978-1-4244-4547-9/09/$26.00 ©2009 IEEE TENCON 2009

This work was supported in part by a grant from the Australian Department of Infrastructure, Transport, Regional Development and Local Government.

Signal Processing Approach to Road Roughness Analysis and Measurement

Reyhaneh Hesami (IEEE Member) Centre for Sustainable Infrastructure Swinburne University of Technology

Hawthorn, Australia RHesami@swin.edu.au

Kerry J. McManus Canter for Sustainable Infrastructure Swinburne University of Technology

Hawthorn, Australia KMcManus@swin.edu.au

Abstract— In this paper, the application of signal processing for the analysis of discretely sampled road profile data of sealed bituminous (or flexible) pavements is considered. Firstly, road profile data is modelled as a function of time. Then power spectrum density (PSD) analysis is briefly introduced as a tool for estimating the distribution of energy of various wavebands embedded in the road elevation. It is shown that this analysis can be used to discriminate the deterioration modes in pavement structure by identifying features that contribute to the roughness. Moreover, wavelet analysis has been introduced as an alternative signal processing method for road profile analysis. It not only verifies road roughness features, but it is also able to locate high frequency defects such as cracks and potholes. Two experimental low traffic volume highway sections of 100 m in length (one smooth and one rough) are selected to examine PSD and wavelet analysis. Results show that wavelet based road profile analysis can be used as a better diagnostic tool than PSD for multi-resolution analysis and measurement of pavement roughness.

Keywords- signal processing, wavelet analysis, PSD analysis, road profile, road roughness, road profile characterizing

I. INTRODUCTION Characterizing road profiles in terms of roughness as an

indication of road users’ comfort and safety has been of interest to the road authorities for years. Early attempts at road roughness assessment were based on the judgment of ride quality by a panel of experts riding over the road on a test vehicle [1] [2] [3]. This was followed by a technique using the measurement of the accumulated vertical travel of an axle of a vehicle travelling over a fixed distance [4]. The most common road roughness index that has been adapted by most road authorities around the world is called the International Roughness Index (IRI) [5]. This index is found from the response of a computer generated ‘quarter-car model’ vehicle which travels across pavement profile. IRI is calculated by dividing the sum of the absolute values of the vertical suspension deflection by the travelled distance. Typically, IRI is used as a guide to the road performance at the network level [6]. At the project level, notice is taken of user assessment and observation of condition, since the IRI is not seen as sufficiently discriminatory at this level.

Due to the fast growth in laser measurement technology, road profile data acquisition equipment has advanced significantly. Accurate dense elevation data can now be

produced at relatively high speed and a reasonable price. As a result, production of an automated and accurate spatial model of pavement network is emerging as one of the viable applications of such data. Such a model facilitates automatic evaluation of pavement performance at both network and project level. Availability of accurate pavement spatial data leads to the use of new numerical methods of roughness measurement which can be more detailed and accurate as they are independent from users’ perception or vehicle characteristics.

The aim of this paper is to use longitudinal profile data obtained by accurate laser profilometer measurements to investigate how signal processing methods such as power spectrum density (PSD) analysis and wavelet transform (WT) can be used as numerical tools to analytically characterize road profiles. The main emphasis is an application of discrete wavelet transform (DWT) in road roughness analysis. In Section II, interpretation of the longitudinal profile as a signal is described and the embedment of information in such data is examined. Section III explains the usage of signal processing for pavement data analysis. The results of the experiments on wavelet analysis of selected flexible highway profile sections are presented in Section IV. Section V concludes the paper.

