# Measurement Model of Full-Width Roughness Considering Longitudinal Profile Weighting
## 3 Equivalent Longitudinal Profile and Equivalent Elevation  
The current prevalent method for calculating the IRI involves using the elevation of a single longitudinal profile at the wheel trajectory zone. This results in a large amount of elevation point cloud data being wasted, and using this IRI to evaluate the performance of the entire lane is also not rigorous. This study presents an innovative approach that utilizes the weighting model \(f_{\rm norm}(x)\) to assign weights to multiple elevations within each cross-section. This is followed by summation, scaling, and other processing techniques to obtain equivalent elevations for the entire roadway. In this scenario, the roadway and its vast point cloud of elevations can be represented by a single equivalent longitudinal profile and its corresponding equivalent elevations. The following are the detailed steps for calculating the equivalent elevations based on point cloud data from an inspected roadway:  
Step (1): Determine the key parameters \(L\) and \(\mu\) based on the specific type of inspected roadway. Generate the data sequence for the longitudinal profile weighting model \(f_{\rm norm}(x_{i})\) according to the lateral point cloud density, where \(x_{i}\) ranges from \([-L/2,L/2]\). Step (2): Along the direction of travel, sequentially perform point-by-point multiplication between the elevation point cloud \(P_{1}(x_{i},n)\) of the \(n\)-th cross-section and the longitudinal \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Passing Lane** & **Travel Lane** & **Slow Lane** & **Mixed Lane** \\ \hline \(L\) (m) & 1.6 & 1.8 & 2.0 & 1.8 \\ \(\mu\) (m) & 0.8 & 0.9 & 1.0 & 0.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Suggested \(\mu\) and \(L\) values for different lane types. profile weighting model sequence \(f_{\text{norm}}(x_{i})\) to generate the weighted elevation point cloud \(P_{2}(x_{i},n)\). The calculation process is shown in Equation (8). \[P_{2}(x_{i},n)=P_{1}(x_{i},n)\cdot f_{\text{norm}}(x_{i}),n=1,2,\ldots,N \tag{8}\]  
In the Equation, \(n\) represents the \(n\)-th cross-section along the direction of travel, and \(N\) denotes the total number of cross-sections in the point cloud. Step (3): Perform the cumulative summation of elevations \(P_{2}(x_{i},n)\) within the same cross-section to obtain the accumulated elevations \(P_{3}(n)\) for each cross-section. The calculation process is shown in Equation (9). Elevations \(P_{3}(n)\) integrate the elevations \(P_{1}(x_{i},n)\) of various points within the \(n\)-th cross-section along with their corresponding weights\(f_{\text{norm}}(x_{i})\). \[P_{3}(n)=\sum_{i=0}^{F_{S}}P_{2}(x_{i},n) \tag{9}\]  
Step (4): Scale the accumulated elevations \(P_{3}(n)\) according to Equation (10) to obtain the equivalent elevation of a specific roadway, denoted as \(P_{4}(n)\). \[P_{4}(n)=a\times P_{3}(n)/Fs \tag{10}\]  
In the Equation, the parameter \(Fs\) represents the number of sampling points in the cross-section, and the parameter \(a\) is set to 6, which is derived by rounding up the double of the reciprocal of the ratio 0.3226. The origin of the parameter 0.3226 is described below. In theory, the IRI can be calculated using a general roughness calculation method once the accumulated elevations \(P_{3}(n)\) are obtained. However, as evident from the multiplication process described in Equation (8), the weighting model of longitudinal profile \(f_{\text{norm}}(x)\) has an impact on the amplitude of the elevation point cloud data. While this process introduces weight distribution, it also alters the amplitude of the elevations. Furthermore, according to the superposition process outlined in Equation (9), numerous elevations are aggregated to compute the equivalent elevations. Although this process provides a consolidated representation, it may excessively emphasize the aggregation effect, thereby obscuring certain critical detail information.