## 2 Materials and Methods
### PS-InSAR Processing and Vertical Velocity Decomposition  
With a deformation estimation of millimeter precision, PS-InSAR allows for detecting miniature surface deformation surrounding collapsed buildings before the incident. The PS-InSAR technique relies on the identification of objects smaller than the resolution cell that remained coherent and highly scattered in all the stack's images. The processing of PS-InSAR overcomes temporal decorrelation and phase unwrapping problems by forming a network to separate signals of the atmospheric phase screen (APS), topographic errors, and deformation by spatiotemporal filtering [4].  
In this study, we used the SAR PROcessing tool by periZ (SARPROZ) [22]. The process started by selecting an appropriate master image. In this step, we avoid any extreme weather conditions and select a master image with the smallest possible normal baseline. After extracting slave images based on the master one, the coregistration process started. We used a standard PS-InSAR approach, with a threshold of 0.2 for the amplitude stability index, but with a flowered tree graph connection for APS estimation. The estimation of the linear trend was limited between \(\pm\)160 mm and the estimation of the residual topographic error between \(\pm\)220 m. We selected the reference point in a stable area based on our background knowledge of the study area, and based on estimated parameter graphs during APS extraction. We also tried to select close reference points for LOS deformation estimation for an effective vertical velocity decomposition process. Finally, to define stable PS points in our study, we selected those points with a temporal coherence of 0.7 or higher.  
We used the residual LOS velocity of each image in ascending and descending orbits to decompose a vertical deformation time series. Decomposing vertical velocity by integrating two cross-heading orbits avoids the limitation of the LOS measurements and gives a chance to distinguish vertical and horizontal motion effectively [23]. Each image of an orbit was integrated into its closest image in the other orbit. The time difference was four days in the case of the same S1 sensor (S-1A or S-1B) and two days when using both sensors. The LOS residual velocity of image points were used to interpolate the residual LOS velocity in the city area on that day based on a grid with a pixel size of five meters, which produced a detailed surface of deformation for the city, then integrated to the closest interpolated LOS residual velocity of the other orbit to decompose vertical velocity by using the matrix presented in Equations (1) and (2) [24].  
\[\left[\begin{array}{c}d_{asc}\\ d_{desc}\end{array}\right]=A\left[\begin{array}{c}d_{vert}\\ d_{horiz}\end{array}\right], \tag{1}\]  
where  
\[A=\left[\begin{array}{cc}\cos\theta asc.&\sin\theta asc/\cos\theta\Delta \alpha\\ \cos\theta desc.&\sin\theta desc\end{array}\right], \tag{2}\]  
\(d\) is the deformation along LOS for ascending \(d_{asc}\) and descending \(d_{desc}\), \(d_{vert}\) is the vertical velocity, \(d_{horiz}\) is the projection of horizontal deformation in descending azimuth look direction, \(\theta\) is the incident angle, and \(\Delta\alpha\) is the satellite heading difference between ascending and descending mode.