Yoshifumi Yamamoto and Jack W. Baker Stanford …bakerjw/Publications/Yamamoto...• The regression...
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-2 0 2 -2 -1 0 1 2 QQPlot [cor=0.99769] quantiles of input Sample Standard Normal Quantiles 10 -1 10 0 0.5 1 correlation of epsilon T1 ρ Simulations(0.2s) Baker and Jayaram(0.2s) Simulations(1s) Baker and Jayaram(1s) Simulations(3s) Baker and Jayaram(3s) 10 0 10 2 0 0.2 0.4 0.6 0.8 1 Rjb(km) sd of log(Sa) PGA 10 0 10 2 0 0.2 0.4 0.6 0.8 1 Rjb(km) sd of log(Sa) Sa(0.2s) 10 0 10 2 0 0.2 0.4 0.6 0.8 1 Rjb(km) sd of log(Sa) Sa(1.0s) 10 0 10 2 0 0.2 0.4 0.6 0.8 1 Rjb(km) sd of log(Sa) Sa(3.0s) 10 0 10 2 10 -2 10 0 Rjb(km) Sa(g) PGA 10 0 10 2 10 -2 10 0 Rjb(km) Sa(g) Sa(0.2s) 10 0 10 2 10 -2 10 0 Rjb(km) Sa(g) Sa(1.0s) 10 0 10 2 10 -2 10 0 Rjb(km) Sa(g) Sa(3.0s) Stochastic model for earthquake ground motions using wavelet packets Yoshifumi Yamamoto and Jack W. Baker Stanford University, Dept. of Civil & Environmental Engineering Summary of results to date • A stochastic ground motion model with time-frequency nonstationarity has been developed. A wavelet packet based model is used to allow easy control of time-frequency nonstationarity • The model is empirically calibrated and captures important characteristics of observed ground motion recordings. Simulations reasonably match the em- pirical ground motion prediction models for spectral accelerations, Arias In- tensity, duration, mean period and epsilon. • Simulations have parameters that vary with earthquake magnitude, distance and local site conditions. This differs from some stochastic simulations which reproduce properties of only a recorded seed ground motion. • The regression and wavelet packet transform calculations are extremely com- putationally inexpensive, so simulating large numbers of ground motions for engineering use is easy. A desktop PC can produce 1000 simulations in one hour. • The high freqency portion of these stochastic simulations could be combined with low frequency physics-based simulations to produce broadband simu- lated ground motions. Wavelet packet transform The wavelet packet transform is an extended version of the discrete wavelet transform. By controlling wavelet packets at varying scales and locations, the time-frequency character- istics of the time series can be determined. The following figures show the relationship between a time series its Fourier spectrum, and its wavelet packets. Abstract For performance-based design, non-linear dynamic structural analysis for various types of input ground motions is required. Since the number of ground motion recordings is lim- ited for such analysis, synthetic ground motions can be used as additional input. Here a stochastic ground motion model with time-frequency nonstationarity is developed using wavelet packets. The wavelet packet transform can decompose time-series data into wavelet packets with constant resolution in the time and frequency axes, and it can also reconstruct a time- series data from wavelet packets. This allows time series to be modeled intuitively in the time-frequency domain using the wavelet packets. Ground motions siumulated using the proposed approach reasonably match ground motion recordings in several respects, including elastic and inelastic spectral acceleration, duration, bandwidth, cumulative energy, and time-frequency nonstationarity. In addition, the median and log standard deviation of spectral accelerations from the simulated ground motions match those of empirical ground motion prediction models. These re- sults suggest that the synthetic ground motions generated by our model can be used as realistic inputs