the wavelength of the satellite is represented by \(\lambda\), and \(\Delta\varphi_{j}^{topo}(x,\ r)\) is the phase component of topography, which is defined as follows:  
\[\Delta\varphi_{j}^{topo}(x,\ r)\approx\frac{4\pi}{\lambda}\frac{B_{\perp j} \Delta z(x,\ r)}{rsin\theta}, \tag{4}\]  
where \(B_{\perp j}\) is the perpendicular (normal) baseline of the two SAR acquisitions, \(\theta\) is the incidence angle of the SAR images, which is approximately \(28^{\circ}\) for PALSAR-2 images used in this study, and \(\Delta z(x,\ r)\) is a topographic artifact that can be reduced by a digital elevation model (DEM). Here, thermal noise is assumed to be negligible, and according to Equation (3), the topography and atmosphere are non-displacement phase components that must be reduced or removed. Thus, the number of processed pairs and their temporal and spatial baselines should be selected under predefined criteria. Since the L-band PALSAR-2 images can keep high phase correlation over time, we assume that all SAR images are eligible for SBAS analysis, despite the large gaps between image acquisitions.