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Normalized truncated Levy walk applied to flexible pavement performance

Maria Cristina Mariani a, Alessandra Bianchini b,, Paola Bandini c

a Department of Mathematical Sciences, University of Texas at El Paso, 500 University Ave., Bell Hall 124, El Paso, TX 79968-0514, USAb US Army Engineer Research and Development Center, 3909 Halls Ferry Road, Vicksburg, MS 39180, USAc Civil Engineering Department, New Mexico State University, Espina St., Hernandez Hall, Room 244 Las Cruces, New Mexico 88003-8001, USA

a r t i c l e i n f o

Article history:

Received 1 November 2010

Received in revised form 17 January 2012

Accepted 18 January 2012

Keywords:

Levy flight

Levy distribution

Pavement performance

Serviceability

Flexible pavement

a b s t r a c t

Government agencies and consulting companies face the challenge of pavement manage-

ment aiming to maintain pavements in serviceable condition in the long term. This paper

analyzed the evolution of the performance of flexible pavements to forecast the change in

the serviceability level offered by the structure. Data from the Minnesota Road Research

Project (MnROAD) road test corresponding to the low-volume traffic loop were used in

the study. The data analysis centered on the normalized truncated Levy walk model. It

was concluded that the truncated Levy distribution optimally describes the decrease of

pavement serviceability caused by traffic and climatic conditions specific to the infrastruc-

ture location and traffic. The Truncated Levy Flight (TLF) provided a better approximation

over the full range of the normalized pavement serviceability index than the Gaussian dis-

tribution for dataset considered.

Published by Elsevier Ltd.

1. Introduction

Many government agencies and private consulting companies must deal with the issue of pavement maintenance. The

ability to maintain an in-service pavement structure in acceptable condition from the structural and functional points of

view is related to many factors, which are often not explicit and change with time. Although the maintenance strategies

for highway pavements also depend on human experience, data interpretation and agencys policies, a tool capable of pre-

dicting pavement serviceability is a desirable feature that can further support the implementation of maintenance and bud-

geting plans.

The stochastic approach been implemented in geomechanics to describe the soil complex system and its interactions. It

has been applied to evaluate the nonlinear soil behavior, soilstructure interaction (e.g.,Fenton, 1999; Feda, 2002; Manolis,

2002), liquefaction phenomenon in granular soils (Ural, 2002), and foundation settlements (Marinilli and Cerrolaza, 1999). In

the field of the pavement analysis, the stochastic approach has been also analyzed (e.g.,Butt et al., 1992; Vennalaganti et al.,

1994; Hong and Wang, 2003). In particular, Li et al. (1996) and Yang et al. (2006) investigated different aspects of the Markovchain process related to the prediction of the overall pavement condition.

Within the stochastic field, the Econophysics (Mantegna and Stanley, 1994, 1999) proved to be one of the new resources

in the analysis of the financial models. Econophysics was introduced in 1995 by Stanley et al. (1996, 1999) and deals with the

application of mathematical tools, usually applied to physical models, to the evolution of market indices. In the study of

financial indices, the statistical properties of the temporal series provided an important contribution to the modeling of these

economic systems. In fact, the Brownian motion (or random walk) is employed in the representation of the option price

evolution. The model assumes that the increment in the logarithm of the process follows a diffusive process with Gaussian

distribution (Willmott et al., 1993).

0968-090X/$ - see front matter Published by Elsevier Ltd.http://dx.doi.org/10.1016/j.trc.2012.01.006

Corresponding author. Tel.: +1 601 634 5379; fax: +1 601 634 4128.

E-mail address: [email protected](A. Bianchini).

