Differences Evaluation of Pavement Roughness Distribution Based on Light Detection and Ranging Data

Abstract

Pavement roughness serves as a crucial indicator for evaluating road performance. However, traditional measurement methods, such as laser detection vehicles, are limited to providing roughness values for a single profile, failing to capture the overall pavement condition comprehensively. To address this limitation, this study utilized high-precision light detection and ranging technology (LiDAR) to acquire three-dimensional point cloud data for a 25 km road section in Shanghai. Road elevations were extracted from different lateral survey lines. Subsequently, variance analysis and the Kruskal–Wallis non-parametric test were conducted to evaluate the differences in the lateral distribution and longitudinal variability of the pavement roughness. The findings revealed significant differences in the international roughness index (IRI) among the survey lines within the road section. Moreover, the observed variations in the lateral distribution of pavement roughness were influenced by the characteristics of the road section itself. Roads exhibiting discrete roughness patterns displayed a higher likelihood of significant detection disparities. Additionally, it was discovered that the discrepancy between the detection length and the actual road length introduced volatility in repeated detection results, necessitating a limitation of this discrepancy to 30 m. Consequently, it has been recommended to consider the lateral distribution of pavement roughness and to regulate the detection length in road performance evaluations to enhance reliability and facilitate more accurate maintenance decision making. The study highlights the importance of incorporating comprehensive assessment approaches for pavement roughness in road management practices.

1. Introduction

Pavement roughness constitutes a crucial attribute of roads that has significant implications for driving safety, comfort, and road longevity [1,2]. It denotes the fluctuations in road elevation and serves as a key performance indicator for evaluating road performance. Delanne [3] claimed that unevenness in the pavement could lead to a fuel consumption increase of up to 6%. Chesher and Harrison [4] found that a 1 m/km rise in the international roughness index (IRI) can result in a fuel cost increase of approximately 6% for a typical car in India, and 5.5% for an equivalent vehicle in Brazil, respectively. Further, according to the National Center for Asphalt Technology (NCAT), every 1 m/km escalation in the pavement roughness index costs an additional $340 million per year on maintenance. Hence, pavement roughness is a critical determinant of road performance and demands accurate evaluations to enhance the road service quality.

Sayer et al. introduced the international roughness index (IRI) as an indicator for quantifying the pavement roughness [5], which has since become the most widely used pavement roughness evaluation index due to its high stability and transferability. According to its fundamental principles, measurements of pavement roughness are categorized into three methods: subjective rating panels, direct profile measurements, and response-type road roughness measurements (RTRRMS) [6]. The subjective rating panels adopt expert grading, which is highly subjective, and is used primarily as a reference for road evaluation. Direct profile measurements detect the elevation variation of the road surface along the driving track of a vehicle. Precise lasers, such as laser profilers [7] and 3D LiDAR [8], are deployed on specialized detection vehicles, with the IRI determined directly from the longitudinal profile elevation. This method exhibits a high accuracy and is the most widely used for evaluating the pavement roughness. As to the RTRRMS, it is used to measure the pavement roughness by collecting the vibration responses of vehicles, such as the bump integrator [9], the Bureau of Public Roads (BPR) roughometer [10], accelerometers [11], and smartphones [12]. This method has a low cost and a high efficiency, thereby making it a helpful supplement to direct profile measurements. However, it is sensitive to sudden changes and susceptible to the vehicle parameters, driving environment, and operating state. As mentioned above, these three measurements calculate the pavement roughness by detecting the elevation variations of one or several survey lines [13]. However, they cannot fully capture the overall condition of the road’s surface. While road elevation variations follow a statistically zero-mean normal distribution [14], the lateral distribution of the elevations is not uniform due to the effects of vehicle rolling, road cross and longitudinal slope, and weather conditions. When different detection vehicles traverse a given road section, the detection track of the laser probes is difficult to replicate completely, resulting in a low consistency of the IRI detection results, which ultimately impacts the evaluation of large-scale pavement roughness measurements.

In recent times, light detection and ranging (LiDAR) technology has been rapidly developing and finding more frequent use in pavement detection [15]. LiDAR is an optical ranging method that helps obtain observed objects’ position, distance, angle, and other parameters represented through a 3D point cloud output [16]. Unlike linear lidar, 3D LiDAR is less affected by light and has a broader detection range, allowing for converting 2D detection results into a 3D characterization. It improves the data dimension of pavement detection but dramatically increases the difficulty and consumption of data processing. Therefore, data processing has been the focus of the research conducted in this field. Sun et al. adopted point cloud mapping and quadratic polynomial fitting to effectively separate the pavement boundary from the surrounding buildings and realized the automatic extraction of point clouds in the pavement area [17]. Based on the calculation results of road alignment, Diaz-Vilarino et al. developed a road slice segmentation method and a K-means clustering method to extract a survey line at a distance of 1 m from the outer road edge. Following this, they calculated the root-mean-square value, skewness, and kurtosis to evaluate the pavement roughness [18]. To obtain a broader range of LiDAR data, Gao et al. collected high-altitude LiDAR data through the unmanned aerial vehicle (UAV). Moreover, the IRI results were calculated using the digital surface model (DEM) extracted from the point cloud data. However, due to the high altitude of these UAVs, the resolution of the pavement measurement points was found to be low, and the model accuracy was only about 75% [19]. With regard to roughness distribution, there are few studies. Barbarella et al. used point cloud data and the DEM model to calculate airport runways’ roughness and refined the elevation and slope variations of multiple survey lines [20]. Although this study considered the lateral influence of roughness distribution, it did not analyze whether the roughness index obtained under multiple survey lines made a significant difference. Recent advancements have demonstrated the reliability of LiDAR technology in processing point cloud data for pavement roughness measurements. However, due to the layout conditions of LiDAR and the principles governing point cloud data generation, obtaining direct elevation variations at the wheel track positions is challenging. Existing studies often calculate IRI results based on the elevation of the road centerlines or fixed-width survey lines from the centerline, thereby disregarding the influence of lateral roughness distribution. Nevertheless, it was found that lateral pavement roughness distribution could affect the vehicle excess fuel computation by up to 11% [21]. Therefore, a comprehensive evaluation of the lateral distribution of the pavement roughness is warranted.

