Updating Pavement Deterioration Models Using the Bayesian Principles - PowerPoint PPT Presentation

Updating Pavement Deterioration Models Using the Bayesian Principles and Simulation Techniques 1 st AISIM University of Waterloo, August 6, 2005 Feng Hong Jorge A Prozzi The University of Texas at Austin 1

Outline � Background � Data Considerations � Model Specification � Estimation Approach � Results Implications � Performance Forecast � Conclusions 2

Pavement Performance � Distresses • Rutting • Cracking • Pothole • Faulting • Spalling � Subjective indicators • Present Serviceability Rating (PSR) 3

Performance Modeling Approach � Empirical (CBR, AASHTO 93) • Data fit based on regression � Mechanistic (in theory) • Material response and pavement performance through mechanistic analysis � Mechanistic-Empirical (upcoming AASHTO Guide) • Material response (mechanistic) • Correlate response to performance (empirical) 4

Data Considerations � Commonly used data sources • In-service sections: PMS, LTPP • Test sections: AASHO, Westrack, LTPP, NCAT • APT: HVS, ALF, MLS � Data characteristics • Cross sectional data • Panel data: incorporating both cross section and time series information 5

The AASHO Road Test � Ottawa, IL, 1958-1960 • One environment, one subgrade � Structural variables (factorial design) • Varying thickness of surface, base, and subbase � Traffic • Around 1 million repetitions • Axle configurations • Single axle with single wheel • Single axle with dual wheels • Tandem axle 6

Pavement Deterioration Model � Original AASHO Model ⎛ ⎞ α W ( ) ⎜ ρ ⎟ p p p p = − − t f ⎝ ⎠ 0 0 Where serviceability at time t p : t p : serviceability at time t = 0, i.e. initial serviceability 0 p : terminal serviceability f W : accumulated axle load repetitions until time t ρ : accumulated axle load repetitions until failure α : regression parameter determining the curvature of performance model � General deterioration model c p a bN = − t t 7

Determining term b � b = b S b E • Structural variables { } { } b S H H H = λ γ + γ + γ exp exp 1 1 2 2 3 3 • H 1 , H 2 , H 3 : thickness for surface, base and subbase layers • Environmental variable b G = ϕ exp{ } E t • G t : Frost penetration gradient 8

Incremental Model � Taylor expansion p p p dN e N ∆ = − = ∆ + ε t t t t − 1 − 1 Where d , e : parameters to be estimated N : some measure of accumulative traffic until the one-period t − 1 backward of time t ∆ : projected incremental traffic during the time period between t -1 N and t ε : error term { } { } { } d H H H G ' = λ γ + γ + γ ϕ exp exp exp t 1 1 2 2 3 3 9

Final Specification β p H H H G N N 5 ∆ = β + β + β + β + β ∆ + ε exp{ } it i i i t i t it it − 0 1 1 2 2 3 3 4 , 1 ∑ t − N 1 N = ∆ i t il − l , 1 = 1 ⎧ ⎫ β β ⎛ ⎞ ⎛ ⎞ β ⎪ ⎪ FA ⎛ SA ⎞ TA 8 8 8 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ N n A B ⎨ i i i ⎬ ∆ = + + ⎜ ⎟ ⎜ ⎟ il il i i ⎝ ⎠ ⎪ β β ⎪ ⎝ ⎠ ⎝ ⎠ 18 18 18 ⎩ ⎭ 6 7 Where p serviceability loss in pavement section i during time period t ∆ : it N cumulative traffic until the one-period backward of time t : i t , − 1 N incremental traffic during the time period between l -1 and l ∆ : il n : traffic volume during the time period between l -1 and l il FA : front axle load magnitude i SA : single axle load magnitude i TA : tandem axle load magnitude i A : number of single axles on one vehicle i B : number of tandem axles on one vehicle i 10

Estimation Approach for Panel Data � Ordinary Least Square (OLS) � Fixed Effects � Random Effects � Random Coefficients • Generalized Least Square (GLS) • Bayesian Approach 11

Bayesian Principle ( ) ( ) p X Y p θ θ , ( ) ( ) ( ) p X Y p X Y p θ = ∝ θ θ , , ( ) ( ) p X Y p d ∫ θ θ θ , Where ( ) p X Y denotes the posterior distribution θ , θ consists of the parameters ( ) p X Y denotes the likelihood of the observed data given the parameters θ , θ ( ) p denotes the prior distribution of the parameter set θ θ 12

