a parametric statistical generalization of uniform...

A parametric statistical generalization of uniform-hazard earthquake ground motions

Bryce W. Dickinson, A.M.ASCE1 and Henri P. Gavin, M.ASCE2

ABSTRACTSets of ground motion records used for seismic hazard analyses typically have intensity measures cor-

responding to a particular hazard level for a site (perhaps conditioned on a particular intensity value andhazard). In many cases the number of available ground motions that match required spectral ordinates andother criteria (such as duration, fault rupture characteristics, and epicentral distance) may not be sufficientfor high-resolution seismic hazard analysis. In such cases it is advantageous to generate additional groundmotions using a parameterized statistical model calibrated to records of the smaller data set. This studypresents a statistical parametric analysis of ground motion data sets that are classified according a seismichazard level and a geographic region, and which have been used extensively for structural response andseismic 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. Asystematic fitting of parameterized pulse functions to the individual ground motion records, of parameter-ized envelopes to individual instantaneous ground motion amplitudes, and ofparameterized power spectraldensity functions to averaged power spectra result in probability distributions for ground motion parame-ters representative of particular seismic hazard levels for specific geographical regions. This methodologypresents a means to characterize the variability in a set of ground motions records in terms of physicallymeaningful parameters.

INTRODUCTIONEarthquake engineers and engineering seismologists have appreciated the importance of uncertainty

in earthquake ground motions for over sixty years (Housner 1947). Earthquake ground motion data sets,even when scaled to a common intensity measure, exhibit significant variability inspectral and temporalcharacteristics (Housner 1947), (Cornell 1968), (Conte and Peng 1997). Despite the growing availability ofground motion records, ground motion characteristics having an intensity witha certain return period at asite can be known only approximately due to natural variability in earthquake source mechanisms, seismicwave propagation and local site effects. Consequently, expectations about the potential ground motioncharacteristics at a site are suitably interpreted on a probabilistic basis.

Procedures to generalize and simulate strong ground motions may be generally divided into three mainclassifications: computational geophysical models, empirical attenuation models, and empirical stochasticmodels. Computational geophysics employs finite difference or finite element methods to numerically solveelastodynamic wave equations in order to simulate wave propagation through amodeled media (Graves1998). Attenuation models attempt to relate peak ground motions and spectral response values to parametersdescribing the earthquake source, the path of the wave propagation, and the local and regional geophysicalconditions through empirical relationships (Power et al. 2008). Stochasticmodels are employed to describeand reproduce the stochastic characteristics of a set of observed ground motions without necessarily con-sidering the geophysical properties of the event (Shinozuka and Deodatis 1988). Researchers have furtherdeveloped methods to combine attenuation and stochastic process models (Boore 2005b).

Probabilistic Seismic Hazard Analysis (PSHA) combines local seismic hazard models, attenuation mod-els, and site effects to determine the spectral accelerations and other ground motion characteristics of a givenreturn period at a site (Petersen et al. 2008). Performance-based design of structures with nonlinear behav-ior, on the other hand, requires realistic ground motions representative of the hazard designated by the PSHA

1Rutherford & Chekene, 55 Second Street, Suite 600, San Francisco, CA 941052Dept. Civil & Environ. Eng’g, Duke Univ., Box 90287, Durham NC 27708

(McGuire 1995). Considering the dispersion in intensity measures, even when conditioned on an intensitymeasure with a specific hazard level, a large number of records may be necessary for a high-resolutionperformance-based design.

To this end we present a general approach toward the statistical characterization of ground motions froma sample of representative ground motion records. Parameter values thatdescribe the stochastic content, thelong-period velocity pulse, and the time-modulating envelope function are determined individually for eachrecord. For this study we make use of sets of ground motion records that have been widely appropriated andwhich remain in use in by many researchers. This methodology may be applied as easily to other sets ofground motions.

BACKGROUND, LITERATURE REVIEW, AND MOTIVATIONMethods for simulating earthquake ground motions typically describe strong ground motions as nonsta-