The present study introduces a novel method for measuring the pavement roughness using point cloud data, allowing for accurate evaluations of the international roughness index (IRI) of road surfaces and the characterization of their spatial roughness distribution. Further, parametric and non-parametric tests were conducted to examine for the differences in roughness distribution among multiple survey lines.

The remainder of this paper is organized as follows. Section 2 present the data acquisition and processing details. The calculation method of the pavement roughness is introduced in Section 3. Section 4 illustrates the evaluation results of lateral pavement roughness. And Section 5 indicates the volatility evaluation results. Finally, Section 6 offers a summary of this research, and draws some implications for road maintenance.

2. Data Acquisition and Processing

2.1. Data Acquisition

The study utilized the LiMobile XT2 for the 3D scanning of the pavement environment, a high-precision device with an integrated inertial navigation and encoder. The device has a point cloud accuracy of ±1 cm, a ranging error of ±1 mm, a 360° horizontal perspective, and a vertical perspective of approximately 30°. When mounted on a vehicle, the longitudinal line density interval on the pavement is 220 mm, which meets the sampling requirement of 250 mm specified in the relevant standard. Additionally, the effective measurement range of the device was 600 m. To ensure the accuracy of the acquired point cloud data, only point cloud results within a radius of 25 m around the device were selected for this study. The single-line scanning step was 0.009°, which satisfied the density requirements of data acquisition. Figure 1 presents the LiDAR device used in this study.

Field test roads were located in the Jiading District, Shanghai, with a total length of approximately 26 km, including ten road sections, namely the Bao’an Highway section, the Boyuan Road section, the Cao’an Highway Section, the Jiasong North Road section, the Lianqun Road section, the Lvhuan Road section, the Lvyuan Road section, the Xiangfang Road section, the Yining Road section, and the Zhongbai Road section, respectively. Field tests were conducted on the road section without stagnant water and height limitations to ensure the reliability of the obtained results and avoid the effects of the surrounding vehicles. Data collection was performed between 23:00 to 03:00 while maintaining a detection speed ranging from 30 to 40 km/h, respectively. The vehicle remained in the middle or right lane of each road section during the test period.

2.2. Data Processing

The point cloud data acquired from the LiDAR detection system is prone to volatility, discreteness, and uncertainty due to the influence of the device movements and the environment. The system is commonly mounted on vehicles and operates at a fixed height and viewing angle, which may result in blockages caused by obstructing objects, such as trees and vehicles. As a consequence, it becomes challenging to obtain the complete 3D scanning data of the scene. To ensure the reliability of the acquired data, data processing is therefore required. This processing entails a sequence of steps, including LiDAR angle correction, coordinate transformation, point cloud registration, and data filtering before further calculations can be performed. The details of these steps are elaborated upon in subsequent sections.

2.2.1. LiDAR Angle Correction

During LiDAR acquisition, vehicle vibration may cause deflections in the LiDAR device. This angular deflection can be classified into three types: yaw angle, pitch angle, and roll angle. Specifically, the yaw angle corresponds to the angular deflection resulting from rotation around the Z-axis of the coordinate system. In contrast, the pitch angle corresponds to the angular deflection resulting from rotation around the Y-axis, and the roll angle corresponds to the angular deflection resulting from rotation around the X-axis, respectively. These definitions are illustrated in Figure 2a. If there is a roll angle between the LiDAR device and the vehicle, the road cross-section at that location has an inclination angle, as shown in Figure 2b. The red dotted line represents the road cross-section in the point cloud data, while the solid blue line represents the actual road cross-section. This causes the road cross-section in the point cloud to have an inclination angle, and the farther it is from the LiDAR scanning center, the greater the height error. Figure 2c illustrates the correction diagram, and the theoretical correction is provided in Formula (1):

$$\{\begin{array}{l}\mathrm{\Delta }l=\frac{h\left(\sin \left(\theta +\alpha \right)-\sin \theta \right)}{\cos \left(\theta +\alpha \right)}\\ \mathrm{\Delta }h=h\cdot \left(\frac{\cos \theta }{\cos \left(\theta +\alpha \right)}-1\right)\end{array}$$

(1)

where Δh and Δl indicate the height error and the lateral distance error, respectively, h is the height of the LiDAR laser transmitter, θ is the angle between the incoming LiDAR laser ray and the vertical direction, and α is the roll angle of the LiDAR scanner.

When the LiDAR device is inclined with a pitch angle, the angle between the LiDAR scan lines and the ground surface increased. This inclination causes the collected point cloud data from the rear of the vehicle to be higher than the original position. As shown in Figure 2d, the red dot above represents the position of the road cross-section in the obtained data, while the blue dot below represents the actual position of the road cross-section. Since the theoretical equation cannot correct the pitch angle error, it is typically compensated by adding a pitch angle to each frame of the point cloud data. Similarly, when the LiDAR device has a yaw angle, the scan line is no longer perpendicular to the road direction, resulting in an error that the theoretical equation cannot correct. This is illustrated in Figure 2e, wherein the pavement elevation of the red dashed line is not aligned with the actual elevation of the straight line. Therefore, during field tests, the joints were strengthened when installing the LiDAR device to minimize the angular deflection.

