Issue 
Acta Acust.
Volume 5, 2021



Article Number  52  
Number of page(s)  13  
Section  Noise Control  
DOI  https://doi.org/10.1051/aacus/2021040  
Published online  03 December 2021 
Technical & Applied Article
Statistical tyre/road noise modelling based on continuous 3D texture data
^{1}
AIT Austrian Institute of Technology, 1210 Vienna, Austria
^{2}
TU Wien, 1040 Vienna, Austria
^{*} Corresponding author: reinhard.wehr@ait.ac.at
Received:
23
April
2021
Accepted:
24
September
2021
In this article, statistical modelling approaches for tyre/road noise levels (according to the CPX method) based on continuous 3D texture measurements are presented. The main focus is to estimate if viable correlations of interpretable 3D texture parameters with frequencydependent CPX levels can be found. Therefore, a set of descriptive 3D texture parameters is introduced. Two different modelling approaches, focussing on linearity in order to gain interpretability, are shown. First, a linear model is calculated based on the first principal components of the texture parameters, and second, a random forest regression approach is performed on the direct texture parameters. As a principal component analysis of the CPX measurement data reveals three highly correlated frequency ranges, both approaches are calculated independently for low, mid and high frequency bands. Both models show good performance in the low and mid frequency range, whereas the accuracy in the high frequency range declines. Due to the focus on linearity, both approaches result in comparable statistical benchmark parameters.
Key words: Tyre/road noise / CPX method / 3D road surface texture / Statistical modelling
© R. Wehr et al., Published by EDP Sciences, 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Road noise can be seen as main source of noise pollution in Europe [1]. Fifteen percent of the European population are reported as to be exposed to high levels of road traffic noise during the nighttime period, with an even higher percentage of 20% during the dayeveningnight period. In [2], the reduction of road noise to a level of 53 dB (L_{den}) resp. a level of 45 dB (L_{night}) is strongly recommended by both reductions at the source as well as on the transmission path. In [3], the WHO reports an estimated loss of disabilityadjusted lifeyears of 1.0–1.6 million DALYs in western Europe. This substantiates the need for reducing road noise.
Generally speaking, points of leverage to tackling road noise can be summarized in four categories:
traffic volumes,
speed limits,
noise generation at the source,
noise propagation/transmission path.
The first two categories can be influenced by legislative and/or political resolutions. For the remaining two categories, technical solutions may help in noise reduction at the source and the propagation path.
Reducing traffic noise in the propagation path is widely accomplished by the installation of noise barriers adjacent to main traffic corridors. A general overview of the use of noise barriers along European roads can be found in [4].
Reductions at the source may be the most straightforward approach towards noise control. Here, from approx. 40–50 km/h upwards until approx. 130 km/h (where noise from air turbulences of the vehicle body begin to dominate), tyre/road noise is the main source for noise generation [5], although this crossover is changing towards lower speeds due to the increase in electric vehicles. Different generation mechanisms, mainly tyre vibrations, aeroacoustic effects such as air pumping, and stick/slip effects, lead to complex frequencydependent interactions in noise generation. These mechanisms are strongly dependent on the road surface, where the surface macrotexture as well as the void contents for porous and semiporous road surfaces exhibit an important influence [5].
Measurement methods for determining the rolling noise of different tyres are e.g. described in [6] and performed on a special test road surface [7]. Unfortunately, it is indicated that a direct transfer of the results obtained on such a test track to typical road surfaces may not be possible [8]. For assessing the influence of different road surface types, the standardized measurement methods ISO 118191 for passby noise and ISO 118192 (CPX method) for the continuous onboard determination of tyre/road noise [9, 10] exist. A study analyzing the relationships of the two methods can be found in [11]. The CPX method, which was used in the work presented furtheron, is based on a reference test tyre [12] in order to focus on the impact of the road surface on the noise emission. Influences of the tyre and environmental conditions are adjusted by correction procedures, and were recently studied extensively [13–17]. Measurement uncertainties due to the interrelations between tyre and road temperature as well as tyre shore hardness were analyzed in [18], and results of roundrobin tests can be found in [19–21].
In order to understand the generation mechanisms of tyre/road noise, approaches to simulate a rolling tyre were performed over the last 20 years. General research on the topic of the tyre/road texture contact with focus on the contact forces by small roughnesses can e.g., be found in [22, 23]. The influence of the contact stiffness on tyre vibration induced noise was studied in [24] by means of simulations. Approaches to physically model the vibrations of a radiating tyre on an ISO surface are e.g., introduced in [25]. A statistical approach to correlate the contact forces with lowfrequency tyre/road noise was performed in [26]. There, short sections of 3dimensional texture measurements were used to characterize different road surface types and thereby estimate tyre/road noise in onethird octave bands. High levels of the correlation coefficients are reported, whereas the standard deviation is indicated as above 1 dB below the frequency of 1250 Hz.
In this article, a more direct approach towards statistical modelling of tyre/road noise emissions based on 3D texture measurements is presented. In order to characterize the road surface, a highresolution texture scanner was used to continuously measure the road surface directly in front of the tyre of a CPX measurement system. Thereby, the actual input texture geometry was measured and furtheron used within the modelling process. Due to the large amount of texture data, the focus of the research work is not on the physical contact modelling, but on the determination of meaningful and interpretable 3D texture parameters in order to estimate the potential of modelling approaches of tyre/road noise based on 3D texture measurements.
2 Description of the dataset
The measurement locations used for the modelling approach dataset consist of more than 50 stone mastic asphalt (SMA) sections on Austrian motorways. Information regarding the road surface type as well as its age and traffic loads were merged with the measurement locations. On the measurement sections, typically 500 m long measurements of CPX levels and 3D texture were performed. All road surfaces used for the analysis were in good condition, viz. not exhibiting cracks or apparent degradation.
