Repeatable Path Effects on The Standard Deviation for Empirical Ground Motion Models Po-Shen Lin (Institute of geophysics, NCU) Chyi-Tyi Lee (Institute of applied geology, NCU) Norman Abrahamson (PG&E) Melanie Walling (UCB) Brian Chiou (Caltran) The Next Generation of Research on Earthquake-induced Landslides: An International Conference in Commemoration of the 10th Anniversary of the Chi-Chi Earthquake

Outline Introduction Introduction Ground motion prediction equations (GMPE) for Taiwan Ground motion prediction equations (GMPE) for Taiwan Variability of empirical ground motion prediction equation Variability of empirical ground motion prediction equation Decomposition of the variability of empirical ground motion prediction equation Decomposition of the variability of empirical ground motion prediction equation Conclusions Conclusions

What ’ s is Ground Motion Prediction Equations (GMPE) “ Was ” “ Was ” Attenuation relationship Attenuation relationship Attenuation model Attenuation model Attenuation equation Attenuation equation It is an equation that can be used to predict the possible ground-motion value during a future earthquake. It is an equation that can be used to predict the possible ground-motion value during a future earthquake. Most of then are “ empirical ”, and was developed from a set of ground-motion data with proper physical meaning Most of then are “ empirical ”, and was developed from a set of ground-motion data with proper physical meaning

Motivation Several recent studies have estimated the reduction in the standard deviation for single sites and for single sites with the earthquakes restricted to small source regions: Chen and Tsai (2002), Atkinson (2006), and Morikawa et al. (2008) Several recent studies have estimated the reduction in the standard deviation for single sites and for single sites with the earthquakes restricted to small source regions: Chen and Tsai (2002), Atkinson (2006), and Morikawa et al. (2008) The standard deviation of the ground motion has a large impact on the seismic hazard at long return periods (e.g. Restrepo-Velez and Bommer, 2003) The standard deviation of the ground motion has a large impact on the seismic hazard at long return periods (e.g. Restrepo-Velez and Bommer, 2003) A possible approach for improving the models of the standard deviation is to remove the ergodic assumption A possible approach for improving the models of the standard deviation is to remove the ergodic assumption

The goal of this research To estimate the reduction in the aleatory variability if the ergodic assumption is removed To estimate the reduction in the aleatory variability if the ergodic assumption is removed To model the spatial correlations of the repeatable path and source effects. To model the spatial correlations of the repeatable path and source effects.

Ground motion Prediction equation for Taiwan MwR F NM F NM F NM F RV F RV F RV C 1 C 8 H Mw is moment magnitude ， R is the closest distance to the fault plane ， F NM for indicate normal fault( when rake angle -120~-60 degree ， F NM =1 ， other angle F NM =0) ， F RV for indicate reverse fault( when rake angle between 30~150 degree ， F RV =1 ， other angle F RV =0) ， C 1 ~C 8 and H were coefficient from regression alaysis 。 Distance scaling Magnitude scaling Site condition Focal mechanism

The results of GMPE for Taiwan Different focal mechanism and magnitude Different magnitude and Vs30

Variation of the standard deviation in the SA equation with periods from sec Total sigma Inter-event sigma (τ) Intra-event sigma (σ)

` Station Earthquake epicenter

Station Earthquake epicenter

The notations i → event i → event k → station k → station l → region l → region Residuals Residuals Intra-event, inter-event Intra-event, inter-event Inter-station, intra-station Inter-station, intra-station Inter-path, intra-path Inter-path, intra-path Inter-region, intra-region Inter-region, intra-region Standard deviations Standard deviations Intra-event, inter-event Intra-event, inter-event Inter-station, intra-station Inter-station, intra-station Inter-path, intra-path Inter-path, intra-path Inter-region, intra-region Inter-region, intra-region Following the notation of Walling (2009)

