Open AccessArticle

Measurement Model of Full-Width Roughness Considering Longitudinal Profile Weighting

Yingchao Luo

Huazhen An

Xiaobing Li

Jinjin Cao

Na Miao

and

Rui Wang

Research Institute of Highway Ministry of Transport, Beijing 100088, China

Author to whom correspondence should be addressed.

Appl. Sci. 2024, 14(22), 10213; https://doi.org/10.3390/app142210213

Submission received: 1 October 2024 / Revised: 4 November 2024 / Accepted: 5 November 2024 / Published: 7 November 2024

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Browse Figures

Figure 1
Distribution frequency of vehicle centerlines and its fitting results. "> Figure 2
Lateral distribution models of wheel trajectories f(x). "> Figure 3
The ratio of cumulative sum S of fnorm(x) to the value of Fs. "> Figure 4
Calculation process of full-width roughness. "> Figure 5
Experimental testing roads with various specifications of standard blocks. "> Figure 6
Original road elevation point cloud data P1. "> Figure 7
The weighted elevation point cloud data P2. "> Figure 8
Curve of cumulative elevation P3. "> Figure 9
Elevation comparison between equivalent profile VF and profile V1. "> Figure 10
Comparison of the IRI based on 3 longitudinal profile elevations. "> Figure 11
Comparison of Cv based on 3 longitudinal profile elevations. ">

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Abstract

This study proposes and establishes a roadway longitudinal profile weighting model and innovatively develops a process and method for evaluating road surface roughness. Initially, the Gaussian model is employed to accurately fit the distribution frequency of vehicle centerlines recorded in British Standard BS 5400-10, and a generalized lateral distribution model of wheel trajectories is further derived. Corresponding model parameters are suggested for different types of lanes in this study. Subsequently, based on the proposed distribution model, a longitudinal profile weighting model for lanes is constructed. After adjusting the elevation of the cross-section, the equivalent longitudinal elevation of the roadway is calculated. Furthermore, this study presents a new indicator and method for assessing the roughness of the entire road surface, which comprehensively considers the elevations of all longitudinal profiles within the lane. To validate the effectiveness of the proposed new method and indicator, a comparative test was conducted using a vehicle-mounted profiler and a three-dimensional measurement system. The experimental results demonstrate significant improvements in measurement repeatability and scientific rigor, offering a new perspective and evaluation strategy for road performance assessment.

Keywords:

full-width roughness of road surface; elevation point cloud data; Gaussian model; lateral distribution model of wheel trajectories; longitudinal profile weighting model

1. Introduction

In modern transportation systems, highways utilize traffic signs and markings to scientifically delineate lanes, effectively regulating vehicle travel and enhancing traffic capacity and the driving experience [1,2]. However, with continuous increase in traffic load, the road surface experiences uneven subsidence and wear, altering its elevation and operational performance [3]. From the perspectives of road maintenance, improving passenger comfort, and reducing dynamic loads on vehicles and roadways, evaluating road roughness is a crucial area of research [4].

The International Roughness Index (IRI) stands as a prevalent metric in the global evaluation and management of road systems [5]. Considerable research efforts have been dedicated to this indicator by scholars worldwide. Loprencipe et al. [6] compared and analyzed three different methods for evaluating road roughness and explored the influence of vehicle speed on the roughness threshold. Mirtabar et al. [7] monitored road conditions using cost-effective MEMS accelerometers and GPS sensors, calculating the IRI from road surface elevation data. Zhang et al. [8] introduced new IRI thresholds based on riding comfort, comprehensively assessing the impact of road roughness on driving comfort using three distinct evaluation techniques. Kaloop et al. [9] proposed an IRI prediction model leveraging multiple linear regression analysis and artificial neural networks (ANNs) for flexible road condition forecasting, achieving high prediction accuracy. Similar predictive work on IRI models was also conducted by Abdelaziz et al. [10]. Sidess et al. [11] developed a deterioration model for IRI, integrating empirical mechanical methods with regression-based approaches. These investigations highlight the effective application of IRI in assessing road roughness, thereby providing valuable data support for road management departments in formulating maintenance strategies. From the perspective of data acquisition and processing, early studies typically relied on devices like laser profilometers [12] and 3D LiDAR [13] to record road surface elevation changes [14], enabling road roughness evaluation. It is worth noting that these IRI calculations were primarily based on elevation data from single or a few longitudinal profiles, without establishing direct connections to other profiles.