In general, no further details regarding the exact road surface type, viz. maximum aggregate size or void content of the SMA, were known. Regarding the maximum aggregate size, nevertheless, with regard to the Austrian building regulations, a value of 11 mm can be assumed throughout the dataset. Also, this quantity should be implicitely treated by the model, as 3D texture information is its main input parameter. Concerning the void content, SMA surfaces in Austria are categorised in 2 relevant groups (SMA S1 with a void content of 1.5–4.0% and SMA S3 with a void content of 6.0–12.0%). These groups will be treated as one in the modelling process, but will be distinguished throughout this article as to further discuss possible influence factors.
CPX levels were measured and evaluated according to the ISO standard using the P1 [12] tyre. As measurement speed, 60 km/h was chosen, as this was the maximum speed of the 3D texture measurement system used. Only the right wheel track and only the rear microphone position was used for the modelling approach. This is due to the 3D texture sensor installed in front of the right measurement tyre. Thereby, it could be ensured that the measured road surface texture worked as the input function into the tyre/road interaction. By choosing the rear microphone, possible influences, viz. acoustic reflections on the sensor, could be diminished. This precaution may not be necessary with regard to the modelling approach, as the measurement system was not altered during the data acquisition, and said reflections can be assumed as a constant additive disturbance altering the measurement levels, but not the modelling approach, as long as the road surface type is kept constant.
The 3D texture measurements were performed with an inhouse developed measurement system (see Sect. 2.2, [27]) with a spatial resolution of approx. 75 μm in all directions and a recording width of approx. 7.5 cm. The sensor was installed directly in front of the right measurement tyre of the CPX trailer. Due to the high amount of storage usage, section lengths were chosen as approx. 500 m. At this length, it is assumed that a representative part of the section is captured by the measurement.
2.1 Principal component analysis of CPX data
As different generation mechanisms of tyre/road noise take effect in different acoustic frequency areas, a frequencydependent approach for identifying the texture influence on the noise emission must be chosen. As CPX levels are typically measured as overall or onethird octave band levels, a thirdband resolution may be selected. Thereby, models for the CPX frequency span of 315–5000 Hz would be determined. In order to reduce the amount of models, and thus gaining clarity, the onethird octave band CPX dataset at first is analysed with regard to its interrelations. This is accomplished via a principal component analysis (PCA) of the onethird octave band levels of the whole dataset. As can be seen from its visualization of the loading strength of the first three principal components in Figure 1 as well as the described variance by principal component (Fig. 2), the main information is already included in the first three principal components.
Figure 1 Loading vectors of the first three principal components of the CPX onethird octave band PCA, the loading strength is colorencoded. 
Figure 2 Importance of the principal components of the CPX frequency bands. 
A second approach to identify frequency areas exhibiting a comparable behaviour is by calculating their Pearson correlation matrix (Fig. 3). Both methodologies result in three adjoined frequency areas. Thus, statistical modelling will be performed for these three frequency spans (see Tab. 1). In order to aggregate the onethird octave bands, energetic summation within the groups is performed. Thereby, the model inputs within the three frequency areas are dominated by the louder bands under the assumption of comparable variation within each frequency band.
Figure 3 Pearson autocorrelation matrix of the CPX onethird octave bands. 
Center frequencies of the CPX onethird octave band limits for the three model frequency ranges.
It needs to be kept in mind that this grouping of the onethird octave bands is only valid for the specific dataset used. Especially other road surfaces, where noise generation mechanisms may be excited with different weights, may exhibit different frequency groups. Also, measurements at different measurement speeds may result in shifts of frequency components to different onethird octave bands.
2.2 3D texture
The 3D texture scans that are used in this analysis were obtained with the 3D texture measurement system that is described in [27]. This system is a linescan stereo system which captures the texture as well as the depth of a scanned object and is mounted on the CPX trailer (Fig. 4). As the sensor is installed directly in front of the CPX measurement tyre, it is ensured that the surface texture measured can be interpreted as the input function of the tyre. Although the sensor might cause additional acoustic reflections towards the CPX microphones, the transfer path can be seen as constant. Influences from the sensor therefore will not alter the correlations shown in the next sections.
Figure 4 Photo of the texture scanner mounted in front of the CPX measurement tyre (AIT/Zinner). 
The system consists of two verged linescan cameras and an illumination module. The optical axes of the setup are verged to obtain a larger overlapping region. Given the two images from these cameras that were taken from the slightly shifted viewpoints (resp. angles), a 3D model can be estimated by determining pixel correspondences between the images [27, 28] (Fig. 5).
Figure 5 Example of a 3D texture measurement: 3D view of single aggregates (the height is colorencoded). 
Compared to conventional area cameras, linescan cameras are able to acquire objects continuously and at highspeeds – two indispensable properties when scanning road surfaces while driving. Using a high optical resolution of approx. 75 μm, the max. acquisition speed of the 3D texture scanner is limited to 60 km/h. The setup covers a stripe of 7.5 cm in front of the CPX tyre, which corresponds to a line length of 1024 pixels.
2.2.1 3D texture data processing
When trying to characterise the road surface with 3D texture measurements, multiple data processing steps have to be performed in order to avoid calculation uncertainties and information loss of the soughtafter texture parameters. First, the size of the single surface texture patch needs to be considered. This should, depending on the purpose of the further analysis, cover multiple wavelengths of the longest influencing spatial wavelength. In most cases relating to effects based on the tyre/road contact zone (as tyre/road noise for smooth road surfaces in good maintenance condition, as used in the present dataset), this will be the maximum aggregate size of the road surface. Here, a texture patch size of approx. 75 × 75 mm was chosen covering 6 wavelengths of a typical road surface with 11 mm maximum aggregate size.
Secondly, as the measured road surface may be tilted relatively to the sensor due to side slopes, in order to prevent errors in gradientrelated calculations, some kind of surface levelling has to be applied to the measured surface. Here, multiple possibilities exist, e.g., a linewise levelling in longitudinal or lateral direction, levelling via fitting of a horizontal plane, etc. In the current dataset, this levelling was performed by subtracting a gaussian blur filter working on the height dimension of the texture from the original texture. This is equivalent to the subtraction of a lowpass filter, where the cutoff frequency is determined by the standard deviation of the Gaussian function, i.e., the geometric area of the filter. The cutoff frequency has to be chosen with care, as if the blurring area of the gaussian filter is chosen too low, information in the size range of e.g., the surface aggregates may be lost, if chosen too high, the sought after levelling may not be applied properly.
