Empirical-Mechanistic Model for Estimating Pavement Roughness

Prozzi and Madanat 1 EMPIRICAL-MECHANISTIC MODEL FOR ESTIMATING PAVEMENT ROUGHNESS by Jorge A. Prozzi1 and Samer M. Madanat2 1 Assistant Professor, Department of Civil Engineering, The University of Texas at Austin, ECJ 6.112, Austin, TX 78712, Phone: (512) 232-3488, Fax: (512) 475-8744, email: [email protected] (corresponding author) 2 Professor, Civil and Environmental Engineering, University of California at Berkeley, 114 McLaughlin Hall, Berkeley, CA 94720, Phone: (510) 643-1084, Fax: (510) 642-1246, email: [email protected] Total Number of Words: 7,434 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat 2 ABSTRACT The objective of this paper is to demonstrate the development of pavement performance models by combining multiple data sources. The form of the model is formulated mechanistically and the estimation of the parameters is done empirically following a two step approach. In the first step a riding quality model based on serviceability is developed using the data of the AASHO Road Test. Due to the experimental nature of this data set, some of the estimated parameters of the model may be biased when the model is to be applied to predict performance in the field. Hence, in the second step, the original model is updated by applying joint estimation with the incorporation of a field data set. This data set was collected through the Minnesota Road Research Project (MnRoad). The updated model, referred to as the joint model, can be used to predict the roughness of in service pavement sections. Joint estimation allowed for the full potential of both data sources to be exploited. First, the effect of variables not available in the first data source were identified and quantified. In addition, the parameter estimates had lower variance because multiple data sources were pooled, and biases in the parameters of the experimental model were estimated and corrected. The development and application of the methodology demonstrated that different indicators of riding quality can be incorporated into the model by using a measurement error model. TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat 3 INTRODUCTION The accurate prediction of pavement performance is important for efficient management of the transportation infrastructure. By reducing the prediction error of pavement deterioration, agencies can obtain significant budget savings through timely intervention and accurate planning (1). Pavement deterioration models are not only important for highway agencies to manage their road network, but also in road pricing and regulation studies. Both the deterioration of the pavement over time and the relative contribution of the various factors to deterioration are important inputs into such studies. Useful models should be able to quantify the contribution to pavement deterioration of the most relevant variables. The objective of this paper is to demonstrate the development of pavement performance models by combining experimental and field data. The approach followed in this paper combines data of in-service pavements with data from experimental studies, by using joint estimation, a statistical method that was designed specifically for the purpose of parameter estimation with multiple data sets. Joint estimation is particularly well suited to problems in which different data sources have different levels of precision, or where they are suspect of suffering from one or more types of bias. Data obtained from field studies are likely to be less precise than those obtained from controlled experimental tests, due to correlation among explanatory variables. On the other hand, experimental data are likely to suffer from biases, as they do not represent the true deterioration mechanisms of pavements. The method of joint estimation has been applied successfully to a variety of problems in transportation planning and engineering including the estimation of travel demand models (2). The method of joint estimation has recently been used to develop rutting progression models by combining data from two experimental data sets (3). The present paper focuses on investigating the feasibility and desirability of applying joint estimation with experimental and field data to predict pavement roughness progression of in-service pavement sections. Data from actual in-service pavement sections subjected to the combined actions of highway traffic and environmental conditions are the most representative of the actual deterioration process. All other data sources produce models that are likely to suffer from some kind of bias unless special considerations are taken into account during the estimation of the parameters of the model. However, data from in-service pavements (field data) may suffer from several limitations. The most common problems encountered in models developed from field data are caused by the presence of multi-collinearity between explanatory variables, unobserved events typical of such data sets, and the problem of endogeneity bias caused by the use of endogenous variables as explanatory variables. The problem of multi-collinearity is typical of time-series pavement performance data sets. Variables such as pavement age and accumulated traffic are usually collinear. Hence, the estimated models usually fail to identify the effects of both variables simultaneously. While multi-collinearity does not introduce biases in the model parameter estimates, it lowers confidence in their significance. Experimental data do not usually suffer from this problem, because the application of traffic loading to the pavement sections can be accelerated, thus reducing the correlation with pavement age. Data gathering surveys during experimental tests are usually of limited duration. Thus, if only the events observed during the survey are considered in the statistical analysis (ignoring the information of the after and before events), the resulting models would suffer from truncation bias. If the censoring of the events is not properly accounted for, the model may suffer from censoring bias (4). TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat 4 Another common problem is endogeneity bias. Pavements that are expected to carry higher levels of traffic during their design life are designed to higher strengths. The bearing capacity of these pavements is higher than those designed