of Uniform Hazard Earthquake Ground Motions

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Henri Gavin, Ph.D., P.E., Associate Professor
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This site provides information and software useful to generate a large number of earthquake ground motions representative of a few particular hazard levels for a few particular geophysical areas of the United States. While these results are based on a data set of highly-regarded and widely-used earthquake ground motion records, the method may be applied to any sample of ground motions.

Bryce W. Dickinson and Henri P. Gavin,

"A parametric statistical generalization of uniform hazard earthquake ground motions,"

Sets of ground motion records used for seismic hazard analyses typically have intensity measures corresponding to a particular hazard level for a site (perhaps conditioned on a particular intensity value and hazard). In many cases the number of available ground motions that match required spectral ordinates and other criteria (such as duration, fault rupture characteristics, and epicentral distance) may not be sufficient for high-resolution seismic hazard analysis. In such cases it is advantageous to generate additional ground motions using a parameterized statistical model calibrated to records of the smaller data set. This study presents a statistical parametric analysis of ground motion data sets that are classified according a seismic hazard level and a geographic region, and which have been used extensively for structural response and seismic hazard analyses. Parameters represent near-fault effects such as pulse velocity and pulse period, far field effects such as velocity amplitude and power spectral attributes, and envelope characteristics. A systematic fitting of parameterized pulse functions to the individual ground motion records, of parameterized envelopes to individual instantaneous ground motion amplitudes, and of parameterized power spectral density functions to averaged power spectra result in probability distributions for ground motion parameters representative of particular seismic hazard levels for specific geographical regions. This methodology presents a means to characterize the variability in a set of ground motions records in terms of physically meaningful parameters.

Henri P. Gavin and Bryce W. Dickinson,

"Simulation of uniform hazard earthquake ground motions,"

This paper presents statistical models for the generation of bi-axial earthquake ground motion time histories with spectra that match those from samples of ground motion records. The model parameters define near-field characteristics such as pulse velocity and pulse period, far-fault characteristics such as velocity amplitude and power spectral density, and envelope characteristics. The samples of ground motions used in this study were previously selected and scaled to be representative of particular hazard levels in particular geographical regions. A companion paper presents the fitting of the model to samples of ground motion waveforms. In this paper, the new concept of a parameter-response correlation spectrum establishes the period-dependence of the correlation between the response spectrum and ground motion parameters. Parameters that correlate to variability of the response spectra are retained as random variables and are then fit to mean and mean-plus-standard deviation of bi-axial response-spectra of the samples of historical records. Parameter statistics also include correlations between velocity amplitudes and pulse periods.

quake_SAC2d.m --- Generate an artificial (synthetic) earthquake ground motion record with a consistent acceleration, velocity, and displacemnt pulse using ground motion parameters representative of a SAC ground motion set.

[ time, quake_data, X ] = quake_SAC2d ( quake_set, delta_t, f_lo, f_hi, seed ) Generate an artificial (synthetic) earthquake ground motion record with a consistent acceleration, velocity, and displacemnt pulse using ground motion parameters representative of a SAC ground motion set. Input Variable Description -------------- ----------- quake_set one of ... 'nrfault' near fault - LA 10 percent in 50 year 'la10in50' LA 10 percent in 50 year (default) 'la2in50' LA 2 percent in 50 year 'se10in50' Seattle 10 percent in 50 year 'se2in50' Seattle 2 percent in 50 year delta_t the time step constant, sec f_lo low spectral frequency value (default = 0.10 Hz) f_hi high spectral frequency value (default = 10.0 Hz) seed a seed for the random phase generation Output Variable Description -------------- ----------- time : time record ... ( time = [1:P]' * delta_t; ) quake_data : earthquake ground motion data ... P rows, six columns [ accelNS velocNS displNS accelEW velocEW displEW ] X : vector of 13 ground motion parameters used in simulation [ VpNS VpEW Tp Nc Tpk phi VrNS VrEW tau1 tau2 tau3 fg zg ]SAC_stats2d.m --- Return ground motion paramters and parameter statistics for a particular bi-directional SAC earthquake scenario.