In actual roadways, vehicle trajectories exhibit relative randomness, and the distribution of road surface elevations is unpredictable. Based on the computational principles of IRI, variations in elevation across different longitudinal profiles can lead to disparities in IRI values. Hence, selecting longitudinal profiles scientifically and reasonably is crucial for accurately assessing the roughness of a given roadway. However, there is currently no unified standard for choosing longitudinal profiles [15], and existing methods often lack scientific rigor [14]. Peter Múčka [15] provided a comprehensive overview and tabular presentation of the differences in the number (1, 2, or multiple) and position (0.75 m, 0.79 m, 0.8 m, 0.9 m, 1.0 m, 1.2 m, etc. from the centerline of the roadway) of longitudinal profiles. Under actual road conditions, the elevation of each profile may vary. Additionally, the trajectories of inspection vehicles and other road users are not always confined to the same longitudinal profile. Consequently, relying solely on elevation data from a few longitudinal profiles to evaluate the performance of the entire roadway raises questions about scientific validity and accuracy. This limitation is primarily attributed to the technical constraints of early instrumentation, particularly the limited number of ranging sensors available [16], which hindered large-scale and wide-ranging road surface measurement efforts. Therefore, to more accurately evaluate the performance of a given roadway, comprehensive measurement and analysis of road surface elevations are necessary to obtain precise IRI values.

With the advancement of science and technology, numerous three-dimensional point cloud measurement devices have been applied in the field of road surface elevation measurement [2,17,18]. These devices, typically utilizing laser scanning technology, are capable of rapidly acquiring large amounts of high-precision point cloud data, enabling accurate measurement of road surface elevation and precise calculation of the IRI. Díaz-Vilariño et al. [19] extracted and computed road surface roughness at 1 m from the outer edge of the road using road slice segmentation and K-means clustering methods. De Blasiis et al. [20] processed three-dimensional point cloud data using ordinary Kriging and inverse distance weighting methods to extract road surface elevation changes for calculating IRI values. Kumar et al. [21] proposed determining roughness from point cloud data by establishing the standard deviation of elevation relative to an interpolated planar surface. Tran and Taweep [13] developed a segmentation algorithm based on point cloud voxelization to estimate road surface roughness. Alhasan et al. [22] utilized point cloud data acquired from three-dimensional stationary terrestrial laser scanning (STLS) to process and extract road surface roughness characteristics. Prosser-Contreras et al. [23] used unmanned aerial vehicles (UAVs) and photogrammetry to obtain the IRI of roads. These studies demonstrate that three-dimensional measurement technology has proven to be a reliable means in the field of road surface roughness detection.

Despite technological advancements, existing research still exhibits deficiencies in data analysis methods [20]. Specifically, numerous studies predominantly focus on calculating the IRI through single longitudinal profile elevation assessments, neglecting the lateral distribution variations of road surface roughness [14]. While this singular or fixed measurement line selection simplifies data processing, it fails to fully utilize the rich information provided by three-dimensional point cloud data, resulting in limitations for comprehensive road surface roughness evaluations. Gao et al. [14] employed UAVs equipped with LiDAR technology to collect road surface elevation data. Through an analysis of the lateral distribution of the IRI, they discovered significant differences among various measurement lines. Based on experimental results, they advocated for consideration of the lateral distribution of roughness. Barbarella et al. [24] made progress in evaluating airport runway surface roughness using three-dimensional point cloud data, taking a step forward by refining the lateral impact of multiple measurement lines on roughness distribution. These studies reveal that, compared to traditional measurement tools, methods based on point cloud data offer more diversified research perspectives for road surface IRI analysis [14]. However, without in-depth application of these massive elevation point cloud data, their potential value cannot be fully exploited. Therefore, there is an urgent need to adopt more scientific and reasonable data analysis techniques to enhance the accuracy and completeness of road surface roughness evaluations.

Given the randomness of vehicle trajectories and the fact that contact points with the road surface are distributed across the entire lane, it is scientifically reasonable to incorporate elevation data from the full lane into IRI calculations. However, efficiently integrating elevation data from the entire roadway width poses a new challenge. Fortunately, British Standard BS 5400-10 [25] records the frequency of lateral distribution of vehicle centerlines based on bridge traffic observation data. Although these data are specific to bridges, they provide a valuable reference for constructing new roughness measurement models. Gao et al. [14] found that the lateral distribution of asphalt pavement roughness follows a normal distribution. These research findings reinforce the idea of establishing a lateral distribution model of wheel trajectories in this study.

Based on the frequency distribution recorded in BS 5400-10, this study establishes a lateral distribution model of wheel trajectories in arbitrary lanes under different track configurations. Recommended model parameters are provided for passing lanes, driving lanes, slow lanes, and mixed lanes. Furthermore, utilizing this model, the study proposes and develops a longitudinal profile weighting model for lanes. By integrating this model with elevation point cloud data, the equivalent elevation and corresponding IRI of the lane are calculated. To distinguish it from traditional methods based on single longitudinal profile elevations, this study named the result obtained based on all elevation point cloud data as the full-width roughness, which will be characterized by IRI_F in the following text.

The structure of the rest of this paper is organized as follows. In Section 2, based on the distribution frequency values of vehicle centerlines recorded in BS 5400-10, we construct a lateral distribution model and a longitudinal profile weighting model for vehicle wheel trajectories. Section 3 introduces the concept of equivalent longitudinal profiles and elaborates on the calculation method for the equivalent elevation. Section 4 provides a detailed explanation of the calculation steps for the newly proposed road performance indicator, namely IRI_F, along with relevant precautions. Section 5 demonstrates the superiority of the new evaluation method and indicator through practical case studies. Finally, the conclusion of the paper is presented in Section 6.

