Spatial Correlation of Ground Motions
inSeismic Hazard Assessment

Taojun Liutliu82@uwo.ca
Department of Civil & Environmental Engineering
University of Western Ontario
London, Ontario, Canada1

Outline• Introduction• Spatial Correlations of Ground-Motion Parameters– Estimation of spatial correlations– Observed spatial correlations of ground motion parameters• Simulation of Spatial Correlated Ground Motions– Test using stochastic finite-fault program (EXSIM)– Potential simulation method• Summary2Introduction• What is spatial correlation?
The similarity of ground motions at two sites.

• Spatial correlation of ground motion parameters
Ground Motion Prediction Equation:=ln Aij f ( M i , Rij , θij ) + η j + ε ijηjand εij represent the interevent variability and theintraevent variability, respectively.

• Spatial correlation is important for seismic hazard
assessment of spatially distributed structures and/orstructures with large foot print, such as highway andpipeline.
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Simulation-Based Seismic Risk Analysis Framework

Spatial Correlated
Ground Motions

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Estimation of Spatial Correlations• The total correlation coefficient (between ηj(T1)+εij(T1) andηj(T2)+εkj(T2)):
ρη (T1 , T2 )σ η (T1 )σ η (T2 ) + ρε (∆, T1 , T2 )σ ε (T1 )σ ε (T2 )ρT (∆, T1 , T2 ) =σ T (T1 )σ T (T2 )σ T (T1 )] + [σ T (T2 ) ] − [σ d (∆, T1 , T2 ) ][ρ ( ∆, T , T ) =2orT

1222σ T (T1 )σ T (T2 )2where [σ d (∆, T1 , T2 )] : variance of [ηj(T1)+εij(T1)]-[ηj(T2)+εkj(T2)](Boore et al., 1997)25

Estimation of Spatial Correlations• The intraevent spatial correlation coefficient (i.e., spatial correlation
within a single event) can be expressed as:

σ d (∆, T1 , T2 ) ][ρε (∆, T1 , T2 ) =1−
2σ ε (T1 )σ ε (T2 )2• If T1 and T2 equal to T, we have:σ d ( ∆, T ) ][ρε (∆, T ) =−12
2 [σ ε (T ) ]2• Empirical relation for ρε(∆,T):
ρε ( ∆, T=) exp ( −α∆ β )6Observed Spatial Correlations

California records

Chi-Chi records
7Simulation of Spatial Correlated Ground Motions• EXSIM stochastic finite-fault program (Motazedianand Atkinson (2005)), a freely-available and widelyused program that has been validated under a range ofconditions.
• Chi-Chi Earthquake8

Intraevent Spatial Correlation Characteristics of EXISM• Ground Motion Prediction Equations (GMPEs):
 VS 30 ln A =c0 + c1 ln R + c2 R + c3 ln +ε
 Vref 2
where A is the ground-motion parameter;=RRrup+ h 2, whereRrup is the closest distance from the recording site to the faultrupture plane and h is an “added depth” term that builds innear-source saturations due to finite-fault effects. Vref = 760
m/s, c0, c1, c2 and c3 are regression coefficients to be
determined, and ε is the intraevent residual.

9Model Parameters in EXSIMParameter
Fault orientation (strike/dip)
Fault dimensions along strike and dip (km)Depth of the upper edge of the fault (km)Mainshock moment (dyne∙cm)
Subfault dimensions (km)Stress parameter Δσ (bar)Radiation-strength factor
Number of subsources summed
Q(f)Geometric spreading

Windowing function
Kappa (sec)
Crustal amplification
Crustal shear-wave velocity (km/sec)
Rupture velocity (km/sec)
Crustal density (g/cm3)

Parameter value
5°/34°
110 by 40
02.8 × 1027
5×5
100
1.0
176
117∙f0.77
1/R for R < 50 km
1/R0 for 50 km ≤ R < 150 km
1/R0.5 for R ≥ 150 km
Cosine-tapered boxcar
0.06
Boore and Joyner (1997) western North
America generic rock site
3.2
0.8 × (shear-wave velocity)
2.710

Intraevent Spatial Correlation Characteristics of EXISM

Model bias (log10[Obs./Sim.]) of EXSIM simulations

Model bias

0.5

0.0-0.5 -1
10010Frequency (Hz)101

Averaged model bias at 389 stations.
95% confidence interval of the mean is shown in dashed lines.
11Intraevent Spatial Correlation Characteristics of EXISM

