LIST OF EQUATIONS
4. Chapter 4 Methodology
4.7. Deterministic Model Approach
George, Rajagopal and Lim (1989) defined the deterministic model as comprising structural performance, functional performance, damage models and initial response. Anyala, Odoki and Baker (2014) outlined deterministic models as
“predicting condition as precise values using mathematical functions”. Determining the pavement performance should be a function of a set of random variables that do not change over time. The deterministic approach is used in this research to model pavement performance indicators (IRI and PCI) under the HDM-4 model. The following tools are used to analyse the received data.
4.7.1. Numerical Analysis
To date various methods have been developed and introduced to model pavement performance indicators (IRI and PCI). The numerical analysis method is one of the more practical ways of examining a deterministic model. This approach has a number of attractive features such as descriptive statistics. Descriptive statistics are used to handle the process of data organising, summarising and presenting in order to achieve a very convenient and informative set of data (Keller 2009). In terms of data analysis, Mubaraki (2010) stated that estimating a parameter for the distribution, to characterise the spread or variability, is an essential task in exploratory data analysis.
For example, the most accepted measure of the central tendency of data distribution is the mean. Other parameters use the median, which defines the midpoint of a distribution. Standard deviation is the best method to rate the variability of a distribution. Further characterisation of the data including skewness and kurtosis can also be introduced to define the lack of symmetry and whether the data are peaked or flat with respect to a normal distribution respectively. All these analyses and tests are used in this research. To ease the process of analysis, SPSS software is utilised.
4.7.2. Non-linear Regression Modelling
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For a statistical model, three major elements structure the principle of modelling, and these are the response variable (y), the mathematical function (x) and random errors (e). Regression methods are a widely accepted method in modelling.
Regression is very powerful in cases regarding the description of the association between a response variable and one or more explanatory variables. In this research, such analysis is used to define the relationship between different variables in the HDM-4 model. Also, the relationship between IRI and PCI is determined.
Furthermore, a similar method is used to analyse the results generated from the survey questionnaire. The results can be reported based on statistically significant (p-values).
4.7.3. Estimating Total Change in Roughness in the Default HDM-4 Model
The determination of the roughness model structure based on default equation and coefficients for the default HDM-4 model is conducted. Collecting, sampling and analysing the data, which are presented in Chapter 5, are used in this model.
Assumptions are made that all road sections are built according to the standard. The coefficients and equations of change in roughness based on structural, rutting, cracking and environmental components for HDM-4 are studied. Once the model is built based on the HDM-4 equation and variables, the inputs of different climate change scenarios are tested and the results for the years 2013 (current weather data), 2020, 2040 and 2060 (future climate change scenarios) are recorded.
4.7.4. Estimating Total Change in Roughness on the Modified HDM- 4 Model
There are different methods to develop pavement performance models. One of the most accepted methods is ‘Statistical Regression Analysis’ (Amador-Jiménez and Mrawira 2012). To develop new model coefficients and equations, descriptive statistical analysis with the help of the SPSS software program is used. Data preparation and cleaning such as removing any invalid data or outliers is carried out to obtain the most solid sample. The descriptive statistical analysis in this section includes measurement aspects of non-linear and linear regression analysis. The primary objective is to investigate the most reliable coefficient based on the available
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data gathered from the Ministry of Public Works in the UAE. Goodness fit test (ANOVA test), correlations test and parameter estimates test for change in roughness based on structural, rutting, cracking and environmental components is conducted.
Three different experiments are carried out to compare the original HDM-4 model, observed data and the modified HDM-4. Two deficiencies are reported, and improvement is applied accordingly. The calibration process involves introducing the adjustment factors, which are linear multipliers for adjusting the predictions to meet the conditions of the selected area and are adapted in this study by following Thube’s (2013) approach. Once the model is built based on the new, modified HDM-4 (equations, coefficients and calibration), it will be tested with different inputs of climate change scenarios (pavement temperature). These scenarios are for the year 2013 (current weather data), 2020, 2040 and 2060 (projected weather data).
4.7.5. Pavement Condition Index (PCI) Model
The Pavement Condition Index (PCI) ranges from zero to 100, where 100 represents an excellent pavement condition. Mubaraki (2016) stated that the PCI values are obtained based on many elements, which are the pavement distress type, severity of distress and assessment of collected distress through visual inspection. To obtain PCI value, the International Roughness Index is used in many model forecasts.
This study tests the relationship between the PCI and the IRI; the author follows Park, Thomas and Lee’s (2007) approach using available data gathered from Al Ain City Municipality. The SPSS software program is utilised. The data preparation is carried out to clean the data and remove any invalid data or outliers. This descriptive statistical analysis includes two different measurement aspects: correlation and regression analysis. A correlation test is used to represent how and to what extent two variables are associated. Regression analysis is used to predict one variable from existing information on one or more variables and to find significant relationships (residual square), since linear and non-linear regressions are used to estimate the result of a dependent Pavement Condition Index. More details are presented in Chapter 6 section 6.5.
