Spatial Cross-correlation Models for Vector Intensity Measures (PGA, Ia, PGV and Sa s) Considering Regional Site Conditions

Transcription

1 Spatial Cross-correlation Models for Vector Intensity Measures (PGA, Ia, PGV and Sa s) Considering Regional Site Conditions Gang Wang and Wenqi Du Department of Civil and Environmental Engineering Hong Kong University of Science and Technology March 5, 23 PEER GMSM Working Group Meeting

2 Outlines Introduction Spatial Correlation of Scalar Intensity Measures Strong Motion Database and Regional Site Conditions Spatial Cross correlation of Vector Intensity Measures A Site dependent LMC Model for [PGA, Ia, PGV] A Site dependent LMC Model for Sa(T) Applications and Conclusions The presentation is based on Wang and Du (23), Spatial Cross correlation Models for Vector Intensity Measures (PGA, Ia, PGV and Sa s) Considering Regional Site Conditions, submitted to Bulletin of Seismological Society of America, under review. 2

3 Introduction Modeling spatial variability of ground motion intensity measures (IMs) is essential for the seismic hazard analysis and risk assessment of spatially distributed infrastructure, such as lifelines, transportation systems and structure portfolios; Spatial correlation is not accounted for by GMPEs; It is necessary to consider the simultaneous occurrence of multiple intensity measure (Vector IM); It is important to consider the influence of regional geological features on the correlation structures. 3

4 Introduction Chi Chi h Ground motion prediction equation: ( ) lny = ln Y M, R, θ + η + ε Measured data ij ij i ij Median prediction Inter event residual Intra event residual Epicenter Epicenter and distribution of record stations for the Chi Chi earthquake Distribution of intra event residuals (from Baker) 4

5 Spatial Correlation of Scalar Intensity Measures A Missing Link Spatial Correlation The joint probability of occurrence of ground motion residuals in space. Empirical semi variograms for intra event residuals can be developed to measure the dissimilarity of data separated by separation distance h. N h γ ( h) = z u i i + h z u = i 2 N h ( ) [ ( ) ( )] ( ) 2 The closer the separation distance, the higher probability they are similar. An exponential model can be used to fit the semivariograms. γ ( h) = a( exp( 3 h/ b)) Semivariogram γ(h) Range Separation distance (km) a: the sill of correlation b: the range of correlation h: the separation distance b = 24 km Sill Robust estimator.2 Manual fitted curve

6 Spatial Correlation of Scalar Intensity Measures An exponential model can be used to fit the semivariograms. γ ( h) = a( exp( 3 h/ b)) Spatial correlation coefficient ρ ( h) = exp( 3 h/ b) The valid spatial correlation matrix is always (symmetric) positive semi definite ρ( h2) ρ( h ) J ρ( h2) ρ( hj) ρ( h2j) Semivariogram γ(h) Range Separation distance (km) a: the sill of correlation b: the range of correlation h: the separation distance Sill Robust estimator.2 Manual fitted curve Valid correlation matrix b = 24 km 6

7 Strong Motion Database Eleven well recorded earthquakes (2686 records) are used to investigate the spatial correlation of PGA, Ia, PGV and Sa(T). Moment magnitude Chi Chi California 5.5 Mexico Japan 5 2 Rupture distance (km) Magnitude and rupture distance distribution of records in the database.

8 Strong Motion Database The trend of residuals versus rupture distance and Vs3 should be corrected to avoid artificial correlation. ( ) ( ) ε = lny ln Y M, R, θ ϕ + ϕ ln( R ) + ϕ ln( V ) corr ij ij ij s 8

9 Regional Site Conditions Chi Chi Epicenter Northridge Heterogeneous Homogeneous 9

10 Regional Site Conditions The correlation range of normalized V s3 values (R Vs3 ) are used to quantify the regional site conditions..5 Northridge Chi Chi Semivariogram.5 Semiviriogram Fitted curve NGA data Separation distance (km) Separation distance (km) R Vs3 = km Heterogeneous R Vs3 = 26 km Homogeneous

11 Regional Site Conditions Influence of inferred Vs3 data. 8 Chino Hills Measured Inferred 8 Anza Measured Inferred V s3 (m/s) 6 4 V s3 (m/s) Rupture distance (km) Anza earthquake Rupture distance (km) Chino Hills earthquake A redistributed procedure is applied considering the uncertainty of Vs3 data to reduce artificial correlation induced by inferred data.