2. Weighting Model of Longitudinal Profile

2.1. 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 N₀ 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 N₀, as shown in Equation (1).

f (x) = N / N_{0}

(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 π} σ_{0}} \exp (- \frac{{(x - μ_{0})}^{2}}{2 {σ_{0}}^{2}})

(2)

The fitting parameters are determined as B₀ = 0.098674, μ₀ = 0, and σ₀ = 0.226058. The good fit indicated by R² = 0.9957 suggests that Equation (2) exhibits high model versatility and can effectively replace the discrete frequencies presented in Figure 1.

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 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)

(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 B₁, B₂, μ₁, μ₂, σ₁, and σ₂, respectively. Here, B₁ and B₂ represent the weights of the two Gaussian models, f₁(x)~N₁(μ₁, σ₁²) and f₂(x)~N₂(μ₂, σ₂²), respectively. Considering the symmetry of the curves in Figure 2, it is reasonable to assume that B₁ = B₂ = B, μ₁ = μ₂ = μ, and σ₁ = σ₂ = σ. 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 π} σ_{1}}) \exp (- \frac{{(x - μ_{1})}^{2}}{2 {σ_{1}}^{2}}) + B_{2} (\frac{1}{\sqrt{2 π} σ_{2}}) \exp (- \frac{{(x + μ_{2})}^{2}}{2 {σ_{2}}^{2}})

(4)

f (x) = B (\frac{1}{\sqrt{2 π} σ}) (\exp (- \frac{{(x - μ)}^{2}}{2 σ^{2}}) + \exp (- \frac{{(x + μ)}^{2}}{2 σ^{2}}))

(5)

As can be seen from Table 1, the fitting parameters B and σ remain largely unchanged under the three different conditions, while the parameter μ varies with changes in the wheel track width L, showing a significant two-fold relationship between them. However, 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).

\{\begin{cases} f_{i} (x) = ω_{i} B (\frac{1}{\sqrt{2 π} σ}) (\exp (- \frac{{(x - 0.5 L_{i})}^{2}}{2 σ^{2}}) + \exp (- \frac{{(x + 0.5 L_{i})}^{2}}{2 σ^{2}})) \\ f (x) = \sum_{i = 1}^{N} f_{i} (x) \end{cases}

(6)

where ω_i represents the weight of the distribution model f_i(x) corresponding to track widths L_i.

2.1.3. General Model Parameters

Based on the generation mechanism of wheel trajectory zones, it can be inferred that the lateral distribution of wheel trajectories is not only related to vehicle type but also influenced by factors such as vehicle traffic volume, traffic density, and vehicle behavior patterns. In mixed traffic environments, particularly within urban traffic flows, the diversity of vehicle types is more pronounced compared to highways, resulting in a higher complexity of the lateral distribution of wheel trajectories. Fortunately, the lane channelization effect has a significant impact on vehicles traveling within each lane, leading to relative uniformity of vehicle types within each lane.

When considering a typical configuration of a three-lane road with one direction, we can assume that the leftmost lane is primarily used for passing at high speeds or overtaking vehicles, where narrow wheel track width vehicles, such as small cars, may predominate. This composition of vehicles tends to result in a lateral distribution curve of wheel trajectories that approximates a wheel track width of L = 1.6 m. The middle lane or travel lane, as the primary path for most vehicles, may contain a more diverse range of vehicle types. If we take medium wheel track width vehicles, such as some trucks, as representative, the lateral distribution curve of wheel trajectories may more closely approximate a wheel track width of L = 1.8 m. As for the rightmost lane, if it primarily serves slower-moving or heavier vehicles, such as wide-track cargo vehicles, the lateral distribution curve of wheel trajectories may trend toward a wheel track width of L = 2.0 m. In mixed traffic conditions, if the distribution of various vehicles across all lanes tends to be uniform, similar to the vehicle mix in the traffic lanes, the lateral distribution curve of wheel trajectories may converge toward an intermediate value, such as a wheel track width of L = 1.8 m, reflecting a more general characteristic of traffic flow. These hypothetical descriptions aim to provide a theoretical framework for understanding the impact of lane channelization effects on the formation of wheel trajectory zones, without being limited to strict classifications of specific vehicle types.

In view of this, this study suggests fixing the parameter μ (or L) when establishing the lateral distribution model of wheel trajectories f(x) for a specific lane, with the specific fixed values referenced in Table 2. The core rationale behind this recommendation lies in ensuring a unified benchmark for different equipment when conducting road surface inspections and processing elevation point cloud data through the fixation of parameters. By doing so, not only can the comparability of data be effectively enhanced, but it also lays a more solid and reliable foundation for subsequent data analysis and decision making.

