Linear and nonlinear time history analyses of the transient responses to earthquakes require recorded or synthesized ground motion time records. Sets of ground motion records used for seismic hazard analyses typically represent a uniform level of seismic hazard, and are customarily classified as design basis ground motions or maximum credible ground motions. The sets of ground motions developed for the SAC steel project are classified according to the level of seismic hazard and specific geographic region, and have been used extensively for structural response and seismic hazard analyses. This study presents a parametric analysis of these ground motion data sets. 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 distribution functions 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 uniform hazard ground motions in terms of physically motivated parameters and also provides data useful for synthesizing ground motion records corresponding to specific hazard levels.

The generation of a set of earthquake ground motions representing a uniform level of seismic hazard for a particular geographic region involves seismic hazard disaggregation studies, source rupture models, calibrated attenuation relationships, and local site amplification models. A companion paper presents statistical parametric analyses of five such data sets which have been used widely in structural response and seismic hazard studies. The ground motion 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 companion paper presents probability distributions for parameters fit to the wave-form characteristics of the ground motion data sets. In contrast, this paper presents the statistics of parameters fit to linear elastic response spectra of the ground motion data sets. The new concept of a parameter-response correlation spectrum establishes the frequency dependence of the sensitivity of the response spectrum to variability in ground motion parameters. Parameters that affect the variability of the response spectra are retained as random variables in a response-spectrum compatible ground motion model that also includes the correlations between velocity amplitudes of orthogonal ground motion components.

[ 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 ]

[ 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)

[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.

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