## 2 Monitoring Method of Bridge Dynamic Deformation
### Digital Close-Range Photogrammetry Technology
By removing the phase component \((\phi_{\text{air},1}-\phi_{\text{air},0})\) occasioned by atmospheric effects and the corresponding noise phase component \(\phi_{strepitus}\), the deformation in the radar line-of-sight direction between t\({}_{0}\) and t\({}_{1}\) can be obtained as follows:  
\[d_{Lo}=\frac{\Delta\phi_{10}^{\prime}\lambda}{4\pi} \tag{5}\]  
In Formula (5), \(\Delta\phi_{10}^{\prime}\) denotes the interference phase of the radar image between t\({}_{0}\) and t\({}_{1}\) after removing the phase component affected by the atmospheric effect and other corresponding noise phase errors. Since the purpose of this monitoring is to obtain the settlement information of the pier, the vertical deformation component in the eye direction should be obtained. This vertical component (i.e., the vertical deformation of the pier) can be calculated using a simple geometric projection, which is expressed as follows:  
\[d_{de}=\frac{d_{Lo}}{\sin(\alpha)}=d_{Lo}\frac{S}{D} \tag{6}\]  
In Formula (6), \(S\) denotes the distance from the radar to the target, and \(D=S\sin(\alpha)\); additionally, \(\alpha\) denotes the angle between the horizontal direction and the radar line of sight. The geometric relationship between visual displacement and vertical deformation is shown in Figure 1.  
(4) Wavelet analysis for denoising  
To obtain more accurate deformation time series, the deformation time series obtained in Step (3) is denoised. First, the deformation time series obtained in Step (3) is denoised using wavelet analysis to further remove the noise component; subsequently, the dynamic characteristic parameters of the bridge are obtained based on the deformation time series after noise removal. (5) IBIS-S measurement accuracy  
The minimum shape variable that the IBIS-S system can detect is \(d_{\text{min}}\), which can be calculated using the following formula:  
\[d_{\text{min}}=-\frac{\lambda}{4\pi}(\phi_{2}-\phi_{1})_{\text{min}} \tag{7}\]  
where \(\lambda\) denotes the wavelength of the electromagnetic wave, and its calculation formula is expressed as follows:  
\[\lambda=\frac{c}{f}=\frac{3.0\times 10^{8}\text{m}/\text{s}}{17 GHZ}=17.649\ \text{m} \tag{8}\]  
\(\left(\phi_{2}-\phi_{1}\right)_{\text{min}}\) denotes the minimum phase difference that can be resolved by the system, and its calculation formula is expressed as follows:  
\[\left(\phi_{2}-\phi_{1}\right)_{\text{min}}=1^{\prime\prime}=\frac{\pi}{648,000} \tag{9}\]  
Therefore, the minimum deformation measured by the IBIS-S technology is 0.000068 mm, and the theoretical measurement accuracy can attain one millionth of a millimeter. However, in actual measurement, due to the role and influence of several factors, the detection accuracy exhibits a certain uncertainty. Therefore, the accuracy of the measurement is limited by the quality of the observation conditions. IBIS-S technology exhibits a nominal accuracy of 0.1 mm in the static monitoring state and 0.01 mm in the dynamic monitoring state. Ground-based radar interferometry technology measures the relative deformation of the subsequent moment relative to the founding moment. During the measurement process, the stability of the measuring instrument should be ensured, and the temperature, air pressure, and wind speed around the instrument should be kept stable, so as to reduce the interference of external environmental factors on the measurement results and improve the measurement accuracy.