استفاده از آمار بیزی برای پیش بینی الگوهای فضایی با قابلیت تکراری

Statistical spatial repeatability (SSR) is an extension to the well known concept of spatial repeatability. SSR states that the mean of many patterns of dynamic tyre force applied to a pavement surface is similar for a fleet of trucks of a given type. A model which can accurately predict patterns of SSR could subsequently be used in whole-life pavement deterioration models as a means of describing pavement loading due to a fleet of vehicles. This paper presents a method for predicting patterns of SSR, through the use of a truck fleet model inferred from measurements of dynamic tyre forces. A Bayesian statistical inference algorithm is used to determine the distributions of multiple parameters of a fleet of quarter-car heavy vehicle ride models, based on prior assumed distributions and the set of observed dynamic tyre force from a ‘true’ fleet of one hundred simulated models. Simulated forces are noted at 16 equidistant pavement locations, similar to data from a multiple sensor weigh-in-motion site. It is shown that the fitted model provides excellent agreement in the mean pattern of dynamic force with the originally generated truck fleet. It is shown that good predictions are possible for patterns of SSR on a given section of road for a fleet of similar vehicles. The sensitivity of the model to errors in parameter estimation is discussed, as is the potential for implementation of the method.

Spatial repeatability is the phenomenon that the pattern of dynamic force applied truck axles to a road pavement is similar for repeated runs at similar speeds. This effect results in a concentration of high dynamic tyre forces at specific locations on a pavement surface and has been observed by several authors both experimentally (Ervin, 1983 and Mitchell, 1987) and in numerical studies (Huhtala et al., 1992). This opposes the traditional assumption that applied dynamic tyre loads are randomly distributed along a pavement length, suggesting that the pavement is uniformly susceptible to damage along its length. Cole and Cebon (1992) performed a numerical investigation of spatial repeatability using an experimentally validated two-dimensional articulated vehicle model. They generated a fleet of 37 leaf sprung vehicle models with similar geometry and eight varying parameters relating to the ride characteristics, identifying repeatable patterns of dynamic tyre forces. The relationship between vehicle velocity and level of repeatability was highlighted. A further experimental study, involving measurement of heavy vehicle tyre forces on a major national route in the UK, was conducted (Collop et al., 1996) which confirmed theoretical predictions of the influence of speed on spatial repeatability of tyre forces. O’Connor et al. (2000) proposed the concept of ‘statistical spatial repeatability’ (SSR). Using data from a multiple-sensor weigh-in-motion (MS-WIM) site in France, they showed that the mean pattern of impact factors is similar for many trucks of the same type. This is illustrated in Fig. 1. Similar patterns were found for different types of truck and even for trucks with different numbers of axles. Full-size image (46 K) Fig. 1. Statistical spatial repeatability of Impact Factor (IF) for gross vehicle weights of nine truck types (from O’Connor et al., 2000). Figure options SSR has great implications for pavement deterioration. Pavement deformation and damage is directly related to impact force and the pattern of SSR is related to the road profile. It seems likely therefore that the process of road pavement deterioration is integrally linked with SSR. Following some initial imperfections, road surface deformations are generated which result in a pattern of SSR. The repeatable forces cause further deformation which may reinforce the existing pattern of SSR or change it. Some research has focused on integrated pavement deterioration models for the calculation of pavement life (Collop and Cebon, 1995), dividing the procedure into four main areas: dynamic vehicle simulation, pavement primary response calculation, pavement damage calculation and profile change and damage feedback mechanisms. Within this context, it is clear that the accurate prediction of applied dynamic forces is necessary for the calculation of long-term pavement performance. This paper describes a method to predict the pattern of SSR. A quarter-car model (Fig. 2) is used to calculate the force applied to a pavement as it travels at a given speed. The pavement surface is modelled as a three-dimensional ‘carpet’ to provide a varied but correlated series of road profiles for the vehicle model. The profile chosen from the 3-D surface depends on the lateral approach position of the vehicle model. Full-size image (2 K) Fig. 2. Quarter-car model. Figure options A range of properties related to the vehicle, its lateral approach position and its speed are assumed to be random variables. Variations in these properties will lead to variations in the applied impact forces. For a given fleet of vehicles, the statistical distributions for the properties, if known, can be used to predict the pattern of SSR. Using Bayesian updating (Lee, 2004), these distributions can be updated through comparisons between calculated and measured impact forces. In this study, the approach is tested using Monte Carlo simulation to generate distributions of impact forces corresponding to a vehicle fleet whose properties have known statistical distributions. With Bayesian Updating, a heavy vehicle fleet model is determined which can be used to predict patterns of SSR.

A method has been presented for the determination of a heavy vehicle fleet model which can be used to predict patterns of statistical spatial repeatability and which can ultimately be used in an integrated pavement deterioration framework. Using a Bayesian statistical inference algorithm, the distributions of multiple parameters of a fleet of quarter-car heavy vehicle ride models are determined inversely based on prior assumed distributions and the set of observed impact factors from a ‘true’ fleet of one hundred simulated models. The impact factors are assumed to be measured at 16 equidistant pavement locations, similar to a multiple-sensor weigh-in-motion site. It is shown that the fitted distributions obtained from the Bayesian statistical inference yield excellent agreement with the true distributions, enabling the prediction of patterns of SSR for multiple vehicles of similar type. The sensitivity of each of the three inferred distributions is discussed and it is noted that for the study in question, the effect of variation in the predicted standard deviations on the RMS error in impact factors is minimal in comparison to error in the predicted means. It was shown that variations in the predicted mean lateral approach could cause notable RMS error in impact factors.