2.2. Calculation Method for Weighting Model of Longitudinal Profile

In addition to characterizing road surface performance, road roughness reflects the dynamic excitation effect exerted by the road surface on vehicle wheels. Across the lateral direction of a roadway, various longitudinal profiles, due to their specific locations, exhibit differences in the weight of excitation imparted to the wheels. To analyze road roughness more accurately and scientifically, it is necessary to assign weights to the point cloud data of each longitudinal profile. Based on the generation mechanism of road wheel trajectory zones, this study selects the lateral distribution model of wheel trajectories f(x) as the theoretical foundation for calculating the weights of different longitudinal profiles. However, considering the amplitude of elevation, it is essential to normalize f(x).

Specifically, the normalization of f(x) should adhere to the following principles: During vehicle operation, the wheel trajectory zone is a critical region that significantly affects the dynamic response of the vehicle. Hence, the elevation in this area needs to be effectively preserved. Conversely, the regions in the middle and on both sides of the roadway have lesser impact on vehicle travel, and the corresponding elevation should be suppressed to minimize its influence on the IRI calculation results. The lateral distribution model of wheel trajectories f(x) can achieve the suppression effect for the latter but is inadequate in preserving the former. Therefore, the overall approach for transformation involves normalizing f(x) to map its range of values to the [0, 1] interval, where 1 represents the maximum weight of the wheel trajectory zone and 0 represents the minimum weight in the middle or on both sides of the roadway. The normalization process of f(x) is presented in Equation (7).

f_{n o r m} (x) = \frac{f (x) - f_{\min}}{f_{\max} - f_{\min}}

(7)

In the Equation, f_max and f_min represent the maximum and minimum values of f(x), respectively. This normalization operation enhances the elevation representation in the wheel trajectory zone area while attenuating the elevation influence in the middle and on both sides of the roadway. Through the normalization process, the lateral distribution model of wheel trajectories f(x) is transformed into the weighting model of longitudinal profile f_norm(x).

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_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 μ based on the specific type of inspected roadway. Generate the data sequence for the longitudinal profile weighting model f_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 P₁(x_i, n) of the n-th cross-section and the longitudinal profile weighting model sequence f_norm(x_i) to generate the weighted elevation point cloud P₂(x_i, n). The calculation process is shown in Equation (8).

P_{2} (x_{i}, n) = P_{1} (x_{i}, n) \cdot f_{norm} (x_{i}), n = 1, 2, \dots, N

(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 P₂(x_i, n) within the same cross-section to obtain the accumulated elevations P₃(n) for each cross-section. The calculation process is shown in Equation (9). Elevations P₃(n) integrate the elevations P₁(x_i, n) of various points within the n-th cross-section along with their corresponding weights f_norm(x_i).

P_{3} (n) = \sum_{i = 0}^{F s} P_{2} (x_{i}, n)

(9)

Step (4): Scale the accumulated elevations P₃(n) according to Equation (10) to obtain the equivalent elevation of a specific roadway, denoted as P₄(n).

P_{4} (n) = a \times P_{3} (n) / F s

(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 P₃(n) are obtained. However, as evident from the multiplication process described in Equation (8), the weighting model of longitudinal profile f_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.

It is worth emphasizing that the choice of sampling frequency in the lane width direction has a crucial impact on the distribution of elevation data on the cross-section. Specifically, when a higher sampling frequency is adopted, a denser point cloud data set can be captured, thereby significantly increasing the sample size of elevation data; conversely, if the sampling frequency is lower, the obtained point cloud data will exhibit a sparser distribution, leading to a corresponding decrease in the amount of elevation data. This characteristic directly determines that different sampling frequencies will result in significant differences in the calculated accumulated elevation values P₃(n) when collecting elevation data for the same road. Furthermore, these differences in elevation data will significantly affect the calculation results of the IRI, causing the IRI values obtained under different sampling frequencies to exhibit significant variation. Therefore, it is inappropriate to directly use P₃(n) for IRI calculations. Based on this, the present study proposes a scaling strategy for the accumulated elevation P₃(n), aiming to map it within a scientifically reasonable numerical range.

Let S represent the cumulative sum of the weighting model function f_norm(x). It is evident that the cumulative sum S is significantly influenced by the cross-sectional sampling frequency Fs. To further analyze this relationship, we define the ratio of S to Fs as the coefficient K, where K = S/Fs. Figure 3 systematically presents the trend of the coefficient K curve under different sampling intervals Δx. Herein, Fs is indirectly represented by the sampling interval Δx. As observed from the figure, despite variations in the sampling interval Δx, the ratio tends to stabilize at a value of 0.3226. Additionally, the standard deviation of this ratio is 0.00455, with a coefficient of variation of only 1.41%, indicating a high consistency in results across different sampling intervals. Based on this significant and stable phenomenon, this study further proposes a scaling approach for equivalent elevation based on observed data. Specifically, Equation (10) is employed to scale the accumulated elevations P₃(n). This scaling process not only helps to eliminate the impact of sampling interval variations on model results but also enhances the model’s generalization ability under different conditions. Furthermore, rounding off the values improves the acceptability of the parameters in practical applications.