to withstand lower traffic levels. Thus, any explanatory variable that is an indicator of a higher bearing capacity will be an endogenous variable that is determined within the model and cannot be assumed to be exogenous. If such a variable were incorporated into the model, the estimated parameters would suffer from endogeneity bias (5). The latter two problems can be addressed using statistical techniques that take into account the presence of truncation or endogeneity or, alternatively, by developing models that are based on experimental data. However, experimental data have their own biases, because they do not represent the true deterioration processes of in-service pavements. Joint estimation with both field and experimental data can be used to correct for these biases. Based on the above considerations a two step approach was used in this research. In the first step a riding quality model based on serviceability was developed using the data of the AASHO Road Test (6). By using data originated from a well-conceived experiment many of the potential problems highlighted in the previous section were avoided. Due to the experimental nature of this data set, some of the estimated parameters of the model may be biased when the model is to be applied to predict performance in the field. Hence, in the second step, the riding quality model (or serviceability model) is updated by applying joint estimation with the incorporation of field data. This data set corresponds to the Minnesota Road Research Project (MnRoad). The final model, referred to as the joint model, can be used to predict the performance of in-service pavement sections. THE AASHO ROAD TEST The AASHO Road Test was sponsored by the American Association of State Highways Officials (AASHO) and was conducted from 1958 through 1960 near Ottawa, Illinois (6). The data from this experiment constitutes the most comprehensive and reliable data set available to date. The site was chosen because the soil in the area is representative of soils corresponding to large areas of the Midwestern United States. The climate was also considered to be representative of many states in the northern part of the country. The test tracks consisted of two small loops and four large loops. A total of 142 flexible pavement sections were built into the various loops. Each section covered two lanes, and each lane was subjected to a different traffic configuration, so the total number of test sections was 284. The riding quality of the various sections was monitored in terms of their serviceability by means of the Present Serviceability Index (PSI). Most of the sections on the loop tangents were part of a complete experimental design. The design factors considered were surface thickness, base thickness, and subbase thickness. The materials used for the construction of the AC surface, base, and subbase layers were the same for all sections. Hence, the effect of the material properties on pavement performance cannot be directly assessed from the data of the main experimental design. Original AASHO Model The first pavement performance model developed was based on the data provided by the AASHO Road. The AASHO equation estimates pavement deterioration based on the definition of a dimensionless parameter g referred to as damage. The damage parameter was defined as the loss in PSI at any given time: TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat p − pt  N t = gt = 0 p0 − p f  ρ gt pt p0 pf Nt ρ, ω : : : : : : 5    ω (1) dimensionless damage parameter, serviceability at time t (in PSI units), initial serviceability at time t = 0, terminal serviceability, cumulative number of equivalent 80 kN single axle loads applied until time t, and regression parameters. This deterioration model was estimated based on data obtained from AASHO Road Test. The parameters ρ and ω were obtained for each pavement test section by applying Equation 1 in a step-wise linear regression approach. The statistical approach used to estimate the model parameters has some inconsistencies. The most serious was the improper treatment of censored observations: pavement sections that had not failed by the end of the experiment were ignored in the estimation of the parameters. Moreover, some of the regression equations were mis-specified because variables with different units were added together. Despite the identified shortcomings of the model specification and the estimation approach, Equation 1 (or subsequent modifications of it) has been used as the basis for pavement design for approximately 50 years (7). SPECIFICATION OF THE MODEL WITH EXPERIMENTAL DATA Basic model The data of the AASHO Road Test was used to develop and estimate the experimental pavement deterioration model (serviceability model). This data set was chosen because load and structural variables were selected following an experimental design, thus avoiding the problems of multi-collinearity and endogeneity discussed in the previous section. As stated earlier, during the AASHO Road Test, the deterioration of the pavement riding quality was determined by the change in the Present Serviceability Index (PSI). In the present study, the loss of serviceability was modeled as follows: p = f (N ) = a + b N c p N a b c : : : : : (2) dependent variable representing pavement serviceability, independent variable representing some measure of traffic, parameter or function that represents the initial serviceability, parameter that represents the rate of change of serviceability, and parameter or function that represents the curvature of the function. For a given pavement structure, pavement serviceability decreases as traffic increases. This condition is represented by the sign of the parameter b. Furthermore, for a given traffic level, pavement serviceability decreases more rapidly for weaker pavements. This is represented by the absolute value of the parameter or function b. From a pavement management perspective, an incremental form of Equation 2 is more beneficial since condition data are usually available on a regular basis and predictions are only desired for the next few time periods. By using a first order Taylor series approximation, Equation 1 can be expressed in an incremental recursive fashion: TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat 6 p t = pt −1 + d N te−1 ∆N t : : : : pt Nt-1 ∆Nt d, e (3) serviceability in PSI at time t, cumulative