prozzi2007

Prévia do material em texto

Effect of Weigh-in-Motion System Measurement Errors on
Load-Pavement Impact Estimation
Jorge A. Prozzi1 and Feng Hong2
Abstract: Weigh-in-motion �WIM� technology has found increasing application in the highway and transportation areas for traffic data
collection for the purpose of highway capacity analysis, aiding enforcement and, most recently, pavement design. The measurement
accuracy of a WIM scale is critical. There are numerous existing studies involving the measurement accuracy per se; however, the
implications and effect of the accuracy in the context of pavement design have been rarely examined. To address this issue, two particular
aspects are considered in this study. First, with traffic data obtained from WIM stations in Texas, axle load spectra are developed from
both statistical data fit and load-pavement impact perspectives. Axle load spectra are shown to be captured best by mixed-lognormal
distributions. Second, based on the first aspect, two scenarios are investigated by incorporating two types of errors into the load
measurements: �1� A random error component due to the WIM scale intrinsic properties; and �2� a systematic error component due to the
improper calibration of WIM system. The relationship between jointly varying measurement error levels and load-pavement impact
estimation errors is established. It is demonstrated �and quantified� that both types of errors contribute to load-pavement impact estimation
inaccuracy. The random error leads to overestimation of load-pavement impact. The results also show that WIM system calibration is of
more importance because load-pavement impact estimation is more sensitive and significantly related to systematic error than random
error. In addition, the estimated load-pavement impact is found to be more sensitive to overcalibration �positive bias� than undercalibration
�negative bias� condition. In summary, the findings in this study provide an effective and efficient approach to evaluate WIM measurement
errors in conjunction with load-pavement impact in pavement design and rehabilitation. The findings in this study could be further applied
for the determination of adequate pavement design reliability standards. Results can also be useful to highway agencies in assessing and
selecting WIM equipment.
DOI: 10.1061/�ASCE�0733-947X�2007�133:1�1�
CE Database subject headings: Pavements; Measurement; Weight; Loads; Errors.
Introduction
Soil and material properties, environmental conditions, and traffic
loads are the three major inputs controlling the design of new and
rehabilitated pavement structures. Typically, material properties
are determined through laboratory and field tests. Environmental
conditions, including temperature and moisture, are available
through established networks of weather stations and databases.
Traffic loads play a critical role in the deterioration of pavement
performance. In the current American Association of State High-
way and Transportation Officials �AASHTO� pavement design
guide the effect of traffic on pavement performance is accounted
for in terms of the accumulated traffic load, measured in equiva-
lent number of single axle loads �ESALs� �AASHTO 1993�. Fur-
ther, in the proposed National Cooperative Highway Research
Program �NCHRP� Mechanistic-Empirical Design Guide, traffic
load will be accounted for by means of axle load spectra �NCHRP
2005�. In this regard, accurate traffic load data collection is of
crucial importance for pavement design and performance
analysis.
Two approaches can be adopted to collect axle load data. One
utilizes static scales and the other uses weigh-in-motion �WIM�
equipment. Both approaches have their advantages and disadvan-
tages. For static weighing, the most appealing characteristic is its
high level of accuracy. However, the disadvantages include a se-
ries of inconveniences and inefficiencies. For instance, the in-
spected vehicle is required to stop while being weighed so that
time loss is a burden for both inspectors and drivers. In addition,
more personnel are required to conduct a static weighing opera-
tion than those necessary to monitor the WIM system. As a result,
static vehicle weighing can only be used to collect limited
samples of axle load data.
These disadvantages, however, can be overcome by applying
WIM technology. Compared with static scale, state-of-the-art
WIM technology has gained popularity due to, in large part, its
ability to collect continuous traffic data without human interven-
tion. Theoretically speaking, as soon as it is set up, a WIM scale
is able to continuously collect and record vehicle information
including date and time of passage, lane and direction of travel,
vehicle class, speed, wheel and axle weight, and axle spacing. At
a given location of interest, it is capable of collecting the popu-
lation data instead of a small sample. Nevertheless, aspects such
as WIM system instability due to sensor technology, environmen-
tal effects, pavement conditions and other factors give rise to the
1Assistant Professor, Dept. of Civil, Architectural and Environmental
Engineering, The Univ. of Texas, ECJ 6.112, Austin, TX 78712.
2Graduate Research Assistant, Dept. of Civil, Architectural and
Environmental Engineering, The Univ. of Texas, ECJ 6.510, Austin, TX
78712.
Note. Discussion open until June 1, 2007. Separate discussions must
be submitted for individual papers. To extend the closing date by one
month, a written request must be filed with the ASCE Managing Editor.
The manuscript for this paper was submitted for review and possible
publication on January 30, 2006; approved on July 26, 2006. This paper
is part of the Journal of Transportation Engineering, Vol. 133, No. 1,
January 1, 2007. ©ASCE, ISSN 0733-947X/2007/1-1–10/$25.00.
JOURNAL OF TRANSPORTATION ENGINEERING © ASCE / JANUARY 2007 / 1
concern over its measurement accuracy. Therefore, WIM system
reliability for collecting accurate data relies heavily on its peri-
odic calibration and maintenance.