2.2.2. Coordinate Transformation and Point Cloud Registration

The point cloud data obtained from LiDAR scanning represents the object positions in the device’s coordinate system. In order to obtain a continuous point cloud space, the collected point cloud data for each frame should therefore be spliced. This process requires a coordinate transformation to convert the data from the device’s coordinate system to the vehicle coordinate system, followed by a second transformation to the geodetic coordinate system. Typically, the relationship between the LiDAR and vehicle coordinate systems can be characterized as a translation and rotation, as shown in Figure 3. The transformation of the point cloud coordinates before and after the conversion can be performed using Formula (2). The second transformation is similar to the first transformation, which is between the vehicle and geodetic coordinate systems.

$$\{\begin{array}{l}\sin \theta =\frac{\sqrt{{\tan }^2\alpha +{\tan }^2\beta }}{\sqrt{{\tan }^2\alpha +{\tan }^2\beta +1}}\\ \cos \theta =\frac{1}{\sqrt{{\tan }^2\alpha +{\tan }^2\beta +1}}\\ \left(n_x,n_y,n_z\right)=\left(\frac{\tan \beta }{\sqrt{{\tan }^2\alpha +{\tan }^2\beta }},\frac{-\tan \alpha }{\sqrt{{\tan }^2\alpha +{\tan }^2\beta }},0\right)\\ T=\left[\begin{array}{ccc}{n}_x^2\left(1-\cos \theta \right)+\cos \theta & n_x{n}_y\left(1-\cos \theta \right)-n_z\sin \theta & n_x{n}_z\left(1-\cos \theta \right)+n_y\sin \theta \\ n_x{n}_y\left(1-\cos \theta \right)+n_z\sin \theta & n_y^2\left(1-\cos \theta \right)+\cos \theta & n_y{n}_z\left(1-\cos \theta \right)-n_x\sin \theta \\ n_x{n}_z\left(1-\cos \theta \right)-n_y\sin \theta & n_y{n}_z\left(1-\cos \theta \right)+n_x\sin \theta & n_z^2\left(1-\cos \theta \right)+\cos \theta \end{array}\right]\\ {\left(x_0,y_0,z_0\right)}^T=T\cdot {\left(x_1,y_1,z_1\right)}^T\end{array}$$

(2)

where ${(x_1,y_1,z_1)}^T$ indicates the position of point cloud data in the LiDAR coordinate system, ${(x_0,y_0,z_0)}^T$ indicates the coordinates converted to the vehicle coordinate system, and α and β represent the rotation angles of the LiDAR coordinate system around the Y-axis and the X-axis of the vehicle coordinate system, respectively. The angle direction is positively clockwise and T is the coordinate rotation matrix, as depicted in Figure 3c. The α and β angles need to be measured or obtained from the manual of the LiDAR device.

The process of point cloud registration involves several steps. Initially, the point cloud data was transformed from the LiDAR coordinate system to the vehicle coordinate system. Subsequently, using the GPS positioning data as the center of the vehicle coordinate system, the rotation parameters, including the yaw angle, roll angle, and pitch angle of the IMU system were applied to adjust the position and direction of the point cloud data for each frame. Ultimately, the registered point clouds were integrated into a single file. The root-mean-square error (RMSE) of the registration results was less than 0.04 m, which was deemed to be acceptable. It should be noted that the GPS positioning time, the IMU system’s recording time, and the point cloud data collection time must be matched, as the data collected at different times cannot be utilized to rectify the point cloud coordinates.

2.2.3. Data Filtering

During the process of data acquisition, multiple factors may lead to ranging errors in the LiDAR device. Firstly, multipath reflection echoes may cause errors. Secondly, errors can arise from the equipment calculation process or from the vibrations generated by the vehicle-mounted structure due to road bumps. These errors manifest as abnormal points that deviate from the object’s surface and may impact the detection results. Therefore, it is necessary to eliminate them as much as possible during the preprocessing. Various denoising methods for point cloud data are available, including the Laplacian denoising algorithm [22], the denoising algorithm based on moving least squares (MLS) [23], the denoising algorithm based on the mean curvature flow [24], and the denoising method based on neighborhood filtering [25,26]. Although each method has its advantages, the total amount of point cloud data collected in this research is too large, typically comprising more than 500,000 points within a range of about 10 m. Consequently, the efficiency of these methods is insufficient to meet the research needs.

Therefore, this study employed a statistical filtering approach for each frame of the point cloud data. Typically, a single frame of the point cloud data consists of no more than 4500 points, with each point lying on the same plane. To circumvent the computational challenges arising from the disorderliness of the point cloud data, the positions of these points were sequenced and calculated accordingly. The proposed approach is based on the following principle: a specified number of adjacent points are sought for each point, and the average distance from each point to its adjacent points is calculated. The mean and standard deviation of these average distances is then computed. Since the distance distribution between two points in the point cloud generally follows a Gaussian distribution, a point can be considered as noise if its average distance to neighboring points exceeds the maximum distance. In such cases, the point is removed from the data. The calculation of the maximum distance is presented in Formula (3):

$$L=M+k\times \sigma$$

(3)

where L is the maximum acceptable distance between two points, M is the mean of the average distance between the points, k is the standard deviation amplification coefficient, and σ is the standard deviation of the average distance between the points. In order to retain accurate data as much as possible, the value of k was taken as 3.

