## 2 Bridge Dynamics in a VBI System and the Data Reconstruction Algorithm
### Signal Processing Methods  
#### 2.3.1 An Intercept Method to Separate Vibration Components  
As previously discussed, the high-frequency free vibration component of the bridge was readily extracted from the raw monitoring data using a high-pass filter, facilitated by the MATLAB(r) (R 2022a) toolbox [26] in this study. However, it is challenging to distinguish between the residual low-frequency responses related to driving forces and those related to temperature changes using this type of high-pass filter, owing to a lack of spectral resolution. To tackle this issue, a novel intercept method is proposed, capitalizing on the observation that the standard deviation of structural responses during vehicle passage is markedly higher compared to periods when no vehicles are present. For instance, Figure 3a illustrates the residual time history of bridge strain with the high-frequency free vibration component removed. The local peaks in Figure 3b denote the instances of vehicle passage. The whole process of the intercept method is listed as follows:  
**Step 1**: The residual data series was segmented into a series of windows, with each window extending to 50% of the subsequent one, as depicted in Figure 4a. When a vehicle passes the bridge, the vehicle-induced strain forms local peaks on the strain time history curve, and the standard deviation of the window is obviously higher than the periods without passing vehicles. The duration of each window should remain uniform, established at \(2T_{0}\), where \(T_{0}\) represents the anticipated period of the first driving-force frequency, which is designed to cover general periods of vehicles passing;  
Figure 4: An intercept method to separate temperature change-related and driving-force-induced structural responses: (**a**) moving window with 50% overlap; (**b**) STD of each window; (**c**) label and pick; (**d**) interpolate; (**e**) approximated temperature strain; (**f**) driving-force-related structural responses. Figure 3: A typical band-pass filtered low-frequency time history of bridge strain: (**a**) time history; (**b**) zoomed time history (the data marked with the rectangle in (**a**) are zoomed in on (**b**)). **Step 2**: The standard deviation (STD) of each window was calculated (as shown in Figure 4b); **Step 3**: The median of all the STDs was calculated, based on the direct observation that during most of the monitoring periods, there were no vehicles on the bridge. **Step 4**: Windows with a standard deviation (STD) exceeding the median value were identified and designated with \"0\", and the remaining windows were assigned \"1\", as illustrated in Figure 4c. If the STD during a window period is higher than the median value, a vehicle is passing the bridge. Otherwise, it is supposed that no vehicles are on the bridge. **Step 5**: All assigned labels were examined and consecutive pairs of windows identified. The first window in each pair was designated as \"1\" and the following window as \"0\". Subsequently, the initial data point in the second window of each pair was referenced as the \"start element\", as demonstrated in Figure 4c. This step marks the start time point of the multiple-vehicle process. **Step 6**: Similarly, the labels were sequentially reviewed and subsequent sets of two adjacent windows selected. The label \"0\" was assigned to the first window and \"1\" to the second window. Then, the first data point in the second window was designated as the \"end element\", as shown in Figure 4c. This step marks the end time point of the process of multiple vehicles. **Step 7**: Subsequently, interpolation between the \"start element\" and the \"end element\" was performed using a spline function to generate a smooth curve. This curve was then used to replace the corresponding segment of the original monitoring data, as depicted in Figure 4d.