In a final preprocessing step, as further parameter calculations were performed on separate 3D texture patches, effects of the patch boundaries need to be treated. This was accomplished by a standard windowing approach, where a flattop hanning window was applied on each texture patch.
With these normalized texture patches, a multitude of different texture parameters may be calculated. Most of these parameters are dependent on the patch area, or even on metaparameters as described in the following example:
One basic parameter used in the following analyses is the number of local maxima found in the texture patch. It can easily be comprehended that this will be strongly dependent on the patch size, as well as the windowing shape and size described in the previous paragraphs. Additionally to this, the measured surface texture, dependent on the spatial sampling rate of the texture sensor, may exhibit a different microscopic roughness. Therefore, in order to only estimate the most prominent peaks of the texture, a lowpass filter needs to be applied to the texture removing unwanted highfrequency roughnesses. Here, the filter parameters may also be chosen dependening on the subsequent data usage. Therefore, the calibration of these metaparameters will strongly influence the usefulness of the texture parameter.
In the field of road surface characterization and tyre/road contact modelling, as automated measurements of 3D texture over long distances are relatively new, no definite normative guidelines for the calculation of texture parameters exist. This is even more complicated, as different surface properties (as e.g., tyre/road noise and skid resistance) may be affected by comparable surface parameters of different spatial extent. In addition, surface characteristics may be only dependent on a part of the texture: in the field of tyre/road noise, the generation mechanism of tyre vibrations is only dependent on the road texture in contact with the tyre, whereas air pumping mechanisms may be primarily dependent on the cavity amount and structure below the tyre/road contact zone.
In the dataset used for the analyses, uncertainties relating to the preprocessing and parameter calculation steps should be of minor influence. This is due to the fact that all measurements were performed with the same measurement system and all calculations were performed within the same software framework. Nevertheless, when comparing the numerical values of the road surface parameters, the origin and processing of external datasets must be kept in mind.
2.2.2 Principal component analysis
In order to describe the road surface, different texture parameters were calculated. It needs to be kept in mind that the texture parameters are partly influenced by hyperparameter finetuning, and also the focus of the article is not laid on their exact definitions. Therefore a precise specification of the texture parameter calculations is not given here. As these parameters are the basis for the modelling in Section 3, one main demand is for them to be meaningful and easily interpretable. Following, a short description of the mainly used surface parameters and their abbreviation used in the article is given:
nmax: number of local maxima per square meter, as calculated from one texture patch. In the scope of this article, the preprocessing steps are adjusted so that this roughly refers to the number of stone aggregates within the patch.
hmax: the median height of the texture at the position of the nmax, relative to the mean height of the texture patch. In combination with hmin this gives an estimate of the visible texture depth.
dmax: the range of the hmax, calculated from 5 to 95%percentiles in order to be robust against outliers. This parameter describes the height variability of the prominent peaks of the texture.
nmin: the number of local minima per square meter, as calculated from one texture patch (see nmax).
hmin: the median height (depth) of the texture at the position of the nmin, relative to the mean height of the texture patch (see hmax).
dmin: the range of the hmin, calculated from 5 to 95%percentiles in order to be robust against outliers.
mpd: mean profile depth, comparable, but not equivalent parameter to the calculation procedure in [29]. The texture patch is split in two halves, and the average of the height (compared to the average level) of the most prominent peaks is calculated. In comparison to the MPD, the spatial wavelength influencing this parameter is considerably smaller.
g: gestaltfactor, comparable, but not equivalent parameter to the 2D calculation procedure [30]. The calculation of this parameter equivalent to the gestaltfactor, but the 3D texture patch is used for the height distribution.
nneigh: mean minimum horizontal distance of each local maximum to its nearest neighbour in the texture patch (nearestneighbourdistance).
mas: maximum aggregate size, as estimated from the nmax and a cubic closest packing sphere model.
grad_{x}: gradient of the texture patch in longitudinal direction. Calculated as integral of the absolute point height difference in longitudinal direction.
grad_{y}: gradient of the texture patch in lateral direction. Calculated as integral of the absolute point height difference in lateral direction.
grad: overall gradient of the texture patch.
It can be assumed that interdependencies and correlations exist inbetween these texture parameters. Therefore, as in Section 2.1 for the CPX data, a principal component analysis of the texture parameters was performed. The importance of the first principal components is shown in Figure 6. Again, the main information gained from the texture measurements can be summarized in the first two principal components. In Figure 7, the loading vectors of the first three principal components are depicted. When interpreting the contributing parameters to the components, a separation of gradientrelated texture parameters in the first and surface structurerelated parameters in the second principal component can be assumed. The first principal component can be seen as description of the depth structure of the surface. Height and depthinformation (hmin, hmax, mpd, dmin, dmax) as well as – via the gradients and the gestaltfactor – the type of depth structure is included, whereas this segregation is represented in the third principal component. The parameters of the second principal component (nmin, nmax, nneigh, mas) describe the aggregate structure with regard to its packing density viz. maximum aggregate size.
Figure 6 Importance of the principal components of the texture parameters. 
Figure 7 Loading vectors of the first three principal components of the PCA, the loading strength is colorencoded. 