Due to the importance of proper calibration, research has
aimed at developing WIM calibration guidelines or quality assur-
ance programs. For instance, Izadmehr and Lee �1987a� sug-
gested the type of trucks and minimum number of vehicles
required for on-site calibration with Texas WIM systems. The
relationship between the tolerance for 95% probability of confor-
mity for axle group or gross-vehicle weight �GVW� and three
vehicle-speed levels was provided in another study conducted by
Izadmehr and Lee �1987b�. Davies and Sommerville �1987� pre-
sented a self-calibration approach and an accuracy “funnel” for
WIM performance appraisal. Based on existing research, a stan-
dard specification was published by the American Society for
Testing and Materials �ASTM 2002� to assist highway agents
with WIM user requirements and test methods.
Apart from the issue of WIM measurement accuracy, the effect
of measurement inaccuracy on pavement performance is of inter-
est from the pavement design perspective. This is especially criti-
cal because the relationship between measurement accuracy and
its effect on pavement performance is complex and nonlinear. If a
WIM scale is not properly calibrated and underestimates axle
load, the expected pavement life would be overestimated resulting
in unexpected premature failures. Overestimated loads by the
WIM system would lead to the design of thicker and more costly
pavement structures than may be actually required. For instance,
the report by the Federal Highway Administration �FFwA� focus-
ing on long term pavement performance test site WIM data sug-
gested that 1% undercalibration �negative bias� of a WIM scale
could produce approximately 3% less ESALs estimation, whereas
1% overcalibration �positive bias� could lead to 4.5% more
ESALs estimation �FHwA 1998�.
The goal of this study is to investigate how the WIM inaccu-
racy modeled by measurement errors will affect the assessment of
load-pavement impact. Two basic scenarios in terms of two types
of WIMmeasurement errors are addressed: random error and sys-
tematic error. In the first scenario, the effect of WIM inaccuracy
on load-pavement impact estimation will be studied based exclu-
sively on the random error. In the second scenario, both the
systematic and random errors are assessed simultaneously. Rec-
ommendations are presented for the relationship between varying
measurement error levels and load-pavement impact estimation
error. Load-pavement impact estimation error is the most impor-
tant aspect for pavement design reliability.
The second section of this paper provides a background of
WIM systems concerning measurement errors and specifications.
The third section describes the traffic data set used in this study
with focus on axle load distributions. The fourth section develops
the methodology to address the effect of measurement errors on
load-pavement impact estimation. The last section summarizes
the main findings.
Background
WIM technology has been extensively used for traffic monitoring
since the early 1950s. A number of WIM system types are avail-
able in terms of the specific sensor technology applied. The fol-
lowing are some of the frequently used WIM scales with their
individual specific sensors �Middleton and Crawford 2001�:
• Bending plates,
• Capacitive weight mats,
• Capacitive strips,
• Hydraulic load cells,
• Piezoelectric cable,
• Quartz-piezoelectric, and
• Strain gauge load cells.
The advantages and disadvantages pertaining to cost, mainte-
nance, and accuracy vary significantly among these sensors.
Among the alternatives, bending plates, piezoelectric and load
cell WIM scales are the most typical and widely used in the
United States.
WIM Measurement Errors
Generally speaking, measurement errors can stem from the fol-
lowing sources: measurement system, inspector, the inspected ob-
jects, or data processing �Humplick 1992�. The focus in this study
is specifically on the first source. Three major factors should be
taken into consideration regarding the accuracy of a WIM scale:
�1� roadway factors, among which pavement smoothness and lon-
gitudinal and transverse profiles play the central role; �2� vehicu-
lar factors, which involve speed, acceleration, tire condition, load,
and body type; and �3� environmental factors, including wind,
water, and temperature �Lee 1998�. In other words, WIM mea-
surement errors result from the joint effect of all these relevant
factors.
A series of studies have been carried out to quantify WIM
measurement error. In those types of studies, test vehicles are
used to obtain their static �or reference� and in-motion weights. A
minimum of three repeated weighings are required to obtain the
static weight, which is calculated as the arithmetic mean of the
three measurements. In addition, a number of in-motion weights
are produced by the underlying WIM scale. The detailed process
of determining the measurement errors is referred to
ASTM E 1318-02 �2002� and Bergan et al. �1997�. Mathemati-
cally, the measurement error of a WIM system can be expressed
in terms of percentage difference �relative error� �Davies and
Sommerville 1987; Izadmehr and Lee 1987b; Bergan et al. 1997�
as
��%� =
WIM weight − static weight
static weight
� 100 �1�
where WIM weight�weight recorded by WIM on one pass of a
given axle load; static weight�axle load weighed by a static
scale.
The measurement error, �, is comprised of two independent
components according to the nature of the error occurrences: ran-
dom and systematic. The random error is described as the statis-
tical fluctuations of measurement �in either direction� from the
truth, and they are intrinsic to the measurement due to the inabil-
ity of the device to determine the truth precisely. On the other
hand, systematic errors persistently generate the inaccuracies
along one direction, which may be due to factors such as faulty
design or inadequate calibration.
