## 3 GB-RAR Deformation Information Estimation Method Considering the Influence of Colored Noise  
The ground-based radar image contains amplitude and phase information, and the phase information in ground-based radar images reflects the geometric distance between radar and target. According to the phase changes between the radar images obtained at different times, the displacement changes between the radar and the target can be inverted by using interferometry technology. Assuming that two radar images (\(t_{B}>t_{A}\)) are obtained at different times \(t_{A}\) and \(t_{B}\), \(\phi_{A}\) and \(\phi_{B}\) are the phase values of the same pixel on the radar image at \(t_{A}\) and \(t_{B}\), respectively. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Description** \\ \hline Radar type & FMCW \\ Bandwidth & 200 MHz \\ Frequency range & 17.1–17.3 GHz (Ku band) \\ Radar antenna inclination & \(10^{\circ}\) \\ Observation distance & 200 m \\ Range resolution & 0.5 m \\ Sampling frequency & 20 Hz \\ Start time & 17 November 2016 06:00 \\ End time & 18 November 2016 18:00 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: IBIS-S system parameter setting. Figure 3: (a) Leveling monitoring site; (b) accurate invar bar code leveling staff for leveling. Figure 2: (a) Monitoring scene of the IBIS-S interferometric radar; (b) ground view of the ELH Bridge from the location of the IBIS-S interferometric radar. The interference phase \(\Delta\phi_{BA}\) with the radar images at \(t_{A}\) and \(t_{B}\) can be derived after image interference processing [39; 40]:  
\[ \begin{array}{ll}\Delta\phi_{BA}&=\phi_{B}-\phi_{A}\\ &=\frac{4\pi(R_{B}-R_{A})}{\lambda}+(\phi_{scatter,B}-\phi_{scatter,A})\\ &+(\phi_{atm,B}-\phi_{atm,A})+\phi_{noise}\end{array}  \tag{1}\]  
where \(\lambda\) is the wavelength of the radar signal; \(R_{A}\) and \(R_{B}\) are the geometric distance between the radar and the target at \(t_{A}\) and \(t_{B}\), respectively; \(\phi_{scatter}\) represents the phase shift caused by the interaction between the microwave and the monitoring target; \(\phi_{atm}\) denotes the phase produced by the influence of atmospheric disturbance; and \(\phi_{noise}\) refers to the random noise phase. The phase component \(\phi_{scatter}\) of the radar image obtained at times \(t_{A}\) and \(t_{B}\) are assumed to be constant. Then, Equation (1) can be reduced as follows:  
\[ \begin{array}{ll}\Delta\phi_{BA}&=\phi_{B}-\phi_{A}\\ &=\frac{4\pi d_{LOS}}{\lambda}+(\phi_{atm,B}-\phi_{atm,A})+\phi_{noise}\end{array}  \tag{2}\]  
where \(d_{LOS}=R_{B}-R_{A}\) is the displacement change of the target in the radar line-of-sight (LOS) direction between \(t_{A}\) and \(t_{B}\). When the phase component (\(\phi_{atm,B}-\phi_{atm,A}\)) and noise phase component \(\phi_{noise}\) produced by atmospheric effect are removed, the deformation occurring in the radar LOS direction can be obtained:  
\[d_{LOS}=\frac{\Delta\phi^{\prime}_{BA}\lambda}{4\pi} \tag{3}\]  
where \(\Delta\phi^{\prime}_{BA}\) denotes the interference phase of the radar images between \(t_{A}\) and \(t_{B}\) after removing the phase errors of atmospheric effect phase component and other corresponding noise. The vertical deformation (i.e., settlement) of the target monitoring point can be solved according to the radar monitoring angle \(\theta\) (the angle between the horizontal direction and the radar LOS direction):  
\[y=d_{LOS}/\sin(\theta) \tag{4}\]  
The settlement deformation time series obtained based on GB-RAR may contain colored noise due to the influence of several factors (such as systematic errors related to the GB-RAR technology). When analyzing the settlement deformation time series, only considering white noise will affect the accurate estimation of the uncertainty of settlement rate. Accordingly, the noise properties of settlement deformation time series must be analyzed.