Again, this categorization should be only seen as valid within the current dataset consisting of SMA surfaces. Especially for road surfaces with diverging grading curves, as asphalt concrete or exposedaggregate cementconcrete surfaces (or even more blatant, anisotropic surfaces as diamondgrinding textures), diverging categorizations may be found. Nevertheless, the principal component analysis substantiates that the basic surface structure can be described with two interpretable groups of texture parameters. The calculation of additional texture parameters may be sought in the future, based on specific problem definitions, although it should be verified that thereby additional information is added to the dataset.
3 Modelling approach
In order to test the prediction capabilities of 3D texture measurements on tyre/road noise, two approaches were pursued.
First, a basic linear modelling of the low, mid, and high frequency range as determined in Section 2.1 was performed using the principal components of the texture parameters (see Sect. 3.1). This model is supposed to give a first glimpse on the potential of identifying correlations between texture and tyre/road noise, without stating physical interpretations of the reasons for these correlations. Also, the linearity confines the model to a narrow range of texture – tyre/road noise interrelations.
Second, a more elaborate modelling approach based on the individual texture parameters utilizing random forest regression models was chosen (Sect. 3.2). Although still operating on the aggregated frequency ranges, nonlinearities due to the random forest regression approach will be allowed.
For both approaches, the texture parameters introduced in Section 2.2.2 will be used. Although also texture data described by the (complex) 2D fourier spectra of the road surface (by e.g., aggregation of thirdoctave bands) or constantQ transforms could be used, intuitive interpretability, which was tried to preserve by the parameters, might suffer thereby.
3.1 Linear modelling via principal components
As first, direct approach to model the interrelations between road texture and CPX levels, a linear univariate modelling approach was performed according to,
where denotes the CPX level of the frequency range f, β the coefficients and PC the principal component used for the model. Based on the low, mid and high frequency CPX aggregates and the first three principal components of the 3D texture, a scatter plot matrix is shown in Figure 8. In the lower triangle, the individual scatter plots of the different dataset components are shown, whereby the different SMA types (SMA S1 and SMA S3) are colorencoded. Supplementary, the coefficients of determination of a linear modelling of the dataset components are shown in the upper triangle (where the font size also resembles the value). Finally, the kernel density estimate of the values is depicted in the scatter plot matrix diagonal. Not surprisingly, the frequencyaggregated CPX levels show no or low correlations, as could be expected from the pearson correlation matrix in Figure 3. The correlations between the principal components, as a direct implication of the PCA, are zero.
Figure 8 Correlations of low, mid and highfrequency CPX levels with the 3D texture principal components. The lower triangle depicts the individual scatter plots, the upper triangle the coefficient of determination of the respective linear regression. The parameter names in the diagonal denote the variables on the axes of the matrix elements, e.g. the lowest row depicts the individual scatter plots of mid, high, and PC1 to PC3 variables on the abscissa vs. the lowfrequency component on the ordinate. In addition, the diagonal elements also show the variables’ kernel density estimate. 
When considering the interrelations of the CPX levels and the principal components, noteworthy correlations appear for all three frequency ranges: For lowfrequency CPX components, a coefficient of determination of 0.70 can be found with the first principal component. The midfrequency CPX aggregate, which strongly influences the overall CPX level, even exhibits a coefficient of determination of 0.78. The highfrequency CPX level correlation shows a value of 0.53. This is insofar interesting, as no information regarding the void content of the road sections is included in the principal components, whereby an influence of this void content may be expected for highfrequency air pumping generation mechanisms. The only parameters possibly interrelating with the void content may be hmax and hmin, which together describe the (visible) depth information of the road surface. Actually, they are loading in the first principal component, which mainly contributes to the highfrequency correlation. An overview of the performance of the linear models is given in Table 2.
Performance parameters for the linear model based on texture PCA data: mean absolute deviation (MAD), root mean squared error (RMSE) and coefficient of determination (R^{2}) for the three frequency ranges.
Moreover, interesting interrelations between the highvalued correlations exist: Where the midfrequency range is solely described by the second principal component (which to some extent resembles the packing density of the road surface), both low and highfrequency ranges of the dataset can be estimated via the first principal component. Here, on the one hand, the gradients of the interrelations show opposing behaviour. On the other hand, the two SMA types within the dataset interchange: where SMA S1 surfaces exhibit lower lowfrequency CPX components, for the high frequency range, SMA S3 surfaces perform slightly better.
Finally, when calculating multivariate models for the low, mid, and high frequency range based on the three principal components according to,
the low and highfrequency models yield significant improvements compared to the univariate linear model. For the lowfrequency range, the coefficient of determination may be enhanced to 0.88 with a mean absolute deviation of 0.36 dB; for the highfrequency model, a coefficient of determination of 0.65 and a mean absolute deviation of 0.30 dB are obtained. Nevertheless, these models should be treated with care, as interaction terms were allowed resulting in a high number of model input parameters and risking model overfitting. Therefore, this model approach will not be followed further on in this article.
As final step in the linear modelling process, a combined model based on the three univariate frequency submodels can be calculated by energetic summation of the submodel predictions:
Its results, performance and comparison with the random forest models introduced in Section 3.2 will be discussed in Section 4.
3.2 Frequencydependent texture parameter model
For modelling the influence of the individual road surface texture parameters on tyre/road noise, a fourstage modelling approach was chosen (see Fig. 9). First, the CPX data was split into three independent, energysummed datasets based on the findings in Section 2.1. This separation, in contrast to a direct modelling of the overall CPX level, is necessary in order to consider multiple tyre/road noise generation mechanisms.
Figure 9 Block diagram of the multivariate modelling approach. 
Second, for each dataset, the choice of the influence parameters was performed according to the following steps:
The dataset was split into training and test data (75% training dataset size).
For each twoparametercombination (to avoid overfitting) of texture parameters (P_{i}, P_{j}), a multivariate model (including the interaction term) was built:
The model validity was checked by calculating the coefficient of determination based on the test dataset.