3. Calculation of the Pavement Roughness Based on 3D LiDAR Data

3.1. Road Profile Extraction

The extraction of the road centerlines and road profiles based on point cloud data is a critical step in road surveying. In order to obtain the road surface area, road centerline extraction should be performed first by extracting the centerline from the road profile, which is parallel to the road direction, followed by the extraction of the road profiles according to a specific sampling spacing. Two main approaches are commonly employed for extracting the road centerline from the point cloud data. The first approach involves the direct fitting of the road centerline through the point cloud data of the road surface. The second approach involves obtaining the road boundary area first using a certain method, and then estimating the median value as the road centerline by fitting the road linearity on both sides of the boundary. However, the accuracy of the first approach is affected by the quality of the collected data. When the detected vehicle does not drive along the road centerline but along the left or right lanes, the fitted centerline tends to be biased towards one side of the road.

Therefore, the second approach was employed, determining the road boundary and extracting the centerline. Unlike the first approach, this approach obviates the need to employ all the point cloud data to compute the road boundary, thereby eliminating the necessity of performing a down-sampling operation on the original point cloud data. However, the problem of road alignment still requires redressal by the method of segmentation statistics. As illustrated in Figure 4a, the road boundary on both sides generally conforms to the maximum or minimum values in the X or Y direction. Based on this assumption, a straight line parallel to the X-axis or Y-axis was moved along the corresponding direction, and the maximum and minimum intersection points with the point cloud data were then recorded. The boundary equation was then derived by fitting the intersection points set of the upper and lower sections. The midline equation was then established, which is a line that is parallel to the upper and lower boundaries and at the same distance from them. As the length of the road cross-sections is generally less than that of the road profiles, the moving line is selected by comparing the length of the road point cloud data in the X and Y directions, with the shorter length deemed as the moving line. In addition, after the maximum and minimum values were extracted, the spacing between each pair and their mean values were calculated for processing the starting and ending areas. Pairs whose spacing deviated from the mean within a certain range were excluded as outliers.

As shown in Figure 4b, this fitting method demonstrated a favorable outcome in extracting the road centerline. The extracted centerline is aligned with the road direction, effectively avoiding the impact of uneven point cloud density, which is a limitation encountered in the first method. In addition, the advantage of using this method was that the segmentation statistics method was adopted to avoid the centerline fitting problem of the non-straight road sections as much as possible. Consequently, the proposed method can be applied to straight and multiple curves, enhancing its applicability in various road alignments.

3.2. 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,

$$\{\begin{array}{l}{m}_s{\ddot{z}}_s+C_s({\dot{z}}_s-{\dot{z}}_u)+k_s(z_s-z_u)=0\\ m_u{\ddot{z}}_u+C_s({\dot{z}}_u-{\dot{z}}_s)+k_s(z_u-z_s)+k_t(z_u-Y)=0\end{array}$$

(4)

$${\ddot{z}}_s=1/m_s\cdot [C_s({\dot{z}}_u-{\dot{z}}_s)+k_s(z_u-z_s)]$$

(5)

$${\ddot{z}}_u=1/m_u\cdot [C_s({\dot{z}}_s-{\dot{z}}_u)+k_s(z_s-z_u)+k_t(Y-z_u)]$$

(6)

$$IRI=\frac{1}{L}{\displaystyle \underset{0}{\overset{L}{\int }}\left|z_s-z_u\right|dx}$$

(7)

where m_s and m_u denote the sprung and the unsprung masses, respectively, k_s indicates the stiffness of the spring, c_s the damping coefficient of the shock absorber, k_t represents the stiffness of the tire, Y is the surface roughness, z_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” (JJG 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 z₀ 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.

3.3. Correlation Test of the Calculated Results

To validate the effectiveness of the proposed method for calculating the pavement roughness using the LiDAR data, a correlation test was conducted to compare the calculated results with those obtained using the standard test method. The standard test method, as prescribed in the “Field Test Methods of Highway Subgrade and Pavement” (JTG 3450-2019) [28], employs precise leveling instruments or total stations to measure the elevation of the road surface. Therefore, the total station was used to measure the road elevation of the test roads, and the IRI was obtained as the reference value. According to the requirements of the specifications [28], 15 test road sections with lengths of no less than 500 m were chosen for the correlation test. In each test road section, two longitudinal profiles located at distances of 0.5 m and 1.5 m from the left edge of the road were extracted as the survey lines, and the sampling interval was 25 cm, as shown in Figure 5.

Due to the disparity between the coordinate systems of the total station and the point cloud data, a coordinate transformation is required before conducting comparisons. This study selected distinctive points that can be clearly identified in the point cloud data and the real environment as feature points. Examples of such feature points include the corner points of road markings and roadside signs. The total station measured the coordinates of these feature points, while their coordinates in the point cloud data were obtained through manual selection. In cases where accurate corresponding points could not be precisely selected due to the sparsity of the point cloud, the average of several nearby points was chosen as a substitute. Subsequently, the feature points were incorporated into Formulas (8) and (9) to solve for the six variables: $\mathrm{\Delta }x,\mathrm{\Delta }y,\mathrm{\Delta }z,{\theta }_x,{\theta }_y,{\theta }_z$.

$$\left[\begin{array}{l}x\\ y\\ z\end{array}\right]=R\left[\begin{array}{l}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right]+\left[\begin{array}{l}\mathrm{\Delta }x\\ \mathrm{\Delta }y\\ \mathrm{\Delta }z\end{array}\right]$$