for non-linear dynamic structural analyses. Time history Wavelet packets distribution Regression analysis to predict model parameters The 12 parameters are predicted by the regression analysis calibrated using 1408 strong ground motion recordings from 25 earthquakes. The predictors of our model are moment magnitude (Mw), rupture distance (Rrup), hypocentral distance (Rhyp) , and Vs30. The fol- lowing equation predicts the Total Energy parameter (Y). Other model parameters have similar equations. η is the interevent residual and ε is the intraevent residual, and other variables are deter- mined by regression analysis. η and ε values for pairs of model parameters are correlatied, and this correlation is tracked using a correlation matrix that is then used for simulations . As seen below, characteristics of the simulated ground motions obtained using these re- gression models reasonably match the characteristics of recorded ground motions as pre- dicted by empirical ground motion prediction (”attenuation”) models. The standard deviation of log spectral accelerations in long period is a little larger than that of the attenuation models. This is because the constant time-frequency resolution of the wavelet packets makes it difficult to control the long-period energy of the simulations. An appealing solution would be to use this technique to produce high-frequency simula- tions, and combine them with physics-based low-frequency simulations. Stochastic model for ground motions In our model, time series are defined by the combination of two types of wavelet packet groups: the major group (which contains the largest coefficients representing 70% of the total ground motion energy) and the minor group (which contains the rest). To determine the amplitudes of the wavelet packets, the following 12 parameters are required for three independent probability density functions. for both wavelet packet groups, we define the following parameters • E(t) - mean time • S(t) - standard deviation of time (related to duration) • E(f ) - mean frequency (related to dominant frequency) • S(f ) - standard deviation of frequency (related to bandwidth) • R(t,f ) - correlation between time and frequency (related to frequency evolution in time) for the major wavelet packets group, we define the following • E(a) - mean amplitude Finally, we specify the total energy in the time series • Energy - total energy 0 2 4 6 8 10 −0.2 0 0.2 Time(s) Amplitude Fourier spectrum 0 1 2 3 4 5 0 1 2 3 4 5 6 7 x 10 −3 Fourier amplitude Frequency(Hz) ) Time(s) Frequency(Hz) 0 2 4 6 8 10 0 2.5 5 Simulated ground motions Spectral Acceleration (Sa) vs Rjb Mw=7, VS30=270m/s Wavelet packets of ground motion recordings from the Northridge Earthquake 0 10 20 0 2 4 6 Time(s) Frequency(Hz) 0 10 20 0 2 4 6 Time(s) Frequency(Hz) Example near field recording Northridge - 17645 Saticoy St Rhyp=18km, Vs30=281m/s Example far field recording Santa Barbara - UCSB Goleta Rhyp=123km, Vs30=339m/s 0 20 40 −0.5 0 0.5 Time(s) 0 20 40 Time(s) 0 20 40 10 0 10 1 0 20 40 0 0.05 0.1 Acc.(G) Amp. 2 Simulation Original Acc.(G) Freq.(Hz) Near field station Northridge - 17645 Saticoy St [Rhyp=18km, Vs30=281m/s] 0 20 40 −0.1 0 0.1 Time(s) 0 20 40 Time(s) 0 20 40 10 0 10 1 0 20 40 0 0.01 0.02 Acc.(G) Amp. 2 Simulation Original Acc.(G) Freq.(Hz ) Far field station Santa Barbara - UCSB Goleta [Rhyp=123km, Vs30=339m/s] Cumulative Squared Acceleration 10 −2 10 −1 10 0 10 1 10 0 10 1 10 2 10 3 1 10 100 1000 10000 0.01 0.1 1 10 100 1000 Displacement(cm) Pseudo Velocity(cm/s) Period(s) Pseudo Acceleration(cm/s 2 ) h=0.05 Spectral Acceleration 10 −1 10 0 10 −2 10 −1 10 0 10 1 Periods(s) Spectral Acceleration(g) Rrup=10km Rrup=20km Rrup=50km 0 20 40 −0.5 0 0.5 0 20 40 −0.5 0 0.5 0 20 40 −0.5 0 0.5 time(s) Rupture Distance = 10km Rupture Distance = 20km Rupture Distance = 50km Acc.