4. Calculation Process of Full-Width Roughness

Figure 4 illustrates the calculation process of full-width roughness involving the following key operational steps: (1) acquiring the raw elevation point cloud data of the road surface; (2) extracting effective elevation point cloud data within the lane, including road boundary recognition, lane outlier removal, and other point cloud data noise processing; (3) obtaining the weighting model of longitudinal profile, including determining lane categories, calculating the lateral distribution model of wheel trajectories, etc.; (4) computing the equivalent longitudinal profile elevations; and (5) calculating roughness indicators such as IRI_F based on the equivalent profile elevations. The primary focus of this study corresponds to steps (3) and (4).

In operational step (1), the raw elevation point cloud data of the road surface can be acquired through vehicle-mounted, airborne 3D laser scanning systems, or other data acquisition devices. It is important to note that the point cloud should cover the entire roadway width to facilitate the subsequent extraction of road edge lines from the data.

During operational step (2), various data processing techniques [14,19,20,21,22,23,24,26,27] can be employed to identify and determine the boundaries of the roadway, thereby defining the scope of the point cloud data. Following this, preprocessing of these raw data is carried out, which includes identifying and eliminating outliers within the lanes. These outliers may arise from equipment errors or foreign objects on the road surface. Ultimately, only the elevation point cloud data within the lanes are retained, serving as the foundation for further analysis of Full-width roughness.

Operational steps (3) and (4) constitute the focal points of this study and have been described in detail in Section 2 and Section 3, respectively.

In operational step (5), the IRI_F is computed using standard roughness calculation methods, such as commercial software [28] or programs written according to publicly available methods. This indicator serves as a key metric for evaluating both road surface roughness and the measurement performance of the instrumentation.

As operational steps (1), (2), and (5) are not the primary focus of this study and have been extensively explored in previous research works [17,28,29], further elaboration on these steps is omitted.

5. Experimental Verification and Analysis

5.1. Experimental Site and Devices

The experimental site selected for this study is located within the Ministry of Transport’s Highway Traffic Test Field in Majuqiao Town, Tongzhou District, Beijing. The chosen test road is a long, straight track specifically designed for experimentation, featuring a two-way, two-lane layout with each lane measuring 3.75 m in width and an effective measurement length exceeding 700 m. The specific experimental roadway within the closed area ensures the continuity and consistency of the tests, while also providing ample space to simulate various traffic environments.

To alter the roughness of the test road, multiple sets of standard test blocks with known dimensions but varying quantities were carefully arranged and laid at different locations along the road during the experiment, as shown in Figure 5 and Table 3. The design and placement of these standard test blocks were aimed at systematically varying the road roughness, thereby simulating different levels of road conditions within specific length ranges. This ensured that they could represent a wide range of road roughness scenarios, from smooth to rough.

During the data collection phase, a domestically produced vehicle-mounted three-dimensional point cloud measurement device was utilized. This equipment offered high-precision measurement capabilities, with both horizontal and vertical resolutions reaching 1 mm, enabling extremely fine measurements of road surface elevations. In the experimental process, the device conducted comprehensive elevation data collection along the right lane from south to north at speeds exceeding 50 km/h. Subsequently, the elevation point cloud data were output at intervals of 100 mm in both the horizontal and vertical directions. To enhance the statistical reliability of the data and minimize potential random errors, the entire measurement process was repeated 9 times. Additionally, a vehicle-mounted laser profiler was employed to measure the elevation at the wheel track trajectory zone of the road in accordance with JJG (Transportation) 075-2010 standards [30]. During road surface elevation collection, both vehicles traveled along the centerline of the lane.

5.2. Experimental Data Analysis

Figure 6 provides a visual representation of the original elevation point cloud data collected along a 600 m section of the tested roadway. Upon observation, it is evident that the difference between the maximum and minimum elevations exceeds 200 mm, and the uneven color distribution indicates significant surface irregularities. The regions with higher elevations are predominantly concentrated on the left side (upper left), while those with lower elevations are located on the right side (upper right). This phenomenon of higher elevation on the left and lower elevation on the right is intentionally designed to facilitate road surface drainage.

Two cross-sections at H₁ = 150 m and H₂ = 450 m were randomly selected for analysis. The cross-slope elevations of these two cross-sections were 103.37 mm and 143.09 mm, respectively, and both were higher on the left and lower on the right. Similarly, two longitudinal profiles at V₁ = −0.9 m and V₂ = 0.9 m were randomly chosen for analysis. The elevation change rates of these two longitudinal profiles were relatively flat, and their slopes met the requirements of high-grade road design specifications. It was assumed that the V₁ and V₂ longitudinal profiles corresponded to the two desired measurement trajectories within the left and right wheel trajectories, respectively. In early road surface roughness measurements, the elevations of these two longitudinal profiles were used to calculate the roughness value of the road. However, this method ignored the road surface elevation data outside the two longitudinal profiles, limiting the comprehensiveness and scientific rigor of the road performance evaluation.

