The Development of Models to Predict Pavement Performance in Frost Conditions JOINT CSHRP/QUEBEC BAYESIAN APPLICATION 6. ROUGHNESS MODEL It is generally accepted that frost action is a major factor influencing roughness progression in frost susceptible pavements. Roughness is a composite index defined as (the deviation of a pavement surface from a planar surface with characteristic dimensions that affect vehicle dynamics, ride quality, dynamic loads and drainage (ASTM, E867820) ). Roughness is composed of surface distortions as well as of irregularities associated with surface distresses such as transverse cracking, rippling, shoving, pot holes, etc. Long wave displacements of the surface (referred to as distortion) is commonly associated with movements (settlement and/or heaving) of the subgrade. Attempts will be made in later versions of the models to dissociate these two components of roughness. However, the first generation model will be a roughness model in which the contribution of base and surface phenomena will have to be dealt with as noise. 6.1 Selection of variables The selection of the type and the number variables to be included in the regression equation is a critical task in the development of the models. The inclusion of a large number of relevant variables is likely to increase the predictive power of the models; however, large models are typically complex and difficult to support. Moreover, simple models (including few variables) usually yield larger errors but are easier to use, more intuitive to practitioners and are less vulnerable to measurement or simplification errors. Good practice suggest to begin with a simple model and increase the level of complexity as needed (stepwise approach). As detailed in section 5, very little was found in the literature about performance models associated with climatic factors and more specifically with frost action. Most consulted references were dealing with general roughness models. Table 1 describes some features of models developed in the past. From the literature review, and based on the mechanics of frost action on pavements, the following potential variables were identified:
After reviewing the potential variables, it was decided to initiate model development using the five variables identified by an asterisk(*). The selection was based on the expected significance of the variables and on their relevance in view of the intended use of the model. Since the model's main application will be in design, the variable selection must allow for the control of structural parameters in a given climatic and soil environment. The selection was supported by the group of experts involved in the model development process. 6.2 Development of the prior In the absence of specific information in the reviewed literature on the evolution of roughness as a function of frost action in pavements, the development of the prior was based on expert judgment. Experts were given access to a few reference values extracted from the provincial pavement inventory database prior to being encoded. Available information on the provincial network outlining: 1) average roughness values by classes of roads and 2) average rate of evolution of roughness by class of roads and by origin of the problem (structure or subgrade) affecting the pavement was summarized in tables and transmitted to the experts in the encoding package. Each expert was then asked to estimate the level of roughness for different sets of conditions by completing three matrices (similar to the one shown in figure 5). Each matrix was specific to a different total thickness of the assumed pavement structure (500, 750 and 1000 mm). Each matrix included a total of 36 cells (conditions) representing 3 different pavement ages (4,8 and 12 years), two precipitation levels (800 and 1200 mm), two freezing index levels (1000 and 1500 °C*day) and three levels of frost susceptibility of the subgrade soil. Responses from 7 experts from different affiliations (university, consulting firm and ministry of transportation) were obtained and encoded. These response were checked and verified for obvious transcription errors or misunderstanding among the experts. No major discrepancies between experts were identified. The responses of the individual experts were then combined to form a (group prior data set) containing 756 (observations). As such, this data set formed a statistical significant sample representative of the collective judgment of the 7 experts on which a (group prior) could subsequently be developed. 6.3 New data Two sources were exploited to gather new data on pavement performance in frost conditions. The main source was the Canadian Long Term Pavement Performance (CLTPP) data base. The CLTPP project was launched in 1987. The project involves the systematic observation of 65 inservice pavement test sections located across Canada. Pavement types under study include asphalt concrete constructed over granular base courses which were overlaid in 1989 or 1990. Only the sites located in a wetfreeze environment were selected for the development of first generation models. A total of 132 observations on 39 test sections were included as new data in the analysis. The second source of data used was the SHRPLTPP (United States) database. The LTPP program has similar goals, objectives and format but has a much broader scope than the Canadian program. It includes nine pavement types and over 2000 test sections distributed across North America. For data analysis purposes, only GPS1 sections (Flexible pavements with granular bases) in a wetfreeze environment and exposed to a minimum freezing index of 200 °C day were considered. Unfortunately, data for many of these sections were not available because it had not yet passed the necessary quality control checks to be released by SHRP. Only 40 observations on 13 test sections were added to the analysis, resulting in a combined field database with 172 observations. 