II. DESCRIPTION OF ROAD PROFILE AS SIGNAL Although the theory of signal processing was originally

developed for electrical signals, it is now widely used in many other area of engineering such as acoustics, biomedical engineering, image analysis and most recently road profile analysis. Similar to the sound, light, voltage and current, the uneven surface of a pavement is also considered as a signal to a profilometer. In this case, a road profile signal can be considered as a collection of waves with different wavebands. While applications such as sound and image that can be expressed as a function of time in their original format are easier to analyze, signal processing of pavement profile as a function of distance is more challenging. In order to analyze such a signal with a signal processing approach, the spatial pavement signal (which is originally a function of distance) is modeled as a function of time. To this purpose, assume x(l) is road profile data which is a discrete function of displacement measured by a profilometer at a very small sampling rate. Also assume that the first measurement x(l1) happened at t1 and similarly the nth measurement happened at tn. Due to the fact

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that there is only one record of distance at a time (meaning that l(t)), road profile data x(l(t)) can be simply considered as function of time, x(t). This assumption is true for elevation data that can be captured by state-of-the-art laser profilometers point-by-point and at a very high frequency. In this case, the time difference between each elevation data (and so as distance

sampling interval) is directly related to the frequency of profilometer and speed of the data acquisition vehicle. So, in the context of road roughness signal processing, time domain can be implicitly considered as distance domain and vice versa. Fig. 1 exemplifies a longitude data of 100 meter length.

0 10 20 30

-4.595

-4.59

-4.585

-4.58

-4.575

-4.57

-4.565

x(l0) , t=t0

x(ln) , t=tn

Figure 1. A selected flexible road profile section (Borung Hwy, Victoria, Australi).

In the road roughness context, an ideal road profile data has a flat shape with no irregularity or unevenness. Any significant deviation from that shape is usually considered to be road roughness caused by a road defects such as potholes, cracks, deformation or surface texture deficiencies. The time representation of the road profile signal, however, cannot readily present the information related to the periodic irregularities embedded in this signal. In order to obtain the distinct information hidden in the frequency content of the data, a transform-base signal processing is needed to break the profile signal into its frequency components (frequency spectrum) and hence to find the strength and weakness of the individual wavebands embedded in the data. It is necessary to note that frequency as an electrical signal is measured in cycle/second or commonly named as “Hertz” but a road profile signal can be more appropriately measured in cycle/meter as there is a one-to-one relationship between time and distance in a longitude function.

There are different methods of time-frequency transform available in the literature such as Wigner distribution [7], Hilbert transform [8], Fourier transform and short time Fourier transform [8] and [9]. Every transform technique has its advantages and limitations with a specific area of application. Among the transform methods, the Fourier transform is the most popular method to analyze a time signal for its frequency content. Wavelet analysis is a relatively new method that is strongly linked to the Fourier transform. Fourier transform is a two-sided transform that allows the signal to be represented in either time or frequency domain. This makes the Fourier transform a suitable choice for applications with stationary signal. A stationary signal is defined as a quality of a signal in which the statistical parameters (mean and standard deviation) of the signal do not change with time [10]. As a result, it is not necessary to have both frequency and time information simultaneously for such a signal as frequency contents of the signal do not change in time. Fourier transform can be used to analyze non-stationary signals where only the frequency

components are needed, without referencing to the time of occurrence. Recent statistical analysis on the properties of road profile data proved that this data is a highly non-stationary and a non-Gaussian signal that contains transients [11], [12] and [13]. Transients or road irregularities are hard to locate when profile data are analyzed in a time domain itself. In the following section the road profile features, such as interpretation of wavelengths expressed as frequency contents and type of irregularities are explained in more detail.

A. Profile Features A road profile signal consists of a range of wavelengths,

varying from fractions of millimetres to hundreds of meters. The road roughness usually covers the wavelength range of 0.1 m to 100 m. The long wavelengths are normally caused by environmental processes as a result of pavement layer properties including poor drainage, swelling soils, and other influences such as climate change and growth of vegetation in the shoulder of highways. The short wavelengths or irregularities relate to structural; performance of the pavement and are often caused by localized pavement distresses including potholes, cracks, deformation and texture deficiencies. In order to analyze and characterize of such data, it is necessary to decompose the profile signal into a range of wavelengths to extract different types of features.