(8)

$$R=\left[\begin{array}{ccc}\cos {\theta }_y\cos {\theta }_z-\sin {\theta }_x\sin {\theta }_y\sin {\theta }_z& -\cos {\theta }_y\sin {\theta }_z-\sin {\theta }_x\sin {\theta }_y\cos {\theta }_z& -\sin {\theta }_y\cos {\theta }_x\\ \cos {\theta }_x\sin {\theta }_z& \cos {\theta }_x\cos {\theta }_z& -\sin {\theta }_x\\ \sin {\theta }_y\cos {\theta }_z+\sin {\theta }_x\cos {\theta }_y\sin {\theta }_z& -\sin {\theta }_y\sin {\theta }_z+\sin {\theta }_x\cos {\theta }_y\cos {\theta }_z& \cos {\theta }_x\cos {\theta }_y\end{array}\right]$$

(9)

where x, y, z represents the coordinates of the feature points in the point cloud data, while x′, y′, z′ represents the coordinate values of the feature points in the total station data, R denotes the rotation matrix mapping the total station coordinate system to the point cloud coordinate system, $\mathrm{\Delta }x,\mathrm{\Delta }y,\mathrm{\Delta }z$ is the differences in translation between the coordinates of the total station and the point cloud, and ${\theta }_x,{\theta }_y,{\theta }_z$ indicates the rotation angles in the three directions between the total station coordinate system and the point cloud coordinate system.

After obtaining the reference value and calculated results of the IRI, the correlation test method in Appendix C of the specification [28] was used for validation. The calculation equations are presented in Formulas (10)–(13):

$$Y=AX+B$$

(10)

$$A=\frac{n{\displaystyle \sum xy}-{\displaystyle \sum x}{\displaystyle \sum y}}{n{\displaystyle \sum x^2}-{({\displaystyle \sum x})}^2}$$

(11)

$$B=\overline{y}-A\overline{x}$$

(12)

$$R=\frac{\overline{xy}-\overline{x}\overline{y}}{\sqrt{(\overline{x^2}-{\overline{x}}^2)(\overline{y^2}-{\overline{y}}^2)}}$$

(13)

where Y represents the calculated results using the point cloud data, denoted as $IR{I}_{calculated}$, X corresponds to the reference value of the IRI obtained from the total station measurements, denoted as $IR{I}_{reference}$, A is the slope derived from the linear regression formula, B signifies the corresponding intercept, and R denotes the correlation coefficient.

Figure 6 indicates the correlation test results, with a correlation coefficient of R² = 0.9690 (>0.95). This high correlation coefficient demonstrates a strong relationship between the IRI results calculated using the point cloud data and those obtained from the total station measurements. Therefore, the method proposed in this study for calculating road roughness using point cloud data is reliable.

4. Lateral Distribution Evaluation of the Pavement Roughness

4.1. Difference Analysis of the Pavement Roughness among Multiple Survey Lines

As stated previously, the variation in the road elevation among multiple survey lines may be influenced by various factors, such as the road slope and the vehicle axle load. Therefore, this study utilized the centerline of the field test roads as a reference line, and shifted the survey lines by a distance of l in turn to determine the elevation variation, as illustrated in Figure 7. The red line represents the road centerline, while the yellow line depicts the survey line, with a distance of l from each other. Since the straight sections were selected as the field test sections, a survey line can be obtained by connecting the starting and ending points with a vertical distance of l from the centerline.

This study evaluated the variance of pavement roughness among 10 lateral survey lines extracted from a field test road. The spacing between the survey lines ranged from 0.2 m to 1.2 m, respectively, capturing the road profile comprehensively. The survey lines were obtained from different positions, including the lane lines, the wheel tracks, and the center of the lane, facilitating comparisons with the traditional approach.

Figure 8 illustrates the IRI results obtained from different survey line spacings, revealing significant disparities in roughness measurements among different profiles within one road. The observed range of IRI values spanned from 1.12 m/km to 10.16 m/km, respectively, indicating a substantial difference of 9.04 m/km. Such variations can lead to a downgrade in the overall assessment by up to three grades. These discrepancies can be attributed to the specific location of the field test roads in the Jiading District of Shanghai, with a large presence of large and medium-sized trucks. Consequently, the pavement roughness was deemed to be relatively poor, with many road sections exhibiting IRI values exceeding 6 m/km.

Table 1. The variance of IRI results under different survey line spacing.

Spacing l (m)	Variance (m2)
0.2	0.4364
0.4	0.7278
0.6	1.2621
0.8	3.0781
1.0	7.7535
1.2	4.1404

presents the variance of IRI results obtained under different survey line spacing. The results indicate that as the spacing increases, the variance also increases, which aligns with the findings observed in Figure 8, where the IRI results from closer spacing survey lines exhibit less fluctuation. It was also uncovered that road profiles with larger distances between survey lines, particularly between different lanes, show significant differences in roughness. Of note, when the spacing was 1 m, that is, the survey lines were distributed at the wheel track (considering a standard lane width is 3.5 m, and the left and right wheelbase of the vehicle are about 1.8 m respectively), the variance of IRI results reached its maximum. Hence, it has recommended to assess the extent of road damage and consider the necessity of obtaining roughness values from multiple survey lines when conducting road roughness detection.

A pseudo-color map depicting the distribution of longitudinal (approximately 1500 m) and lateral (four lanes) roughness is presented in Figure 9. The IRI was calculated at a longitudinal interval of 100 m. The map highlights variations in roughness distribution across different spatial locations on the road. Notably, the inner lane exhibited higher IRI values compared to the outer lane. Moreover, areas with a poorer roughness were mainly concentrated around 3 m from the centerline, which is consistent with the observations in Figure 9. This can be attributed to the road section’s characteristics, where the widened portion has undergone significant deterioration over time compared to the original road, leading to a poorer performance. In contrast, the outer side of the road, which has undergone widening, exhibited a comparatively better performance.