6.4 Quality control of the data set Inference space: In order to successfully combine prior information and new data, the inference space across each variable for both the field data and expert judgment was designed to be compatible (where appropriate). As illustrated in figure 6, there is a reasonable agreement, with respect to the inference space, across each variable for the two data sources. There is however a significant difference between inference spaces for the freezing index. It was necessary to expand the inference space of this variable in order to collect sufficient field data to support the Bayesian analysis. The implication of the poor match will have to be assessed The implementation of the model will likely require calibration for the specific climatic context of the province of Quebec. Expert judgment: Reasonableness and coherence of the estimates given by the experts were verified by performing a sensitivity analysis on linear prior models derived for each expert. Prior models were used to assess the variation in roughness predictions. The results of the sensitivity analysis is illustrated in figure 7. The analysis shows that all experts are in relatively good agreement on signs and magnitude of the coefficients (sign and steepness of the slope on the graphic) for each variable. Precipitation is considered by most experts to be the variable with the least influence on roughness. 6.5 Functional form of the model The selection of the functional form was guided by the following constraints and considerations: 1. Independently of the magnitude of all other factors, the roughness value must converge to the same initial value (R0) at time = 0. (R0) can be either specified by the designer or obtained statistically for different classes of roads. 2. The slope of the model must be >= 0 (roughness can not decrease with time). 3. It is believed that roughness has a curvilinear relationship with time. Based on these constraints, the first regression iterations led to the following general form of the model:
6.6 Regression analysis Equation (1) needed to be transformed in order for the model to confirm the known boundary conditions and to be linear in the coefficients. Thus, the following manipulations were executed:
In equation (2), ln(Rt (R0) becomes the transformed Y and a, b, c, d, e, and n are the coefficients to be determined through multivariate Bayesian regression analysis. Using estimates from several experts increases the level of confidence in prior information, but it causes some practical problems. Each set of estimate is based on the expert's specific experience and affected by his level of confidence. There is no formal way to deal with the different levels of certainty of different experts nor any basis for excluding data which does not reflect the general consensus. In fact, there is valuable information in the estimates of each expert. Good practice in Bayesian analysis to generate one model for each expert involved and select the model considered the most appropriate for further development or for implementation. The authors felt this approach also creates some practical problems because it introduces more subjectivity in the analysis (selection of the most appropriate model) and it may introduce biases associated with expert personality. Therefore, it was decided to combine estimates into one (data set) to reflect a group prior. Classical multivariate regression was performed on the grouped data set to generate the statistics of the prior model required as input to XLBayes, namely: the vector of coefficient means, the variance/covariance matrix, the degree of freedom and the standard error of the residuals. The field data set (i.e. SHRP and CSHRP) was then introduced into (XLBayes) ) to be combined through Bayesian regression with the prior thereby generating the posterior model. The probability distribution of regression coefficients for each variable of the prior, data and posterior models is illustrated in Figure 8. The posterior model developed was as follows:
The precision of the model can be quantified using the standard error of the residual (Se). The models described in this paper were obtained by manipulating the equation in order to keep it linear in the coefficients. The calculated Se of 0,82 therefore applies to the transformed (Y) in equation (2). A more meaningful way to represent the precision of the model is to generate the corresponding 68% confidence interval for predicted roughness in given conditions. For example, if all variables were set to the average value of the inference space defined for encoding expert judgment (i.e. A=8, T=750, P=1000, FI=1250, FS=200) then, predicted roughness would be 2,62 with a nonnormal 68% confidence interval between 1,77 and 4,55. Figure 9 illustrates the relationship between (measured) and predicted roughness. In this figure, the expert estimates and the field data are combined to form the (measured) data set. 6.7 Discussion As expected, increasing thickness reduce the rate of deterioration (roughness) associated with frost action. Age, precipitation, freezing index and frost susceptibility of subgrade soil also show the expected detrimental influence on roughness. For a 95% confidence interval, (false rejection of the null hypothesis one time in twenty), all variables were found to be statistically significant. The most significant variable is frost susceptibility (T=16,16) followed by freezing index (14,41), age (13,96), thickness (12,97), and precipitation (marginal T value of 2,44). The model described is the result of several regression iterations but it remains a (first generation) model. It's predictive capability is limited by the relatively small field data set used to update expert judgment. Subsequent iterations will be based on more field data from both the Canadian and the American LTPP projects and should yield a more robust model. A decision was made early in the project to exclude the influence of traffic in the development of the model in order to keep the model simple and focus on environmental factors. However, it is clear from previous work reviewed that traffic has a significant influence on roughness. Another phenomena which contributes significantly to roughness and is not specifically dealt with by the model is frost action along transverse cracks. Water and salt infiltration through transverse cracks typically induce heaving resulting in a sharp bump along the crack at the surface of the pavement. Improvements of the model will require either a modification of the dependent variable (long wave distortions instead of roughness) or the inclusion of a variable describing the influence of traffic and frost action within pavement layers. Frost susceptibility of subgrade soils was found to be a variable with relatively high explanatory power. It is however believed that better characterization of frost susceptibility and more specifically it's longitudinal variability along an highway corridor would significantly improve the model. |