tionary stochastic processes. A number of methods have been proposedfor generalizing and simulating thenonstationary behavior of earthquake ground motions (Grigoriu 1995),(Conte and Peng 1997), (Michaelovet al. 1999), (Shinozuka et al. 1999), (Iyengar 2001), (Mukherjee and Gupta 2002), (Wen and Gu 2004),(Gu and Wen 2007), (Gu and Wen 2007). The more common of these stochastic process descriptors involvetime-dependent spectral parameters and a time modulating envelope function.The well-known Kanai-Tajimiground motion spectrum (Kanai 1957), (Tajimi 1960) models ground motion as the response of a dampedlinear oscillator excited by broadband acceleration. The power spectraldensity function for in this modelcontains two physically meaningful parameters: ground frequency and damping. To achieve nonstationarity,such a stationary stochastic process may be multiplied by a deterministic envelopefunction (Jennings et al.1968), (Iyengar and Iyengar 1969), (Liu 1970), (Lin and Yong 1987). Shinozuka and Deodatis compre-hensively reviewed stochastic ground motion simulation methodologies including the filtering of Gaussianwhite noise, the filtering of Poisson processes, the spectral representation of stochastic processes, and simu-lation based on stochastic wave theory (Shinozuka and Deodatis 1988). Acompanion review paper (Kozin1988) discusses autoregressive moving average (ARMA) models to parametrically match target ground mo-tions. Many of these models efficiently simulate the evolving amplitude and frequency content of a recordedground motion and contain parameters that depend on the records chosenfor model calibration. The PointSource Stochastic Simulation Model is a commonly-used numerical procedurefor simulating ground mo-tion records (Boore 2003), (Boore 2005b). This method makes use of adeterministic envelope functionapplied to a filtered white noise process, in which the power-spectral density and duration of the record arerelated empirically to the earthquake fault rupture, path effects, and site conditions. This method has beenshown to be effective in simulating the stochastic content of ground motions (Boore 2003) and has bridgeda gap between geophysical attenuation relationships and stochastic process models.

Rather than attempting to model the range of ground motions that a given structure may experience overits lifetime, critical excitation methods have been introduced (Takewaki 2005)which assume there is a singleworst case input excitation that will cause the most significant response ina given structure. This criticalexcitation is based on the properties of the elastic or inelastic structural system using the input energy and theenergy input rate as the measures of criticality. With this method, a building may bedesigned to withstandthe worst case input, which is certainly important for critical structures. This method is not amenable toestimating exceedance rates or other probabilistic measures.

Computational geophysical models enable simulation of ground motions with characteristics that maybe under-represented in catalogs of recorded ground motions and associated empirical models. Such char-acteristics include forward directivity effects, residual ground displacement, basin effects, and fault-zonechanneling. These methods involve the coupling of source models with wave propagation models in or-der to simulate ground motion activity throughout a geophysical region (Graves 1998) (Bielak et al. 2005)(Olsen et al. 2008).

Increased awareness of the destructive capability of near-fault velocity pulses has motivated several

2 B.W. Dickinson and H.P. Gavin

studies examining the response of structures and their contents subjected tothis type of forcing (Berteroet al. 1978), (Rao and Jangid 2001), (Zhang and Iwan 2002a), (Zhang and Iwan 2002b), (Mavroeidis andPapageorgiou 2003), (Mavroeidis et al. 2004), (Tian et al. 2007). The energy wave released upon the slipof a fault can exhibit a number of characteristics including forward directivity, permanent translation, andradiation effects (Mavroeidis and Papageorgiou 2003). When the faultrupture velocity is near the shear wavevelocity, seismic energy accumulates at the wave front as the rupture propagates; ground velocity records atsites that are close to the rupture and located ahead of the slip exhibit a distinct high-amplitude pulse in thefault-normal component. Such pulses are typically observed near the beginning of the record (Somerville2003), (Tothong and Baker 2007). Mavroeidis and Papageorgiou have modeled near-fault earthquake groundmotion records containing long-period velocity pulses (Mavroeidis and Papageorgiou 2003). This model isparameterized by the period, the amplitude, the number of cycles, and the phase of the pulse, (all of whichhave a clear physical interpretation) and the ultimate generation of synthetic near-fault ground motions isaccomplished through the superposition of a coherent, long-period velocity pulse with incoherent seismicradiation. A number of subsequent studies have examined the mechanisms and modeling of near-faultground motions in order to study the resulting structural response (Somerville 2003), (Bray and Rodriguez-Marek 2004), (Tian et al. 2007). A quantitative measure of pulse-like characteristics determined that 91 ofthe 3551 records in the PEER-NGA database contains a pulse (Baker 2007). The destructive potential ofground motions containing a strong velocity pulse indicates a need to include a pulse simulation model instochastic process models for the simulation of demanding ground motions.

Stochastic methods for the simulation of ground motion records must thereforecontain several key char-acteristics. The process model must be capable of simulating the nonstationarity of strong ground motions.It must be capable of the inclusion of both long-period velocity pulses as well as incoherent radiation ef-fects. Ideally, the model will be simple with a small number of physically meaningful parameters. Currentmethods in simulating ground motions that include a near-fault velocity pulse andstochastic content havenot yet considered the probability distributions of the model parameters or the relation of these probabilitydistributions to particular seismic hazard levels or geographical regions.