Transportation Research Part C 24 (2012) 18

Contents lists available at SciVerse ScienceDirect

Transportation Research Part C

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e/ t r c
http://dx.doi.org/10.1016/j.trc.2012.01.006mailto:[email protected]://dx.doi.org/10.1016/j.trc.2012.01.006http://www.sciencedirect.com/science/journal/0968090Xhttp://www.elsevier.com/locate/trchttp://www.elsevier.com/locate/trchttp://www.sciencedirect.com/science/journal/0968090Xhttp://dx.doi.org/10.1016/j.trc.2012.01.006mailto:[email protected]://dx.doi.org/10.1016/j.trc.2012.01.006

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Nevertheless, some studies have shown that the Brownian motion does not describe accurately the evolution of financial

indices near a crash (e.g., Mandelbrot, 1963; Negrea, 2000; Mariani and Liu, 2007) or the predictionof high crop loss (Mariani

et al., 2009). The solution to this problem was initially proposed by Mandelbrot (1963)with the development of a stochastic

model for the evaluation of the cotton price by using a stable non-Gaussian Levy process. With regard to the financial mar-

kets, the Truncated Levy Flight (TLF), which is a derivative of the Levy methodology, was suggested as the solution (Mante-

gna and Stanley, 1994, 1999; Negrea, 2000). The truncated Levy distribution is characterized by the sum of random variables

that converges to a normal random variable with Gaussian characteristic distribution. The random variables are named Levy

flights because they are distinguished by the Levy distribution.

2. Objective and application

The objective of this paper is to analyze the stochastic distribution of the Levy theory as an approximation method to rep-

resent pavement performance and compare such analysis with the typical Gaussian distribution. Based on the positive re-

sults of past experimental studies, this paper aims to study the application of the TLF distribution to represent the

evolution of the pavement performance in terms of serviceability. The database from the MnROAD test track containing cur-

rent and past pavement conditions for low-volume roads was used in this analysis.

From the applicative standpoint, an analysis to describe long-term pavement performance represents a valuable tool for

pavement management agencies when planning at the network level. The network level planning takes into account the

agencys short- and long-term budget needs to identify and prioritize future maintenance and construction projects. Unlike

previous studies that used different numerical techniques centered on the application of neural networks or neuro-fuzzy

models to evaluate pavement performance (e.g., Eldin and Senouci, 1995; Saltan et al., 2002; Loizos and Karlaftis, 2006; Bian-chini and Bandini, 2010), this paper shows that the stochastic theory offers a new perspective in modeling complex systems

related to practical engineering applications such as the performance of a pavement structure. The TLF represents an alter-

native tool to describe the complex phenomenon of pavement performance and can be implemented to identify future crit-

ical pavement situations that agencies need to take into account for budget and project planning.

3. Pavement performance

The term performance refers to the structural and functional responses of the pavement structure to the actions of traffic

and environmental factors which change the serviceability condition of the pavement. The changes in the pavement perfor-

mance and serviceability offered to the traveling public are important elements that agencies managing highway pavement

networks need to take into consideration for efficiently maintaining the infrastructure at a high level of serviceability.

The concept of serviceability comprises the structural and functional competences of the pavement structure in support-

ing the designed traffic. The structural competence relates to the load carrying capacity, whereas the functional competenceconsists of how well the infrastructure serves the user and relates to riding comfort and ride quality. The structural and func-

tional aspects cannot be analyzed or defined separately. The appearance of a specific type of distress mainly caused by the

traffic loads creates irregularities on the pavement surface that directly influence the ride quality.

The experimental results of the AASHO Road Test (Carey and Irick, 1960, 1962) led to expressing the pavement service-

ability in terms of the present serviceability index, PSI, which represents the current ability of a pavement to serve the design

traffic (Carey and Irick, 1962). Originally defined on users rating, the index ranges from 5 (excellent road) to 0 (impossible

road).

Subsequent studies (e.g.,Uzan and Lytton, 1982; Hall and Muoz, 1999; Prozzi and Madanat, 2003 ) on the AASHO Road

Test database proposed formulations of the PSIthrough parameters related to the conditions of the pavement surface. The

study ofHall and Muoz (1999)provided aPSIestimate through the international roughness index, IRI. TheIRIis a standard

roughness index calculated from the records of a specific type of sensors installed on vehicles or trailers. For flexible pave-

ments,Hall and Muoz (1999)proposed the following equation to estimate the PSIfromIRImeasurements:

PSI 5 0:2397x4 1:771x3 1:4045x2 1:5803x; 1

wherexlog1 SVandSVis the slope variance calculated as SV2:2704IRI2.The pavement performance is quantified by the PSI, which is the random variable whose behavior is analyzed by the TLF

developed in this study. The reason supporting this choice resides in the two types of information contained in the definition

of the index about pavement structural and functional capabilities.

