## 2 Materials and Methods
### Mathematical Modeling of InSAR Phase Components  
Interferometric Synthetic Aperture Radar (InSAR) calculates surface deformation and topographic information by analyzing the phase difference between two SAR images acquired at different times from slightly different satellite positions [25]. The radar signals received from the ground are recorded as complex numbers containing both amplitude and phase, forming Single Look Complex (SLC) images [17; 26]. When two SLC images are acquired from different positions, the difference in the distance between the satellite and the ground target introduces a phase difference. This phase difference is directly related to the surface displacement and is influenced by the baseline distance between the satellite positions, the radar wavelength, and the geometry of acquisition [27]. The interferometric phase is derived by multiplying one image with the complex conjugate of the other. This phase consists of contributions from ground deformation, topography, orbital errors, atmospheric delays, and noise. To isolate the deformation phase, these error components are removed through correction procedures [15]. The simplified interferometric phase can be expressed as a function of the perpendicular baseline, target height, radar wavelength, slant range, and incidence angle. Ultimately, the target elevation and deformation are calculated using these geometric and physical parameters. The final phase model is fundamental for generating Digital Elevation Models (DEMs) and monitoring ground deformation with high precision [16]. In Figure 3, the SAR data received by antennas A\({}_{1}\) and A\({}_{2}\) are represented as S\({}_{1}\) and S\({}_{2}\). Here, S\({}_{1}\)(R) and S\({}_{2}\)(R) are the complex numbers of the radar signal, \(\phi\) (R) and \(\phi\) (R + \(\Delta\)R) are the phase components of the radar signal, and u\({}_{1}\)(R) and u\({}_{1}\)(R + \(\Delta\)R) are the amplitude components of the radar signal. \[S_{1}(R)=u_{1}(R)\cdot\exp(i\phi(R)) \tag{1}\]  
\[S_{2}(R)=u_{2}(R+\Delta R)\cdot\exp(i\phi(R+\Delta R)) \tag{2}\]  
The received data contain phase information obtained from repeat-pass orbits, along with error phases caused by noise, scattering, and other factors. The phase components arg(u\({}_{1}\)) and arg(u\({}_{2}\)) represent the error phases induced by scattering and noise. Due to the baseline distance B, a difference in incidence angles occurs between the two acquisitions, resulting in discrepancies between the two SAR images. \[\psi_{1}=2\bigg{(}\frac{2\pi}{\lambda}\bigg{)}R+\arg\{u_{1}\} \tag{3}\]  
\[\psi_{2}=2\bigg{(}\frac{2\pi}{\lambda}\bigg{)}(R+\Delta R)+\arg\{u_{2}\} \tag{4}\]  
By multiplying the two images with their complex conjugates, an interferometric image containing phase difference information can be obtained, where the phase values are represented as real numbers. Assuming that the irregular phase values of the two images are identical, the phase of the interferometric image is determined by \(\Delta\)R.  
\[\phi=-\bigg{(}\frac{4\pi}{\lambda}\bigg{)}\Delta R+2\pi N\ \ \text{where}\ \ N=0,\ \pm 1,\ \pm 2,\ \ldots \tag{5}\]The phase value of the interferometric image is represented as the sum of the absolute phase value and residual noise. To derive the absolute phase, undergoing the absolute phase restoration process is necessary to eliminate noise. By simplifying this equation, excluding the \(\Delta\)R\({}^{2}\) term, the resulting expression is as follows:  
\[\sin(\theta_{0}-\alpha)=\frac{\left(\left(\text{R}+\Delta\text{R}\right)^{2}- \text{R}^{2}-\text{B}^{2}\right)}{\left(2\text{RB}\right)} \tag{6}\]  
\[\text{z}=\text{H}-\text{R}\text{ cos }\theta_{0} \tag{7}\]  
In general, the value of \(\Delta\)R is extremely small, and thus \(\Delta\)R\({}^{2}\) can be considered negligible and omitted. In the case of SAR sensors mounted on satellites, the baseline distance B between antennas is significantly smaller than the distance R between the satellite and the target point on the Earth's surface.