In the next section two signal processing tools are examined which can be used to divide road profile signal into its components. One of them can also be used to simultaneously locate the pavement section irregularities.

III. SIGNAL ANALYSIS OF ROAD PROFILE DATA In the following subsections, first the usage of PSD analysis

as a measure of road roughness is briefly explained and then discrete wavelet transform is introduced as a better approach over Fourier transform for pavement roughness measurement.

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A. PSD Analysis Power Spectral Density (PSD) analysis is a method

available for evaluating the distribution of roughness type in a pavement. It uses Fourier transform to analyze the road profile as a series of wavebands [14] and [15].

The international standard on mechanical vibrations from road surface profiles (ISO 8608:1995) suggests that the roughness or smoothness of a road pavement could be ranked based on the Fourier analysis of profile measurement. To this purpose, ISO 8608 can be used as a template to categorize road profile data into a range of smooth to very rough roads. In this method, displacement PSD verses spatial frequency of its longitudinal profile is calculated and plotted on a log-log scale. Then, linear regression is used to fit a straight line to the plot and the resulting line compares with the ISO 8608 template to obtain the closest ranking of the road roughness. Fig. 2 shows a plot of elevation data and the distribution of roughness via a PSD of slope verses wavelength for two selected rough (IRI = 4.17 m/Km) and smooth (IRI = 1.36 m/Km) profile sections (of 100 m in length). The plot of PSD shows that profile data with the smallest IRI (meaning smoother road which is illustrated in

red) has smaller amplitude, however the smoother road has some irregularities in short wavebands (λ = 0.1 to 0.3m). As illustrated in this figure, due to the fact that the PSD is not always a straight line, more information can be obtained by the displacement PSD (or the RMS displacement) in the different octave bands. This would make it possible to classify a road for every octave band in an appropriate class, thus providing a better insight into the most appropriate maintenance strategy to be employed.

To highlight the importance of analyzing a road pavement for separate wavebands, it was reported by Mann et al. [14] that two pavements of near identical IRI values could possess quite different roughness distributions. Therefore, it would be expected that both of these pavements are deteriorating differently, and hence must be evaluated differently for maintenance purposes. Overall, increased roughness in the longer wavelengths was expected to be due to deep seated movements within the pavement subgrade, whereas increased roughness in the shorter wavelengths was expected to be due to seal deterioration and strength issues in the upper base materials (i.e. potholing).

Figure 2. (top) Plot of elevation data and (bottom) plot of slope PSD analysis of outer wheel path of two road profile section

B. Wavelet Analysis Wavelet analysis is a powerful tool for numerical analysis

of non-stationary signals [16] (e.g. longitudinal road profile data) in both time (distance) and frequency (wave number) domain simultaneously at multiple resolutions. It calculates the correlation between the signal and a wavelet function. The similarity between the signal and the wavelet function is computed separately for different wavebands, resulting in a two dimensional representation. A wavelet function is a small wave, designed to discriminate between different frequencies while the following mathematical criteria are satisfied [17]: (a) A wavelet must have finite energy, (b) the wavelet energy is the integrated squared amplitude of the wavelet function, (c) the Fourier transform of the wavelet function has to have non-zero frequency component (this condition implies that the mean of the wavelet function should be equal to zero); and for complex wavelets, the Fourier Transform must be real and vanish for negative frequencies. In general, the wavelet transform is an improvement on the Short Time Fourier Transform (STFT) to provide a time-frequency representation

of a signal. In the context of road roughness analysis, wavelet transform represents distance-wave number of the elevation (longitudinal) signal to extract useful information from road profile data. Wavelet analysis can be performed in two ways: continuous wavelet transform (CWT) and discrete wavelet transform (DWT). By definition, the continuous wavelet transform is a convolution of the input signal with a set of wavelets generated from the wavelet function called ‘mother wavelet’. Continuous wavelet function Ψ(t) is defined by [17]:

)(1)(, abt

atba

−= ψψ (1)

The continuous wavelet transform (CWT) is then defined as:

∫−= dt)a

bt()t(xa

1)a,b(CWT ψ

(2)

where x(t) is a continuous integrable signal. The CWT performs a multi-resolution analysis by scaling (contraction (0<a<1) and dilatation (a>1)) of the wavelet function while

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moving over the signal by time step b Є R (translation). The discrete wavelet transform (DWT) is defined as:

∑ −= )()(1),( kxnak

anaaDWT ii

ii ψ

(3)

where Ψ is wavelet base and x is the original signal. On the other hand, the DWT uses multi-resolution filter banks and special wavelet filters for analysis and synthesis of the original signal. To simplify DWT, a can be assigned equal to two to compute data points on a dyadic grid. The dyadic grid has half of the number of data points at each consecutive lower octave. This assumption limits the application of DWT to the signals without frame synchronous data such as data of Hidden Markov Models. To overcome this limitation and to be able to compute CWT using computers, discretised/sampled CWT is introduced which is sampling the time-scale plane (SCWT). It is necessary to note that the discretised version of CWT is not equal to DWT as it uses discretised versions of scale and dilatation axes to split the signal instead of filter banks in DWT. The SCWT is defined as:

∑ −= )()(1),( kxa

nka

naSCWT ii

i ψ

(4)

If a = 2, wavelets will be an octave space apart. In this paper, DWT has been chosen to use for wavelet analysis of road profile data because it is easy to implement and reduces the computation time and resources required for wavelet transform. As mentioned above, in the case of DWT, a filter bank containing filters to decompose signal into frequency bands is used. An example of a multi-resolution three-level filter bank is shown in Fig. 3.

Figure 3. Three level wavelet decomposition filter bank

The low-pass (L) and high-pass (H) filter separate the frequency content of the profile signal x(n) into smaller frequency. In order to avoid redundancy, the signal needs to down-sampled after each filter level without loss of information. In the first level, the coefficients d1(n) (detailed component) and a1(n) (coarse component) are the highest and lowest half of the frequencies in x(n). Then, they are down-sampled to double the frequency resolution. In turn, the time resolution is halved as every half of the samples are presented in d1(n) and a1(n). In the second level of signal decomposition, a1(n) of the first level again filtered through the low and high pass filter and down-sampled. After each level, the output of the high-pass filter represents the highest frequency (here-wave number) content of the low-pass filter of the previous level, this leads to a pass-band. The decomposition process for L level is mathematically described as (5) and (6):

∑=

−=N

njj nxikhka

1)()2()( (5)

∑=

−=N

njj nxikgkd

1

)()2()( (6)

where j indicates the level of decomposition, N is number of data points, x is elevation signal, h is low-pass filter; and g is high-pass filter. After decomposition, the elevation signal x(n) can be expressed as:

121 ...)( ddddanx LLL +++++= − (7) Note that the wavelet filters are designed specifically for

DWT based on the continuous mother wavelets which are supported and characterized by low-pass and high-pass analysis. Some generally used filter families for DWT are Daubechies [18], Coiflets [19], Symlets [18] and Biorthogonals [20]. The quantitative parameter corresponds to each of the decomposed signal can be defined by its energy (E) calculated as [21, 22]:

∑=

Δ=N

1n

2s )n(xE (8)

where s is frequency sub-band signal, Δ is sampling interval of elevation profile data (approx. 50 mm for pavement sections analyzed in this study); an N is total number of data points. For frequency sub-band aL, dL, dL-1,...,d2, d1, their corresponding energy is EaL, EdL, EdL-1,..., Ed2 and Ed1.