equivalent traffic up to time t-1, equivalent traffic increment from time t-1 to time t, and parameters or functions to be estimated. By applying the recursive Equation 3 from the beginning of the life of the pavement, the following equation is obtained: t pt = p0 + ∑ d N le−1 ∆N l l =1 : p0 (4) initial serviceability in PSI at time t = 0. Specification for aggregate traffic A generalization of the traditional approach of aggregating all traffic into its equivalent number of standard 18,000 lb (80 kN) single axle loads is used in this research. This number is usually referred to as the number of Equivalent Single Axle Loads (ESALs). All axle load configurations are converted into their equivalent number of ESALs by means of a load equivalence factor (LEF). The most commonly used form for the determination of the LEF is the so-called power law: LEF  L =   18  LEF L η : : : η (5) load equivalence factor, axle load in kips (1,000 lbs), and parameter that is usually assumed to be between 4.0 and 4.2. Different power-laws for the different axle configurations present in the experimental data set were used because local and foreign research indicates that different standard loads (denominator of the power-law) are necessary to transform different axle configurations into ESALs. The approach proposed in the present research takes these considerations into account by the Equivalent Damage Factor (EDF) concept. The equivalent damage factor is defined as a number that depends only on the configuration and load characteristics of the truck. When the EDF is multiplied by the number of trucks the equivalent number of standard axles is obtained as follows: EDF  FA =   18 λ1 EDF FA SA TA λ1,λ2,λ3 : : : : :     λ2  SA  + m1    18  λ2  TA + m2   18 λ 3     λ2 (6) equivalent damage factor, load in kips (1,000 lb) of the front axle (single axle with single wheels), load in kips of the single axle with dual wheels, load in kips of the tandem axles with dual wheels, parameters to be estimated, and TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat m1, m2 : 7 number of single and tandem rear axles per truck, respectively The equivalent traffic is obtained by multiplying the equivalent damage factor (EDF) of each truck configuration by the actual number of truck passes during time period t: ∆N t = nt EDF nt ∆Nt : : (7) number of truck passes during period t, and number of ESALs during period t. Finally, the cumulative equivalent traffic (Nt) at time t is obtained by: t N t = ∑ ∆N l l =0 (8) Specification for structural strength The function d in Equation 4 is a decreasing function of the strength of the pavement: for stronger pavement structures, serviceability decreases slower than for weaker pavements. The specification of the function d is based on the concept of thickness index. The thickness index is the weighted sum of the thicknesses of the various layers of the pavement structure (6). In this research, an alternative designation is proposed to differentiate the present specification from the specification developed during the initial analysis of the AASHO Road Test. Thus, the function d is considered to be dependent on the equivalent thickness (ET) according to the following specification: d = ET d 0 = (1 + d1 H 1 + d 2 H 2 + d 3 H 3 ) H1, H2, H3 d0-d3 ET : : : d0 (9) thickness of surface, base and subbase layers, respectively, set of parameters to be estimated, and equivalent thickness. Since the value of the function d decreases as the pavement strength increases, the parameter d0 is expected to be negative. The parameters d1, d2, and d3 in Equation 9 represent the contribution of the asphalt surface, base, and subbase to the total pavement strength. Environmental considerations During the AASHO road test, the most relevant environmental factor was the effect of the freeze-thaw cycles. To account for this effect, an environmental factor was developed that augments or diminishes the structural resistance of the pavement depending on the prevailing environmental conditions. Three distinctive deterioration phases were observed in the pavement sections of the AASHO Road Test as characterized by their loss of serviceability (Figure 1): (i) A normal phase characteristic of the summer and fall periods during which the serviceability decreases at a fairly uniform rate. TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat 8 (ii) A stable phase characteristic of the winter period during which the riding quality of the test sections remained very stable - the serviceability did not decrease significantly. (iii) A critical phase during which the rate of deterioration increased significantly and rapidly compared to the previous two phases. This phase corresponded to the spring months. Furthermore, it was observed that the three phases described above corresponded to the periods of zero frost penetration, increasing depth of frost penetration, and decreasing depth of frost penetration, respectively. Hence, the frost penetration gradient variable was included to capture the effect of environmental conditions on pavement deterioration. The frost penetration gradient in period t, Gt, is defined as the ratio between the change in the depth of frost penetration during period t and the length of period. This is accounted for in the specification by the introduction of an environmental factor (Fe) that multiplies the value of the function d in Equation 4. Although this variable is not often available to the agencies, it can be easily estimated by using the enhanced Integrated Climatic Model developed by FHWA. The expression for Fe is: Fe = exp{g Gt } Gt g : : (10) frost penetration gradient, and parameter to be estimated. Specification for initial serviceability The initial value of serviceability of actual in-service flexible pavement sections does never reach the theoretical value of 5.0 PSI. The initial value (p0 in Equation 4) depends, inter-alias, on the total thickness of the surface layer. As the thickness of the asphalt surface layer increases, it is usually constructed in various sub-layers or lifts. Each lift provides additional support and improved working conditions for the construction equipment, leading to a better riding quality of the finished surface. Thus, it is believed that the initial serviceability could be represented as an increasing function of the asphalt layer thickness as follows: p 0 = u + v exp{w H 1 } u, v, w : H1 : (11) parameters to be estimated, and total thickness of the asphalt surface layer. Final specification of the serviceability model In this section the full specification is given taking into account that time series and cross sectional data are available simultaneously. From Equations 4 and 10, the complete specification becomes: t −1 pit = pi 0 + ∑ d i exp{g Gl }N ile ∆N i ,l +1 l =0 (12) Where pit is the serviceability at any given time, based on the initial serviceability of the section plus the summation of the changes in serviceability from the beginning of the experiment until the period of interest. The first subscript, i, indicates the pavement test section (i = 1, …, S), and S is the total number of pavement test sections. The second subscript, t, indicates the time period (t = 1, …, Ti). TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat 9 For the final formulation all the parameters are renamed as follows: t −1 p it = β 1 + β 2 exp{β 3 H 1i } + ∑ (1 + β 4 H 1i + β 5 H 2i + β 6 H 3i ) β7 exp{β 8 Gl }N ilβ 9 ∆N i ,l +1 l =0 Where N il = ∑ l q =0 (13a) ∆N iq , and ∆Niq represents the traffic increment expressed in the number of ESALs for period q. The number of ESALs is obtained by multiplying the equivalent damage factor of section i (EDFi) by the actual number of truck passes during period q. ∆N iq =   FA  β12 i  niq    β 18  10    niq : m1i, m2i : FAi : SAi : TAi : β1-β12 :  SA  + m1i  i   18  β12  TAi   + m2i    β 11 18  β12     (13b) actual number of truck passes for section i at time period q, number of rear single axles and tandem axles per truck, respectively, load in kips of the front axle (single axle with single wheels), load in kips of the single axle with dual wheels, load in kips of the tandem axles with dual wheels, and set of parameters to be estimated using a non-linear optimization method. PARAMETER ESTIMATION Nonlinear estimation The model described in the previous section is intrinsically nonlinear, or nonlinear in the parameters. In this sense, the term nonlinear refers to the procedure required to estimate the parameters of the specification rather than to the specification form. A general form of the nonlinear regression model can be represented as follows: ( ) y i = h xi , β + ε i yi xi β εi h : : : : : (14) dependent or explained variable, vector of independent or explanatory variables, vector of parameters, and random error term, and a nonlinear function of β. If the assumption is made that the εi in Equation 14 are normally distributed with mean zero and constant variance σ2, then the value of the parameters that minimize the sum of the squared deviations will be the maximum likelihood estimators as well as the nonlinear least squares estimators (8). Unlike linear regression, the first order conditions for least squares estimation are nonlinear functions of the parameters. Panel data The data set corresponding to the AASHO Road Test data set consists of panel data (time series and cross sectional data). Several approaches can be followed to undertake estimation with panel data. If the TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat 10 parameters of the deterioration model are believed to be constant across sections and along time, efficient parameters can be estimated by combining all data into a single regression. Under this assumption, the most popular estimation technique is ordinary least-squares (OLS) estimation. In this case, the intercept term is assumed to be the same for all sections. For a controlled experiment, this assumption is reasonable because it considers that the deterioration of all pavements is the result of the same process and only depends on the variables that are observed. However, unobserved heterogeneity is often present as a result of unobserved section-specific variables. Some of the most commonly techniques used to deal with unobserved heterogeneity are: the dummy variable approach (or fixed effects approach), the error component approach (or random effects), and the random coefficients approach. The former two approaches make the assumption that the unobserved heterogeneity can be captured by means of the intercept term. The latter approach addresses the problem by assuming that one or more of the slope parameters are random rather than constant. The random effects (RE) approach makes the assumption that the intercept term is randomly distributed across cross-sectional units. That is, instead of assuming that there is one intercept term β1i for each section (as the fixed effect approach does), it assumes that β1i = β1 + ui, where ui is a random disturbance which is a characteristic of the section i that remains constant through time. Thus, the regression model becomes: y it = h( β , x it ) + u i + ε it (15) ESTIMATION RESULTS The parameters of the serviceability deterioration model were estimated using OLS and RE approaches. The large sample size grants the use of asymptotic theory. The estimated parameters and the asymptotic t-statistics are given in Table 1. The parameter estimates are significantly different from zero at a five percent level and all the parameters have the expected sign. The estimate of the standard error of the OLS regression is σ̂ ε = 0.248 PSI, which is approximately half of the value of the standard error of the original linear model (Equation 1). It should be emphasized that this improved accuracy was achieved using the same data source and the same number of explanatory variables. This is the result of a better-specified model. Table 1 illustrates the difference in the estimates obtained between the OLS approach and the RE approach. The estimates of the variance of the error components for the random effect approach were 0.142 and 0.126 for the overall error (εit) and the section specific error (ui), respectively. Both values are of the same order of magnitude, indicating that heterogeneity should not be ignored. The parameters for the determination of the equivalent layer thickness (β4, β5, and β6) are different from the parameters that were developed during the original analysis of the AASHO Road Test for the determination of the thickness index. The relative values, however, are comparable. For instance, in the new model the ratios β4/β5 and β5/β6 are approximately 4 and 1.2, respectively. The equivalent ratios obtained from the original model are 3 and 1.3, respectively. The equivalent thickness is