Provided the WIM system is properly calibrated and installed
in a sound road structure and subjected to normal traffic and
environmental conditions, only the random errors occur. Typi-
cally, random errors exhibit a normal distribution with zero mean
�Davies and Sommerville 1987; Izadmehr and Lee 1987b; Bergan
et al. 1997�. The standard deviation of the underlying normal
distribution ���� is a measure indicating the WIM accuracy or
precision �Bergan et al. 1997�. In this paper, �� is defined and
used as the WIM accuracy indicator. Fig. 1 illustrates the distri-
2 / JOURNAL OF TRANSPORTATION ENGINEERING © ASCE / JANUARY 2007
butions of random error for weighing GVW by three typical types
of WIM equipment: single load cell ���=1.5% �, bending strain
���=5% �, and piezo ���=10% �.
The systematic error arises from the calibration bias. The cali-
bration bias may be due to the initial improper calibration or the
WIM system falling out of calibration during service. A depiction
of WIM systematic error is presented in Fig. 2. It is shown that
the shift of the random error distribution �in this case ��=5% is
fixed� leads to the systematic error. In the example of −10% bias,
the WIM system is undercalibrated, whereas +10% bias is an
example of overcalibration. In the case of ideal calibration, only
random error occurs. In addition, it is implied that when calibra-
tion bias occurs, both random and systematic errors exist.
WIM Specifications
Due to the existence of varying WIM accuracy levels, a review of
WIM specifications with focus on accuracy is necessary. ASTM
has issued the most recent specifications on highway WIM sys-
tems in ASTM E1318-02, where four types of WIM systems—
Type I, Type II, Type III, and Type IV—are presented to meet the
user’s need based on different applications �ASTM 2002�. Types I
and II are used in the data collection procedure, while Types III
and IV are for the purpose of enforcement. Type IV is still in the
conceptual development process. Past research suggested that the
accuracy of WIM is not adequate enough to be employed as a
legal basis to write a weight-violation ticket, although recent de-
velopments may change this situation �Belfield and Souny-Slitine
1999�. Currently, concerning the ASTM specification for WIM
functional performance, the tolerance for 95% probability of con-
formity in terms of single axle load are ±20, ±30, and ±15% for
Types I, II, and III WIM systems, respectively. The corresponding
values in terms of the axle group load are ±15, ±20, and ±10%;
and ±10, ±15, and ±6% for the GVW. However, the accuracy that
these tolerances can provide for pavement design remains in
question, which is addressed in this paper together with some
recommendations.
Data Set
Traffic data in this study were collected from the various WIM
systems deployed in Texas. Currently, there are 20 WIM stations
in Texas. This research is based on traffic load from the available
data set from Texas WIM systems. The case study focus on the
data from a randomly drawn WIM Station D516, located on In-
terstate Highway 35 �I-35� in San Antonio. The traffic sample
from this particular WIM station covers the data collection period
from January 1998 to March 2002.
The raw data were collected and stored in binary form on a
daily basis, and converted into ASCII format after extraction by
PAT software REPORTER5 �PAT 1997�. These data were stored
in a project database for further analysis. Each passing vehicle is
represented by a record containing the vehicle’s detailed traffic
information. Overall traffic volume can be obtained by querying
the amount of records. Traffic classification information can be
found directly from the corresponding database field. Axle load
distribution can be obtained from the load counts for the indi-
vidual load ranges of each axle set on each traffic class.
Traffic Classification
Each, vehicle is categorized into one of 13 classes according to
the axle number and configuration, and Class 4 to Class 13 are
defined as trucks �Traffic Monitoring Guide, FHwA 2001�. For
this study, the PAT classificationsystem is preferred in order to
avoid conversion into FHwA’s 13-class scheme, thus minimizing
error sources. The PAT WIM system used in Texas produces a
15-class-based classification scheme, with truck classes ranging
from Class 4 to Class 15. In addition, Class 15 includes the ve-
hicles not identified by any of the previous traffic classes and
records due to system errors. Fig. 3 presents the traffic classifica-
tion scheme employed in the current study.
Axle Load Distribution
Based on axle spacing and configurations, truck axles are divided
into five groups: steering axle, single axle with dual wheels �re-
ferred to hereafter as single axle�, tandem axle, tridem axle, and
quadruple �quads� axle. Tridem axles exist on Class 7 and Class
11 trucks. Quads exist on Class 11 trucks only. With the indi-
vidual axle load magnitude obtained from each WIM station, it is
easy to obtain the number of loads �frequency� falling in each bin
�discrete load weight interval� for each axle type on a given truck
class. The counts for each axle type are normalized in terms of
percentages �normalized frequencies� to obtain the discrete axle
load distribution. Axle load distribution is also referred to as axle
load spectrum. After examining all axle load distributions for all
truck classes across the 20 WIM stations, it was found that axle
load spectra feature multimodal patterns, with the number of pro-
nounced peaks ranging from one to two. The axle load spectra for
single and tandem axles of Class 10 �18-wheel truck, Class 9 in
FHwA’s 13-class scheme�, is illustrated in Figs. 4 and 5. The
Fig. 1. Effect of WIM system accuracy �random error� on
measurement error distribution
Fig. 2. Effect of WIM system biases �systematic error� on
measurement error distribution
JOURNAL OF TRANSPORTATION ENGINEERING © ASCE / JANUARY 2007 / 3
histograms represent the axle load spectra directly from observa-
tion counts. The lines in Figs. 4 and 5 represent the fitted distri-
bution functions, which are now discussed.
