# Differences Evaluation of Pavement Roughness Distribution Based on Light Detection and Ranging Data
## 3 Calculation of the Pavement Roughness Based on 3D LiDAR Data
### Calculation Method of the IRI  
The international roughness index (IRI) is a widely used index for evaluating the pavement roughness, and it is currently incorporated as an evaluation parameter in the highway specifications of China. The IRI has been established to provide a standardized quantitative indicator of the pavement roughness under different measuring devices and modes across various countries. Therefore, the IRI can correlate with various response-type roughness indicators, which can be unified under the same standard, such as the standard deviation of roughness and the power spectral density. Based on the 1/4 vehicle model, the IRI is defined as the ratio of the cumulative vertical displacement of the vehicle's suspension system to the vehicle's forward distance when driven at a steady speed of 80 km/h on the test road. This definition has been represented mathematically in Formulas (4)-(7), which simulates the vehicle's response to the pavement roughness under the average corrected road slope,  
\[\left\{\begin{array}{l}m_{\text{s}}\dot{z}_{s}+C_{\text{s}}(\dot{z}_{s}-\dot{ z}_{u})+k_{\text{s}}(z_{s}-z_{u})=0\\ m_{\text{u}}\dot{z}_{u}+C_{\text{s}}(\dot{z}_{u}-\dot{z}_{s})+k_{\text{s}}(z_{ u}-z_{s})+k_{\text{f}}(z_{u}-Y)=0\end{array}\right. \tag{4}\]  
\[\ddot{z}_{s}=1/m_{s}\cdot\left[C_{\text{s}}(\dot{z}_{u}-\dot{z}_{s})+k_{\text{ s}}(z_{u}-z_{s})\right] \tag{5}\]  
\[\ddot{z}_{u}=1/m_{u}\cdot\left[C_{\text{s}}(\dot{z}_{s}-\dot{z}_{u})+k_{\text{ s}}(z_{s}-z_{u})+k_{\text{f}}(Y-z_{u})\right] \tag{6}\]  
\[IRI=\frac{1}{L}\int\limits_{0}^{L}|z_{s}-z_{u}|dx \tag{7}\]  
where \(m_{s}\) and \(m_{u}\) denote the sprung and the unsprung masses, respectively, \(k_{\text{s}}\) indicates the stiffness of the spring, \(c_{\text{s}}\) the damping coefficient of the shock absorber, \(k_{\text{f}}\) represents the stiffness of the tire, \(Y\) is the surface roughness, \(z_{\text{s}}\) and \(z_{u}\) are the displacements of the sprung and the unsprung masses, respectively, \(L\) is the distance of the roughness measurement, and \(dx\) is the sampling interval of the roughness statistics.  
The traditional calculation of the international roughness index involves utilizing a road laser profilometer to gather pavement elevation changes at the wheel track location. The IRI value is then computed using the approach outlined in Appendix A of the \"Vehicle Bearing Road Laser Profilometer\" (JG 075-2010) [27]. LiDAR point cloud technology can achieve the same level of precision as a linear laser. It enables the acquisition of not only the elevation changes of a fixed survey line, but also the relative 3D coordinates of various positions within a 3D environment through point cloud processing. Cross-sectional data can be extracted at different survey line positions to obtain the same elevation data as the road laser profilometer. During the detection process, the collected road elevation \(z_{0}\) is converted into IRI results referring to the calculation method in the professional standard [27]. LiDAR data's ranging function is used to extract points equidistant from the centerline, which is then connected to obtain a specific road profile's 3D point cloud data. By minimizing the impact of road slope, the elevation variation of a survey line can thereby be determined.