Among the wavelet families, Daubechies wavelets are mathematically more tractable and well established among numerous applications. There exists a trade-off between the order of wavelets and computation time. Higher order wavelets are smoother and are better able to distinguish between the different frequencies, however they require more computation time. In this experiment, Daubechies wavelet of the order of nine (DB=9) with ten level of decomposition (L=10) to most satisfy the accuracy and computational cost of the analysis. Generally, Lth order filter produces a delay of N samples [23]. As mentioned previously, DWT is computed on a dyadic grid (a = 2), hence the number of data points is half at each lower octave band. As a result, maximum level of decomposition (L) of a road profile data with N number of samples (/data points) can be calculated from 2L ≤ N. Consequently, sub-band frequencies and wavelengths for approximation band (cA) and detail bands (cDl) can be calculated by:

]2,0[ 1s

LcA ff −−= (9)

]2,2[ 1s

ls

lcD fff

l

−−−= (10) where fs is the sampling rate of the profile data. The profile

data used for this study captured in samples of about 50 mm apart, so for a typical section of 100 m; there is about 2000 data points. As a result, maximum level of decomposition for such signal is L = 10.

In general, road roughness analysis with discrete wavelet transform is not only capable of separating long and short wavebands, but it is also able to find local road roughness features. Next section examines application of discrete wavelet transform on real road profile data to analyze road roughness.

IV. EXPERIMENTAL RESULTS AND ANALYSIS In this section, the two experimental low volume highway

profile sections used are presented in Fig. 2. The sections are

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100m in length and are samples of a smooth and rough section of road, both located in Victoria, Australia. Fig. 4 and 5 illustrate the result of wavelet decomposition on selected smooth and rough profiles and their raw profile data, respectively. The percentage of energy that each waveband contributes to the profile signal is calculated (EA10, ED10-ED1 in the right side of each waveband). Comparison of smooth and rough profile analysis shows that, in the rough profile analysis, the contribution of relative energy is more than the similar wavebands in smooth profile analysis. result can be concluded from PSD analysis presented in Fig. 2. Observation of Fig. 4 shows that there is an irregularity existing at the first 10 m of the profile. This irregularity is diagnosed by the wavelet analysis as it appears in D4 to D7 which represents wavebands of 0.8 to 12.8 meters. Observation of Fig. 5 shows that two irregularities exist in about 37 and 68 m of rough profile data which are clearly diagnosed in D6 (waveband of range 3.3 and6.5 meters). The periodic trend of both profile data is located in D9 and D10, the longest waveband (range of 25 to 102 meters).

V. CONCLUSION The main objective of this paper was to investigate

application of signal processing in road roughness measurement and analysis. Two pilot sections of smooth and rough profile pavements from the state of Victoria were identified for inclusion in the study. Power spectral density analysis and discrete wavelet transform was chosen as analyzing tools. The following conclusions were drawn from this study:

• Road profile analysis with signal processing tools is advanced the conventional approach because it is independent the passengers’ perception of ride and vehicle characteristics. tudy shows while these methods have a good correlation with the traditional method (some of them are well established, e.g. IRI) [6], they provide more accurate analysis with more detail information about the pavement state.

• Although there is a strong link between both signal processing approaches, wavelet analysis outperforms PSD analysis. The advantage of wavelet analysis is the ability to perform local analysis. The multi-resolution nature of this analysis makes it possible to analyze trends as well as breakdown points, etc from the original signal.

REFERENCES [1] A. Scala and D. W. Potter, "Measurement of Road Roughness," 1977. [2] K. F. Porter, "The Development of a Road Rating System for Australian Conditions," in 16th ARRB Regional Symposium Tamworth, 1979 [3] W. N. Carey and P. E. Irick, "The Pavement Serviceability Performance Concept," Highway Research Bulletin 250, Highway Research Board, pp. 40-58, 1960. [4] W. R. Hudson, D. Halbach, J. P. Zaniewski, and L. Moser, "Root-Mean-Square Vertical Acceleration as a Summary Roughness Statistic," in