(G) Acc.(G) Acc.(G) Simulated ground motions Mw=7, Vs30=270m/s 10 −1 10 0 10 −2 10 −1 10 0 10 1 Periods(s) Spectral Acceleration(g) sample 1 sample 2 sample 3 0 20 40 −0.5 0 0.5 0 20 40 −0.5 0 0.5 0 20 40 −0.5 0 0.5 time(s) Acc.(G) Acc.(G) Acc.(G) sample 1 sample 2 sample 3 Simulated ground motions Mw=7, Rrup=10km, Vs30=270m/s ij i S W W V d R c M b M b a Y ε η + + + + + + = ) ln( ) ln( ) ln( ) ln( 30 2 2 1 2 2 h R R rup + = rup hyp dif R R R − = ∑∑ − = = − ⋅ = 1 2 0 2 1 , ) ( ) , ( ) ( p p n j k p k j p f t k j W t f ψ : Time history : Wavelet packet coefficients : Scaling function ) , ( k j W p f ) ( , t p k j ψ ) (t f j k : Frequency order (frequency axis) : Scale factor (time axis) : Depth (band width of each level j) : Total number of data p n 2 Major wavelet packets 0 20 40 −0.5 0 0.5 Time(s) 0 20 40 10 0 10 1 Original Acc.(G) Freq.(Hz) Observed ground motion 0 20 40 10 0 10 1 0 20 40 −0.5 0 0.5 Time(s) Acc.(G) Original Freq.(Hz) 0 20 40 −0.5 0 0.5 Time(s) 0 20 40 10 0 10 1 Original Acc.(G) Freq.(Hz) Minor wavelet packets 0 20 40 10 0 10 1 0 20 40 −0.5 0 0.5 Time(s) Acc.(G) Freq.(Hz) Simulation Simulated ground motion 0 20 40 −0.5 0 0.5 Time(s) 0 20 40 10 0 10 1 Acc.(G) Freq.(Hz) Simulation 0 20 40 −0.5 0 0.5 Time(s) 0 20 40 10 0 10 1 Acc.(G) Freq.(Hz) Simulation parameters parameters How to simulate ground motions Selected simulations Median of simulations Sigma of simulations Observation Simulations Near field Far field 0 10 20 30 40 0 0.2 0.4 0.6 0.8 1 Time(s) Cumlative squared acc Near field observation Near field simulation Far field observation Far field simulation Standard deviation of log(Sa) vs Rjb Mw=7, VS30=270m/s 10 0 10 1 10 2 10 0 10 1 10 2 Rup(km) Duration(s) 10 0 10 1 10 2 0 0.2 0.4 0.6 0.8 1 Rrup(km) Standard deviation of ln(dur) Abrahamson and Silva(1996) Simulated ground motions 5-95% duration vs Rrup Mw=7, Rrup=Rhyp=10km, VS30=270m/s 10 0 10 2 10 -2 10 0 Rrup (km) Arias Intensity (m/s) 10 0 10 2 0 0.2 0.4 0.6 0.8 1 Rrup (km) Standard deviation of ln(Ia) Travasarou et al.(2003) Simulated ground motions Arias Intensity vs Rrup Mw=7, Rrup=Rhyp=10km, VS30=270m/s 10 0 10 1 10 2 10 -1 10 0 Closest Distance (km) Mean Period Tm(s) 10 0 10 1 10 2 0 0.2 0.4 0.6 0.8 1 Rrup (km) Standard deviation of ln(Tm) Rathje et al.(2004) Simulated ground motions mean Period(Tm) vs Rrup Mw=7, Rrup=Rhyp=10km, VS30=270m/s characteristics of epsilon Mw=7, Rrup=Rhyp=10km, VS30=270m/s Abrahamson and Silva(2008) Boore and Atkinson(2008) Campbell and Bozorgnia(2008) Chiou and Youngs(2008) simulated ground motions Abrahamson and Silva(2008) Boore and Atkinson(2008) Campbell and Bozorgnia(2008) Chiou and Youngs(2008) simulated ground motions
Transcript of Yoshifumi Yamamoto and Jack W. Baker Stanford …bakerjw/Publications/Yamamoto...• The regression...
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Stochastic model for earthquake ground motions using wavelet packetsYoshifumi Yamamoto and Jack W. Baker Stanford University, Dept. of Civil & Environmental Engineering
Summary of results to date
• A stochastic ground motion model with time-frequency nonstationarity has been developed. A wavelet packet based model is used to allow easy control of time-frequency nonstationarity
• The model is empirically calibrated and captures important characteristics of observed ground motion recordings. Simulations reasonably match the em-pirical ground motion prediction models for spectral accelerations, Arias In-tensity, duration, mean period and epsilon.