Components of Variability In most modern empirical ground motion studies, the total variability is separated into inter-event and intra-event components, the observed ground motion can be written as In most modern empirical ground motion studies, the total variability is separated into inter-event and intra-event components, the observed ground motion can be written as The standard deviations of the intra-event residuals, , and the inter-event residuals, , are σ and τ, respectively. The intra- event and inter-event residuals are independent so the total standard deviation, σ T, can be written as The standard deviations of the intra-event residuals, , and the inter-event residuals, , are σ and τ, respectively. The intra- event and inter-event residuals are independent so the total standard deviation, σ T, can be written as

Components of the Intra-Event Standard Deviation If we have multiple recordings at each site, then we can separate out the median site-specific amplification for each site and the intra-event residual can be written as If we have multiple recordings at each site, then we can separate out the median site-specific amplification for each site and the intra-event residual can be written as where is the inter-site residual for the k th site and is the record-to-record (intra-site) residual from the i th earthquake at the k th site after the median site-specific amplification has been removed. where is the inter-site residual for the k th site and is the record-to-record (intra-site) residual from the i th earthquake at the k th site after the median site-specific amplification has been removed.

Components of the Intra-Event Standard Deviation The intra-event standard deviation can then be separated into the standard deviation of the differences in the median site-specific amplification factors ( ) and the standard deviation of the record-to-record variability at a single site ( ). The term is the standard deviation of the and the term is the standard deviation of the The intra-event standard deviation can then be separated into the standard deviation of the differences in the median site-specific amplification factors ( ) and the standard deviation of the record-to-record variability at a single site ( ). The term is the standard deviation of the and the term is the standard deviation of the

Components of the Intra-Event Standard Deviation The record-to-record residuals include the variability due to differences in the wave propagation for different ray paths. If we have recordings from multiple earthquakes in a single small source region, then we can separate out the differences in the median path-specific effects from the record-to-record residuals. The subscript l is used to identify the region. The record-to-record residuals include the variability due to differences in the wave propagation for different ray paths. If we have recordings from multiple earthquakes in a single small source region, then we can separate out the differences in the median path-specific effects from the record-to-record residuals. The subscript l is used to identify the region.

Components of the Intra-Event Standard Deviation The term can be separated into the standard deviation of the median path-specific effects,, and the remaining unexplained intra-event variability,. The term is the standard deviation of the and the term is the standard deviation of the The term can be separated into the standard deviation of the median path-specific effects,, and the remaining unexplained intra-event variability,. The term is the standard deviation of the and the term is the standard deviation of the

Components of the Inter-Event Standard Deviation If we have recordings from multiple earthquakes from a single source region, then we can separate out the regional differences in the median event terms. The inter-event residual can be written as If we have recordings from multiple earthquakes from a single source region, then we can separate out the regional differences in the median event terms. The inter-event residual can be written as

Components of the Inter-Event Standard Deviation The inter-event standard deviation,, can be separated into the standard deviation of the median inter-event residuals for small source regions,, and the standard deviation of the inter-event residuals from earthquakes within a small source region, The inter-event standard deviation,, can be separated into the standard deviation of the median inter-event residuals for small source regions,, and the standard deviation of the inter-event residuals from earthquakes within a small source region,

Decomposition of the variability of empirical ground motion prediction equation - conclusion Inter-eventIntra-event Inter-site (site-term) record-to-record path-to-path small source region Single-path sigma Single-site sigma (single-station sigma) unexplained intra-event variability unexplained inter-event variability

Data Set A total of 63 crustal earthquakes recorded at sites with at least 20 recordings were compiled from the TSMIP data set. A total of 63 crustal earthquakes recorded at sites with at least 20 recordings were compiled from the TSMIP data set. The resulting strong motion database contains 2054 recordings from 63 crustal earthquakes recorded at 84 sites in Taiwan The resulting strong motion database contains 2054 recordings from 63 crustal earthquakes recorded at 84 sites in Taiwan