4.2. Difference Test for the Lateral Distribution of Pavement Roughness

To effectively characterize the lateral roughness distribution in a large-scale road network, this study analyzed the variation in the IRI distribution among multiple survey lines and road sections using a dataset covering 26 km of road. Parametric and non-parametric tests were conducted to examine significant differences in the IRI distribution. To ensure data quality, the collected data were divided into 100 m intervals, and only straight pavement sections were included, while turning road sections were excluded. Generally, the roughness distribution of asphalt pavement exhibits a normal or skewed pattern. However, in cases with insufficient data samples, the roughness distribution may deviate from normality according to the law of large numbers. Skewed distributions can affect the validity of the parameter tests, leading to the rejection of null hypotheses. Therefore, the Box–Cox transformation was applied to adjust the data distribution to a normal form and to maintain a constant variance of the dependent variables. The transformation involves converting the original values using a specific function to bring the new data distribution closer to normality. Figure 10a demonstrates the skewed distribution of the pavement roughness prior to the Box–Cox transformation, while Figure 10b shows the normal distribution after the transformation. The formula is shown in Formula (14):

$$IR{I}_{after}{}^{(\lambda )}=\frac{IR{I}_{before}{}^{\lambda }-1}{\lambda }$$

(14)

where $\lambda$ is related to the distribution of the IRI, which is determined by the maximum likelihood estimation method. Assuming that the transformed IRI follows a normal distribution, $\lambda$ is the value of $\lambda$ when the likelihood function value is at its maximum.

Upon converting the roughness samples of different survey lines to conform to a normal distribution, variation analysis (ANOVA) can be utilized to investigate significant differences in the roughness distribution. It is important to note that despite multiple survey lines being obtained from the same road section, each profile constitutes an independent sample. The calculation of the IRI involves resampling a road profile, treating each survey line as an independent sample.

Table 2. ANOVA test results of the pavement roughness distribution under different survey line spacings.

Spacing l (m)	p-Value
0.1	0.2909
0.2	0.3487
0.3	0.4266
0.5	0.2556
1.0	0.4356
1.5	0.2027
2.0	0.9469
2.5	0.0311

presents the p-value obtained from the ANOVA test for IRI distribution among different survey line spacings. These results revealed that only when the spacing was 2.5 m, the p-value was below the significance level of 0.05, indicating a significant variation in roughness distribution. In other cases, no significant variation was observed. Furthermore, these results revealed a significant difference in the roughness distribution across different lanes, while no significant difference was found among the survey lines within the same lane. Notably, the ANOVA test can only shed light on the absence of significant differences in the overall roughness distribution. It cannot illustrate that the IRI results of different survey lines within one road section do not differ significantly. The ANOVA test presupposes that the sample conforms to a normal distribution. Nevertheless, in the case of small sample sizes, the roughness data may display a stronger skewed trend. Therefore, the Kruskal–Wallis test (KW test) was conducted to analyze the roughness difference among multiple survey lines with small samples.

The KW test is a widely employed non-parametric test method that is used to assess differences among multiple independent samples of continuous data. Unlike the ANOVA test, which primarily focuses on determining whether the means of different distributions are equal, the KW test emphasizes the ranking of multiple independent samples and their collective contribution to the overall sample. As a result, the KW test provides a superior evaluation of the inherent variability within the distributions. The KW test statistics are shown in Formulas (15)–(17):

$$K{W}_c=KW/C$$

(15)

$$KW=\frac{12}{n\left(n+1\right)}\cdot {\displaystyle \sum _{i=1}^k\frac{T_i^2}{n_i}-3\left(n+1\right)}$$

(16)

$$C=1-\frac{{\displaystyle \sum \left({\tau }_i^3-{\tau }_i\right)}}{n^3-n}$$

(17)

where T_i denotes the rank sum of each sample, k represents the total number of compared data samples, n indicates the total individual number after mixing the samples, and τ_i is the number of ith tied values (ranks with the same value). In this paper, the field test roads were divided into 10 sections by intersections, including the Bao’an Road section, the Boyuan Road section, the Cao’an Road section, the Jiasong North Road section, the Lianqun Road section, the Lvhuan Road section, the Lvyuan Road section, the Xiangfang Road section, the Yining Road section, and the Zhongbai Road section, respectively. The roughness values of the different survey lines were extracted at intervals of l = 0.1 m, 0.5 m, 1.0 m, 1.5 m, 2.0 m, 2.5 m, and 3.0 m, respectively. The KW test results are shown in

Table 3. KW test results of the pavement roughness distribution under different roads and survey line spacings.

Spacing l (m)	0.1	0.5	1	1.5	2	2.5	3
Bao’an Highway section	<0.0001	0.0002	0.0007	0.0163	0.0035	0.0335	0.0118
Boyuan Road section	0.0002	0.0088	0.0013	0.5093	0.4464	0.6793	0.4317
Cao’an Highway section	0.9547	0.9805	0.8293	0.6707	0.7544	0.6237	0.3974
Jiasong North Road section	0.0011	0.1498	0.0892	0.0464	0.408	0.4983	0.1648
Lianqun Road section	0.9915	0.9984	0.97	0.9512	0.7567	0.632	0.7761
Lvhuan Road section	0.0003	0.0225	0.0815	0.0407	0.8434	0.0154	0.0733
Lvyuan Road section	0.0033	0.0748	0.0565	0.2409	0.0236	0.5043	0.2725
Xiangfang Road section	0.9733	0.963	0.8559	0.7526	0.7472	0.601	0.8537
Yining Road section	0.0159	0.0773	0.0721	0.1258	0.0952	0.3167	0.1525
Zhongbai Road section	0.9199	0.9944	0.8625	0.822	0.5448	0.3886	0.4462

Bold numbers indicate significant results, which are less than 0.05.