Measuring Road Roughness and Its Effects on User Cost and Comfort, T. D. Gillespie and M. W. Sayers, Eds., 1985. [5] M. W. Sayers, "On the Calculation of International Roughness Index from Longitudinal Road Profile," Transportation Research Record, Transportation Research Board (TRB), No. 1501, pp. 1-12, 1995. [6] R. Hesami, K. J. McManus, R. P. Evans, and R. Hassan, "A Comparative Study of Roughness Indices for Monitoring the Performance of Thin Seal Flexible Pavements subjected to Low Traffic Volumes in Australia," in The Twelfth International Conference on Civil, Structural and Environmental Engineering Computing, Maderia, Portugal, September 2009. [7] S. Qian and D. Chen, "Joint Time-Frequency Analysis: Methods and Applications," N.J: Prentice Hall, 1996. [8] A. Papoulis, The Fourier Integral and Its Applications McGraw-Hill, 1962. [9] A. V. Oppenheim, Discrete-Time Signal Processing: Prentice Hall, E.Cliffs, 1989. [10] M. B. Priestley, Spectral Analysis and Time Series: Academic Press, 1981. [11] C. Milton, G. David, and S. T. C., "Some Statistical Tests in the Study of Terrain Modelling," International Journal of Vehicle Design, vol. 36, no. 2-3, pp. 132-148, 2004. [12] B. Bruscella, V. Rouillard, and M. Sek, "Analysis of Road Surface Profiles " Journal of Transportation Engineering, vol. 125, no. 1, pp. 55-59, 1999. [13] V. Rouillard, B. Bruscella, and M. Sek, "Classification of Road Surface Profiles," Journal of Transportation Engineering, vol. 126, no. 1, pp. 41-45, 2000. [14] A. Mann, K. J. McManus, and R. P. Evans, "The Use of Power Spectral Density Analysis to Determine Deterioration Modes in Pavement Structures," in 9th Road Engineering Association of Asia and Australasia Conference. vol. 2 Wellington, New Zealand, 1998, pp. 263-268. [15] J. J. Pont and A. Scott, "Beyond Road Roughness - Interpreting Road Profile Data," Road and Transport Research, vol. 8, no. 1, pp. 12-28, 1999. [16] M. Misiti, Y. Misity, G. Oppenheim, and J.-M. Poggi, "Wavelet Toolbox, User’s Guide, MathWorks," 1996. [17] P. S. Addison, The Illustrated Wavelet Transform Handbook: Applications in Science, Engineering, Medicine and Finance: Institute of Physics Publishing, 2002 [18] I. Daubechies, Ten Lectures on Wavelets. Philadelphia: Society for Industrial and Applied Mathematics, 1992. [19] M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, "Image Coding Using Wavelet Transform " IEEE Transaction on Image Processing, vol. 1, no. 2, pp. 205-220, 1992. [20] S. Mallat, A Wavelet Tour of Signal Processing: Academic Press, 1999. [21] L. Wei, T. F. Fwa, and Z. Zhe, "Wavelet Analysis and Interpretation of Road Roughness," Journal of Transportation Engineering, vol. 131, no. 120, 2005. [22] L. Wei and T. F. Fwa, "Characterizing Road Roughness by Wavelet Transform," Journal of the Transportation Research Board, no. 1869, pp. 152-158, 2004. [23] M. G. E. Schneiders, Wavelets in Control Engineering, Master thesis, Eindhoven University of Technology, 2001.

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99.7 99.71 99.72 99.73 99.74 99.75 99.76 99.77 99.78 99.79 99.8-1

-0.9-0.8

prof

ile (

m)

DWT decomposition for raw profile data -BI010997- using -db9- (outer wheel path)

