## 2 Bridge Dynamics in a VBI System and the Data Reconstruction Algorithm
### Characteristics of Bridge Dynamics in the VBI System  
Firstly, we analyzed the response components of the direct measurement data. Typically, long-term monitoring data encompass responses induced by temperature fluctuations as well as those related to driving forces and free vibrations. Although these components have distinct frequencies, vehicle loading does not influence the temperature-related responses. In this study, we concentrate solely on the responses induced by vehicle loading, which are related to the driving force and free vibrations. [PERSON] et al. [24], after examining 898 highway bridges, concluded that the first natural frequency generally lies between 2 Hz and 5 Hz. He and [PERSON] [25] observed that the ratio of the first driving-force frequency to the first natural frequency is minimal, usually not exceeding 0.15 at vehicle speeds of 30-60 km/h, indicating that resonance is unlikely to be triggered by driving-force-induced responses. Furthermore, since temperature fluctuations occur much more slowly than the vehicle operation process, the high-frequency free vibrations of the bridge can be effectively isolated from the monitoring data using a high-pass filter. The remaining long-term monitoring data should primarily consist of low-frequency components related to driving forces and temperature variations, as depicted in Figure 2. Subsequently, the separation of driving-force and temperature-related components was carried out using the intercept method described in Section 2.3.1. The transfer matrix for responses at different cross-sections was then investigated. Assume that there are three specific locations, \(x_{1}\), \(x_{2}\), and \(x_{3}\), on the simply supported beam. According to previous research [19] (a detailed derivation is listed in Appendix A), the transfer matrix corresponding to the low-frequency free vibration from \(x_{1}\), \(x_{2}\) to \(x_{2}\), \(x_{3}\) can be derived from Equation (9) as follows:  
\[ \begin{bmatrix}u_{b,1,x_{2},t_{0}}&u_{b,1,x_{2},t_{1}}&\cdots&u_{b,1,x_{2},t_{N }}\\ u_{b,1,x_{3},t_{0}}&u_{b,1,x_{3},t_{1}}&\cdots&u_{b,1,x_{3},t_{N}}.\end{bmatrix} = \boldsymbol{T_{b,1}} \begin{bmatrix}u_{b,1,x_{1},t_{0}}&u_{b,1,x_{1},t_{1}}& \cdots&u_{b,1,x_{1},t_{N}}\\ u_{b,1,x_{1},t_{0}}&u_{b,1,x_{2},t_{1}}&\cdots&u_{b,1,x_{2},t_{N}}.\end{bmatrix}  \tag{12}\]  
where \(t_{0}\), \(t_{1}\), and \(t_{N}\) are the \(N+1\) time points, and the transfer matrix \(\boldsymbol{T_{b,1}}\) is:  
\[\boldsymbol{T_{b,1}}= \begin{bmatrix}sin\frac{\pi x_{2}}{L}&sin\frac{2\pi x_{2} }{L}\\ sin\frac{\pi x_{3}}{L}&sin\frac{2\pi x_{3}}{L}\\ \end{bmatrix}\begin{bmatrix}sin\frac{\pi x_{3}}{L}&sin\frac{2\pi x_{3}}{L}\\ sin\frac{\pi x_{3}}{L}&sin\frac{2\pi x_{3}}{L}\\ \end{bmatrix} ^{-1} \tag{13}\]  
The second row of the left-side matrix of Equation (12) is the actual responses at \(x_{3}\) at \(N+1\) time points (\(t_{0}\), \(t_{1}\), \(\ldots\), \(t_{N}\)), as the target output. While the inputs are the measured responses at \(x_{1}\) and \(x_{2}\), which are the first and second rows of the left-side matrix of Equation (12), respectively. Through the multiplication between \(T_{b,1}\) and the input displacement matrix, the displacement at \(x_{3}\) can be then calculated with Equation (12). It is important to note that only the first two vibration modes, corresponding to the first and second free vibration frequencies, were taken into account, due to their significant contributions to the overall high-frequency bridge responses. These two modes are also the global vibration modes, whose responses can typically be measured along the length of the bridge. Higher vibration modes, which are primarily associated with local deformation of the bridge, were not considered in this study. It is assumed that for these local vibration modes, the selection of input observation points must be carried out with great care. Otherwise, a low signal-to-noise ratio (SNR) could hinder the achievement of accurate reconstruction results.