# Surface Subsidence Monitoring in Kunming City with Time-Series InSAR and GNSS
## 2 Methods  
### Fundamental Principle of SBAS-InSAR Technique  
SBAS-InSAR is an InSAR time series analysis method proposed by [PERSON] [24], which improves the problems of spatial and atmospheric incoherence existing in D-InSAR method. Compared with the Permanent Scatterer (PS) InSAR, it reduces the requirement of SAR data volume and has the advantage of continuous deformation sequence in space monitoring. The core idea is to use a reasonable spatio-temporal baseline to set one of the scene images as the main image, and the other slave images are registered with the main image to obtain several interference pairs. Then the singular value decomposition (SVD) method of matrix is used for joint solution [34], so as to obtain the deformation information of the study area within the time span.  
Assuming that \(N+1\) SAR images (\(t_{1},t_{2},\ldots,t_{N}\)) are acquired within the time range of the same region, \(M\) differential interferograms can be generated. \(M\) satisfies the following formula:  
\[\frac{N+1}{2}\leq M\leq\frac{N(N+1)}{2} \tag{1}\]  
If \(t_{1}\) is set as the initial time, then for values in \(t_{A}\), \(t_{B}\) (\(t_{A}<t_{B}\)) time obtained image generated interferogram, the phase of any point (\(x,\ r\)) is as follows:  
\[\begin{array}{c}\delta\varphi_{j}(x,r)=\varphi(t_{B},x,r)-\varphi(t_{A},x,r) \\ \approx\Delta\varphi_{flat}+\Delta\varphi_{topo}+\Delta\varphi_{def}+\Delta \varphi_{atmo}+\Delta\varphi_{noise}\end{array} \tag{2}\]  
In the formula, \(\Delta\varphi_{flat},\Delta\varphi_{topo},\Delta\varphi_{def},\Delta\varphi_{ atmo},\Delta\varphi_{noise}\) respectively represent the flat phase, terrain phase, surface deformation phase, atmospheric phase and noise phase. \(\varphi(t_{B},\ x,\ r),\varphi(t_{A},\ x,\ r)\) represent the \(t_{A}\), \(t_{B}\) the two images in (\(x,\ r\)) point of phase [35].  
\(\Delta\varphi_{atmo}\) is separated from the interferometric phase by a time high pass filter and a space low pass filter. After removing the atmospheric component, the interferometric phase is mainly composed of the surface deformation phase and the residual elevation component. Suppose that a rate vector can be used to represent a linear model. In order to obtain surface deformation information and residual elevation component, the surface deformation phase \(\Delta\varphi_{def}\) in Equation (2) can be converted into the product of average phase rate and adjacent time interval:  
\[[v]^{T}=\left[v_{1}=\frac{\varnothing_{2}}{t_{2}-t_{1}},\ldots,v_{N-1}=\frac{ \varnothing_{N}-\varnothing_{N-1}}{t_{N}-t_{N-1}}\right] \tag{3}\]  
\[\Delta\varphi_{topo}=\frac{4\pi}{\lambda}\frac{B_{\perp j}\Delta z}{Rsin\theta} \tag{4}\]  
In the formula, \(\lambda\) represents radar wavelength, \(B_{\perp j}\) represents the perpendicular baseline of the interference on \(j\), \(\theta\) is the incident angle, \(\Delta z\) is the height error, Matrix \(C\) is defined as follows:  
\[C(M\times 1)=\left[\frac{4\pi}{\lambda}\frac{B_{\perp 1}}{Rsin\theta},\ldots, \frac{4\pi}{\lambda}\frac{B_{\perp M}}{Rsin\theta}\right] \tag{5}\]  
Then the observation equation is as follows:  
\[C_{v}=D\cdot\Delta z=\delta\varnothing \tag{6}\]  
where \(D\) corresponds to the \(M\times(N-1)\) coefficient matrix, and the deformation rate can be extracted from Equation (6) by SVD method. Finally, the corresponding time series can be calculated according to the time interval between image acquisition.