Priori � Prior joint distribution ⎛ ⎞ ⎛ − ⎞ τ τ ( ) 8 ⎜ ( ) ⎟ { } p ⎜ ⎟ 2 θ ∝ k k k k β − β η φ − − ςη ∏ , exp , 1 exp ⎝ ⎠ k uk ⎝ ⎠ π 2 2 k = 0 ( ) N β β Λ ~ , u [ ] T β = − − − − − − 1 , 0 . 44 , 0 . 14 , 0 . 11 , 0 . 1 , 0 . 5 , 0 . 5 , 2 , 4 u 1 Λ τ = : Three precision scenarios: τ = 0.1, 1; CoV =1 σ 2 N ε η ~ ( 0 , ) it gamma η φ ς ~ ( , ) 13

Likelihood Function ⎧ ⎫ ⎛ − ⎞ η η T n ( ( ) ) ( ) i p X Y ⎜ ⎟ p g X ⎨ 2 ⎬ θ = ∆ − β ∏∏ , exp , ⎝ ⎠ it it ⎩ ⎭ π 2 2 i t = 1 = 1 Where n is the number of pavement sections T is the number of time periods when data were collected on pavement i section number i is the observed serviceability loss during time period t on pavement p ∆ it section i η , as in Equation 11 ( ) ( ) g X E p X H H H G N β N 5 β = ∆ = β + β + β + β + β ∆ , exp{ } it it i i i t i t it 0 1 1 2 2 3 3 4 , − 1 14

Posterior Prior ⎧ ⎫ ⎛ − ⎞ η η ( ) T ( ( ) ) n p X Y i p g X ⎜ ⎟ ⎨ 2 ⎬ θ ∝ ∆ − β , ∏∏ exp , ⎝ ⎠ ⎩ ⎭ it it π 2 2 i = t = 1 1 ⎧ ⎫ ⎛ − ⎞ τ τ ( ) { } 8 ⎜ ⎟ ⎨ 2 ⎬ × β − β η − ςη k k k k φ − ∏ , exp , 1 exp ⎝ ⎠ ⎩ k uk ⎭ π 2 2 k = 0 Likelihood 15

Markov Chain Monte Carlo Simulation � Gibbs sampling ( ) f U U U • Joint pdf: , ,..., m 1 2 • Starting with arbitrary values: U U U U ( 0 ) ( 0 ) ( 0 ) ( 0 ) , , ,..., m 1 2 3 • First draw ( ) U f U U U U ( 1 ) ( 0 ) ( 0 ) ( 0 ) ← , ,..., m 1 1 2 3 ( ) U f U U U U ( 1 ) ( 1 ) ( 0 ) ( 0 ) ← , ,..., m 2 2 1 3 ... ... ( ) U f U U U U ( 1 ) ( 1 ) ( 1 ) ( 1 ) ← , ,..., m m m − 1 2 1 ( ) U r U f U r ( ) ⎯ ⎯→ → ∞ ~ d • With , s s s 16

Estimation Results Precision τ = 0.1 τ = 1 CoV=1 Parameters Mean StdDev. Mean StdDev. Mean StdDev. β -5.781 0.029 -5.432 0.028 -5.457 0.027 0 β -0.454 0.044 -0.485 0.044 -0.479 0.048 1 β -0.157 0.015 -0.157 0.016 -0.157 0.016 2 β -0.151 0.014 -0.153 0.014 -0.151 0.014 3 β -0.117 0.006 -0.121 0.006 -0.120 0.006 4 β -0.244 0.031 -0.265 0.030 -0.265 0.028 5 β 0.723 0.172 0.686 0.168 0.768 0.259 6 β 2.164 0.125 2.229 0.190 2.220 0.228 7 β 3.030 0.238 3.222 0.238 3.167 0.241 8 17

Posterior Distributions 18

Performance Forecast � Inspection frequency: once every two years 5 4 3 PSI 2 1 0 0 10 20 30 40 50 60 Time (Weeks) Obs 50 percentile 60 percentile 19

Performance Forecast � Inspection frequency: once per year 5 4 3 PSI 2 1 0 0 10 20 30 40 50 60 Time (Weeks) Obs 50 percentile 60 percentile 20

Performance Forecast � Inspection frequency: once per 6 months 5 4 3 PSI 2 1 0 0 10 20 30 40 50 60 Time (Weeks) Obs 50 percentile 60 percentile 21

Conclusions � Methodology • Bayesian approach to update pavement performance model by accounting for heterogeneity of parameters. � Findings • Heterogeneity is significant, and should be addressed in performance modeling • Not all parameters are normally distributed • Large uncertainty is found in performance forecast; forecast confidence interval is highly sensitive to inspection frequencies. 22

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Thank You 24