The characteristics of multimodal distribution of axle load
spectra lead to the adoption of mixed lognormal distribution to fit
the data. The motivations to apply mixed-lognormal distribution
are as follows �Prozzi and Hong 2005�:
1. The nature of axle load being positive is captured by the fact
that a lognormal random variable is nonnegative. Hence, the
sum of the linear combination of lognormal distributions as-
signed with positive weights is guaranteed to be nonnegative;
2. The individual peaks of a load distribution can easily be
captured by lognormal distributions through meaningful pa-
rameters;
3. More important, as shown in the following discussion, the
load-pavement impact can be described succinctly and effec-
tively through the moment statistics of lognormal
distribution.
Mixed Lognormal Distribution Function
Assume a random variable X has a lognormal distribution
X � ln��,��, X � 0 �2�
The probability density function �pdf� is
f�x;�,�� =
1
�2�x�
exp�− 1
2
� ln�x� − �
�
�2� �3�
where � and ��parameters for lognormal distribution.
Fig. 3. PAT vehicle classification scheme used in Texas
Fig. 4. A representative of single axle load spectrum Fig. 5. A representative of tandem axle load spectrum
4 / JOURNAL OF TRANSPORTATION ENGINEERING © ASCE / JANUARY 2007
Thus, the pdf of the mixed lognormal distribution representing
a multimodal load spectrum is given by
f�x;�k,�k,Wk� = �
k=1
K
Wk
�2�x�k
exp�− 1
2
� ln�x� − �k
�k
�2� �4�
where �k and �k�parameters for each lognormal distribution; k�
kth piece of lognormal distribution, referred to as mode or peak;
Wk�weight of the kth mode, and �k=1
K Wk=1.
Parameter Estimation
The first step is to determine the number of modes, K, of lognor-
mal distributions to fit the individual axle load spectrum. For
instance, with regard to load spectrum with pronounced bimodal
patterns, two lognormal distributions are the minimum require-
ment for capturing the peaks. If the error by fit function is
sufficiently small �e.g., R2�99%� with two mixed lognormal dis-
tributions, two distributions are adopted. In other cases, it is found
that although the two mixed lognormal distributions can capture
the peaks, the central segment �between the two peaks� is not well
fitted and may result in significant errors. In such cases, a third
lognormal distribution is added as the transition between the two
pronounced peaks. The reason for imposing a high fit precision
requirement is that load-pavement impact is sensitive to the fit
error. For parameter estimation, the nonlinear least square
�NLLS� technique is applied to fit the discrete distribution. Con-
straints are imposed on the weights with Wk�0 and �k=1
K Wk=1.
For example, for a fitting function incorporating three lognormal
components, its eight parameters can be obtained simultaneously
via NLLS.
As representatives, the estimated parameters and data fit sta-
tistics for truck tandem classes with the four largest sample sizes
are shown in Table 1. The four truck classes are 10, 6, 4, and 9 in
descending order of sample size. The parameters for the load
spectrum based on tandem load data for all truck classes are pre-
sented in the last row of Table 1. The cells without data mean the
third peak is not necessary. It is shown that the underlying func-
tions fit the data well, with all R2 larger than 99%. As examples,
the fitted results are illustrated by the solid lines in Figs. 4 and 5.
To date, studies on fitting load distribution have focused on
data fit exclusively, such as the work by Mohammadi and Shah
�1992�, and Timm et al. �2005�. However, the performance fit
error in terms of load-pavement impact estimation is of more
interest from the perspective of pavement design and rehabilita-
tion. As shown in the next section, load-pavement impact can be
estimated through the moment statistic of axle load distribution.
In this regard, the error is obtained as the relative difference be-
tween the fourth moment from fitted lognormal distribution and
that from the discrete load distribution �estimation minus obser-
vation�. The corresponding results are presented in the last col-
umn in Table 1. It is found that the errors are acceptable for
pavement design purposes. For instance, the error for all truck
tandems by fitted function is 2.19%.
In summary, axle load spectra can be described effectively by
mixed lognormal distributions. Not only is good data fit obtained
but, most important, it is also determined that fitted functions and
their parameters have physical meaning.
Methodology
Impact of Axle Load on Pavement
It was established through the analysis of the American Associa-
tion of State Highway Officials �AASHO� Road Test results that
the impact of each individual axle load on flexible pavement in
terms of serviceability loss can be estimated according to the
fourth power law �AASHTO l993; Huang 2003�. The fourth
power law implies that pavement damage caused by passing ve-
hicles increases exponentially with the increase of their axle load.