99.7 99.71 99.72 99.73 99.74 99.75 99.76 99.77 99.78 99.79 99.8-1000-900-800

A10

EA10 =99.9798λ >102.1446 meter/cycle

99.7 99.71 99.72 99.73 99.74 99.75 99.76 99.77 99.78 99.79 99.8-50

050

D10

ED10 =0.01856151.0723< λ <102.1446 meter

99.7 99.71 99.72 99.73 99.74 99.75 99.76 99.77 99.78 99.79 99.8-20

020

D9 ED9 =0.0013592

25.5362< λ <51.0723 meter

99.7 99.71 99.72 99.73 99.74 99.75 99.76 99.77 99.78 99.79 99.8-10

010

D8 ED8 =0.00021283

12.7681< λ <25.5362 meter

99.7 99.71 99.72 99.73 99.74 99.75 99.76 99.77 99.78 99.79 99.8-505

D7 ED7 =3.9887e-005

6.384< λ <12.7681 meter

99.7 99.71 99.72 99.73 99.74 99.75 99.76 99.77 99.78 99.79 99.8-202

D6 ED6 =3.7614e-006

3.192< λ <6.384 meter

99.7 99.71 99.72 99.73 99.74 99.75 99.76 99.77 99.78 99.79 99.8-101

D5 ED5 =6.4665e-007

1.596< λ <3.192 meter

99.7 99.71 99.72 99.73 99.74 99.75 99.76 99.77 99.78 99.79 99.8-101

D4 ED4 =6.0419e-007

0.798< λ <1.596 meter

99.7 99.71 99.72 99.73 99.74 99.75 99.76 99.77 99.78 99.79 99.8-0.5

00.5

D3 ED3 =2.2217e-007

0.399< λ <0.798 meter

99.7 99.71 99.72 99.73 99.74 99.75 99.76 99.77 99.78 99.79 99.8-0.5

00.5

D2 ED2 =1.6585e-007

0.1995< λ <0.399 meter

99.7 99.71 99.72 99.73 99.74 99.75 99.76 99.77 99.78 99.79 99.8-101

D1(

mm

)

ED1 =2.7222e-0070.099751< λ <0.1995 meter

Distance (Km)

(a)

61 61.02 61.04 61.06 61.08 61.1 61.121

1.52

prof

ile (

m) DWT decomposition for raw profile data -BE161000- using -db9- (Outer wheel path)

61 61.02 61.04 61.06 61.08 61.1 61.12100015002000

A10 EA10 =99.9879

λ >104.3886 meter/cycle

61 61.02 61.04 61.06 61.08 61.1 61.12-50

050

D10

ED10 =0.009467352.1943< λ <104.3886 meter

61 61.02 61.04 61.06 61.08 61.1 61.12-50

050

D9 ED9 =0.0021539

26.0972< λ <52.1943 meter

61 61.02 61.04 61.06 61.08 61.1 61.12-20

020

D8 ED8 =0.00042629

13.0486< λ <26.0972 meter

61 61.02 61.04 61.06 61.08 61.1 61.12-10

010

D7 ED7 =2.3922e-005

6.5243< λ <13.0486 meter

61 61.02 61.04 61.06 61.08 61.1 61.12-505

D6 ED6 =8.1376e-006

3.2621< λ <6.5243 meter

61 61.02 61.04 61.06 61.08 61.1 61.12-505

D5 ED5 =3.117e-006

1.6311< λ <3.2621 meter

61 61.02 61.04 61.06 61.08 61.1 61.12-505

D4 ED4 =1.1523e-006

0.81554< λ <1.6311 meter

61 61.02 61.04 61.06 61.08 61.1 61.12-202

D3 ED3 =3.8827e-007

0.40777< λ <0.81554 meter

61 61.02 61.04 61.06 61.08 61.1 61.12-101

D2 ED2 =8.5182e-008

0.20388< λ <0.40777 meter

61 61.02 61.04 61.06 61.08 61.1 61.12-0.2

00.2

D1(

mm

)

ED1 =5.2556e-0090.10194< λ <0.20388 meter

Distance (Km)

(b)

Figure 4. (a) Wavelet decomposition of a smooth (IRI = 1.36) profile data. (b) Wavelet decomposition of a rough (IRI = 4.7) profile data.

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