[ X, Ex, Vx, Cz ] = SAC_stats2d(quake_set,N) Return ground motion paramters and parameter statistics for a particular bi-directional SAC earthquake scenario http://nisee.berkeley.edu/data/strong_motion/sacsteel/ground_motions.html INPUT DESCRIPTION quake_set one of ... 'nrfault' LA near fault ground motion 'la10in50' LA 10 percent in 50 year 'la2in50' LA 2 percent in 50 year 'se10in50' Seattle 10 percent in 50 year 'se2in50' Seattle 2 percent in 50 year N maximum allowable variability in Vp, Vr, and Tp ... 2 or 3 X will be less than Ex + N * sqrt(Vx); OUTPUT DESCRIPTION X ground motion parameters Ex ground motion parameter mean values Vx ground motion parameter variances Cz ground motion parameter correlation matrix for standardized random variables where X(1) ... Vp peak velocity of coherent pulse, cm/s NS X(2) ... Vp peak velocity of coherent pulse, cm/s EW X(3) ... Tp period of coherent pulse, s X(4) ... Nc cycles in coherent pulse X(5) ... Tpk time to the peak of the pulse X(6) ... phi phase angle of the pulse X(7) ... Vr peak velocity of incoherent ground motion, cm/s NS X(8) ... Vr peak velocity of incoherent ground motion, cm/s EW X(9) ... Tau1 envelope rise time, s X(10) ... Tau2 constant time, s X(11) ... Tau3 envelope decay time, s X(12) ... power spectrum central frequency, Hz X(13) ... power spectrum bandwidth factor, Ground motion parameters are generated according to the lognormal probability distribution function. A default case is also provided in which the ground motion paramters are similar to those of la10in50, but without any cross-correlation. (Tp - 0.8) is lognormal-distributed with mean Ex(3) and variance Vx(3) (Nc - 0.5) is lognormal-distributed with mean Ex(4) and variance Vx(4) (phi + 2*pi) is lognormal-distributed with mean Ex(6) and variance Vx(6) (Vr - 10.0) is lognormal-distributed with mean Ex(7) and variance Vx(7) NS (Vr - 10.0) is lognormal-distributed with mean Ex(8) and variance Vx(8) EW (Tau1 - 1.0) is lognormal-distributed with mean Ex(9) and variance Vx(9) (Tau3 - 1.0) is lognormal-distributed with mean Ex(11) and variance Vx(11)pulseV.m --- Generate a velocity pulse with the corresponding acceleration and displacement pulses.

[TIME,PULSE] = PULSEV(Vp,Tp,Nc,Tpk,phi,P,delta_t) Computes an earthquake-like acceleration, velocity, and displacemnt pulse INPUT DEFAULT ========= ======= Vp - max velocity of pulse 1.0 Tp - time period of pulse 1.0 Nc - number of cycles in pulse ... approximate 1.0 Tpk - location of peak pulse in time axis 1.0 phi - phase of the pulse ... between -pi and +pi 0 P - number of points in the pulse record 1000 delta_t - time step value 0.005 OUTPUT ====== The first column of PULSE contains an 'acceleration' record. The second column of PULSE contains a 'velocity' record. The third column of PULSE contains a 'displacement' record.ftdsp.m --- Fourier-transform based digital signal processing for filtering, differentiation, and integration

y = ftdsp(u,ni,flo,fhi,sr) band-pass filter and integrate a discrete-time signal, u u : the discrete-time signals to be filtered/integrated, in column vectors ni : the number of integrations (may be zero or negative for differentiation) flo : the low frequency limit for the bandpass filter ( > = 0 ) fhi : the high frequency limit for the bandpass filter ( < = sr/2 ); sr : the sample rate

This material is based upon work supported by the National Science Foundation under Grant No. NSF-CMMI-0704959 (NEES Research), and Grant No. NSF-CMS-0402490 (NEES Operations). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundatioon.

© 2008 Henri P. Gavin; Updated: 5-2-2008, 3-9-2010