important in the specification because it dictates the rate at which deterioration (in terms of serviceability loss) progresses. This is illustrated graphically in Figure 2. As TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat 11 expected, the rate of deterioration decreases as the strength of the pavement increases. The rate of serviceability loss depends also on the cumulative traffic. The rate of deterioration decreases with cumulative traffic. This is represented by the sign of parameter β9 in the specification, which is negative. Other parameters that deserve special attention are the parameters corresponding to the aggregate traffic specification. That is, β10, β11, and β12. These parameters facilitate the estimation of the equivalent damage factors (EDFs) and the determination of the equivalent axle loads for different axle configurations. The axle load corresponding to an EDF of one determines the equivalent axle load for the given configuration. The estimated equivalent load for a single axle with single wheels is approximately 10,000 lbs., while the equivalent load for a tandem axle with dual wheels is about 33,000 lbs. These values are obtained by multiplying the parameters β10 and β11 by the standard axle load (18,000 lbs). MINNESOTA ROAD RESEARCH PROJECT (MNROAD) To update the initial model by applying joint estimation, a second data source was incorporated. This data source is the Minnesota Road Research Project (MnRoad). The facility is located parallel to Interstate 94 (I-94) in Otsego (Minnesota), - approximately 40 miles northwest of the Minneapolis-St. Paul metropolitan area. The test set up comprises both experimental test sections and in-service pavement sections (field sections). The field data set consists of 3 miles of two-lane interstate (also referred to as the High Volume facility). The experimental data set consists of a 2.5 miles closed-loop test track (also referred to as the Low Volume facility). This facility is subjected to controlled experimental loading consisting of a single vehicle circling the two-lane test track. The inside lane is trafficked four days a week with a legally loaded truck while the outside lane is trafficked only one day a week with a 25 per cent overloaded truck. The interstate portion of the test facility has been divided into two parts, referred to as the 5-Year and the 10-Year Mainline. These interstate sections have been designed for an estimated five- and ten-year design life, respectively. Both the five- and the ten-year mainline sections have PCC and AC test cells. However, only the data corresponding to the flexible pavement cells are used for the estimation of the deterioration models in this research. One of the main advantages of the MnRoad project data set is that it combines both experimental data (Low Volume Road) and field data from in-service pavement sections subjected to actual highway traffic (High Volume Facility). This is perfectly suited to the objective of this paper and can be fully exploited by the application of joint estimation. Another advantage is that the field data consisted of specially built pavement sections, and thus did not suffer from the problem of endogeneity in the explanatory variables. JOINT ESTIMATION METHOD Assuming two different data sources (experimental (E) and field data (F)), the joint estimation approach can be formulated as follows: rE = h(θ , x,θ E , x E ) + ε E (16a) rF = h(θ , x,θ F , x F ) + ε F r E, r F x : : (16b) riding quality from the experiment and the field, respectively, explanatory variables shared by the experimental and field data sources, TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat θ xE θE xF θF ε E, ε F : : : : : : 12 vector of parameters shared by both models, vector of variables unique to the experimental model, vector of parameters corresponding to xE, vector of variables unique to the field model, vector of parameters corresponding to xF, and random error terms for the experimental and field model, respectively. In general, parameter estimation results from the optimization of a particular objective function with respect to that set of parameters. In the case of joint estimation, the objective function is the sum of the objective functions of the individual data sources. This summation is reasonable under the assumption that the error terms of the two data sources are uncorrelated. For the AASHO Road Test data set and the MnRoad Project data the error terms can be safely assumed to be uncorrelated. Some of the advantages of joint estimation are (3): (i) Identification: by incorporating a new field data source, variables that were not observed during the experiment can now be observed in the field and their effect can be incorporated into the specification and estimated from the pooled data. (ii) Bias correction: it may be reasonable to expect that the model estimated with the experimental data set could produce biased parameter estimates for the prediction of the performance of field sections. Joint estimation enables such potential biases to be estimated and corrected. (iii) Efficiency: if the deterioration process described by the set of Equations 16 is believed to be the same for the different data sources, efficient parameter estimation cannot be achieved by estimating the parameters of the equations separately. Only joint estimation with the pooled data would produce efficient parameter estimates. It is reasonable to expect that the specification of the deterioration model based on MnRoad Project data will be different than the one based on the AASHO Road Test data. The reasons for riding quality deterioration, however, remain the same. MEASUREMENT ERROR MODEL The necessary condition for the application of joint estimation is that both models represented by Equations 16 have at least one parameter in common. This condition is satisfied because the AASHO Road Test and the Low Volume Road of the MnRoad Project make use of controlled experimental traffic. The main difference lies in the fact that the layer materials used at AASHO and at MnRoad have different strength characteristics. The common parameters make joint estimation feasible, while uncommon parameters enable the identification of the effect of new variables. A second necessary condition for the applicability of joint estimation is that the observed dependent variable be equivalent. Riding quality observations