The present study aims to quantify the statistical variation of ground motion parameters fit to samplesof ground motion records and to generate additional ground motions containing these same statistical char-acteristics with a model that includes the superposition of a long-period velocity pulse and a nonstationarystochastic acceleration record. Several investigators have assembled samples of ground motion records rep-resentative of a given hazard level as designated by a PSHA (Iervolino and Cornell 2005), (McGuire 1995),(Somerville et al. 1997), (Tothong and Baker 2007). In this study models are calibrated to the ground motiondata sets developed for the SAC steel project (Somerville et al. 1997). The five data sets explored in thisstudy were developed to represent the hazard probabilities of exceedance of 2% in 50 years and 10% in 50years in Los Angeles and Seattle. A set containing 40 records with near fault effects is also explored. Thesedata sets have been regarded as statistical samples for the potential ground motion for a given hazard leveland specific geographical region criteria. As a result, the SAC ground motions have been used widely forseismic analysis and performance-based design (Chang et al. 2002), (Lee and Foutch 2002), (Wen and Song2003), (Yun et al. 2002), (Morgan and Mahin 2008).

Statistical analysis of the model parameters for each of the near-fault, LosAngeles, and Seattle data setsallow for waveform parameters to be drawn from distributions with means, variances, and parameter corre-lations calibrated to samples of historic ground motion records. In a companion study (Gavin and Dickinson2010), correlations between input model parameters and spectral responses are plotted as “correlation spec-tra.” These spectra are used to reduce the number of input parameters toonly those which associate withvariability in structural response.

GROUND MOTION PARAMETERIZATIONIn this study synthetic earthquake ground motions are generated through the superposition of a stochastic


ground velocity record with a single, coherent, long-period velocity pulse. The stochastic ground motionvelocity record is generated by first simulating anenveloped andunscaled stochastic ground accelerationrecord,au(t),

au(ti) = e(ti; τ0, τ1, τ2, τ3)N+1∑

k=1

[

S(fk; fg, ζg)]1/2

cos(2πfkti + θk) , (1)

whereS(fk; fg, ζg) is the power spectrum of the stochastic content of the acceleration recordparameterizedby ground frequency,fg, and a ground damping,ζg,

S(fk; fg, ζg) =(2ζgfk/fg)

2

(1− (fk/fg)2)2 + (2ζgfk/fg)2, (2)

shown in Figure 1, ande(t; τ0, τ1, τ2, τ3) is an envelope function. For digital simulation, discrete frequencies

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

spec

tral

pow

er

frequency, Hz

g2ζ

fg

fg

FIG. 1. Power spectrum parameters.

fk are uniformly spaced fromflo to fhi; fk = flo+(k−1)(fhi−flo)/N . The phase angle at each frequency,θk, is a uniformly distributed random variable between 0 and2π (Shinozuka and Jan 1972; Shinozuka andDeodatis 1991). In this study,N = 1024 and the frequency limits,flo andfhi, depend on the type of groundmotion to be simulated. The envelope function used here separates the ground motion into four time periods(Jennings et al. 1968): a time of no ground acceleration,τ0, the time the ground motion takes to reach itspeak acceleration,τ1, the time the acceleration is at the plateau peak region,τ2, and an exponential decaytime constant,τ3,

e(ti; τ0, τ1, τ2, τ3) =

((ti − τ0)/τ1)2 for τ0 ≤ t ≤ τ0 + τ1

1 for τ0 + τ1 ≤ t ≤ τ0 + τ1 + τ2exp[−(ti − τ0 − τ1 − τ2)/τ3] for τ0 + τ1 + τ2 ≤ t

, (3)

as shown in Figure 2.An unscaled stochastic ground velocity recordvu(t) is calculated fromau(t) by integrating frequency

components betweenflo andfhi in the frequency-domain (Boore 2005a), (Dickinson 2008). The scaledstochastic ground velocity record isvs(t) = vu(t)(Vs/max |vu(t)|), whereVs is a parameter denotingthe peak of the stochastic ground velocity record. The artificial ground velocity record is then found bycombining the scaled stochastic ground velocity with a velocity pulse,

v(ti) = vs(ti) + vp(ti). (4)


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0.8

1

0 2 4 6 8 10 12 14

enve

lope

time, s

τ 1 τ 2

τ 3

FIG. 2. Envelope function parameters.

The pulse model has five parameters: the peak pulse velocity,Vp, the period of the pulse,Tp, the number ofcycles in the pulse,Nc, the location of the pulse,Tpk, and the phase of the pulse,φ,

vp(ti;Vp, Tp, Nc, Tpk, φ) = Vp exp

−π2

4

(

t− Tpk

NcTp

)2

cos

(

2πt− Tpk

Tp

− φ

)

, (5)

as shown in Figure 3. This function is essentially a single Gabor wavelet in which the scale and shift

−50−40−30−20−10 0

10 20 30 40 50

0 2 4 8 10 12 14

velo

c, c

m/s

time, s

NTp c

Tp

pV

Tpk

FIG. 3. Pulse parameters for φ = 0.

parameters are continuous and have a clear physical interpretation. Theartificial ground acceleration recordag(t) is found by differentiatingvg(t) from frequencies of 0 tofhi in the frequency-domain.