The steps 1–3 were repeated 1000 times, based on different training/test random samples of the dataset.
For each cycle, the bestperforming twoparameter combination (P_{i}, P_{j}) was chosen.
Based on the frequency distribution of this crossvalidated grid search, the twoparameter combination with most occurences as bestperforming combination for each frequency range was chosen for further modelling.
Third, based on the parameter choice, in order to allow for slight nonlinearities, for each frequency range a randomforest regression model was calculated:
with B denoting the number of crossvalidated grid search runs,
the single regression tree, c_{m} the coefficients, a unity matrix describing the splits of the regression tree, P the two texture parameters and R_{m} the partitions of the feature space [31]. Here, due to the linear modelling during the parameter choice, no strong nonlinearities are to be expected. This focus on linearity, on the one hand, was chosen in order to avoid overfitting (which although was not expected within the parameters’ data range). On the other hand, interpretability of the model parameter choice was sought, which is easier established for steadily rising/falling model relationships.
Finally, the outputs of the three submodels were combined by energetic summation to predict the overall CPX level as in (3). Here, the overall model performance can be calculated by a reevaluation of the assembled submodels. A separation in test/training datasets is not meaningful at this stage, as the model optimization procedure was performed for the frequencydependent submodels, where the three models were trained independently from each other and the calculation of the overall value.
3.2.1 Parameter selection
As described above, the parameter selection for the three CPX frequency ranges was performed by testing different multivariate linear twoparameter models for multiple training/test data sets. For each training/test dataset slice, the best model parameter selection was kept. The distribution of the associated coefficients of determination is depicted in Figure 10 for the three frequency CPX levels. In a predominant part of the model test runs, a satisfactory value for the coefficient of determination is reached. Especially for the lowfrequency range, the crossvalidated grid search results in a maximum of the distribution of above 0.8, meaning that for each dataset slice, a highly performant twoparameter model can be found.
Figure 10 Distribution of the best twoparameter model coefficients of determination resulting from the 1000 crossvalidated grid search runs. 
The occurence of the best parameter combinations for low frequencies is shown in Figure 11. In approx. 175 of 1000 cases, the best parameter choice consists of the number of local maxima per square meter nmax in combination with the texture gradient in driving direction grad_{x}. This parameter selection is followed by nmax combined with grad_{y}; as grad_{x} and grad_{y} are highly correlated texture parameters for isotrope road surfaces, the combination of nmax and a directional grad_{x/y} can be considered as best model choice in over 25% of the crossvalidated grid search runs. Not surprisingly, the first seven parameter combinations consist of texture parameters assigned to different loadings in the principal component analysis of Section 2.2.2. As first singlePCA combination, the parameters of hmin and mpd both load on the first principal component, which was found to be the bestcorrelated principal component to the lowfrequency CPX levels in the linear modelling approach of Section 3.1.
Figure 11 Histogram of the occurences of the best choices of the low frequency twoparameter model feature selection process. 
Alike, the feature selection process was performed for the midfrequency CPX levels. Here, also with comparable levels of the coefficient of determination distribution, as most frequent combination the nmax and the nneigh parameters are responsible for more than 25% of the best parameter combinations. This is insofar interesting, as both nmax and nneigh mainly load the second principal component of the 3D texture PCA. As secondbest combination, the nneigh are identified with the dmax parameter in approx. 15% of the crossvalidated grid search runs. nneigh can even be found in the first five parameter combinations and is thereby selected in approx. 70% of the crossvalidated grid search combinations as relevant parameter. This may lead to the assumption, that not only the number of local maxima per square meter, but even more their spatial distribution has a strong influence on midfrequency CPX levels.
For the highfrequency CPX levels, the distribution of the coefficients of determination results in a slightly poorer model accuracy estimation. The maximum of the distribution is broadened in comparison to the low and midfrequency calculations, and although the maximum is still at approx. 0.75, a relevant amount of the crossvalidated grid search runs results in a best coefficient of determination of below 0.5. As mainly selected texture parameter, hmax, describing the visible depth structure of the road surface, appears in the first four linear model selections accounting for more than 50% of the crossvalidated grid search runs. The best parameter choice is identified as hmax in combination with nmax, followed by the combination of hmax and nneigh.
3.2.2 Random forest regression
Based on the parameter selection in the previous section, three independent random forest regression models were built [32]. The results of the models are shown in Figures 12–14. The model evaluations are colorencoded in the plots according to the predictor values. Additionally, the density of the input dataset is depicted via the dashed contour lines. Thereby, strongly supported areas of the modelling can be distinguished from areas where extrapolation occurs. The results of the performance measures is given in Tables 3 and 4.
Figure 12 Visualization of the low frequency random forest predictions; the estimated CPX levels are colorcoded, the dashed contour lines denote the kernel density estimates of the input data, thus marking the areas of model inter and extrapolation. 
Outofbag performance parameters for the random forest model based on texture parameters: mean absolute deviation (MAD), root mean squared error (RMSE) and coefficient of determination (R^{2}) for the three frequency ranges.
Reevaluation performance parameters for the full random forest model based on texture parameters: mean absolute deviation (MAD), root mean squared error (RMSE) and coefficient of determination (R^{2}) for the three frequency ranges.
For the lowfrequency CPX range, the model is supported for a broad parameter range both for the nmax and grad_{x}. The approximately round shape of the input density distribution is a direct result of noncorrelating modelling parameters loading on different principal components in Section 2.2.2. The model accuracy can be described by the root mean squared error of 0.7 dB. This is no further improvement compared to the linear modelling using the principal components in Section 3.1 (which was not to be expected due to the parameter selection based on linear models), but increases interpretability: high lowfrequency CPX levels are mainly found for high values of grad_{x} and low amounts of nmax. The extrapolation of the model suggests that by increasing the nmax towards the highest values found in the measured input dataset and simultaneously keeping the grad_{x} below a threshold of 0.5, lowfrequency CPX levels can be reduced by approx. 2 dB from the mean value found in the dataset and approx. 4 dB from the highest occuring levels. Here, it has to be kept in mind that the model input dataset consists of SMA surfaces of the same categorical maximum aggregate size. Therefore, only slight variations within the grading curve, as well as ageing effects restructuring the texture/tyre contact area, result in significantly higher lowfrequency CPX levels.
