## 3 Methodology
### The Calculation Model and Algorithm
#### 3.2.2 Model and Algorithm  
Since bridgehead settlement mainly occurs at the junction of road and bridges, the research section should cover the distance before and after the bridge joint. Therefore, the mark should be made for the case without bridge joints to identify the junction of road and bridge on the pavement images. Moreover, after the road profile line and the road image position correspond, the bridge joint is located on the line. At this time, the starting point of the road profile line is the acquisition starting point, and the joint bridge position is reset to the horizontal zero point of the profile line, as illustrated in Figure 4.  
The calculation model is suitable for the bridgehead settlement of the entering bridge and exiting bridge. Take the entering bridge as an example. As shown in Figure 4a, the bridge joint point is located on the profile line, and the starting point of the transition \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Speed V/(km/h)** & **\textgreater{}60**, \textless{}80 & **\textgreater{}40**, \textless{}60 & \textless{}40 \\ \hline Threshold of Longitudinal Slope Difference & 3\% & 3.5\% & 4\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Regulation of longitudinal slope difference threshold under different speeds. Figure 3: Left column: original road surface image. Right column: preprocessed road surface image. section is located at the elevation mutation point on the line. The horizontal distance between the bridge joint point and the transition section starting point is defined as \(D\). The profile line at \(-D\)-\(D\) in the horizontal direction is intercepted. Moreover, the points are evenly selected along the horizontal direction with \(\Delta d\) as the interval. Inspired by [PERSON]'s research [22], as shown in Figure 4b, the distance \(d_{0}\) (\(d_{0}>\Delta d\)) from the front and rear of each point is drawn along the horizontal direction. Since \(d_{0}\) is greater than \(\Delta d\), only part of the points in Figure 4a participates in the determination of whether they are settlement points. Unlike [PERSON]'s study [22], we optimize the calculation of the absolute value of the difference between the fore-slope and the back-slope. These points are called effective calculation points, and their number is defined as \(n_{e}\), and the sort of points is \(k\). The value rules of the above parameters will be described in the following test and data analysis. The calculation model of non-staggered-step bridgehead settlement is as follows:  
_Step 1_: The absolute value \(\Delta S_{k}\) (%) of the difference between the fore-slope and the back-slope of each effective calculation point after the slope drawing is calculated:  
\[\Delta S_{k}=\left|\frac{H_{k+1}-H_{k}}{d_{0}}-\frac{H_{k}-H_{k-1}}{d_{0}} \right|,k=1,2,3,\ldots,n_{e} \tag{11}\]  
where \(H_{k}\) denotes the elevation (m) of the \(k^{\text{th}}\) effective calculation point; \(H_{k-1}\) is the elevation (m) of the \(k-1^{\text{th}}\) effective calculation point, namely, the starting point at the front slope of the \(k^{\text{th}}\) effective calculation point; \(H_{k+1}\) is the elevation (m) of the \(k+1^{\text{th}}\) effective calculation point namely the ending point at the back slope of the \(k^{\text{th}}\) effective calculation point; while \(H_{0}\) is denoted as 0, and \(H_{n_{e}+1}\) is the elevation of the ending point of the profile line. The meanings of other parameters are the same as before. Among them, the number of effective points \(n_{e}\) is calculated by Formula (12), where the result is an integer. \[n_{e}=\frac{2(D-d_{0})}{\Delta d}+1 \tag{12}\]  
_Step 2_: The longitudinal slope difference threshold, as seen in Table 1, denoted as \(\Delta S_{0}\), is determined according to the test speed. If \(\Delta S_{k}\) exceeds the threshold \(\Delta S_{0}\), then the  
Figure 4: Bridgehead settlement calculation model diagram. (**a**) Diagram of uniform points taking. (**b**) Diagram of slope drawing.