.

As illustrated in Table 3, the IRI results of various survey lines within one road section were found to be significantly influenced by the road characteristics. Specifically, the Bao’an Highway section exhibited p-values below 0.05 for all spacing intervals, which indicates substantial differences in the roughness distribution among multiple survey lines. Conversely, the Cao’an Highway section and the Lianqun Road section displayed p-values above 0.05 across all spacing intervals, suggesting no significant differences in their roughness distribution. In addition, notable variations in p-values were observed for different spacing intervals, suggesting that the IRI distribution was influenced by the spacing l between the survey lines in most road sections.

Furthermore, the analysis included the examination of the IRI distribution along the centerline of each road section and its correlation with the roughness results. Figure 11 indicates that road sections with a higher substantial dispersion of roughness, such as the Bao’an Highway section, the Lvyuan Road section, and the Jiasong North Road section, exhibited a higher likelihood of significant differences in the roughness results. Conversely, road sections with lower dispersion of roughness, such as the Boyuan Road section and the Cao’an Highway section, showed a relatively low probability of significant differences in the roughness results. This observation can be attributed to the fact that roads with a strong IRI dispersion undergo frequent maintenance and repair activities, leading to changes in their material and structural properties. Consequently, there is an increased likelihood of significant variations in their lateral distribution characteristics.

5. Volatility Evaluation of the Pavement Roughness under Different Detection Lengths

The volatility of the pavement roughness can be categorized into two aspects: natural volatility and artificial volatility. The former is associated with the lateral variation of pavement roughness that is influenced by various environmental factors, such as the road material condition, traffic volume, and climatic conditions. On the other hand, the latter refers to the longitudinal variation in the repeated roughness detection results, which is mainly influenced by the data collection methodology. While the previous section discussed the analysis of lateral volatility, this section will primarily concentrate on investigating artificial volatility.

Since the detection vehicle encounters a red light at the intersection, requiring it to decelerate to a stationary state, the IRI results around the intersection area significantly differ from those of the other areas. Therefore, the invalid detection results of the low-speed area were generally excluded from further analysis. However, given that a red light is a random event, a discrepancy existed between the detection length and the actual road length in multiple detections, with variations of up to 70 m having been observed in the detection data. This discrepancy arises from the fluctuation of the starting and ending points during each IRI calculation, leading to the repeated detection results volatility. Thus, this study provides insights into the influence of the variation in the starting and ending points on the roughness results, which is essential for ensuring the reliability of the results under different detection lengths.

In accordance with the repeatability test method specified in the standard [27], the variation coefficient was chosen to evaluate the roughness volatility. It was calculated as shown in Equation (9) and is required to be within 5%. To further assess the significance of the results, the Mann-Whitney U test was employed. It is a non-parametric test that is typically utilized to determine whether two independent sequential data samples originated from the same population. Since minimal roughness volatility was expected within one road section during detection, the original hypothesis assumed that the variation coefficient was less than or equal to the maximum threshold C_v0 (C_v0 = 5%). The hypotheses and test statistics of the one-tailed test are presented in Formulas (18)–(20) [29]:

$$\{\begin{array}{l}S=\sqrt{\frac{{\displaystyle \sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2}}{n-1}}\\ C_v=\frac{S}{\overline{x}}\cdot 100\%\end{array}$$

(18)

$$H_0:C_v\le C_{v0};H_1:C_v>C_{v0}$$

(19)

$$\mu =\frac{\sqrt{n-1}\left|C_v-C_{v0}\right|}{C_{v0}\sqrt{0.5+C_{v0}^2}}$$

(20)

where S is the standard deviation of repeatability, x_i is the ith roughness result, $\overline{x}$ is the average of roughness results, n is the total number of samples, C_v0 is the variation coefficient calculated for the sample, and μ is the test statistics. At a 95% confidence level, if μ is less than 1.96, the result is considered insignificant, and the original hypothesis H₀ is rejected. If μ is greater than or equal to 1.96, the result is considered statistically significant, and the original hypothesis H₀ can thereby be accepted, and when μ is greater than 2.58, the result is considered very significant.

The primary length range of the road section fell between 100 and 500 m, respectively. Given that the statistical interval was generally taken as 100 m, the shorter road was not conducive to analysis. Therefore, a 672 m long road section on the Cao’an Highway was utilized as the test section. The starting and ending points were offset to calculate the roughness for analyzing the influence of the different detection lengths. In order to ensure comparability, the pavement roughness was calculated using the elevation data obtained from the road wheel track. The calculated results are shown in

Table 4. IRI results under different starting and ending point offsets.