In the midfrequency CPX range (Fig. 13), the input parameter density distribution of nmax and nneigh shows a deviating behaviour compared to low frequencies: the strongly directional density distribution is a result of the highly correlated input parameters, both loading on the second principal 3D texture component. High amounts of nmax are mainly correlated to low values of nneigh, which for anisotropic road surface textures is to be expected. Only for inhomogeneous spatial distributions of the stone aggregates, deviations from this direct correlation may be expected. With an outofbag root mean squared error of approx. 0.5 dB, the model accuracy can be estimated as useful for the present objectives. The interpolated CPX level range for mid frequencies is slightly lower than for the low frequency range. Low amounts of nmax result in a CPX level reduction of approx. 2 dB from the highest occurring values, whereas the linearity in the populated dataset, seen via the diagonal in the Figure, combined with the good model accuracy, suggests further potential in reducing CPX levels by increasing the amount of nmax towards the model extrapolation area.
Figure 13 Visualization of the mid frequency random forest predictions; the estimated CPX levels are colorcoded, the dashed contour lines denote the kernel density estimates of the input data, thus marking the areas of model inter and extrapolation. 
In Figure 14, the visualization of the highfrequency CPX level random forest model is depicted. The modelling parameter density distribution exhibits a wide variety of the nmax and hmax (as both load on different principal components), extrapolation occurs mainly for high amounts of nmax and high values of hmax in the graphic. The random forest regression model exhibits an outofbag mean squared error of 0.9 dB, indicating it as the frequency model with the poorest benchmark parameter. As suspected from the linear modelling based on the principal components, the main influence factor on the highfrequency CPX levels is the hmax parameter. High values of hmax, representing a large (visible) depth structure in the road surface texture, may lead to low highfrequency CPX levels. The limitation to the visible depth structure here rests upon the measurement principle of the 3D texture, where an optical method is used that is incapable of measuring porous structures of the road surface. This may be a reason for the lowest benchmark parameter of the three frequencydependent random forest regression models, although the input dataset road surfaces exhibit mainly low void contents.
Figure 14 Visualization of the high frequency random forest predictions; the estimated CPX levels are colorcoded, the dashed contour lines denote the kernel density estimates of the input data, thus marking the areas of model inter and extrapolation. 
4 Combined overall CPX level model
Based on the frequencydependent submodels, by energetic summation of the predictions an overall model can be developed. In contrast to the linear modelling approach, were all input datapoints were used for a single model creation, this will be done both for the outofbag as well as the inbag random forest predictions separately. The results of the performance measures are given in Table 5.
Reevaluation performance parameters for the modelling approaches: mean absolute deviation (MAD), root mean squared error (RMSE) and coefficient of determination (R^{2}) for the three frequency ranges.
The results of the linear modelling reevaluation are depicted in Figure 15. As the performance measures suggest, a suitable recalculation of the overall levels could be achieved. The maximum prediction error can be calculated to approx. 1.5 dB. Interestingly, the slope of the prediction clearly deviates from the diagonal.
Figure 15 Reevaluation of the combined overall linear principal component CPX model. The black line shows the linear regression model of the prediction. 
As final step in the random forest model building, the submodel outputs can be combined to calculate predicted overall CPX levels. Reevaluating the model output with the full dataset by calculating a linear model based on the outofbag prediction (see Fig. 16), the overall performance of the modelling process can be rated. Although the coefficient of determination of the full model is relatively low with a value of 0.56, the root mean squared error exhibits a promising value of 0.44 dB. Compared to an estimated repeatability of the CPX measurements of approx. 0.15 dB according to [18] and an expanded measurement uncertainty of 1.0 dB according to [10] for a coverage probability of 95%, the modelling approach results in suitable prediction errors. The maximum prediction error is found as the same data point as in the linear model with a value of 1.4 dB. In contrast to the linear model, the slope of the correlation is nearly one. As seen from the calculations in Section 3.2.2, main contributions to these errors originate from the highfrequency range with a root mean squared error of 0.9 dB (high frequency range), 0.7 dB (low freq. range) and 0.5 dB (mid freq. range). These values are of similar size as in the approach based on the principal components, but obtained with the use of only four independent 3D texture parameters.
Figure 16 Reevaluation of the combined overall random forest CPX model based on the outofbag prediction. The black line shows the linear regression model of the prediction. 
As the linear model approach was – due to the univariate approach – calculated directly on the full dataset instead of the feature selection via crossvalidated grid search and bagging for the random forests, a reevaluation of the final random forest models is given in Figure 17. This results in a more comparable intermodel comparison between the linear and random forest approach. Here, we see that the random forest allows for corrections for nonlinearities without losing the nearly diagonal slope in the prediction. The performance measures exhibit a clearly better value as the linear model with a maximum prediction error reduced to approx. 0.6 dB, although for the overall model performance, the benchmark parameters of the outofbag predictions should be considered.
Figure 17 Reevaluation of the combined overall random forest CPX model based on the final model prediction. The black line shows the linear regression model of the prediction. 
Generally speaking, when building statistical models for the purpose of predicting rolling noise levels based on texture parameters, more robust estimations of the model performance will be reached by use of a separate test data set. In the work described here, this was not implemented due to two reasons: once, due to the high cost of the data acquisition, only a limited dataset could be gathered. Second, the overall prediction performance was of subordinate interest. As described in Section 5, mainly the contributing texture parameters to the generation of rolling noise were sought to be identified.