Station Earthqauke Data Set A total of 63 crustal earthquakes recorded at sites with at least 20 recordings were compiled from the TSMIP data set. A total of 63 crustal earthquakes recorded at sites with at least 20 recordings were compiled from the TSMIP data set. The resulting strong motion database contains 2054 recordings from 63 crustal earthquakes recorded at 84 sites in Taiwan The resulting strong motion database contains 2054 recordings from 63 crustal earthquakes recorded at 84 sites in Taiwan

Data Set Histogram of the number of recordings per station Histogram of the V S30 distribution for the 89 sites

Period dependence of the data set of usable response spectral values Number of Recordings Number of Earthquakes PGA T= T= T= T= T=

Approach for Quantifying Site, Path, and Source Effects In the first step, the residuals of the TSMIP data are computed from a standard empirical ground motion prediction equation. In the first step, the residuals of the TSMIP data are computed from a standard empirical ground motion prediction equation. In the second step, using a random effects regression, the residuals are fit to the following form In the second step, using a random effects regression, the residuals are fit to the following form The intra-event residuals were then modeled using a second random effects regression to estimate the inter- site residuals: The intra-event residuals were then modeled using a second random effects regression to estimate the inter- site residuals:

Results Event Random Effect   Record Random Effect C1C1  C2C2 pp rr  PGA T= T= T= T= T=

Approach for Quantifying Site, Path, and Source Effects A normalized record-to-record residual,, is defined as A normalized record-to-record residual,, is defined as For each site, we compute the difference between the normalized record-to-record residuals for all pairs of earthquakes recorded at that site and divide by sqrt(2) so that the difference has unit variance. The normalized difference in the record-to-record residuals between the i th and j th earthquakes recorded at the k th site is given by For each site, we compute the difference between the normalized record-to-record residuals for all pairs of earthquakes recorded at that site and divide by sqrt(2) so that the difference has unit variance. The normalized difference in the record-to-record residuals between the i th and j th earthquakes recorded at the k th site is given by

Approach for Quantifying Site, Path, and Source Effects Instead of define specific regions (l subscript) and assuming that the path effects are the same for all sources in a region, we use a region-less approach using a parameter that describes the similarity of two source-site paths Instead of define specific regions (l subscript) and assuming that the path effects are the same for all sources in a region, we use a region-less approach using a parameter that describes the similarity of two source-site paths

The path closeness index (CI) Instead of define specific regions (l subscript) and assuming that the path effects are the same for all sources in a region, we use a region-less approach using a parameter that describes the similarity of two source-site paths as shown

Approach for Quantifying Site, Path, and Source Effects Using this definition of the closeness index, we evaluate the dependence of the standard deviation of on the closeness index to develop a model for the spatial correlation of the path effect and the contribution of the path effect to Using this definition of the closeness index, we evaluate the dependence of the standard deviation of on the closeness index to develop a model for the spatial correlation of the path effect and the contribution of the path effect to

Results

Coefficients for the CI dependence of the record-to-record standard deviation b1b1 b2b2 b3b3 PGA T= T= T= T= T=

At CI=0, the paths are the same so =0. Therefore, the standard deviation of the remaining unexplained intra-event variability,, is given by the standard deviation of at CI=0 multiplied by to remove the normalization: At CI=0, the paths are the same so =0. Therefore, the standard deviation of the remaining unexplained intra-event variability,, is given by the standard deviation of at CI=0 multiplied by to remove the normalization:

Approach for Quantifying Site, Path, and Source Effects A similar approach is used for the inter-event standard deviation. A normalized inter-event residual,, is defined as A similar approach is used for the inter-event standard deviation. A normalized inter-event residual,, is defined as The normalized difference in the inter-event residuals between the i th and j th earthquakes recorded is given by The normalized difference in the inter-event residuals between the i th and j th earthquakes recorded is given by

Approach for Quantifying Site, Path, and Source Effects We then evaluate the dependence of the standard deviation of on the distance between hypocenters,, and develop a model for the spatial correlation of the inter-event residuals. We then evaluate the dependence of the standard deviation of on the distance between hypocenters,, and develop a model for the spatial correlation of the inter-event residuals.