The factors causing the decrease in the serviceability level are multiple and include traffic, climatic changes ( e.g., freeze

thaw cycles, rainfalls, and seasonal temperature variations), pavement materials and construction practices. Their influences

cannot be explicitly quantified from the deterministic standpoint. This type of information is contained in the numerical val-

ues that the pavement serviceability indicator assumes at any moment during the life of the structure ( Ovik et al., 2000;

Chadbourn, 2001; Prozzi and Madanat, 2003).

These considerations led to the conclusion that factors and variables influencing the performance of the pavement struc-

ture and the interactions among them can be defined as a complex system (Mantegna and Stanley, 1999). The pavement per-

formance system cannot be represented by objective and deterministic mathematical laws; only specific numerical analysis

2 M.C. Mariani et al. / Transportation Research Part C 24 (2012) 18

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approaches are able to provide an approximate description. In the past, empirical performance equations were determined

from linear regression using experimental data. The pavement complex system is influenced by parameters that may have

normal, or log-normal, variability; however, these parameters lose the normalcy aspect due to their interaction with time

and assume the characteristics typical of life-cycle type distributions. With this perspective, the stochastic model and, in par-

ticular, the TLF can be employed for the best description of a complex system solely explained by data resulting from the

compound effects of multiple causes.

4. The truncated Levy flight

The goal of this paper is to evaluate the possibility of representing the performance of a pavement system through a sto-

chastic approach with the intention to overcome the uncertainty about the factors that mostly determine pavement deteri-

oration. The literature review presented different applications of stochastic theory to market analysis, which is influenced

quite often by qualitative factors, such as sociological and political trends. The Levy distribution, and in particular the Trun-

cated Levy Flight (TLF), showed promising results in economics. For this reason, the evaluation of the TLF in pavement engi-

neering applications has the potential to offer an additional tool to describe pavement performance that is deemed necessary

for agencies pavement management.

The TLF is characterized by the sum of random variables that converges to a normal random variable with Gaussian char-

acteristic distribution (Mantegna and Stanley, 1999; Negrea, 2000). The functional form characterizing all stable distribu-

tions was proposed byKhintchine and Levy (1936) and Levy (1937). The characteristic functionuq representing theentire class of stable distributions is defined as

lnuq ilq cjqj2 1 ib q

jqjtan p

2a

h i a 1

lq cjqj 1 ib qjqj

2p ln jqj

h i a 1

8>: 2

whereq is a parameter that can assume any real value, i is equal toffiffiffiffiffiffiffi

1p

,a is a stability parameter (0 < a 6 2), also calledcharacteristic exponent in this paper, c is a positive scale factor, l is any real number, and bis an asymmetry parameter thattakes values in the interval [1,1]. The reader is referred toMantegna and Stanley (1999)for an extensive discussion. Theanalytic form of a stable Levy distribution is only known for specific values ofa and the asymmetric parameter b. In partic-ular,a0:5 and b1 generate the LevySmirnov distribution; a1 and b0 generate the Lorentzian distribution; theGaussian distribution is obtained fora2.

In developing the Levy distribution and the TLF, only symmetric stable distributions (withb0) with a zero mean (l = 0)are considered. These assumptions define the characteristic function in the form

uq ecjqja

: 3

The probability density function is consequently derived by the inverse Fourier transform of(3). For a stable distribution

of the exponenta and scale factorc, the probability density functionPL is defined as