0.80.6SA at 0.3 s
SA at 0.3 s fitted
SA at 1.0 s
SA at 1.0 s fitted
SA at 3.0 s
SA at 3.0 s fitted

0.4
0.2
0.004030
20
10Separation distance, ∆ (km)50(b) 1.0Intraevent correlation, ρε(∆,T)

Intraevent correlation, ρε(∆,T)(a) 1.00.80.6

SA at 0.3 s
SA at 0.3 s fitted
SA at 1.0 s
SA at 1.0 s fitted
SA at 3.0 s
SA at 3.0 s fitted

0.40.20.0010203040
Separation distance, ∆ (km)50

Estimated intraevent spatial correlation coefficient ρε(∆,T) samples and their fitted
curves of SAs for T equal to 0.3, 1.0 and 3.0 sec: (a) using the Chi-Chi records; (b)using simulation records (one trial).
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Intraevent Spatial Correlation Characteristics of EXISM
0SA at 1.0 s (g)

10-1

10-2

100.1

Chi-Chi records
EXSIM simulation (rock)
EXSIM simulation (soil)
1

10100

Rjb (km)Ground motions of Chi-Chi records and simulation (one trial) versus Rjb (closesthorizontal distance from site to surface projection of the rupture) in units of g for
SA at 1.0 sec.
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Intraevent Spatial Correlation Characteristics of EXISM
We introduce an additional intraevent variability (i.e., an error
term εE(T)) to the ground-motion parameter from the EXSIM
simulations, εsim(T), whose standard deviation is σεsim(T), suchthat the new error term from the simulations after this postprocessing, ε’sim(T), is given by,′ε=ε sim (T ) + ε E (T )sim ( T )where εE(T) is a normal variate with zero mean and standard
deviation of σE(T) such that the variance of ε'sim(T), σ2ε(T), equals
the variance obtained from the actual records:=σ ε2 (T ) σ ε2sim (T ) + σ E2 (T )14

Intraevent Spatial Correlation Characteristics of EXISM

After post-processing:
0SA at 1.0 s (g)

10-1

10-2

100.1

Chi-Chi records
EXSIM simulation (rock)
EXSIM simulation (soil)
1

10100Rjb (km)15

Intraevent Spatial Correlation Characteristics of EXISM

σ d ( ∆, T ) ][1ρε (∆, T ) =−22 [σ ε (T ) ]2

Comparison of the intraevent spatial correlation coefficient of the Chi-Chi
records, EXSIM simulation and modified simulation with added variability for
SA at 0.3 and 3.0 sec (only fitted curves are shown)16

Potential Simulation Method for Spatially Correlated
Ground Motions• Coherency,

γ jk (ω ) =S jk (ω )S jj (ω ) S kk (ω )is a measure of “similarity” in the seismic motions, andindicates the degree to which the data recorded at thetwo stations are related by means of a linear transferfunction.

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Summary• Spatial correlation is important in seismic hazard
assessment• Spatial correlations of ground motion parameters(PGA, SAs) are investigated• The widely-used stochastic simulation technique fails
to reproduce observed spatial correlations• Potential simulation method: coherency-based
method18

Acknowledgements
I would like to acknowledge the contributions ofDr. Hanping Hong,Dr. Katsuichiro Goda,
Dr. Gail M. Atkinson andDr. Karen Assatourians
for the presented information.

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Detailed information on this talk can be found in:
Taojun Liu, Gail M. Atkinson, Hanping Hong, and Karen Assatourians (2011)
Intraevent Spatial Correlation Characteristics of Stochastic Finite-Fault
Simulations, submitted to Bulletin of the Seismological Society of America.
Goda, K. and Hong, H.P. (2008) Spatial correlation of peak ground motions
and response spectra, Bulletin of the Seismological Society of America, 98, 1:
354-365.
Goda, K. and Hong, H.P. (2008) Estimation of Seismic Loss for SpatiallyDistributed Buildings, Earthquake Spectra, 24, pp. 889-910.
Hong, H.P., Goda, K. and Davenport, A. G. (2006) Seismic hazard analysis: acomparative study, Canadian Journal of Civil Engineering, 33(9), 1156-1171.

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Thank you!

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