5 Discussion of the model results
When comparing the two modelling approaches – linear modelling via principal components and random forest regression modelling based on direct 3D texture parameters – similar values for benchmarking parameters could be achieved. Nonlinearities were prohibited in the univariate PCAapproach resp. avoided for the random forest regressions in order to maintain easy interpretability.
Where the univariate linear approach suggests strong influences of the depth structure for lowfrequency CPX levels, the random forest regression approach reveals both influences of the surface gradient as well as – to a lesser extent within the dataset range – the amount of local maxima nmax. In an attempt to physically interpret these results, road surfaces with low gradients in combination with high amounts of nmax may yield in a more constant contact deformation of the tyre rubber within the tyre/road contact zone, thus leading to lower tyre vibrations. To further substantiate this assumption, calculations of the rubber deformations may be suitable. Subsequently, the texture parameters can be recalculated for the deformed tyre rubber, and thereby only describing the actual contact between the tyre and the road surface.
For the midfrequency CPX levels, as primary contribution to the models, the aggregate structure is most relevant. The amount of local maxima nmax as well as its packing density via the nearestneighbourdistance nneigh show strong influence on the CPX levels. This is most interesting with regard to the dataset consisting only of SMA surfaces with a maximum aggregate size of 11 mm. The main reason for the variation within the parameters in the dataset is based on different installation qualities and ageing effects of the road sections. Here, the strongly relevant nmax should be further analysed. In its current form, this parameter directly counts the number of local maxima within the texture patch with the potential of finding multiple local maxima per stone aggregate. A further improvement of the parameter calculation, e.g., a split into the number of aggregates and the dominant rubber deformation locations may help in finding more suitable surface parameters.
When analysing the model qualities for the highfrequency CPX levels, lower rates of the benchmark parameters are found for both approaches. Where the linear modelling of principal components suggests an opposing behaviour of the first principal component with regard to lowfrequency emissions, the random forest regression approach points towards a more differentiated view. There, mainly the texture depth, represented via the hmax is found to be influencing the highfrequency CPX levels. Therefore, a combined optimization of the road surface texture for all frequency ranges may be possible. High amounts of local maxima, leading to a constant rubber deformation in the tyre/road contact, with simultaneous deep structures within the road texture, are a viable approach to lownoise road surfaces with low void content.
All of these considerations – although not novel – are insofar interesting, as they are drawn from a dataset with very limited input data ranges. Correlations of single texture parameters within a narrow class of road surfaces are not expected to be a sure success. This substantiates the approach of finding causes of rolling noise emissions by combined measurements of road surface texture and tyre/road noise. Even more, effects of small variations within the road surface texture may be further on analysed. In the dataset presented, reasons for ageing effects within the SMA dataset could be revealed by use of easily interpretable surface texture parameters.
3D texture measurements in combination with statistical analyses based on different modelling approaches can be seen as a tool to further interpret the interrelationships between the road surface texture and rolling noise emissions. Physical modellings of the tyre and/or rubber deformations are expected to further improve the models and thereby verify the parameter validity. Thereby, modelling benchmarks can also be used as test of physical models, as by incorporating additional parameters in the statistical modelling process, their validity on tyre/road noise emissions is expected to result in models with higher prediction accuracy.
6 Summary
In this article, statistical modelling approaches of tyre/road noise CPX levels, based on continuous 3Dtexture measurements, where introduced. Different approaches, based on the direct texture parameters as well as on a beforehand performed principal component analysis, were presented. It could be shown that, with use of tangible texture parameters, significant correlations to CPX levels were found. Due to the focus on linearity, both approaches presented in this article result in similar model qualities. By use of strongly nonlinear models, higher prediction accuracies, on the cost of interpretability of the interrelations, may be found.
The relevance of continuous 3D texture measurements, combined with rolling noise measurements, on the analysis of the tyre/road contact, could be shown. Also, a generalization of the approach on different road parameters, as skid or rolling resistance, seems viable. Further improvements may be expected by incorporating texture parameters based on the calculation of tyre rubber deformations resp. the contact forces in the tyre/road contact zone.
Conflict of interest
Authors declared no conflict of interests.
Acknowledgments
The work presented in this article was performed in the scope of the project ADURA, funded by the Austrian Research Promotion Agency (FFG) in the DACH Kooperation Verkehrsinfrastrukturforschung 2017 – DACH 2017 (project number 863065). The authors would like to thank their project partners from PMS Consult, Grolimund + Partner AG, TU Braunschweig and Müller BBM as well as their project advisor Karl Gragger from ASFINAG.
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Cite this article as: Wehr R. Fuchs A. & Breuss S. 2021. Statistical tyre/road noise modelling based on continuous 3D texture data. Acta Acustica, 5, 52.
All Tables
Center frequencies of the CPX onethird octave band limits for the three model frequency ranges.
Performance parameters for the linear model based on texture PCA data: mean absolute deviation (MAD), root mean squared error (RMSE) and coefficient of determination (R^{2}) for the three frequency ranges.
Outofbag performance parameters for the random forest model based on texture parameters: mean absolute deviation (MAD), root mean squared error (RMSE) and coefficient of determination (R^{2}) for the three frequency ranges.
Reevaluation performance parameters for the full random forest model based on texture parameters: mean absolute deviation (MAD), root mean squared error (RMSE) and coefficient of determination (R^{2}) for the three frequency ranges.
Reevaluation performance parameters for the modelling approaches: mean absolute deviation (MAD), root mean squared error (RMSE) and coefficient of determination (R^{2}) for the three frequency ranges.
All Figures
Figure 1 Loading vectors of the first three principal components of the CPX onethird octave band PCA, the loading strength is colorencoded. 