PLx 1

p

Z 10

ecjqja

cosqxdq; 4

wherex is the independent and identically distributed (i.i.d.) random variable. The asymptotic behavior of the distribution

for large values ofjxj approximates a power law given by

PLjxj c C1 a sinpa=2

pjxj1a jxj

1a; 5

and the value ofPLfor x = 0 is

PLx 0 C1=apac1=a

: 6

The power-law asymptotic behavior for large values ofjxj determines a distribution characterized by infinite variance thatis typical of all Levy stable processes witha l

cPlx l< x < l

0 x< l

8>: 7

whereP1xis a symmetric Levy distribution, cis a normalization constant, and l is a scale factor. The symmetric Levy dis-tribution is stable. The TLF distribution is not stable but its variance is finite and converges slowly to the Gaussian distribu-

tion. Also, the TLF distribution presents an abrupt cut in the tail. Koponen (1995)proposed a TLF with a cut function that is

M.C. Mariani et al. / Transportation Research Part C 24 (2012) 18 3

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exponentially decreasing and characterized by a fictitious parameter l. The model proposed by Koponen (1995) allowed eval-

uating the convergence of the Levy process towards the Gaussian. The choice ofKoponen (1995)was dictated by his deri-

vation of an analytic expression of the TLF characteristic equation in agreement with the work previously done by

Mantegna and Stanley (1994), which represents the basis of this paper. In Koponen (1995), the characteristic function of this

distribution is defined as

uq exp c0c1q2 1=l

2a=2

cospa=2 cosa arctanljqj( ); 8

wherec1is a scale factor given by

c1 2p cospa=2aCa sinpa

At; 9

whereA is a fictitious parameter, tis the time scale andc0 is

c0 l

a

cospa=2c1

2paCa sinpa

Ala

t: 10

The variance of the TLFs distribution in (7)is directly calculated from the characteristic function as

r2t @2uq@q2

q0

t 2Ap1 aCa sinpa

l2a

; 11

The random process given by the sum of a finite number of i.i.d. random variables is defined partitioning the time interval

into Nsteps Dt, so that the time lag isTNDt. Thus, the characteristic function originated from the sum ofNi.i.d. stochasticvariables is

uq;N exp c0Nc1Nq2 1=l

2a=2

cospa=2 cosaarctanljqj

( ): 12

For small values ofN, the return probability will be very similar to the stable Levy distribution in(6).

5. Methodology

This paper analyzes the application of the TLF to represent the behavior of the pavement performance and its variability

when the pavement structure is subjected to traffic and climatic factors over time. The evaluation of the pavement degra-dation trend is an important aspect for agencies managing the highway infrastructure. The MnROAD test site (Albertville,

Minnesota, USA) provided a consistent set of IRIvalues that allowed expressing the performance of conventional flexible

pavements and quantifying the level of serviceability of the structure through the PSIas defined in(1)according toHall

and Muoz (1999).

TheIRImeasurements from six different units (cells) of asphalt pavement of the MnROAD test site were analyzed in this

study. TheIRIdata were from June 1997 to April 2006 and taken approximately every 6 or 4 months. The traffic imposed on

these cells was in average 445,000 vehicle passes per year. The pavement cells were characterized by an asphalt layer of

thickness varying from 3.1 to 5.1 in. and a base course layer of thickness varying from 4 to 12 in. This type of pavements

is typical of urban and rural roads and represents one of the main assets that local and state agencies manage and maintain.

The contribution of this paper may improve the agencies efficiency in infrastructure management and, at the same time,

enhance the applicability of those methods traditionally considered as pure theoretical to problem solving strategies con-

nected to real practice.

Starting from Eq.(1)ofHall and Muoz (1999)to calculatePSIfrom IRIvalues, the analyzed stochastic variable is the PSIvariability (DPSIt), defined as the difference of the logarithm of two consecutive measurements of serviceability:

DPSIt logPSIt logPSItT; 13

whereTis the time lag between the values PSItandPSItT. The time lag refers to the interval length given by the number ofmeasuredPSIvalues belonging to the measurement sequence. To compare the PSIvariability for different values ofT, the

parameter DPSItwas normalized with the variance as

gtDPSIt hDPSIiT

r ; 14

wherer2 hDPSI2iT hDPSIi2Tand the function hiTdenotes the arithmetic average over the temporal series of scale T. Table 1includes mean, variance, skewness, and kurtosis of the data. To apply Koponens model, further normalization of the char-

acteristic function was needed. Noting that


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r2 @2uq@q2

q0

15

and

@2uq=r

@q2

q0

1

r2@2uq@q2

q0

1; 16

the normalized model results:

lnuNq lnu q

r

;

c0c1q=r2 1=l2a=2

cospa=2 cos a arctan ljqj

r

;

