geostatistics for modeling of soil spatial variability in ...bakerjw/publications/baker_(2006... ·...

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Eidgenossische Technische Hochschule ZürichSwiss Federal Institute of Technology Zürich

Institute of Structural EngineeringGroup Risk and Safety

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Geostatistics for modeling of soil spatial variability in Adapazari, Turkey

Jack W. BakerMichael H. Faber

Institute of Structural Engineering (IBK)Group Risk and Safety

ETH - Zürich



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Practical evaluation of liquefaction occurrence

• Obtained from empirical observations

• Useful for practical evaluations

• Applied only at single locations

(from Seed et al.)

An approach is proposed here for incorporating spatial dependence of soil properties to evaluate the potential spatial extent of liquefaction

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Proposed procedure for evaluating liquefaction extent

Note that soil property values may be available for a few locations



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A demonstration site in Adapazari, Turkey

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Sampled N1,60 values near the demonstration site in Adapazari, Turkey

Test site Sampled soil values



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Sampled N1,60 values near the demonstration site in Adapazari, Turkey

( )( ) 0.85 1 exp 5000hhρ ⎡ ⎤= −⎢ ⎥⎣ ⎦

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Simulation of N1,60 values, conditional on observations (Sequential Gaussian Simulation)

1. Transform data so that it is normally distributed: ( )1 ( )z F y−=Φ

Empirical CDF of N1,60 values Standard normal CDF



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2. Estimate spatial dependence of soil properties

( )( ) 0.85 1 exp 5000hhρ ⎡ ⎤= −⎢ ⎥⎣ ⎦

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3. Simulate one additional point (Z1), conditional upon observed values

1 12

21 22

0 1,

orig

ZN⎛ ⎞⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎝ ⎠

ΣZ 0 Σ Σ∼ ( ) ( )1 1

1 12 22 12 22 21| , 1origZ N − −= ⋅ ⋅ − ⋅ ⋅Z z Σ Σ z Σ Σ Σ∼

( )( ) 0.85 1 exp 5000hhρ ⎡ ⎤= −⎢ ⎥⎣ ⎦



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4. Repeat step 3 for each location, treating previously simulated points as fixed

1 12

21 22

0 1,

orig

ZN⎛ ⎞⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎝ ⎠

ΣZ 0 Σ Σ∼ ( ) ( )1 1

1 12 22 12 22 21| , 1origZ N − −= ⋅ ⋅ − ⋅ ⋅Z z Σ Σ z Σ Σ Σ∼

( )( ) 0.85 1 exp 5000hhρ ⎡ ⎤= −⎢ ⎥⎣ ⎦

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4. Repeat step 3 for each location, treating previously simulated points as fixed

5. Transform the simulated Gaussian variables back to the original distribution

1 12

21 22

0 1,

orig

ZN⎛ ⎞⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎝ ⎠

ΣZ 0 Σ Σ∼ ( ) ( )1 1

1 12 22 12 22 21| , 1origZ N − −= ⋅ ⋅ − ⋅ ⋅Z z Σ Σ z Σ Σ Σ∼

( )( ) 0.85 1 exp 5000hhρ ⎡ ⎤= −⎢ ⎥⎣ ⎦



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Conditional simulations of N1,60

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Mean and coefficient of variation of conditional N1,60simulations

Mean CoV



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Comments on Sequential Gaussian Simulation

• Common in petroleum engineering and mining

• Requires jointly Gaussian random variables– Partially achieved using the normal score transform– This numerical transform is applicable for any probability distribution

• Can also be used for vectors of dependent parameters (e.g., SPT blow count and fines content)

• It is not necessary to condition on all previous points when simulating

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Procedure for evaluating liquefaction extent



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0.85

0.85

0.0198 if 12m

12 0.0198 if >12mrd

d d

dεσ

⎧ ⋅ ≤⎪=⎨⋅⎪⎩

'0.65 veq d

v

PGACSR rg

σσ⎛ ⎞⎛ ⎞

= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

*,12m

*,12m

*max ,12m

0.341( 0.0785 7.586)

*max ,12m

0.341(0.0785 7.586)