from the AASHO Road Test and the MnRoad Project are, at first sight, incompatible. During the AASHO Road Test, riding quality was assessed as serviceability (PSI). Riding quality for the MnRoad Project is assessed in terms of roughness by means of the International Roughness Index (IRI). An empirical relationship between IRI and serviceability was developed during the International Road Roughness Experiment conducted in Brazil in 1982 (9). That relationship is: TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat  5.0    p  r = 5.5 ln r p : : 13 (17) roughness in m/km IRI, and serviceability in PSI. This relationship is especially valid for the serviceability observed during the AASHO Road Test, where ninety five percent of the serviceability is explained by the variance of the surface profile (10). The simultaneous estimation of bias in the parameters and the estimation of the measurement error model are not feasible when only two data sets are available. However, by jointly estimating the deterioration model with AASHO and MnRoad data, three different data sets are in fact used. The procedure is as follows: the model is specified in terms of roughness based on the AASHO Road Test. Since roughness was not observed during the AASHO Road Test, the observed serviceability is used as the dependent variable. An error is thus introduced into the model. This error is referred to as the measurement error (11). This measurement error cannot, in general, be determined and produces parameter estimates that are unbiased but not efficient. However, by incorporating a second data source (MnRoad Low Volume facility) and applying joint estimation, the magnitude of the measurement error can be estimated. The following relationship can be established for the AASHO data: y1 = h( X ,θ ) + ε 1 (18) Where y1 is the observed roughness (in m/km IRI) during the AASHO Road Test. Accordingly, for MnRoad data: y 2 = h( X , θ ) + ε 2 (19) Where y2 is the observed roughness at MnRoad. The assumption is made that the error terms ε1 and ε2 are both normally distributed with zero mean (E(ε1) = E(ε2) = E(ε) = 0) and constant variance (σ12 = σ22 = σ2). However, during the AASHO Test y1 (roughness) was not observed but y1* (which represent the calculated roughness as a function of the observed serviceability by using Equation 17), so: y1* = y1 + ε * (20) The error term ε* is also assumed to be normally distributed with zero mean and constant variance (σ*2). The final assumption is that the independent explanatory variables (X) in Equation 18 are uncorrelated with ε*. Under this assumption the final joint model is: y1, 2 = h ( X , θ ) + (ε + ε * ) (21) Under these assumptions, both error terms (ε and ε*) are present when considering the AASHO Road Test data, while only one component (ε) is present when considering the MnRoad project data. TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat 14 Specification of the joint model The joint model specification is based on the specification of the serviceability model described earlier, and the relationship given by Equation 17. However, the joint specification for riding quality is given in terms of roughness rather than serviceability. It should also be noted that the pavement strength is given by the equivalent asphalt thickness (EAT). Six different layers are now considered in the specification. The first three correspond to the surface, base and subbase layers used at the AASHO test, while the last three correspond to the surface, base and subbase layers used at MnRoad. Taking into account these two aspects, the specification for the roughness is given by: t −1 θ rit = θ 1eθ 2 H1i + ∑ θ 3 EATi 9 eθ10Gl N ilθ11 ∆N i ,l +1 l =0 (22a) EATi = 1 + H 1i + θ 4 H 2i + θ 5 H 3i + θ 6 H 4i + θ 7 H 5i + θ 8 H 6 i rit EAT Hj G θj : : : : : Where N il = (22b) roughness (in m/km IRI), equivalent asphalt thickness, layer thickness, frost gradient, and parameters to be estimated. ∑ l q =0 ∆N iq and ∆Niq represents the traffic increment in ESALs for period q. In the cases of AASHO and MnRoad Low Volume facility, the number of ESALs is obtained by multiplying the equivalent damage factor by the actual number of truck passes during period q: ∆N iq =   FA θ14 i  niq    β 18  12    niq : m1i, m2i : FAi : SAi : TAi : θ14  SA  + m1i  i   18   TAi   + m 2i    β 13 18  θ14     (22c) actual number of truck passes for section i at time period q, number of rear single axles and tandem axles per truck for each section, respectively, load in kips of the front axle, load in kips of the rear single axle, and load in kips of the rear tandem axles. Traffic on the High Volume facility consists of actual highway traffic. Only aggregate information in terms of ESAL is available. Therefore, the equivalent traffic is determined by converting the calculated ESALs by means of a multiplicative bias correction factor: ∆N iq = β 15 ∆ESALMiq (22d) Where ∆ESALMiq is the calculated number of ESALs for section i and period q at the MnRoad High Volume Road facility. The estimation of ∆ESALMiq is based on the AASHO approach (7), while the determination of ∆Niq is based on the concept of the equivalent damage factor introduced in this research (Equation 22c). TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat 15 RESULTS OF THE JOINT MODEL The parameters were estimated using the random effects approach, taking into account the measurement error model. The estimated parameters and their asymptotic statistics are given in Table 2. The estimated variances of the two error components are σˆ ε2 = 0.380 (overall error) and σˆ u2 = 0.368 (section specific error). The estimate of the error of the measurement error model is σˆ *2 = 0.793. If the measurement error were ignored, some of the estimated parameters would not be significantly different from zero at the five percent significance level. The estimated standard error of the regression ( σˆ 2 ε ) + σˆ u2 is 0.865 m/km IRI. The improved accuracy of the nonlinear model developed in this research is attributed to an appropriate specification form and the use of adequate estimation techniques. It should be emphasized that both models made use of the same explanatory variables. Observed and predicted deterioration of two different pavement sections are illustrated in Figure 3. It should be noted that the data of the AASHO sections represented in Figure 3 were not used for the estimation of the