2pAlataCa sinpa

1 ql

r

2 1

!a=2cos a arctan

ql

r

0@1A:

17

Eq.(17) represents the normalized Levy model applied in this paper. The values ofA, the fixed arbitrary scale parameter l,

and the characteristic exponenta were adjusted to best fit the cumulative function.

6. Application and discussion of the results

The analysis in this paper provided a new way to evaluate the decrease of serviceability typical of conventional asphalt

pavements of low-volume traffic roads. The empirical study of the statistical behavior of the pavement serviceability em-

ployed the normalized TLF model. The implementation of the normalized model allowed the comparison of the statistical

properties of the pavementPSIunder different timeframes. The change in serviceability is influenced by many different fac-

tors such as traffic and climatic conditions. These factors act together and simultaneously, but the proportional amount of

their individual influences on the pavement condition changes with time and cannot be univocally determined. The overall

result of these combined effects becomes visible on the structure surface after a certain time, which is quantifiable in terms

of years. Thus, the definition of the random variable DPSItaimed at extracting the underlying pattern that characterizes the

change in pavement serviceability. The random variable normalization was intended to provide uniformity among the mea-

sured observations on different pavement cells belonging to the same category (e.g., conventional asphalt pavement of low-

volume traffic roads).

In adjusting the free parameters of(17)to best fit the cumulative function, the scale parameter l was set equal to 0.5. A

heuristic search was used to evaluate the time lag Tand the characteristic exponenta.Table 2summarizes the values ofaandTtested in the evaluation of the best distribution for the TLF. Fig. 1shows the Levy and Gaussian distribution plots ob-

tained with four pairs ofa and Tvalues and the data point distribution, which almost resembles a continuous line due to theclose spread of the data points. With regard to the time lag, the valueT= 32 provided the best approximation of thePSIdata

(and the normalized serviceability indexgt) by the normalized distribution curve. For lower values ofT, the Gaussian distri-

bution was the best fit, but for T = 32, the Levy distribution was the best fit. In the definition of the stochastic variableDPSItas

given by (13), the time lag Thad a remarkable influence on the data distribution to be approximated by the TLF. Longer time

lags had the effect of flattening and leveling the abrupt discontinuities appearing in the data sequence. In an analogy with the

Table 2

Values of the exponentafor four values of the time lag Tusedto compute the distributions shown inFig. 1.

T Exponenta

1 1.25

4 1.20

8 0.70

32 0.90

Table 1

Mean, variance, skewness, and kurtosis of the data.

Mean 0.4244Variance 0.2898

Skewness 0.6611Kurtosis 0.4519


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Fig. 1. Loglog plots of the cumulative distribution of the normalized pavement serviceability gt(Eq.(14)) for four different values of time lag T. The solid

line is the best fit of the Levy distribution. The dashed line is the Gaussian distribution. The thicker line resulted from the close spread of the data points.

Fig. 2. Loglog plots of the cumulative distribution of the normalized pavement serviceability gt(Eq.(14)) for four values of the characteristic exponent a.

The solid line is the best fit of the Levy distribution. The dashed line is the Gaussian distribution. The thicker line resulted from the close spread of the datapoints.


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signal analysis theory, the time lag may be compared to the sampling frequency applied to a signal and the consequent Fou-

rier components with which the initial signal is approximated. High-frequency sampling describes more in detail the signal

variability that also includes noise, whereas a low-sampling frequency captures a more stable pattern of the signal itself. In

other words, choosing the time lagTmay be configured as a low pass filter, eliminating the disturbances in the data consid-

ered as high-frequency noise.

