LIST OF EQUATIONS
3. Chapter 3 Theoretical Methods for Modelling Pavement Deterioration Deterioration
3.4. Modelling Pavement Indicators
3.4.8. Modelling the HDM-4 Environmental Component of Roughness Roughness
ΔRIc Incremental change in roughness due to cracking during analysis year, in m/km IRI
Kgc Calibration factor for the cracking component of roughness=1
a0 The coefficient values =0.0066
ΔACRA Incremental change in area of total cracking during analysis year, in per cent
3.4.8. Modelling the HDM-4 Environmental Component of Roughness
The HDM-4 environmental component of roughness is calculated based on the following equation presented in Table 3-7.
Table 3-7: Environmental component of roughness (ΔRIe) Main Equation 4 Variable Description ΔRIe = Kgm m RIa
m = 0.197+ 0.000155 TMI
ΔRIe Incremental change in roughness due to the environment during analysis year, in m/km IRI
Kgm Calibration factor for the environmental component (default = 1.0)
m Environmental coefficient
RIa Roughness at the start of the analysis year, in m/km IRI TMI Thornthwaite Moisture Index
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3.5. Pavement Deterioration Probabilistic Modelling 3.5.1. Modelling of Markov Chain
The prediction deterioration model is a mathematical approach that can be applied to forecast how the future pavement is going to deteriorate. A model entirely depends on the existing pavement condition, deterioration factors and previous maintenance (OCED1987). Arimbi (2015) classified the prediction deterioration model into two categories: the deterministic model (predicts as an exact value based on mathematical functions from observed deterioration) and probabilistic model (predicts the pavement condition as the probability of occurrence of a range of possible outcomes) (Ortiz-García, Costello and Snaith 2006). In theory, probabilistic models are applied for pavement evaluation at the network level, while the deterministic model is the only appropriate tool for project-level performance.
Moreover, Haas, Hudson and Zaniewski (1994) stated that Markov chains are the most accurate techniques for prediction models since the future state of the model element is estimated solely for the current state of the component.
The Markov method requires transition probability matrices (TPMs) to express the transition from one pavement condition state to another. For the case of pavements, in order to conduct a Markov chain, it is crucial to estimate the probability of shifting from one condition state to another, which is usually done by expert judgement or based on the analysis of available previous information. The fundamental rule in Markov chains on the probability of shifting from one state to another is independent of an item’s prior condition history (Black, Brint and Brailsford 2005; Austroads 2012). Parzen (1987) expressed a discrete-time Markov technique that future process only depends on the present and not on the past.
Ha et al. (2017) stated that the Markov chain approach is a broadly accepted tool for deterioration modelling of infrastructure. Nevertheless, they added that such a tool has some limitations in the validity of its assumptions, according to Thomas and Sobanjo (2016). One of the critical limitations of the Markov chain approach depends on the probability of changing from one phase to another (independent of time). This theory is questionable because it indicates that changes in the future, such
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as environmental impact (weather condition), will not affect the transition probabilities (Lytton 1987).
3.5.1.1. Homogenous Discrete-time Markov Chain
State probabilities and state transition are the two major components in the discrete-time Markov model. State probabilities are estimated based on pavement condition states (e.g.: very good, good, poor… etc.) (Abaza 2016). The transition probability represents the probability of the pavement changing from one condition state to another during one specified period (Abaza 2016). The time interval is defined as a discrete period, which is in practice (for pavement assets) taken every one or two years, and each period represents one transition. In practice, quantifying state probabilities needs at least one cycle of pavement distress assessment, which is surveyed by the road and highway agencies. Typically, the pavement project is divided into small pavement sections to gather a better condition survey and reflect the certain present condition of the pavement. The numbers of pavement sections (Ni) assigned to various deployed condition states can then be used to estimate the state probabilities (Si), as defined in equations 3-11 and 3-12.
N Si Ni
Equation 3-11: Equation for estimating the state probabilities by Abaza (2016)
m
i
Ni
N
1
Equation 3-12: Equation for estimating total number of pavement sections used by Abaza (2016)
Where,
S
i= is the ith state probability
N
i= is the number of pavement sections assigned to the ith condition state
N= is the total number of pavement sections used in the study (i.e. sample size)
and m is the number of deployed pavement condition states.
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In this research a homogenous Markov chain is adopted. It is assumed that the transition probabilities remain constant over time (steady-state condition). In order to forecast the state probabilities associated with the use of Equation 3-13, the model is built on the state and transition probabilities. Theoretically, the initial (i.e. present) state probabilities and transition probabilities are known. The initial state probabilities for new pavement can be assumed to take on the values of 1, 0, 0, . . . , 0.
Equation: 3-13: Equation for estimating state probabilities based on steady-state condition by Abaza (2016)
Where,
S
(k)= is the row vector representing state probabilities after k transitions
S
()0= is the row vector representing initial state probabilities
P
(k)= is the transition matrix raised to the kth power m is the number of deployed pavement condition states
n is the number of deployed discrete-time intervals (transitions)
The transition matrix is a square matrix (m × m) comprising all estimated transition probabilities. The matrix entries (Pi,i) represent the probability of pavements remaining in the same condition states after the elapse of one transition or the likelihood of the portion of the network in the state ‘i’ moving to state ‘j’ in one duty cycle. There are two scenarios for transitions. First is the deterioration transition probability (Pi,j; j > i), which indicates the probability that the pavement condition is transitioning to a worse condition state. This assumption is based on the fact that there is no maintenance and rehabilitation works. The other scenario is an improvement in
m
i k i
m k m k
k k k
k k
s
S S
S S S
S S
S S S
n k
P S S
1 ) (
) 0 ( )
0 ( 3 ) 0 ( 2 ) 0 ( 1 ) 0 (
) ( )
( 3 ) ( 2 ) ( 1 ) (
) ( ) 0 ( ) (
, 0 . 1
), ,...
, , (
), ,...
, , (
) ,..., 2 , 1 (
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the transition probabilities. The main diagonal (Pi,j; j < i) indicates the probability that the pavement condition is transitioning to a better condition state after one transition.