parameters. Another important aspect of the nonlinear model is its ability to predict the critical phase (step in Figure 3). The critical phase corresponds to the thawing period characteristic of the spring months. Several of the parameter estimates of the joint model (roughness model) have an equivalent counter part in the serviceability model described earlier. It is important to note that the corresponding equivalent parameter of both models have very similar estimated values. For instance, the parameters corresponding to the aggregate traffic specification in the serviceability model are β10, β11 and β12 (.552, 1.85, 4.15), while the corresponding parameters in the roughness model are θ12, θ13 and θ14 (.523, 1.85, 3.85). The largest difference in the estimated values of these three parameters is approximately seven percent. This corresponds to the exponent of the power law. Although the difference seems to be negligible, it may have important implications when determining the design ESALs for a given pavement section. The value 4.15 allocates more weight to the higher traffic axle loads (greater than 18 kips), while the value 3.85 places more weight on the lighter traffic axle loads (smaller than 18 kips). Another important difference relates to the formulation of the equivalent thickness. In the serviceability model, the equivalent thickness (ET) is expressed relative to the subgrade protection against loss in serviceability. This approach is compatible with the traditionally used thickness index developed during the original analysis of the AASHO Road Test. In the roughness model, the equivalent asphalt thickness (EAT) is expressed in terms of the effectiveness of the asphalt layer to protect the pavement against damage due to roughness. Hence, the absolute values of the parameters β4, β5 and β6 (serviceability model) bear no direct relationship to the absolute value of parameters θ4 and θ5. (roughness model). However, their relative values β5/β4 and β6/β4 are 0.237 and 0.195, which compare favorably with the estimated values for θ4 and θ5, respectively. Joint estimation allows the estimation of the layer strength coefficients for materials that were not available during the AASHO Road Test. Three new strength coefficients were estimated (θ6, θ7, and θ8) which correspond to the asphalt surface, base and subbase materials used for the construction of the Mn/Road test sections. In the MnRoad Project, two asphalt binders were used for the surface layer, and four different untreated granular materials for the base and subbase layers (Class 3 to Class 6 according to TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat 16 MnRoad specifications). Due to the lack of data, however, it was decided to group the materials together following current MnRoad practice. The estimated parameters for these three material groups are 1.82, 0.288 and 0.236 (Table 2). According to these estimates, the specific asphalt mixtures used in MnRoad are 82 percent more effective than the asphalt mixture used in the AASHO test in terms of protecting the pavement structure against roughness damage. Accordingly, one inch of base and subbase quality materials is approximately 29 and 24 percent as effective as one inch of the original asphalt mixture. The estimation of a multiplicative bias parameter (θ15) to correct for the ESALs determined at High Volume facility is made possible by the joint estimation technique. This value indicates that the current method to estimate ESALs underestimates traffic. This discrepancy is partially attributed to the fact that the current procedure for the estimation of equivalent traffic is based on the AASHO approach, which does not necessarily applies. According to the estimated model, the rate at which the roughness of a given pavement section increases is a function of the equivalent asphalt thickness (EAT), the gradient of frost penetration (G), and the cumulative traffic (N). This relationship is represented graphically in Figure 4 for three EAT values and three values of G. CONCLUSIONS AND RECOMMENDATION This research has highlighted the benefits of using joint estimation for the development of pavement performance models. A nonlinear serviceability model was developed using the same data set and the same variables as the equivalent existing linear model. The prediction error of the new nonlinear model was, however, half that of the existing model. By halving the prediction error, highway agencies in charge of the management of the road network can obtain significant budget savings by timely intervention and accurate planning. The serviceability model was then updated to estimate riding quality in terms of roughness (IRI). It should be noted that in the estimation, no restrictions were imposed on the parameters, traditionally used values were not assumed. All the parameters of the updated model were jointly estimated with the data from the AASHO Road Test and the MnRoad Project. Joint estimation allows for the full potential of both data sources to be exploited. The main advantages of joint estimation were: (i) (ii) (iii) (iv) The effect of variables not available in the first data source were identified and quantified. The parameter estimates had lower variance because multiple data sources were pooled. Bias in the parameters of the experimental model were identified and corrected. Different indicators of the same property were incorporated by using a measurement error model. Like any other deterioration model, the model developed in this research is only an approximation of the actual physical phenomenon of deterioration. There is a prediction error associated with the model. However, unlike deterministic predictions characteristic of mechanistic approaches, this error can be estimated to assess the confidence of the predictions. Although the prediction capabilities of the developed models are superior to most existing models, a number of limitations have been identified and should be further researched. The two data sources used for the joint estimation are from the States of Illinois and Minnesota. Environmental conditions at these locations are similar, especially in terms of weather and soil conditions. The developed model is thus conditional on such conditions, and might produce biased predictions in regions of markedly different characteristics. A possible approach to overcome this limitation would TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat 17 consist of obtaining another data source (corresponding to the new regions) and updating the models by applying joint estimation once again. The data collected as part of the Long-Term Pavement Performance (LTPP) studies of the Federal Highway Administration could be used for this purpose. By using inservice pavement data, a large number of new variables could be incorporated into the deterioration model, and important potential biases could be determined and corrected. An attempt was made to use the data contained in DataPave3 but a large enough set of reliable data could not be found. The most common problems were those related to the availability of traffic information, and the length of the time series performance in terms of roughness. Finally, these limitations are a characteristic of the specific model. However, this research ultimately aimed at showing the feasibility and advantages of using joint estimation to develop pavement deterioration models rather than the advantages of the model itself. As indicated above, most of these limitations can be overcome by repeatedly applying joint estimation to more data sources. ACKNOWLEDGEMENT Funding for this research was provided by the University of California Transportation Center. The authors thank John Harvey for his assistance and to Benjamin Worel for providing MnRoad data set. RERERENCES 1. Madanat, S. M. Incorporating inspection decisions in pavement management. Transportation Research B, Vol. 27B, 1993, pp. 425-438. 2. Ben Akiva, M., and T. Morikawa. Estimation of switching models from revealed preferences and stated intentions. Transportation Research A, Vol. 24A, No. 6, 1990, pp. 485-495. 3. Archilla, A. R., and S. M. Madanat. Estimation of rutting models by combining data from different sources. ASCE Journal of Transportation Engineering, Vol. 127, No. 5, 2001, pp. 379-389. 4. Prozzi, J. A., and S. M. Madanat. Using duration models to analyze experimental pavement failure data. In Transportation Research Record 1699, TRB, National Research Council, Washington, D.C., 2000, pp. 87-94. 5. Madanat. S. M., S. Bulusu, and A. Mahmoud. Estimation of infrastructure distress initiation and progression models. ASCE Journal of Infrastructure Systems, Vol. 1, No. 3, 1995, pp. 146-150. 6. Highway Research Board. The AASHO Road Test. Special Reports No. 61A-E, National Academy of Sciences, National Research Council, Washington, D.C., 1962. 7. American Association of State Highways and Transportation Officials. AASHTO Guide for Design of Pavement Structures. AASHTO, Washington, D.C., 1993. 8. Greene, W. H. Econometric Analysis. Prentice Hall, Fourth Edition, New Jersey, 2000. 9. Sayers, M. W., T. D. Gillespie and C. A. V. Queiroz. The International Road Riding Quality Experiment: establishing correlation and a calibration standard for measurements. Technical Paper 45, World Bank, Washington, D.C., 1986. 10. Haas, R., W. R. Hudson, and J. Zaniewski. Modern Pavement Management. Krieger Publishing Company, Malabar, Florida, 1994. 11. Humplick, F. Highway pavement distress evaluation: modeling measurement error. Transportation Research B, Vol. 26B, No. 2, 1992, pp. 135-154. TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat 18 LIST OF TABLES TABLE 1: Parameter estimates and asymptotic t-values for the OLS and RE estimation. TABLE 2: Parameter estimates of the joint model and corresponding t-values. LIST OF FIGURES FIGURE 1: Averaged observed effect of the frost depth on deterioration at AASHO. FIGURE 2: Deterioration rate as a function of strength and traffic. FIGURE 3: Observed versus predicted performance by the linear and the nonlinear models for a section not used in the estimation sample. FIGURE 4: Variation of the rate of roughness increase as a function of traffic, pavement strength and environmental conditions. TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat 19 TABLE 1: Parameter estimates and asymptotic t-values for the OLS and RE estimation. Parameter OLS estimate Asym. t-value RE estimate Asym. t-value β1 β2 β3 β4 β5 β6 β7 β8 β9 β10 β11 β12 4.45 -1.47 -0.555 2.28 0.775 0.546 -2.67 -0.186 -0.473 0.790 1.72 3.57 57.1 -16.5 -6.2 14.1 10.8 11.3 -29.5 -49.0 -39.8 22.3 101.2 46.0 4.24 -1.43 -0.856 1.39 0.329 0.271 -3.03 -0.173 -0.512 0.552 1.85 4.15 165.4 -8.9 -8.4 17.6 14.4 15.2 -35.2 -47.7 -49.5 29.6 109.4 54.6 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat 20 TABLE 2: Parameter estimates of the joint model and corresponding t-values. Parameter Estimated value t-value θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8 θ9 θ10 θ11 θ12 θ13 θ14 θ15 1.58 -0.126 0.787 0.237 0.204 1.82 0.288 0.236 -3.77 -0.157 -0.374 0.523 1.85 3.85 4.27 45.8 -28.0 15.7 56.3 54.5 22.7 8.6 11.7 -70.2 -77.3 -50.7 45.2 170.5 92.9 4.4 TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat 21 5 .0 90 4 .5 80 A C T U A L D E T E R IO R A T I O N 70 3 .5 60 AVERAGE D E T E R IO R A T I O N 3 .0 50 2 .5 40 2 .0 30 F R O S T G R A D IE N T 1 .5 20 1 .0 10 0 .5 0 0 .0 FROST DEPTH (inches) SERVICEABILITY (PSI) 4 .0 -1 0 N D J F M A M J J A S O N D J FIGURE 1: Averaged observed effect of the frost depth on deterioration at AASHO. TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat 22 DETERIORATION RATE (PSI/1,000 ESALS) 1 .00 0 ET = 2 0 .10 0 ET = 4 0 .01 0 ET = 6 ET = 8 0 .00 1 0 2 0 0 ,0 0 0 4 0 0 ,0 0 0 6 0 0 ,0 0 0 80 0 ,0 0 0 1 ,0 0 0 ,0 0 0 E Q U IV A L E N T T R A F F IC (E S A L s) FIGURE 2: Deterioration rate as a function of strength and traffic. TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat 23 6 5 ROUGHNESS (m/km IRI) DATA 4 NONLINEAR MODEL 3 2 ORIGINAL AASHO MODEL 1 0 0 200,000 400,000 600,000 800,000 1,000,000 1,200,000 AXLE REPETITIONS FIGURE 3: Observed versus predicted performance by the linear and the nonlinear models for a section not used in the estimation sample. TRB 2003 Annual Meeting CD-ROM Paper revised from original submittal. Prozzi and Madanat 24 RATE OF ROUGHNESS (m/km per 1,000 ESALs) 1 G = -2 inches/day G = 0 inches/day G = +2 inches/day 0.1 EAT = 4 0.01 EAT = 6 EAT = 8 0.001 0 200,000 400,000 600,000 800,000 1,000,000 EQUIVALENT TRAFFIC (ESAL) FIGURE 4: Variation of the rate of roughness increase as a function of traffic, pavement strength and environmental conditions. TRB 2003 Annual Meeting CD-ROM View publication stats Paper revised from original submittal.