12 Regional Site Conditions V s3 (m/s) Chino Hills Measured Inferred Rupture distance (km) Anza earthquake Redistributed V s3 (m/s) Chino Hills Rupture distance (km) Randomized data Original V s3 (m/s) Anza Measured Inferred Rupture distance (km) Chino Hills earthquake Redistributed V s3 (m/s) Anza Rupture distance (km) Randomized data 2

13 Regional Site Conditions R Vs3 is closely related to the correlation range of scalar IMs Range of PGA (km) Range of V s3 (km) Niigata Parkfield CHICHI Northridge Tottori Chino hill Alum rock Anza Chuetsu 5 Range of Ia (km) Range of V s3 (km) (Du and Wang, BSSA 23, in press)

14 Spatial Cross-correlation of Vector Intensity Measures IM IM IM IM ( x ) ( x ) 2 ( x ) 2 ( x ) 2 2 Cross correlation matrix IM( x) IM IM( x2) 2( x) IM 2( x2) ρ () ρ 2 () ρ ρ ( h ) 2 ( h ) 2 2 ρ 2 () ρ 22 () ρ ρ ( h ) 2 2 ( h ) 22 2 ρ ρ ( h ) 2 ( h ) 2 2 ρ () ρ 2 () ρ ρ ( h ) 2 2 R() R( h ) 2 R ( h ) R() 2 ( h ) 22 2 ρ 2 () ρ 22 () IM IM ( x ) ( x ) 2 x IM 2 h2 x 2 IM 2 x2 Cross spatial correlation between IM and IM 2 at separation distance h 2 Given an n component vector IM distributed at J sites, the total correlation matrix is [Jxn, Jxn] in dimension R() R( h J ) R( hj ) R() 4 ( x ) ( )

15 Linear Model of Coregionalization for Vector IMs A combination of a short range and a long range exponential basic function are selected to fit empirical data (LMC) 3h 2 3h R( h) = P exp + P exp r r2 As long as P and P 2 are positive semi definite, the total correlation matrix is guaranteed to be positive semi definite regardless of sites considered (a permissible/valid LMC model). P PGA Ia PGV p p p = i i i 2 3 i i i i p2 p22 p23 i i i p3 p32 p33 p.sd. PGA Ia PGV The P and P 2 matrices can be obtained from each earthquake R() R( h J ) R( hj ) R() Valid correlation matrix 5

16 Linear Model of Coregionalization for Vector IMs PGA 3h 2 3h R( h) = P exp + P exp 6 Ia PGV PGA Ia PGV Cross semivariograms and fitted LMC curves obtained for the Niigata earthquake 6

17 Influence of Site Condition on LMC Matrices PGA 3h 2 3h R( h) = P exp + P exp 6 Short range Long range Ia P p p p = i i i 2 3 i i i i p2 p22 p23 i i i p3 p32 p33 PGV PGA Ia PGV 7

18 Influence of Site Condition on LMC Matrices PGA Ia P = P K 2 P = K RVs3 P R Vs K = = PGV PGA Ia PGV 8

19 A Site-dependent LMC Model for [PGA, Ia, PGV] Site dependent LMC model 3h 2 3h R( hr, Vs3) = P ( RVs3) exp + ( RVs3) exp P 6 RVs3 P = P K P R = K 2 Vs3 A permissible LMC model Positive definite for R vs3 25 km Positive definite K = P.9.65 = Positive definite 9

20 A Site-dependent LMC Model for [PGA, Ia, PGV] Site dependent LMC model RVs3 3h RVs3 3h R( hr, Vs3) = P K exp + exp K 6 Examples PGA Ia PGV R( h= 5, R Vs 3 = 2) = R( h= 5, R Vs 3 = ) = PGA Ia PGV Influence of site conditions K = Weaker Stronger 2