This relationship was captured by the load equivalence factor
�LEF� as follows:
LEF = � xr
Ls
�m �5�
where xr�weight of axle load in the rth bin; Ls�load weight on a
standard axle with the same number of axles as xr, usually 18 kip
for the single axle and it is dependent on pavement structure for
the tandem axle, approximately 34 kip; the subscript s represents
“standard,” and m�power denoting the relative damage to the
pavement of a given load xr, typically around 4.
As a result, the load-pavement impact based on a given axle
load spectrum of truck Class j can be obtained by summing the
contributions from all the loads xr’s in the distribution, denoted as
load spectra factor �LSF�, LSFj �under the condition of power
m=4�
LSFj = �
r=1
R � xr
Ls
�4qr,j �6�
where R�total number of load bins and qr,j�normalized fre-
quency of load in the rth bin of a given load spectrum of truck
Class j.
It can be seen that LSFj is the fourth sample moment statistic
�DeGrootand Schervish 2002� divided by Ls
4. As the continuous
distribution function of each axle load spectrum is available, it is
more convenient and equally valid to address the axle load-
pavement impact by employing the population moment from the
pdf. LSF can be thought of as the number of ESALs of one
representative axle in the given spectrum.
Table 1. Data Fit Parameters for Tandem Axles for Trucks
Class
Mixed lognormal distribution parameters
R2
Performance
fit errorW1 W2 W3 �1 �2 �3 �1 �2 �3
4 0.429 0.571 —a 3.192 3.386 — 0.160 0.083 — 0.997 3.12%
6 0.147 0.565 0.288 2.231 2.462 3.404 0.090 0.347 0.173 0.992 1.90%
9 0.269 0.731 — 2.486 2.841 — 0.164 0.350 — 0.998 −4.33%
10 0.424 0.292 0.285 2.733 3.264 3.488 0.325 0.189 0.065 0.995 3.11%
ALL 0.433 0.296 0.270 2.714 3.265 3.487 0.335 0.189 0.065 0.995 2.19%
aParameters not available because only bimodal distribution is applied such as Classes 4 and 9.
JOURNAL OF TRANSPORTATION ENGINEERING © ASCE / JANUARY 2007 / 5
The fourth moment of load spectrum function f�x�, M4, is
defined as
M4 = E�X4� =� x4f�x�dx �7�
Integrating Eq. �7� and the axle load spectra functions, as
shown in Eq. �4�, i.e., summing the contribution from all of the
axle loads according to their distribution, Eq. �6� is equivalent to
LSFj =� � xLs�
4
f j�x�dx
or
LSFj =
� x4f j�x�dx
Ls
4 =
E�X4�
Ls
4 =
M4
C
�8�
where x�axle load weight �kip�; f j�x��axle load spectrum func-
tion of one axle type on truck Class j; Ls
4�constant, C.
In summary, on the basis of the axle load spectrum function,
f�x�, as given in Eq. �4�, axle load-pavement impact can be
equivalently estimated by applying Eq. �8� in terms of the mo-
ment statistic. Thus, what remains to determine the underlying
estimation is the fourth moment of load spectrum function f�x�.
Moment for the Lognormal Distribution
As is shown in Eq. �8�, the fourth moment of axle load spectrum
function is the statistic governing the estimation of load-pavement
impact. The fourth moment for a random variable X with lognor-
mal distribution can be derived as
M4 = E�X4� =� x4 1�2�x� exp�− 12� ln�x� − �� �
2�dx
= exp�4� + 8�2� �9�
where � and � are as in Eq. �3� or �4�.
Thus, with load spectrum fitted by K mixed lognormal distri-
butions, the load-pavement impact of an axle type on truck class
j, LSFj is obtained as
LSFj = �
k
Wk exp�4�k,j + 8�k,j
2 �/Ls
4 �10�
where Wk, �k,j, and �k,j, are as in Eq. �4�, for truck class j. It
should be noted, however, that while the fourth moment captures
the load-pavement impact in terms of serviceability loss or crack-
ing damage, other distress types such as rutting may not be cap-
tured by this specific moment �Carpenter 1992; Archilla and
Madanat 2000�. It has been argued in several studies that the
power may range from around 1 to 4, and not necessary integer
�e.g., Christison 1986; Pont et al. 2002�. In those cases, lognormal
distribution demonstrates its full advantage over other distribu-
tions since its moment statistics do exist and can be easily ob-
tained as in Eq. �9�. However, other competing distributions are
inappropriate in this regard. For example, if normal distribution is
adopted as load spectrum function, under the condition of power
equal to 3.5, it can be shown that its moment does not exist. The
procedure to calculate the LSF under different power values via
moment approach is identical to that presented earlier.
Load-Pavement Impact Estimation under Measurement
Errors
Two scenarios are evaluated to study the effect of WIM measure-
ment errors on load-pavement impact estimation. First, load-
pavement impact estimation is derived under the condition of
ideal calibration �with zero calibration bias and involving random
error only� of a WIM scale. The axle load spectra with mixed
lognormal distributions aforementioned are used as the reference
�without measurement errors�. A comparison is made between the
estimated load-pavement impact with random measurement error
�normal distribution, �� not equal to zero� and that from the ref-
erence. The second scenario investigates the load-pavement im-
pact estimation under the presence of bias resulting from lack of
WIM calibration. In such a case, not only systematic error but
also random error is involved as the latter is unavoidable.