Assuming that the tested road was a mixed traffic lane, L = 1.8 m and μ = 0.9 m were selected as the parameters for the lateral distribution of vehicle trajectories within the lane, and the weighting model of the longitudinal profile f_norm(x) was obtained. Subsequently, the weighted elevation point cloud data P₂(x_i, n) were obtained, presented in Figure 7. The characteristics of the ‘M’ shape indicated that the proposed weighting model f_norm(x) effectively highlighted the elevations within the left and right wheel trajectories while suppressing the elevations along the lane centerline and both boundaries. This outcome aligned with the expectations of the study.

Continuing the elevation analysis of the two cross-sections at specific locations H₁ = 150 m and H₂ = 450 m, the results clearly showed higher peaks within the wheel trajectory zones. Furthermore, the peak values in the left wheel trajectory were higher than those in the right wheel trajectory. This finding indicated that the weighted allocation of cross-sectional point cloud data did not alter the established function of the lane draining toward the right side, thus providing initial validation for the rationality of the weighting model.

Moving forward, elevation analysis of the two longitudinal profiles at V₁ = −0.9 m and V₂ = 0.9 m was conducted. The results revealed that, compared with the curves presented in Figure 6, the elevation amplitude of these two longitudinal profiles was not significantly reduced, and the elevation change trend along the driving direction was consistent. This is because the two longitudinal profiles were at the L/2 position and were least affected by f_norm(x). Notably, within the interval of 500 m to 600 m, multiple points with pronounced elevation fluctuations were observed. These fluctuations may have indicated local variations in road surface roughness. Upon verification, these locations corresponded precisely to the multiple sets of standard test blocks laid within the wheel trajectories during the experiment, further validating the close correlation between our analysis and actual road conditions. However, standard blocks arranged at other locations (longitudinal profiles) could not be displayed by these two longitudinal profiles. This also proved the insufficiency of using elevation values of a single longitudinal profile to represent the performance of the entire lane.

Figure 8 illustrates the cumulative elevation curve P₃(n) for the 600 m long lane. Upon careful observation, several distinct peaks could be identified along the direction of travel, specifically within the ranges of 200 m to 300 m, 300 m to 400 m, and 500 m to 600 m. These peaks corresponded to the multiple sets of standard test blocks placed on the test road. Excluding the peaks induced by the standard test blocks, the cumulative elevation curve P₃(n) remained predominantly within the range of 9750 mm to 11,250 mm. This indicated that, apart from the areas where the test blocks were laid, the overall elevation changes of the lane were relatively stable, with a mean value of approximately 10,645 mm.

From a quantitative perspective, the maximum peaks caused by the standard test blocks in the elevation curve P₃(n) exceeded the average value by more than 750 mm, a significant deviation from the actual thickness of the test blocks, which was less than or equal to 20 mm. This discrepancy could be attributed to the fact that the cumulative elevation P₃(n) is obtained by accumulating all of the weighted elevations P₂(x_i, n) of the n-th cross-section. Additionally, the cumulative elevation P₃(n) is directly related to the sampling density of the point cloud. While it is possible to calculate specific and precise values for road surface roughness indicators such as the IRI directly using the elevation sequence P₃(n) from Figure 8, these values may exceed cognitive ranges and fail to provide accurate and practical guiding suggestions for road management authorities.

Figure 9 illustrates the curves of V₁ longitudinal profile elevation from Figure 6 and equivalent elevations P₄(n) with respect to the driving distance. As is evident from the figure, the amplitude of P₄(n) was significantly reduced, and it aligned with the elevation of the V₁ longitudinal profile. This data processing method effectively suppressed elevation fluctuations, enabling a more accurate representation of the actual terrain characteristics. Furthermore, the scaling operation not only adjusted the elevation data but also skillfully preserved the elevation mutation points caused by the placement of standard test blocks on the actual road. This unique feature is particularly crucial in analyzing the dynamic response of vehicles to road roughness and calculating road roughness indicators, as it ensures that important road feature information is not omitted or mistakenly deleted during data preprocessing. This has profound implications for subsequent engineering analyses, pavement quality evaluations, and optimization of vehicle driving performance.

5.3. IRI Comparison

The vehicle-mounted laser profiler selects the elevation data from the V₁ longitudinal profile for analysis, and the output result is labeled as IRI₀. The vehicle-mounted three-dimensional point cloud measurement device utilizes both the V₁ longitudinal profile and the equivalent longitudinal profile V_F for computational analysis, with the output results denoted as IRI_V and IRI_F, respectively. V_F represents the equivalent longitudinal profile, and its elevation corresponds to P₄(n). It should be noted that although the vehicle-mounted laser profiler aims to use the V₁ longitudinal profile as the measurement trajectory, various factors may cause the actual measurement trajectory to deviate from the expected one, which will ultimately be reflected in the output IRI₀. On the other hand, for the vehicle-mounted three-dimensional point cloud measurement device, the V₁ longitudinal profile is extracted strictly based on the distance from the point cloud data within the lane, providing high certainty and independence from the actual driving trajectory of the vehicle.