In the text 
Figure 2 Importance of the principal components of the CPX frequency bands. 

In the text 
Figure 3 Pearson autocorrelation matrix of the CPX onethird octave bands. 

In the text 
Figure 4 Photo of the texture scanner mounted in front of the CPX measurement tyre (AIT/Zinner). 

In the text 
Figure 5 Example of a 3D texture measurement: 3D view of single aggregates (the height is colorencoded). 

In the text 
Figure 6 Importance of the principal components of the texture parameters. 

In the text 
Figure 7 Loading vectors of the first three principal components of the PCA, the loading strength is colorencoded. 

In the text 
Figure 8 Correlations of low, mid and highfrequency CPX levels with the 3D texture principal components. The lower triangle depicts the individual scatter plots, the upper triangle the coefficient of determination of the respective linear regression. The parameter names in the diagonal denote the variables on the axes of the matrix elements, e.g. the lowest row depicts the individual scatter plots of mid, high, and PC1 to PC3 variables on the abscissa vs. the lowfrequency component on the ordinate. In addition, the diagonal elements also show the variables’ kernel density estimate. 

In the text 
Figure 9 Block diagram of the multivariate modelling approach. 

In the text 
Figure 10 Distribution of the best twoparameter model coefficients of determination resulting from the 1000 crossvalidated grid search runs. 

In the text 
Figure 11 Histogram of the occurences of the best choices of the low frequency twoparameter model feature selection process. 

In the text 
Figure 12 Visualization of the low frequency random forest predictions; the estimated CPX levels are colorcoded, the dashed contour lines denote the kernel density estimates of the input data, thus marking the areas of model inter and extrapolation. 

In the text 
Figure 13 Visualization of the mid frequency random forest predictions; the estimated CPX levels are colorcoded, the dashed contour lines denote the kernel density estimates of the input data, thus marking the areas of model inter and extrapolation. 

In the text 
Figure 14 Visualization of the high frequency random forest predictions; the estimated CPX levels are colorcoded, the dashed contour lines denote the kernel density estimates of the input data, thus marking the areas of model inter and extrapolation. 

In the text 
Figure 15 Reevaluation of the combined overall linear principal component CPX model. The black line shows the linear regression model of the prediction. 

In the text 
Figure 16 Reevaluation of the combined overall random forest CPX model based on the outofbag prediction. The black line shows the linear regression model of the prediction. 

In the text 
Figure 17 Reevaluation of the combined overall random forest CPX model based on the final model prediction. The black line shows the linear regression model of the prediction. 

In the text 
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