• Simulations have parameters that vary with earthquake magnitude, distance and local site conditions. This differs from some stochastic simulations which reproduce properties of only a recorded seed ground motion.
• The regression and wavelet packet transform calculations are extremely com-putationally inexpensive, so simulating large numbers of ground motions for engineering use is easy. A desktop PC can produce 1000 simulations in one hour.
• The high freqency portion of these stochastic simulations could be combined with low frequency physics-based simulations to produce broadband simu-lated ground motions.
Wavelet packet transform
The wavelet packet transform is an extended version of the discrete wavelet transform. By controlling wavelet packets at varying scales and locations, the time-frequency character-istics of the time series can be determined.
The following �gures show the relationship between a time series its Fourier spectrum, and its wavelet packets.
Abstract
For performance-based design, non-linear dynamic structural analysis for various types of input ground motions is required. Since the number of ground motion recordings is lim-ited for such analysis, synthetic ground motions can be used as additional input. Here a stochastic ground motion model with time-frequency nonstationarity is developed using wavelet packets.
The wavelet packet transform can decompose time-series data into wavelet packets with constant resolution in the time and frequency axes, and it can also reconstruct a time-series data from wavelet packets. This allows time series to be modeled intuitively in the time-frequency domain using the wavelet packets.
Ground motions siumulated using the proposed approach reasonably match ground motion recordings in several respects, including elastic and inelastic spectral acceleration, duration, bandwidth, cumulative energy, and time-frequency nonstationarity. In addition, the median and log standard deviation of spectral accelerations from the simulated ground motions match those of empirical ground motion prediction models. These re-sults suggest that the synthetic ground motions generated by our model can be used as realistic inputs for non-linear dynamic structural analyses.
Time history
Wavelet packets distribution
Regression analysis to predict model parameters
The 12 parameters are predicted by the regression analysis calibrated using 1408 strong ground motion recordings from 25 earthquakes. The predictors of our model are moment magnitude (Mw), rupture distance (Rrup), hypocentral distance (Rhyp) , and Vs30. The fol-lowing equation predicts the Total Energy parameter (Y). Other model parameters have similar equations.
η is the interevent residual and ε is the intraevent residual, and other variables are deter-mined by regression analysis. η and ε values for pairs of model parameters are correlatied, and this correlation is tracked using a correlation matrix that is then used for simulations .
As seen below, characteristics of the simulated ground motions obtained using these re-gression models reasonably match the characteristics of recorded ground motions as pre-dicted by empirical ground motion prediction (”attenuation”) models.
The standard deviation of log spectral accelerations in long period is a little larger than that of the attenuation models. This is because the constant time-frequency resolution of the wavelet packets makes it di�cult to control the long-period energy of the simulations. An appealing solution would be to use this technique to produce high-frequency simula-tions, and combine them with physics-based low-frequency simulations.
Stochastic model for ground motionsIn our model, time series are de�ned by the combination of two types of wavelet packet groups: the major group (which contains the largest coe�cients representing 70% of the total ground motion energy) and the minor group (which contains the rest). To determine the amplitudes of the wavelet packets, the following 12 parameters are required for three independent probability density functions. for both wavelet packet groups, we de�ne the following parameters • E(t) - mean time • S(t) - standard deviation of time (related to duration) • E(f ) - mean frequency (related to dominant frequency) • S(f ) - standard deviation of frequency (related to bandwidth) • R(t,f ) - correlation between time and frequency (related to frequency evolution in time)for the major wavelet packets group, we de�ne the following • E(a) - mean amplitudeFinally, we specify the total energy in the time series • Energy - total energy
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Simulated ground motionsMw=7, Rrup=10km, Vs30=270m/s
ijiSWW VdRcMbMbaY εη ++++++= )ln()ln()ln()ln( 30221
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characteristics of epsilonMw=7, Rrup=Rhyp=10km, VS30=270m/s
Abrahamson and Silva(2008)Boore and Atkinson(2008)Campbell and Bozorgnia(2008)Chiou and Youngs(2008)simulated ground motions
Abrahamson and Silva(2008)Boore and Atkinson(2008)Campbell and Bozorgnia(2008)Chiou and Youngs(2008)simulated ground motions
