# Measurement Model of Full-Width Roughness Considering Longitudinal Profile Weighting
## 2 Weighting Model of Longitudinal Profile  
### Lateral Distribution Model of Wheel Trajectories  
#### 2.1.1 Distribution Frequency of Vehicle Centerlines  
When a vehicle travels along a lane, the vertical load generated at the contact points between its tires and the road surface has a significant impact on both the vehicle's driving stability and road surface wear. Let \(N\) denote the number of load applications per unit width of the road by the tires and \(N_{0}\) represent the total number of load applications across the entire lane width. The lateral distribution function of wheel trajectories, denoted as \(f(x)\), can be expressed as the ratio between \(N\) and \(N_{0}\), as shown in Equation (1).  
\[f(x)=N/N_{0} \tag{1}\]  
This ratio directly reflects the probability of wheel compaction on the road and also indicates the excitation effect of the road on vehicles. A high probability value implies that wheels frequently pass through the longitudinal profile, such as in the wheel trajectory zone, where wheel trajectories are most concentrated. These areas, subject to heavy traffic loads,are more prone to wear, damage, and maintenance needs. Conversely, a low probability value suggests fewer wheel passages through the longitudinal profile, as observed in the centerline or side edges of the lane. The frequency of wheels passing through these areas is lower, resulting in less wear and damage, thereby better maintaining the original road surface condition. Significant differences exist in the lateral distribution model of wheel trajectories across various highway grades and lane types. The establishment of a universal, scientific, and precise mathematical model for these distributions is highly desirable. However, achieving this goal poses a substantial challenge due to the multitude of factors that must be considered, including road grade, width, traffic volume, vehicle type, and others.  
British Standard BS 5400-10 records the lateral distribution frequency of vehicle centerlines \(F(x)\) based on annual vehicle flow data, as illustrated in Figure 1. This figure presents the lateral distribution frequency divided into 0.1 m strip widths. As shown, the highest frequency corresponds to the center of the lane, indicating that a considerable number of vehicles tend to strictly maintain their course at the center of the lane. Additionally, the distribution frequencies exhibit a symmetrical characteristic on both sides and gradually decrease as the distance from the lane center increases. This trend suggests that although vehicles primarily travel along the lane center, they also exhibit a certain range of transverse offsets, and these offsets are symmetrically distributed on both sides of the lane. Considering the wide applicability of this standard document and its good interpretability of physical reality, we have adopted it as a standard reference for subsequent models. Incorporating the characteristics of the frequency distribution in the figure, this study fits the 13 frequency values shown in the figure according to a normal distribution model. Finally, the distribution frequency of vehicle centerlines \(F(x)\) can be represented by the curve shown in Equation (2).  
\[F(x)=\frac{B_{0}}{\sqrt{2\pi\sigma_{0}}}\exp(-\frac{(x-\mu_{0})^{2}}{2\sigma_{ 0}^{2}}) \tag{2}\]  
The fitting parameters are determined as \(B_{0}\) = 0.098674, \(\mu_{0}\) = 0, and \(\sigma_{0}\) = 0.226058. The good fit indicated by \(R^{2}\) = 0.9957 suggests that Equation (2) exhibits high model versatility and can effectively replace the discrete frequencies presented in Figure 1.