23.013 2.949 0.999 0.05251

16.258 0.20123.013 2.949 0.999 0.0525

116.258 0.201

s

d

s

w sd V

d rw sV

a M V

era M V

e

ε− + +

+

⎡ ⎤− + + ++⎢ ⎥

⎢ ⎥+⎣ ⎦= +⎡ ⎤− + + ++⎢ ⎥

⎢ ⎥+⎣ ⎦

'

1,60( , ) (1 0.004 ) 13.32ln 29.53ln 3.70ln 0.05 44.97veq L

ag N FC CSR M FC

Pσ

ε= + − − − + + +X u

A probabilistic liquefaction criterion (Cetin et al., 2004)

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0.85

0.85

0.0198 if 12m

12 0.0198 if >12mrd

d d

dεσ

⎧ ⋅ ≤⎪=⎨⋅⎪⎩

'

1,60( , ) (1 0.004 ) 13.32ln 29.53ln 3.70ln 0.05 44.97veq L

ag N FC CSR M FC

Pσ

ε= + − − − + + +X u

'0.65 veq d

v

PGACSR rg

σσ⎛ ⎞⎛ ⎞

= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

*,12m

*,12m

*max ,12m

0.341( 0.0785 7.586)

*max ,12m

0.341(0.0785 7.586)

23.013 2.949 0.999 0.05251

16.258 0.20123.013 2.949 0.999 0.0525

116.258 0.201

s

d

s

w sd V

d rw sV

a M V

era M V

e

ε− + +

+

⎡ ⎤− + + ++⎢ ⎥

⎢ ⎥+⎣ ⎦= +⎡ ⎤− + + ++⎢ ⎥

⎢ ⎥+⎣ ⎦



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0.85

0.85

0.0198 if 12m

12 0.0198 if >12mrd

d d

dεσ

⎧ ⋅ ≤⎪=⎨⋅⎪⎩

'

1,60( , ) (1 0.004 ) 13.32ln 29.53ln 3.70ln 0.05 44.97veq L

ag N FC CSR M FC

Pσ

ε= + − − − + + +X u

'0.65 veq d

v

PGACSR rg

σσ⎛ ⎞⎛ ⎞

= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

*,12m

*,12m

*max ,12m

0.341( 0.0785 7.586)

*max ,12m

0.341(0.0785 7.586)

23.013 2.949 0.999 0.05251

16.258 0.20123.013 2.949 0.999 0.0525

116.258 0.201

s

d

s

w sd V

d rw sV

a M V

era M V

e

ε− + +

+

⎡ ⎤− + + ++⎢ ⎥

⎢ ⎥+⎣ ⎦= +⎡ ⎤− + + ++⎢ ⎥

⎢ ⎥+⎣ ⎦

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Realizations of liquefaction extent



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Probabilistic evaluation of consequences, given a M=7 earthquake causing a PGA of 0.3g

( )( )( | , ) ( | , ) ( )P Y y pga m I h pga m y f d> = >∫ ηη

g η e e

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Probability of liquefaction at the study site, given a M=7.4 earthquake causing a PGA of 0.3g



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The previously computed result for a deterministic loading case can be combined with stochastic loading obtained from

probabilistic seismic hazard analysis (PSHA)

(see Baker and Faber, 2006)

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Comments

• The procedure is useful for organizing the many pieces of relevant engineering data, but the inputs need careful consideration

• Soil random field models are challenging to characterize– Soil layering?– Unidentified local soil lenses?– Homogeneity/Ergodicity assumptions?

• Modeling post-liquefaction behavior is a challenge

• Geotechnical data often comes from several sources of varying quality



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Conclusions

• A framework has been proposed for modeling the spatial extent of liquefaction– Accounts for spatial dependence of soil properties– Incorporates known values of soil properties at sampled locations– Complex functions of the spatial extent of liquefaction can be

evaluated

• Probabilistic seismic hazard analysis can be used to incorporate all possible ground motion intensities– Avoids the use of a scenario load intensity with unknown recurrence

rate– Provides an explicit estimate of annual occurrence probabilities

• This approach potentially allows for the design of projects with uniform levels of reliability

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