Fig. 2shows the Levy and Gaussian distribution plots for T= 32 and four values ofa and the data point distribution fromthe MnROAD test database. The thicker line in the plots ofFig. 2resulted from the close spread of the data points. Table 3

provides four of the variousa values tested with T= 32 for the search of the best approximating TLF distribution. The char-acteristic exponenta = 0.9 was found to guarantee the best fit of the normalized TLF for the dataset used. However, values ofalower than 2 showed that the TLF was a better approximator than the Gaussian distribution curve (Fig. 2). The numericalanalysis also showed the slow convergence of the Gaussian distribution.

During the heuristic search for the optimal values ofTanda, the Gaussian was the distribution to best fit the availabledata, in particular for lower values of the normalized difference of serviceability (gt). The comparison of the plots inFigs. 1

and 2 proved that the Gaussian distribution was not able to describe completely the correct behavior of the normalized index

for values in the neighborhood of 10. Instead, the Levy distribution displayed a good approximation over the full range of

values calculated for the given database, especially when T= 32(Fig. 2). A KolmogorovSmirnov test on the serviceability

data ofT= 32 rejected the null hypothesis and further confirmed that the series was not characterized by normalcy. The

v2-test also confirmed the possibility of rejecting the null hypothesis. As pointed out by Scalas and Kim, 2006, the Kolmogo-

rovSmirnov and v2 hypothesis testing methods may be sensitive to the central part of the distributions or the time lag and

therefore additional analysis on their variability would be necessary.

The normalized dataset showed some outlier points related to a slower decrease in pavement serviceability than the de-

crease measured during the full period covered by the database. These events may be due to particular climatic conditions,

but further investigation is needed to justify these numerical discontinuities. However, outliers were not removed from the

data to avoid misinterpretation of the serviceability trend typical of this type of low-volume traffic pavements located in the

Minnesota region.

7. Conclusions

The numerical results obtained in the analysis led to the conclusion that the Levy distribution, and its normalized model,

can be considered as one of the possible interpretations of the behavior of the pavement serviceability of low-volume roads.

Once the distribution parameters are identified through a heuristic search, the TLF can be immediately applied to evaluate

the trend in pavement performance over time. An increase in numerical accuracy when predicting critical pavement condi-

tion allows improving the efficiency of pavement management in terms of funding allocation and work planning. The

numerical values of critical pavement condition and time gaps, which in the practice can be used to represent maintenance

actions, clearly have influence on those aspects of pavement management that are mainly qualitative rather than quantita-

tive. These aspects include maintenance policies, funding acquisition, and infrastructure assessment.

The use of the TLF offers a promising alternative for representing pavement serviceability trends. In pavement manage-

ment, the typical normal or log-normal distributions of the factors affecting pavement serviceability cannot be linearly

added and used for describing the infrastructure serviceability. Even if a limited amount of pavement condition data col-

lected over a short time can be numerically represented through normal distributions, pavement degradation and decrease

in serviceability do not follow at all a normal distribution. Nevertheless, serviceability is a decreasing function that can be

best represented by the statistical distributions reaching a critical point and showing failure. For this purpose, it is possible

to identify competing distributions that can provide acceptable results based on specific selection criteria; the solution is not

unique but can be sought through a heuristic approach.

The TLF offers an alternative methodology to describe pavement serviceability. The TLF function parameters and time

gaps can be tailored to represent the stochastic characteristics of the factor interaction determining pavement degradation,

maintenance policies, and recurring maintenance programs. Additional investigations using larger databases will allow fur-

ther evaluation of whether this type of stochastic processes is adequate to describe the lifelong response of the pavement

structure to deterioration factors such as traffic and environmental changes.

Table 3

Values of the exponent a for T = 32 used to compute thedistributions shown inFig. 2.

T Exponenta

32 0.40

32 0.90

32 1.20

32 1.40


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Role of the funding source

This analysis was part of a study on the applications of numerical methods to geotechnical and pavement engineering

problems. No sponsor or funding was associated with this work.

Acknowledgement

The authors would like to thank the MnROAD Research Project staff for providing the data used in this study.

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