4.7.6. Model Checking and Validation
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For model checking and validation, the application of an independent T-test is used. Generally, an independent T-test (‘analysis of variance’) is used to study the comparison of the means of different independent groups (either two or more) in order to find out statistically if they are significantly different or not. It is a parametric test.
For the developing model, the analysis is carried out only on the backward direction roads data (see Chapter 5 section 5.2). Model checking and validation is conducted by testing the modified HDM-4 with the calibrated factor in the forward direction data instead of the backward direction. For example, the traffic loading data for a road in the forward direction differs from that for one in the backward direction. A test is conducted to verify the reliability of the model with different variable inputs.
Therefore, a comparison between two means (2013 forward results and 2013 backward results) is conducted.
4.8. Markov Chain Process
To simulate the pavement deterioration process and evaluate variations, the Markov chain method is used. The Markov method requires transition probability matrices (TPMs) to express the transition from one pavement condition state to another. The relevant data sets are from a pavement network of multiple roads and highways under the management of the UAE Ministry of Public Works. The research methodology of the Markov chain consists of four parts. The first part is defining the number of condition states and categories. The second part presents the method to estimate the transition probability matrix in conjunction with the available data. The third part is to determine the current distribution of the pavement condition, and the final part is a simulation and an exploration of the long-run behaviour of the model.
4.8.1. Define Number of Condition States
Many researchers have stated that the most challenging aspect of using the Markov modelling method is the state size. This is because system states need to be comprehensively described. Moreover, simulation of a Markov chain model with numerous condition states shall lead to a very complex system which challenges the
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available computer resources. Therefore, it is crucial to introduce a sufficient number of states. As mentioned in Chapter 4, the ‘Condition Rating System’ is a technique of physical deterioration classification to assess the condition of pavement assets.
Although condition states for the IRI defined by the UAE Ministry of Public Works are rated on a scale of four classifications, five condition states are applied in this research. More details are presented in Chapter 7 section 7.2.
4.8.2. Determine Pavement Deterioration Rate (Transition Probability Matrix)
A transition matrix estimates pavement deterioration rate, which is a change in the existing pavement condition state to the next poor condition state within a specific period, normally one year. The transition probability matrix can be derived from historical condition data. However, estimating transition probability from incomplete data is very difficult (Jin and Mukherjee 2014). Estimation of the transition probabilities can be achieved using two methods. The first method is based on data on pavement condition over several years. Once the available data are obtained, a way of estimating the corresponding deterioration rates can be quickly developed using any of the pavement performance indicators such as PCI or IRI. The second method is to use the subject matter of experts to estimate the transition probabilities. By reviewing the available data (see Chapter 5 section 5.4) from both Al Ain City Municipality and the Ministry of Public Works, it was found that the only data sets available were for three consecutive years (2013, 2014 and 2015). Both methods are tested in this research. More details are presented in Chapter 7 section 7.2.2.
4.8.3. Determine Current Pavement Condition
Distribution of actual pavement condition states for the entire pavement network gives the condition state vector. In this study, the assumption of the initial state vector is (1 0 0 0 0). The assumption is made that the pavement conditions of the study area are in excellent condition state. Once this initial condition state vector is determined, the transient probabilities for the pavement for every year can be
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calculated by multiplying this condition state vector with the deterioration matrix.
More details are presented in Chapter 7 section 7.2.
4.8.4. Model Simulation
In exploring the behaviour of system prediction performance using the Markov chain, it is presumed that at time step zero all pavement conditions are new (excellent condition). Typically, four strategies are involved in this research. Hence, to explore the system’s behaviour under different approaches, four strategies that are linked to different climate change scenarios are examined, as follows: Strategy 1, current weather condition (based on 2013 weather data). Strategy 2, all pavement sections are determined under 2020 predicted conditions. Strategy 3, all pavement sections are determined under 2040 predicted weather conditions. Strategy 4, all pavement sections are determined under 2060-2079 predicted weather conditions. For each case, the transition process is performed in Excel (Microsoft software) and outcomes will be plotted on different graphs (more details are provided in Chapter 7 section 7.3 ). The forecasted period is set for 30 cycles (usual pavement section design life). Each cycle presents a single year. The results of determining IRI for different climate scenarios are listed in Appendix 4.
4.8.5. Markov Model Assumptions
Lytton (1987) and Panthi (2009) described the assumption for the Markov model process which is also adopted by the author:
(1) The transition probabilities rely only on the present condition state.
(2) The probability of transition from one condition state to another is time independent. (The transition process is stationary.)
(3) Increase in condition rating due to maintenance intervention is not considered in this research as the pavement section is assumed to have deteriorated on its own.
Also, condition ratings will be assumed to be constant or decrease with time.
(4) Deterioration process is occurring as a single state condition in one year. More than a single state is not allowed to deteriorate.
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(5) Transition probability matrix is assumed to be homogenous, meaning that the transition probability for deterioration from one year to the next is always the same.
Even though this assumption is generally not valid for pavement conditions, since changes can occur in the weather or the traffic load, the author decided to follow an homogenous Markov chain.