Figure 10 presents the roughness results of IRI₀, IRI_V, and IRI_F, calculated based on the elevation data from two V₁ longitudinal profiles and the equivalent longitudinal profile V_F, with each section measured every 100 m. Initially, comparing the measurement results of the V₁ longitudinal profiles from the two types of equipment, it is evident that the curves of IRI₀ and IRI_V exhibited significant consistency, although there were some local differences. This suggested that there were indeed variations in the elevations collected by the two devices. Due to the absence of a standard IRI, it remained uncertain which measurement result was optimal. However, from the perspective of acquiring longitudinal profile data, the method employed by IRI_V, which determines a fixed measurement line based on the lane edge and subsequently extracts elevation values and calculates roughness, aligned more closely with expectations.

Furthermore, the roughness indicators IRIV and IRIF, both derived from point cloud data, exhibited strong consistency with a correlation coefficient of 0.996. However, it is noteworthy that the IRI_F values were generally higher than the IRI_V values. This phenomenon arose from the more complex and comprehensive composition of the equivalent elevation P₄(n), which encompasses not only the elevations from the V₁ longitudinal profile but also integrates the elevations from multiple other longitudinal profiles within the lane. Therefore, compared to relying solely on elevation data from a single longitudinal profile, the equivalent elevation P₄(n) undoubtedly provides richer and more comprehensive road surface elevation information. This information gain is reflected in the calculation of roughness indicators, resulting in relatively larger IRI values.

The coefficient of variation (Cv) was employed to analyze the repeatability of the IRI, and the results are presented in Figure 11. The Cv indicator has demonstrated effective application in the field of engineering survey [31,32]. When using the vehicle-mounted laser profiler to measure the V₁ longitudinal profile, the results exhibited significant variability, particularly within the road segment of 500–600 m, where the coefficient of variation Cv₀ even exceeded 9%. By contrast, the coefficients of variation for IRI_V and IRI_F based on 3D point cloud data, namely Cv_V and Cv_F, remained at relatively low levels, not exceeding 3.0% and 4.0%, respectively. This comparative analysis strongly supported the feasibility and scientific validity of using point cloud data for road roughness evaluation. It is worth noting that, even when only extracting the elevations of the V₁ longitudinal profile from the point cloud data, stable IRI calculation results could be obtained. This stability was attributed to a key step in point cloud data processing: the precise determination of the V₁ longitudinal profile position within the lane, achieved based on predefined lane boundaries. By comparison, the actual measurement trajectory of the vehicle-mounted laser profiler exhibited a certain degree of randomness during practical operation, which affected the stability and reliability of its measurement results. Therefore, the adoption of 3D point cloud technology for road roughness evaluation demonstrated superior technical advantages and practicality.

Furthermore, the variability of the IRI_F, represented by Cv_F, serves as an indicator of the performance stability and reliability of the measuring device. Specifically, if the 3D measurement system has deficiencies in the collection and processing of road surface elevation data, it will be difficult to ensure a low Cv_F. This is because the equivalent elevation P₄(n) is derived by integrating all elevation data from the n-th cross-section. Abnormal data on this cross-section can alter the accuracy of P₄(n) without affecting the elevation of the V₁ longitudinal profile. Thus, the equivalent elevation P₄(n) provides a comprehensive and detailed description of the road surface morphology. By contrast, the IRI_V indicator, which relies solely on the V₁ longitudinal profile from point cloud data, appeared to be more one-sided. This is because it only reflects the roughness of that specific profile and cannot fully represent the entire road surface condition. Therefore, when conducting road roughness analysis, particularly when the road elevation data comes from 3D point clouds, it is more reasonable to use the full-width roughness IRI_F as an evaluation metric. IRI_F not only provides a more accurate representation of the overall road performance but its variability can also serve as an important reference for evaluating the measurement performance of the instrumentation.

It should be noted that full-width roughness detection methods, while providing more comprehensive pavement information, typically incur higher device costs due to their reliance on advanced technology and complex hardware. Moreover, the processing of extensive point cloud data may not be as computationally efficient as simpler traditional devices, which can quickly produce results albeit with potentially lower accuracy and coverage. Thus, despite its advantages in detail, full-width detection may not rival the economic efficiency and ease of operation of classic, cost-effective detection equipment.