Scenario I
As mentioned previously, under general conditions, the WIM
scale measurement errors exhibit a normal distribution. Assuming
that for, a given axle load �of certain axle type and specific truck
class�, X=x, the observation by a WIM scale with random errors
is a random variable, denoted as, X�, then
�1 =
X� − X
X
� N�0,��
2� �11�
where �1�axle load relative error under Scenario I; the subscript
1 represents Scenario I; and ���indicator of WIM accuracy.
Hence, the variable X� conditional on axle load x also has a
normal distribution
X�	X = x � N�x,x2��
2� �12�
The estimated load-pavement impact by observed load X� con-
ditional on X=x, denoted as LSFX�	X=x, is
LSFX�	X=x =
� �x��4gX�	X�x�	x�dx�
Ls
4 =
E�X�	X = x�4
Ls
4 �13�
where gX�	X�x� 	x��pdf of load observation X� conditional on
X=x, see Eq. �12�; E�X� 	X=x�4�the fourth moment of x� condi-
tional on x; and Ls, see Eq. �10�.
It is shown in Eq. �13� that the moments for the normal distri-
bution are required for the solution, which can be derived on the
basis of moment-generating function �DeGroot and Schervish
2002� of random variable X�
	�t� = EX�„exp�tx��… =� exp�tx�� 1�2�� exp�− �x� − 
�
2
2�2
�dx�
�14�
where 
 and ��parameters of normal distribution of variable X�,
as in Eq. �12�.
As a result, the fourth population moment becomes
Mnormal
4 = E�X�4� = 3�4 + 6�2
2 + 
4 �15�
where the superscript 4 in Mnormal
4 represents the “fourth power,”
the remainders of the superscript represent the power value nu-
merically, and “normal” in the subscript indicates the moment
corresponds to the normal distribution.
The estimated conditional axle load-pavement impact factor
LSFX�	X=x can be determined by substituting Eq. �15� in Eq. �13�
6 / JOURNAL OF TRANSPORTATION ENGINEERING © ASCE / JANUARY 2007
LSFX�	X=x =
�3��
4 + 6��
2 + 1�x4
Ls
4 �16�
As established previously, the variable of axle load X for each
individual axle type follows a mixed lognormal distribution �see
Eq. �4��. Therefore, the estimated axle load-pavement impact
�with random measure errors� based on load spectrum for a given
axle type can be obtained by integrating the contribution from its
overall axle loads, denoted as LSFj
E
LSFj
E =� � x�4gX�	X�x�	x�fXj �x�dxdx�/Ls4
= �3��
4 + 6��
2 + 1��
k
Wk exp�4�k,j + 8�k,j
2 �/Ls
4 �17�
where superscript “E” represents “estimated;” gX�	X�x� 	x� is same
as in Eq. �13�; fX
j �x� is the axle load �without measurement errors�
distribution function of certain axle type on a truck class j, see
Eq. �4�, and the parameters in the summation, see Eq. �10�.
By comparing Eqs. �17� and �10�, it is indicated that
under random measurement error, an additive portion of
�3��
4+6��
2��kWk exp�4�k,j +8�k,j
2 � /Ls
4 is introduced into the load-
pavement impact estimation. Additionally, the always-positive
value in the additive term implies that the random measurement
error results in overestimation of the load-pavement impact. The
extent of overestimation depends on the magnitude of the WIM
accuracy indicator, ��.
Scenario II
When a WIM scale is not properly calibrated �biased�, the axle
load measurement is subject to systematic error and the measured
weight differs from the actual value. In such cases, both system-
atic error and random error should be considered. Assuming that,
for a given actual axle load �of certain axle type on a truck class�,
X=x, the observation by a WIM scale with both errors is also a
random variable, denoted as, X�, then
Fig. 6. Load-pavement impact estimation error versus WIM accuracy
indicator
Fig. 7. Three-dimensional �3D� representation of the load-pavement estimation error under both WIM random and systematic errors
JOURNAL OF TRANSPORTATION ENGINEERING © ASCE /JANUARY 2007 / 7
�2 =
X� − X
X
� N��,��
2� �18�
where �2 is axle load relative error under Scenario II; subscript 2
represents Scenario II; � is the calibration bias, which is 0 if
ideally calibrated; and �� is as in Eq. �11�.
The variable X� conditional on axle load X=x has a normal
distribution under biased WIM calibration condition
X�	X = x � N��1 + ��x,x2��
2� �19�
Hence, the axle load-pavement impact by a given axle type on
certain truck class under the biased calibration condition is esti-
mated using a similar approach as presented in Scenario I, de-
noted as LSFj
E�b�
LSFj
E�b� =� � x�4gX�	X�x�	x�fXj �x�dxdx�/Ls4 = �3��4 + 6��2�1 + ��2
+ �1 + ��4��
k
Wkexp�4�k,j + 8�k,j
2 �
 Ls4 �20�
where the superscript “E�b�” represents “estimated under calibra-
tion bias;” gX�	X�x� 	x� is conditional distribution of observation X,
see Eq. �19�; and the other variables, see Eq. �17�.