21 A Site-dependent LMC Model for [PGA, Ia, PGV] Site dependent LMC model RVs3 3h RVs3 3h R( hr, Vs3) = P K exp + exp K 6 Reduce to local correlation matrix (h=).9.65 R( h = ) R() = P = Compare with Campbell and Bozorgnia (22): ρpgapga, ρpgaia, ρpgapgv, ρpga, Ia ρia, Ia ρ Ia, PGV = ρpga, PGV ρia, PGV ρ PGV, PGV

22 A Site-dependent LMC Model for [PGA, Ia, PGV] Site dependent LMC model RVs3 3h RVs3 3h R( hr, Vs3) = P K exp + exp K 6 Reduce to heterogeneous site conditions (R Vs3 =), 3h R ( h) =R()exp Averaged LMC model for [PGA, Ia, PGV] 3h 2 3h R( h) = Pavg exp + avg exp P 6 P avg = P 2 avg =

23 Model Predictions Northridge Earthquake PGA Semivariogram.5 5 Sepration distance (km) 3h 2 3h R( h) = P exp + P exp 6 Empirical semivariogram Predicted curve using site dependent matrices Eq.(9) Predicted curve using averaged matrices Eq.(26) Ia Semivariogram.5 5 Sepration distance (km) Semivariogram.5 5 Sepration distance (km) PGV Semivariogram.5 5 Sepration distance (km) Semivariogram.5 5 Sepration distance (km) Semivariogram.5 5 Sepration distance (km) PGA Ia PGV 23

24 Model Predictions Chi Chi Earthquake PGA Semivariogram.5 5 Sepration distance (km) 3h 2 3h R( h) = P exp + P exp 6 Empirical semivariogram Predicted curve using site dependent matrices Eq.(9) Predicted curve using averaged matrices Eq.(26) Ia Semivariogram.5 5 Sepration distance (km) Semivariogram.5 5 Sepration distance (km) PGV Semivariogram.5 5 Sepration distance (km) Semivariogram.5 5 Sepration distance (km) Semivariogram.5 5 Sepration distance (km) PGA Ia PGV 24

25 A Site-dependent LMC Model for Sa(T) Site dependent LMC model for Sa(T): 3h 2 3h R( hr, Vs3) = P exp + exp P 7 RVs3 P = PSa KSa 2 2 RVs3 P = PSa + KSa Averaged LMC model (Loth and Baker, 23) 3h 3h R( ) = P exp + exp + 2 P 7 P 2 3 h I h = Loth C., and J. W. Baker (23). A spatial cross correlation model of ground motion spectral accelerations at multiple periods. Earthquake Engineering & Structural Dynamics, (in press). 25

26 A Site-dependent LMC Model for Sa(T) P Sa p.d. RVs3 P = PSa KSa Positive definite for R vs3 25 km 2 P Sa p.d. 2 2 RVs3 P = PSa + KSa Positive definite K Sa p.d. 3h 2 3h R = P exp + exp P 7 Permissible LMC model The model can be interpolated for other periods and still remains 26 valid.

27 Model Prediction Northridge Earthquake 3h 2 3h R( h) = P exp + P exp 6 27

28 Model Prediction Chi-Chi Earthquake 3h 2 3h R( h) = P exp + P exp 6 28

29 Model Applications Random Vector IM Fields (PGA, Ia and PGV are in the natural log scale, in the unit of g, m/s and cm/s, respectively; V s3 is in the unit of m/s)

30 Model Applications Fully Probabilistic Approach using Spatiallycorrelated Vector IMs cf. Du and Wang, Fully Probabilistic Seismic Displacement Analysis of Spatially Distributed Slopes Using Spatially Correlated Vector Intensity Measures, 23 (under review) 3

31 Conclusions Simple permissible spatial correlation models are developed for vector IMs (PGA, Ia, PGV, and Sa) using eleven recent earthquakes. The correlation range of V s3, R Vs3, is found to be a good indicator to characterize the regional geological conditions. In general, the spatial correlations of IMs becomes stronger for a homogeneous regional site condition. The spatial correlation models can be conveniently used in regional specific seismic hazard analysis and loss estimation of spatially distributed infrastructure. 3

32 Questions?