# Measurement Model of Full-Width Roughness Considering Longitudinal Profile Weighting
## 2 Weighting Model of Longitudinal Profile
### Lateral Distribution Model of Wheel Trajectories
#### 2.1.2 Calculation Method for Lateral Distribution Model of Wheel Trajectories  
Vehicle tires leave trajectories on the road surface upon contact and rolling, and the accumulation of numerous trajectories forms the wheel trajectory zone. When the wheel track width is \(L\), the frequency of wheel trajectories generated at \(x-L/2\) and \(x+L/2\) by vehicles with a centerline of \(x\) is the same as \(F(x)\). Therefore, after obtaining the lateral  
Figure 1: Distribution frequency of vehicle centerlines and its fitting results. distribution frequency of the vehicle centerlines \(F(x)\), the lateral distribution model \(f(x)\) for a wheel track width of \(L\) can be derived through data accumulation operations, as shown in Equation (3). \[f(x)=F(x+L/2)+F(x-L/2) \tag{3}\]  
However, the track width \(L\) of vehicles does not adhere to a uniform standard value, with common values ranging from 1.5 m to 2.2 m. Without loss of generality, this study selects 1.6 m, 1.8 m, and 2.0 m as three typical track width values to analyze their corresponding lateral distribution models of wheel trajectories \(f(x)\). The results are presented in Figure 2.  
In Figure 2, the lateral distribution curves of wheel trajectories under three different track widths all exhibit a typical \"M\" shape. These curves can be fitted using a 2-component Gaussian mixture model, as shown in Equation (4). The parameters to be fitted are \(B_{1}\), \(B_{2}\), \(\mu_{1}\), \(\mu_{2}\), \(\sigma_{1}\), and \(\sigma_{2}\), respectively. Here, \(B_{1}\) and \(B_{2}\) represent the weights of the two Gaussian models, \(f_{1}(x)\)-\(N_{1}(\mu_{1}\), \(\sigma_{1}{}^{2})\) and \(f_{2}(x)\)-\(N_{2}(\mu_{2}\), \(\sigma_{2}{}^{2})\), respectively. Considering the symmetry of the curves in Figure 2, it is reasonable to assume that \(B_{1}=B_{2}=B\), \(\mu_{1}=\mu_{2}=\mu\), and \(\sigma_{1}=\sigma_{2}=\sigma\). Under these assumptions, Equation (4) can be further simplified to Equation (5). Finally, the fitting results of the lateral distribution model for wheel trajectories \(f(x)\) are presented in Table 1.  
\[f(x)=B_{1}(\frac{1}{\sqrt{2\pi}\sigma_{1}})\exp(-\frac{(x-\mu_{1})^{2}}{2{ \sigma_{1}}^{2}})+B_{2}(\frac{1}{\sqrt{2\pi}\sigma_{2}})\exp(-\frac{(x+\mu_{2 })^{2}}{2{\sigma_{2}}^{2}}) \tag{4}\]  
\[f(x)=B(\frac{1}{\sqrt{2\pi}\sigma})\left(\exp(-\frac{(x-\mu)^{2}}{2\sigma^{2}} )+\exp(-\frac{(x+\mu)^{2}}{2\sigma^{2}})\right) \tag{5}\]  
As can be seen from Table 1, the fitting parameters \(B\) and \(\sigma\) remain largely unchanged under the three different conditions, while the parameter \(\mu\) varies with changes in the wheel track width \(L\), showing a significant two-fold relationship between them. However, \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(L\) (m) & \(B\) & \(\mu\) & \(\sigma\) & \(\mathbf{R^{2}}\) \\ \hline
1.6 & 0.09867 & 0.800 & 0.226 & 0.9999 \\
1.8 & 0.09867 & 0.900 & 0.226 & 0.9999 \\
2.0 & 0.09867 & 1.000 & 0.226 & 0.9999 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fitting parameters of the lateral distribution model with different track widths \(L\). Figure 2: Lateral distribution models of wheel trajectories \(f(x)\). it must be pointed out that these three track widths do not represent the entire range of widths for vehicles traveling on a specific roadway. In real-world scenarios, the formation of a wheel trajectory zone is the result of collective tire rolling by all passing vehicles within a given traffic lane. This implies that the formation process of the wheel trajectory zone involves vehicles with various wheel track widths. Yet, due to the non-standardized nature of track widths, the specific location and width of the wheel trajectory zone are also subject to variation. In such cases, the lateral distribution model \(f(x)\) corresponding to the lane can be obtained by weighting the lateral distribution frequency models \(f_{i}(x)\) under multiple track widths, as shown in Equation (6). \[\left\{