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

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Updating Pavement Deterioration Models Using the Bayesian Principles and Simulation Techniques

1st AISIM

University of Waterloo, August 6, 2005

Feng Hong Jorge A Prozzi

The University of Texas at Austin

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Outline

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

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Pavement Performance

Distresses

Rutting
Cracking
Pothole
Faulting
Spalling

Subjective indicators

Present Serviceability Rating (PSR)

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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)

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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

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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

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Pavement Deterioration Model

c t t

bN a p − =

Original AASHO Model

( )

α

ρ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = W p p p p

f t

Where

t

p :

serviceability at time t

p : serviceability at time t = 0, i.e. initial serviceability

f

p : terminal serviceability

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

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b = bSbE

Structural variables
H1, H2, H3 : thickness for surface, base and subbase

layers

Environmental variable
Gt: Frost penetration gradient

Determining term b

{ } { }

3 3 2 2 1 1

exp exp H H H bS γ γ γ λ + + =

} exp{

t E

G b ϕ =

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Taylor expansion

ε + ∆ = − = ∆

− −

N dN p p p

e t t t t 1 1

Where

d ,e: parameters to be estimated

1 − t

N : some measure of accumulative traffic until the one-period

backward of time t

N ∆ : projected incremental traffic during the time period between t-1

and t

ε :

error term

Incremental Model

{ }

{ } { }

t

G H H H d ϕ γ γ γ λ exp exp exp

3 3 2 2 1 1 '

+ + =

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Final Specification

it it t i t i i i it

N N G H H H p ε β β β β β

β

+ ∆ + + + + = ∆

−

5

1 , 4 3 3 2 2 1 1

} exp{

∑

− = −

∆ =

1 1 1 , t l il t i

N N ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ∆

8 8 8

18 18 18

7 6 β β β

β β

i i i i i il il

TA B SA A FA n N

Where

it

p ∆ : serviceability loss in pavement section i during time period t

1 , − t i

N : cumulative traffic until the one-period backward of time t

il

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

i

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

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Estimation Approach for Panel Data

Ordinary Least Square (OLS) Fixed Effects Random Effects Random Coefficients

Generalized Least Square (GLS)
Bayesian Approach

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Bayesian Principle

( ) ( ) ( ) ( ) ( ) ( ) ( )

θ θ θ θ θ θ θ θ p Y X p d p Y X p p Y X p Y X p , , , , ∝ =

∫

Where

( )

Y X p , θ

denotes the posterior distribution

θ consists of the parameters

( )

θ Y X p ,

denotes the likelihood of the observed data given the parameters

θ

( )

θ p

denotes the prior distribution of the parameter set θ

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Priori

Prior joint distribution

( )

Λ , ~

u

N β β

[ ]

T u

4 , 2 , 5 . , 5 . , 1 . , 11 . , 14 . , 44 . , 1 − − − − − − = β ) , ( ~ η ε N

it

) , ( ~ ς φ η gamma

( ) ( ) { }

ςη η β β τ π τ θ

φ

− ∏ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛− ∝

− =

exp 2 exp 2

1 8 2 , , k uk k k k k k

p

2

1 : σ τ = Λ

Three precision scenarios: τ = 0.1, 1; CoV =1

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Likelihood Function

( ) ( ) ( )

∏∏

= =

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − ∆ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛− =

n i T t it it

i

X g p Y X p

1 1 2

, 2 exp 2 , β η π η θ Where

n is the number of pavement sections

i

T is the number of time periods when data were collected on pavement

section number i

it

p ∆

is the observed serviceability loss during time period t on pavement section i

η , as in Equation 11

( ) ( )

it t i t i i i it it

N N G H H H X p E X g ∆ + + + + = ∆ =

−

5

1 , 4 3 3 2 2 1 1

} exp{ ,

β

β β β β β β

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Posterior

( ) ( ) ( )

( ) { }

ςη η β β τ π τ β η π η θ

φ

− ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛− × ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − ∆ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛− ∝

− = = =

∏ ∏∏

exp 2 exp 2 , 2 exp 2 ,

1 8 2 , , 1 1 2 k uk k k k k k n i T t it it

i

X g p Y X p Prior Likelihood

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Markov Chain Monte Carlo Simulation

Gibbs sampling

Joint pdf:
Starting with arbitrary values:
First draw
With ,

( )

m

U U U f ,..., ,

2 1 ) ( ) ( 3 ) ( 2 ) ( 1

,..., , ,

m

U U U U

( )

) ( ) ( 3 ) ( 2 1 ) 1 ( 1

,..., ,

m

U U U U f U ←

( )

... ... ,..., ,

) ( ) ( 3 ) 1 ( 1 2 ) 1 ( 2 m

U U U U f U ←

( )

) 1 ( 1 ) 1 ( 2 ) 1 ( 1 ) 1 (

,..., ,

−

←

m m m

U U U U f U

∞ → r

( )

s s d r s

U f U U ~

) (

⎯→ ⎯

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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

1

β

0.454

0.044

0.485

0.044

0.479

0.048

2

β

0.157

0.015

0.157

0.016

0.157

0.016

3

β

0.151

0.014

0.153

0.014

0.151

0.014

4

β

0.117

0.006

0.121

0.006

0.120

0.006

5

β

0.244

0.031

0.265

0.030

0.265

0.028

6

β

0.723 0.172 0.686 0.168 0.768 0.259

7

β

2.164 0.125 2.229 0.190 2.220 0.228

8

β

3.030 0.238 3.222 0.238 3.167 0.241

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Posterior Distributions

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Performance Forecast

Inspection frequency: once every two years

1 2 3 4 5 10 20 30 40 50 60 Time (Weeks) PSI Obs 50 percentile 60 percentile

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Performance Forecast

Inspection frequency: once per year

1 2 3 4 5 10 20 30 40 50 60 Time (Weeks) PSI Obs 50 percentile 60 percentile

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Performance Forecast

Inspection frequency: once per 6 months

1 2 3 4 5 10 20 30 40 50 60 Time (Weeks) PSI Obs 50 percentile 60 percentile

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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.

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