The result in Eq. �20� suggests that, similar to Scenario I, the
estimated load-pavement impact under the biased calibration situ-
ation is also comprised of two components: �1� a multiplicative
term �3��
4+6��
2�1+��2+ �1+��4�, including both systematic and
random errors, and �2� the load-pavement impact obtained with
no measurement errors occurring �as before�. Provided the coef-
ficient term is greater than one, it implies an overestimation of
load-pavement impact; whereas if the term is less than one, the
load-pavement impact is underestimated. Furthermore, the extent
of over- or underestimation is determined by both the magnitude
of WIM accuracy indicator, ��, and calibration bias, �.
To better understand how WIM measurement error affects
load-pavement impact estimation, a numerical illustration is pre-
sented based on the findings just discussed. With respect to WIM
measurement under only random error, a range of the WIM accu-
racy indicator, ��, from 0 to 20% is evaluated to address how
these random errors affect the estimation of pavement perfor-
mance. Fig. 6 shows the relationship between varying WIM ac-
curacy indicator �� �representing random errors� and relative er-
rors of load-pavement impact estimation. It can be seen that the
load-pavement impact estimation error is lower than the WIM
accuracy indicator, ��, for �� below 15%. Above this, the load-
pavement impact estimation errors increase.
When calibration bias occurs, both random and systematic er-
rors should be addressed. Load-pavement impact estimation error
shows a significant variation. Figs. 7 �in three dimensions �3D��
and 8 �in two dimensions �2D�� illustrate the effect of a series of
combinations of WIM accuracy indicator �� and calibration bias
� on load-pavement impact estimation. It is shown that both over-
estimation and underestimation occur. It is also indicated that
load-pavement impact estimation error is more sensitive to the
calibration bias than to WIM accuracy indicator ��, since the
estimation error covers a larger range along systematic errors than
that of random errors.
Next, the sensitivity of load-pavement impact estimation to
calibration bias is examined. A typical WIM scale with accuracy
indicator ��=8% is used to this purpose as shown in Fig. 9. It was
found that 10% overcalibration resulted in 51% overestimation of
load-pavement impact, which is even more significant than previ-
ously reported in FHWA-RD-98-04 �1998�; whereas the 10% un-
dercalibration produced results similar to those in the 1998 report,
approximately 31% underestimation of load-pavement impact.
Conclusions
This research study analyzes the effects of WIM measurement
errors on load-pavement impact estimation. Two types of errors,
random error �represented by the accuracy indicator ��� and sys-
tematic error �represented by the calibration bias �� were identi-
Fig. 8. Two-dimensional �2D� representation of the load-pavement estimation error under both WIM random and systematic errors
8 / JOURNAL OF TRANSPORTATION ENGINEERING © ASCE / JANUARY 2007
fied and evaluated for WIM systems. Axle load spectra for differ-
ent truck classes were established to address the traffic load
characteristics effectively. A methodology based on moment sta-
tistics was developed to represent load-pavement impact under
different WIM measurement error conditions. Traffic data from
WIM systems in Texas were used to demonstrate the varying
load-pavement impact evaluation biases.
To realize the objective of this study, axle load spectra func-
tions were established at first with considerations, not only fo-
cused on statistical data fit but also on load-pavement impact
implications. These functions are mixed-lognormal distributions.
It was found that axle load-pavement impact given a load spec-
trum could be expressed in terms of the moment statistic of log-
normal distributions.
WIM random measurement error was first introduced in Sce-
nario I to analyze its effect on load-pavement impact estimation.
Two moments jointly addressing the random error variable
�normal distribution� and axle load variable �mixed lognormal
distribution� were integrated to represent the estimation of load-
pavement impact. It was found that WIM measurement random
error always leads to overestimation of load-pavement impact.
In Scenario II, both WIM measurement random error and sys-
tematic error were included �representing biased calibration of a
WIM scale�. Calibration bias was accounted for by shifting the
normal distribution of the random error. It was found that com-
pared with the case of random error only, WIM systematic error
contributed more significantly to the error of load-pavement im-
pact estimation. For a WIM scale with a typical accuracy indica-
tor, 10% overcalibration results in as much as 51% overestimation
of load-pavement impact, while it is 31% underestimated by 10%
undercalibration. The results suggest that load-pavement impact
estimation error is more sensitive to overcalibration than under-
calibration of a WIM scale. More important, the findings suggest
that WIM calibration is worthy of more attention for collecting
traffic data for pavement design and rehabilitation.
The results of this study can be applied to evaluate over- or
underdesign of pavement in terms of traffic data provided by
WIM. In addition, the results could be used to aid with selection
of the appropriate WIM system in terms of accuracy and potential
biases from the perspective of pavement design and rehabilitation.
Thus, it could also be used to select the WIM equipment accord-
ing to the requirement of pavement design reliability.