Results The same approach is used for the dependence of the standard deviation of the The same approach is used for the dependence of the standard deviation of the

To develop a model that has a form that is widely applicable, the dependence of the standard deviation of the is modeled using a form that is monotonically increasing with To develop a model that has a form that is widely applicable, the dependence of the standard deviation of the is modeled using a form that is monotonically increasing with The standard deviation of the unexplained inter- event variability,, is given by the standard deviation of at =0 multiplied by to remove the normalization: The standard deviation of the unexplained inter- event variability,, is given by the standard deviation of at =0 multiplied by to remove the normalization:

Coefficients for the ΔH dependence of the inter-event standard deviation Coefficient b4b b5b b6b

The total standard deviation from a typical empirical ground motion model can be written in terms of the components of the standard deviation defined earlier The total standard deviation from a typical empirical ground motion model can be written in terms of the components of the standard deviation defined earlier Removing the three systematic terms from, the aleatory part of is given by Removing the three systematic terms from, the aleatory part of is given by

Single-site standard deviation Some previous studies have removed only the systematic effects related to the site-specific site amplification ( ) from the. In this case, the single-site standard deviation,, measures the variability of ground motions recorded at a single site from earthquakes at multiple source locations. The standard deviation for a single site is given by Some previous studies have removed only the systematic effects related to the site-specific site amplification ( ) from the. In this case, the single-site standard deviation,, measures the variability of ground motions recorded at a single site from earthquakes at multiple source locations. The standard deviation for a single site is given by

Reduction of the inter-event, intra-event, and total standard deviation from the TSMIP data, if the ergodic assumption is removed Single SiteSingle Path TotalIntra- event TotalInter- Event Intra- Event  SS rr  SP 00 00 PGA 0.91  T 0.86  0.54  T 0.69  0.43  T=  T 0.81  0.53  T 0.69  0.42  T=  T 0.86  0.60  T 0.69  0.55  T=  T 0.83  0.61  T 0.69  0.57  T=  T 0.75  0.59  T 0.69  0.51  T=  T 0.69  0.60  T 0.69  0.51 

Correlation of notation differences for components of variability from previous studies with the current study Standard DeviationThis StudyChen & Tsai (2002) Atkinson (2006) Morikawa et al (2008) Total TT  reg  Inter-event EE  (no correction) Intra-event  (no correction) Inter-site SS SS Single site, record-to-record rr rr Inter-path PP Intra-event, Single Path 00  (applied correction) Inter-source region  SR Inter-event, Single Region 00  (applied correction) Single Site (total)  SS ii Single Path (total)  SP  ie (applied correction)

Comparison of single-path standard deviations (σ SP ) as a fraction of σ T This StudyAtkinson (2006)Morikawa et al. (2008) PGA 0.54  T 0.67  T 0.46  T T=  T 0.38  T T=  T 0.68  T 0.44  T T=  T 0.45  T T=  T 0.67  T 0.47  T T=  T 0.47  T

Comparison of single-site standard deviations (σ SS ) as a fraction of σ T. This StudyChen & Tsai (2002)Atkinson (2006) PGA 0.91  T 0.86  T 0.87  T T=  T T=  T 0.91  T T=  T T=  T 0.92  T T=  T 0.93  T

Conclusion The single-site standard deviations are 9-14 % smaller than the total standard deviation; the single-path standard deviations are % smaller than the total standard deviation. The single-site standard deviations are 9-14 % smaller than the total standard deviation; the single-path standard deviations are % smaller than the total standard deviation. The models for the spatial correlation of the source/site- specific effects in the median ground motions for the intra-event and inter-event terms are developed. The models for the spatial correlation of the source/site- specific effects in the median ground motions for the intra-event and inter-event terms are developed. These models can be applied to regions without source/site specific data to account for the increased epistemic uncertainty for seismic hazard studies conducted without the ergodic assumption. These models can be applied to regions without source/site specific data to account for the increased epistemic uncertainty for seismic hazard studies conducted without the ergodic assumption.

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