Offset Position	Offset Distance (m)	IRI Results (m/km)	Mean of the IRI (m/km)	μ Value
No offset	0	5.60, 5.07, 6.14, 5.19, 4.36, 6.51, 3.50	5.196	1.78
Starting point	10	5.50, 5.02, 6.06, 5.15, 4.34, 6.57, 3.85	5.213	1.88
Starting point	20	4.88, 5.03, 6.15, 5.10, 4.35, 6.57, 4.32	5.200	1.93
Starting point	30	4.69, 5.15, 6.04, 5.10, 4.36, 6.67, 4.67	5.240	2.02
Starting point	40	4.09, 5.63, 5.82, 5.30, 4.62, 6.07, 4.85	5.197	2.11
Starting point	50	4.12, 5.74, 6.19, 4.82, 5.33, 5.54, 4.79	5.219	2.13
Starting point	60	5.12, 5.69, 5.25, 4.66, 5.50, 5.26, 5.02	5.214	2.49
Ending point	10	5.34, 5.12, 6.05, 5.13, 4.25, 6.61, 4.25	5.250	1.83
Ending point	20	5.02, 5.13, 6.12, 5.21, 4.22, 6.45, 4.11	5.180	1.89
Ending point	30	5.24, 5.34, 5.94, 4.97, 4.54, 6.21, 4.26	5.214	1.91
Ending point	40	4.95, 5.49, 5.78, 5.24, 5.02, 5.46, 4.52	5.209	1.82
Ending point	50	5.48, 5.61, 6.01, 4.98, 4.51, 5.29, 4.57	5.207	1.87
Ending point	60	4.35, 5.27, 6.06, 5.41, 4.17, 6.48, 4.73	5.210	1.95

Bold numbers indicate significant results, which are greater than 1.96.

and Figure 12.

Table 4 indicates that variations in the starting and ending positions lead to differences in the calculated IRI results within each statistical interval. Figure 12 further illustrates this trend, indicating that as the offset distance increases, the differences in the IRI results within each interval decrease, resulting in a more concentrated roughness distribution. This may be attributed to the fact that the roughness in the middle section is generally less discrete compared to the two extremities. However, it is worth noting that the mean of the IRI results remained relatively constant, suggesting that the conventional method of using the mean value to represent the roughness of the entire road section is reasonably reliable to some extent.

In view of the volatility, it was found that offsetting the starting point has a significant impact on the roughness, while offsetting the ending point has a minimal effect. Specifically, when the offset distance of the starting point exceeded 30 m, the volatility of the roughness became statistically significant (μ > 1.96). This demonstrates that an unreasonable offset of the starting point may obscure the local abnormal areas in these road sections. On the other hand, changing the position of the ending point primarily affected the IRI result of the last statistical interval, resulting in a minimal overall influence. Consequently, the risk of roughness variation caused by offsetting the ending point was deemed to be lower than offsetting the starting point. Therefore, it is advisable to prioritize controlling the recording position of the starting point during the actual detection process.

The analysis of the influence of the starting and ending positions on the roughness revealed that increasing the offset distance significantly increases the volatility of the roughness results. To further evaluate the sensitivity of the road sections with different lengths to the offset distance, 700 road sections with various lengths were selected for analysis. The offset distances that caused significant changes in the roughness results were identified for each road section. In light of the above findings, the volatility evaluation result was used to determine noticeable changes in roughness, considering only the offset of the starting point. These results are presented in

Table 5. Offset distance thresholds when the roughness changes significantly under different road lengths.

Road Length (m)	Average Offset Distance Threshold (m)
100–200	27.86
200–300	26.87
300–400	25
400–500	27
>500	29.47

.

Based on the consistent findings, it can be concluded that the average offset distance threshold causing a significant change in roughness results is less than 30 m, regardless of the variation in the road length. Therefore, for most road sections, when considering the roughness volatility along the longitudinal road distribution, the difference between the detection distance and the actual road length should be limited to 30 m to ensure the reliability of the results.

6. Conclusions

This study utilized a LiDAR device to obtain high-precision point cloud data, covering a distance of over 26 km in field tests. The acquired LiDAR data was processed through angle correction, coordinate transformation, and registration to extract the elevation data from multiple survey lines of the road section. The proposed method for calculating the pavement roughness using point cloud data was validated through a correlation test, comparing the calculated results with those obtained through total station measurements. Moreover, parametric and non-parametric tests were conducted to investigate the differences in the roughness distribution among multiple survey lines. Based on these analyses, the following conclusions can be drawn:

(1)

Significant differences were observed in the IRI results among multiple survey lines within one road section. In particular, roads with an extensive uneven damage or partial widening may exhibit variations in their roughness grades;

(2)

The presence of significant differences in the IRI results among the different survey lines within one road section depends on the characteristics of the road. In summary, roads with a stronger dispersion of roughness tend to exhibit significant differences;

(3)

The difference between the detection length and the actual road length can lead to volatility in the repeated detection results. A significant difference in the roughness results may occur when the offset distance exceeds 30 m. Therefore, it is advisable to restrict the difference to 30 m to ensure the reliability of the results;

(4)

Traditional methods of calculating the IRI based on a single or small number of survey lines may not comprehensively characterize the road performance. Incorporating the IRI distribution attributes of different survey lines in the evaluation indexes is thereby recommended to improve the reliability of road performance assessments.

This study proposed a novel method for measuring the pavement roughness using point cloud data, enabling the calculation of IRI results for any road profile, and providing insights into the spatial distribution of the pavement roughness. It offers road management departments more detailed data to assess driving quality and presents a new technique for road detection. Nevertheless, the analysis was limited to urban road data with IRI values ranging from 3–6 m/km without considering expressways, rural roads, and other scenarios. Future research will expand the data coverage and types into developing a more scientific, comprehensive, and practical model for evaluating the pavement roughness based on the detection results. Moreover, the LiDAR data, with its high precision and comprehensive characteristics, shows promise for further exploration in road performance assessments. In the future, it can be expanded to include the detection of pavement distresses and roadside facilities, providing a more comprehensive understanding of the road infrastructure’s conditions.