Acknowledgments
The writers wish to thank the Texas Dept. of Transportation
�TxDOT� for providing the funds and support for this research
through TxDOT Project No. 0-4510. Joe Leidy and Rich Rogers
from the Construction and Materials Division are particularly ac-
knowledged as well as Dr. German Claros from TxDOT Research
and Implementation Office.
References
American Association of State Highway and Transportation Officials
�AASHTO�. �1993�. AASHTO guide for design of pavement struc-
tures, Washington, D.C.
American Society for Testing and Materials �ASTM�. �2002�. “Standard
specification for highway weigh-in-motion �WIM� systems with user
requirements and test method.” ASTM E 1318-02, Philadelphia.
Archilla, A. R., and Madanat, S. �2000�. “Development of a pavement
rutting model from experimental data.” J. Transp. Eng., 126�4�, 291–
299.
Belfield, K. M., and Souny-Slitine, N. �1999�. “Truck weight limit en-
forcement technology applicable to NAFTA traffic along the Texas-
Mexico border.” Research Rep. No. 167209-1. Center for Transporta-
tion Research, The Univ. of Texas at Austin, Austin, Tex.
Bergan, A. T., Berthelot, C. F., and Taylor, B. �1997�. “Effect of weigh-
in-motion accuracy on weight enforcement efficiency.” International
Road Dynamics, Technical Paper, �http://www.irdinc.com/english/
html/tech-ppr/index.htm� �June 1, 2005�.
Carpenter, S. H. �1992�. “Load equivalency factors and rutting rates for
the AASHO road test.” TransportationResearch Record. 1354, Trans-
portation Research Board, Washington, D.C.
Christison, J. T. �1986�. Pavement responses to heavy vehicle test
program—Part 2: Load equivalency factors, vehicle weights and di-
mensions study, Vol. 9, Roads and Transportation Association of
Canada, Ottawa.
Davies, P., and Sommerville, F. �1987�. “Calibration and accuracy testing
of weigh-in-motion systems.” Transportation Research Record. 1123,
Transportation Research Board, Washington, D.C., 122–126.
DeGroot, M. H., and Schervish, M. J. �2002�. Probability and statistics,
3rd Ed., Addison-Wesley, Boston.
Federal Highway Administration �FHwA�. �1998�. “WIM scale calibra-
tion: A vital activity for LTPP sites.” Techbrief No. FHWA-RD-98-
104, U.S. Dept. of Transportation, Washington, D.C.
Federal Highway Administration �FHwA�. �2001�. Traffic monitoring
guide, U.S. Dept. of Transportation, Washington, D.C.
Huang, Y. H. �2003�. Pavement analysis and design, Prentice Hall, Engle-
wood Cliffs, N.J.
Humplick, F. �1992�. “Highway pavement distress evaluation: Modeling
measurement error.” Transp. Res., Part B: Methodol., 26B�2�, 135–
154.
Izadmehr, B., and Lee, C. E. �1987a�. “On-site calibration of weigh-in-
motion systems.” Transportation Research Record. 1123, Transporta-
tion Research Board, Washington, D.C., 136–144.
Izadmehr, B., and Lee, C. E. �1987b�. “Accuracy and tolerances of
weigh-in-motion systems.” Transportation Research Record. 1123,
Transportation Research Board, Washington, D.C., 127–l35.
Lee, C. E. �1998�. “Factors that affect the accuracy of WIM systems.”
Proc., 3rd National Conference on Weigh-in-Motion, St. Paul, Minn.
Middleton, D., and Crawford, J. A. �2001�. “Evaluation of
TxDOT’s traffic data collection and load forecasting process.” FHWA/
TX-01/1801-1, Texas Dept. of Transportation, Austin, Tex.
Mohammadi, J., and Shah, N. �1992�. “Statistical evaluation of truck
overloads.” J. Transp. Eng., 118�5�, 651–665.
National Cooperative Highway Research Program. �2005�. “Mechanistic-
empirical design of new and rehabilitated pavement structures.”
Fig. 9. Sensitivity of performance estimation on calibration bias
���=8% �
JOURNAL OF TRANSPORTATION ENGINEERING © ASCE / JANUARY 2007 / 9
Research Project No. 1-37A, Washington, D.C.,

http://www.trb.org/mepdg/� �Aug. 8, 2005�.
PAT Traffic Control Corporation, Inc. �1997�. REPORTER, data evalua-
tion software (program description and user’s manual), Chambers-
burg, PA.
Pont, D. J., Steven, B., and Alabaster, D. �2002�. “The effect of mass
limit changes on thin-surface pavement performance.” Proc., 7th Int.
Symp. on Heavy Vehicle Weights and Dimensions, Delft, The Nether-
lands.
Prozzi, J. A., and Hong, F. �2005�. “Optimum statistical characterization
of weigh-in-motion data based on pavement impact.” Proc., 85th An-
nual Meeting of the Transportation Research Board, Washington,
D.C. �CD-Rom�.
Timm, D., Tisdale, S. M., and Turochy, R. E. �2005�. “Axle load spectra
characterization by mixed distribution modeling.” J. Transp. Eng.
131�2�, 83–88.
10 / JOURNAL OF TRANSPORTATION ENGINEERING © ASCE / JANUARY 2007