6. Conclusions

Based on the frequency distribution of vehicle centerlines recorded in BS 5400-10, this study derived a generalized lateral distribution model of wheel trajectories, established a weighted model for the longitudinal profile of the roadway, and introduced a novel indicator for evaluating the roughness of the entire roadway surface along with its calculation process and methodology. This approach integrates elevation data from the entire roadway surface to accurately compute pavement roughness. The conclusions of this study are as follows:

(1).: Utilizing vehicle centerline frequency from the BS 5400-10 standard, this study successfully established a lateral distribution model of wheel trajectories based on a Gaussian mixture model. Furthermore, considering the driving characteristics of vehicles in different lanes, the study recommended suitable model parameters for passing lanes, driving lanes, slow lanes, and mixed lanes.
(2).: Based on the proposed lateral distribution model of wheel trajectories, this study developed a weighting model of longitudinal profile. Specifically, it elegantly preserves the elevation within the wheel trajectory zones while nonlinearly suppressing the elevation in other areas, a feature that aligns closely with vehicle driving patterns under the channelization effect.
(3).: Combining the proposed longitudinal profile weighting model with pavement elevation point cloud data, this study innovatively proposed a new indicator, the full-width roughness IRI_F, to evaluate pavement performance. Compared to traditional methods that rely solely on the elevation of a single longitudinal profile within the wheel trajectory zones, the new indicator proposed in this study more scientifically reflects the actual performance of the pavement.
(4).: Experimental results demonstrate that the numerical values of the full-width roughness IRI_F are similar to those obtained using traditional methods and its repeatability coefficient of variation Cv_F is consistently lower, indicating higher measurement repeatability. Additionally, the fluctuation of the IRI_F indicator reflects the stability of the measurement performance of the 3D measuring instruments.

Overall, the outcomes of this study not only theoretically advance the development of pavement roughness evaluation technology but also provide novel technical support for highway maintenance and management in practice. Future research can further explore the optimization and applicability of model parameters, as well as the effectiveness of the new methods under various climatic and environmental conditions, based on the findings of this study.

Author Contributions

Conceptualization, Y.L., H.A., X.L. and J.C.; Data curation, H.A. and X.L.; Funding acquisition, Y.L.; Investigation, Y.L., H.A., X.L., J.C., N.M. and R.W.; Methodology, Y.L., H.A. and X.L.; Software, H.A., X.L., J.C. and R.W.; Supervision, Y.L.; Validation, Y.L., H.A., X.L. and J.C.; Visualization, H.A., X.L., J.C., N.M. and R.W.; Writing—original draft, Y.L. and H.A.; Writing—review & editing, Y.L. and H.A. All authors have read and agreed to the published version of the manuscript.

Funding

This research was partially funded by the Fundamental Research Funds for Central Public Welfare Research Institutes of China under grant no. 2024-9023 and the Special Fund for Talent Development in 2024—Youth Science and Technology Top Talents of the Research Institute of Highway Ministry of Transport under grant no. 0224RF02ZY1032.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy.

Acknowledgments

The authors thank the anonymous reviewers and editors for their constructive comments on this manuscript.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Distribution frequency of vehicle centerlines and its fitting results.

Figure 2. Lateral distribution models of wheel trajectories f(x).

Figure 3. The ratio of cumulative sum S of f_norm(x) to the value of Fs.

Figure 4. Calculation process of full-width roughness.

Figure 5. Experimental testing roads with various specifications of standard blocks.

Figure 6. Original road elevation point cloud data P₁.

Figure 7. The weighted elevation point cloud data P₂.

Figure 8. Curve of cumulative elevation P₃.

Figure 9. Elevation comparison between equivalent profile V_F and profile V₁.

Figure 10. Comparison of the IRI based on 3 longitudinal profile elevations.

Figure 11. Comparison of Cv based on 3 longitudinal profile elevations.

Table 1. Fitting parameters of the lateral distribution model with different track widths L.

L (m)	B	μ	σ	R²
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

Table 2. Suggested μ and L values for different lane types.

	Passing Lane	Travel Lane	Slow Lane	Mixed Lane
L (m)	1.6	1.8	2.0	1.8
μ (m)	0.8	0.9	1.0	0.9

Table 3. Dimensions and positions in road lanes of different test blocks.

Types	Position Range (m)	Dimensions (mm)
1	50~150	500 × 250 × 30
2	400~500	500 × 500 × 5
3	200~400, 500~600	500 × 250 × 20

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Luo, Y.; An, H.; Li, X.; Cao, J.; Miao, N.; Wang, R. Measurement Model of Full-Width Roughness Considering Longitudinal Profile Weighting. Appl. Sci. 2024, 14, 10213. https://doi.org/10.3390/app142210213

AMA Style

Luo Y, An H, Li X, Cao J, Miao N, Wang R. Measurement Model of Full-Width Roughness Considering Longitudinal Profile Weighting. Applied Sciences. 2024; 14(22):10213. https://doi.org/10.3390/app142210213

Chicago/Turabian Style

Luo, Yingchao, Huazhen An, Xiaobing Li, Jinjin Cao, Na Miao, and Rui Wang. 2024. "Measurement Model of Full-Width Roughness Considering Longitudinal Profile Weighting" Applied Sciences 14, no. 22: 10213. https://doi.org/10.3390/app142210213

APA Style

Luo, Y., An, H., Li, X., Cao, J., Miao, N., & Wang, R. (2024). Measurement Model of Full-Width Roughness Considering Longitudinal Profile Weighting. Applied Sciences, 14(22), 10213. https://doi.org/10.3390/app142210213

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