In summary, twelve individual parameter values are used in this study to completely characterize agiven ground motion record: pulse amplitude,Vp, pulse period,Tp, number of pulse cycles,Nc, pulsearrival time,Tpk, pulse phase,φ, peak velocity of the stochastic content,Vs, central ground frequency,fg,ground damping,ζg, time of no ground acceleration,τ0, time to peak acceleration,τ1, duration of plateauregion,τ2, and acceleration decay time constant,τ3.


PARAMETER ESTIMATIONThis section describes the estimation of the twelve ground motion parameters forfive of the ground

motion data sets developed for the SAC steel project. Estimates of parameter values and their confidenceintervals were computed using the Levenberg-Marquardt method in three related nonlinear least squaresproblems.

The pulse parameters were estimated first. To extract long-period data from the ground motion records,each acceleration record was bandpass filtered between 0.1 Hz and 1.5 Hz and integrated once to givethe velocity record and integrated again to give the displacement record for the long-period content of theground motion. These long-period acceleration, velocity, and displacement records were visually examinedto determine the presence of a coherent pulse. Pulses are typically visible inthe velocity record and are evenmore apparent in the displacement record. Numerical values for the peakpulse velocity,Vp, the period of thepulse,Tp, the number of cycles in the pulse,Nc, the location of the pulse,Tpk, and the phase of the pulse,φ,were identified such that the pulse equation (5) roughly matched the long-period and high-amplitude contentof the velocity record. These values served as the initial guess for a nonlinear least squares fit of the pulseequation to the filtered velocity record,

χ2(Vp, Tp, Nc, Tpk, φ) =1

2

m∑

i=1

[v(ti)− vp(ti;Vp, Tp, Nc, Tpk, φ)]2 . (6)

Parameter constraints were enforced during the fitting process to ensurethat parameter estimates remainedwithin physically-motivated bounds, as shown in Table 1.

TABLE 1. Parameter estimate constraints

Parameter Vp Tp Nc Tpk φ fg ζg τ0 τ1 τ2 τ3

Minimum 5 0.5 0.5 0.1 −π 0.5 0.05 0.1 0.1 0.1 0.1Maximum 500 9.5 5.0 20 π 10 3.0 50 50 50 50

Units cm/s s - s rad Hz - s s s s

Nonlinear least squares methods sometimes converge to local minima and occasionally different ini-tial guesses for the pulse parameters resulted in different parameter estimates. In such cases, decisionswere made as to which estimate was most reasonable. Individual pulse records were designated as “easy,”“medium,” or “difficult” to fit depending on the ease of finding a pulse, the quality of the fit, the speed ofconvergence, and the difficulty of the choices made in the process.

An “easy” to fit record exhibits a single distinct velocity pulse: a clear peakor set of peaks in therecord that is longer in period and higher amplitude than any other surrounding peak. The pulse is often themain oscillatory component in the low-frequency bandpass filtered data set,though some records containadditional higher frequency content. An “easy” pulse to fit typically consists of approximately one cycle,different initial guesses converge to the same parameter estimates within a fewiterations, and the mod-eled pulse displacements are close to the actual displacement record and are occasionally detectable in theacceleration record as well. Examples of records with “easy” to fit pulsesto are shown in Figure 4.

Records which contained both a long-period, low amplitude, pulse and a short-period, high amplitudepulse were fit with “medium” difficulty. Sometimes an extremely high amplitude peak (approximately 150%or higher than other peaks in the record) would be selected as the pulse over some slightly longer-periodbut significantly lower amplitude peaks. Conversely, sometimes a very long-period pulse would be selectedover other peaks with much shorter periods and not significantly larger amplitudes. If additional reasoning isneeded in picking a peak for the pulse fit, the pulse closer to the beginning ofthe record was chosen. Usuallyonce a choice is made as to which peak constituted the desired pulse, the resulting fit would converge to adesirable result within a few iterations. Another reason a fit might be labeled“medium” in difficulty would


0 5 10 15 20 25 30−80−60−40−20

0204060

VE

LOC

, cm

/s (a) nf11

0 5 10 15 20 25 30

−40

−20

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20

40

TIME, s

DIS

PL,

cm

0 2 4 6 8 10 12 14 16−80−60−40−20

0204060

VE

LOC

, cm

/s

(b) la03

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0102030

TIME, s

DIS

PL,

cm

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VE

LOC

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/s (c) la30

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PL,

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LOC

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50V

ELO

C, c

m/s (e) se23

0 1 2 3 4 5 6 7 8 9−40−30−20−10

01020

TIME, s

DIS

PL,

cm

FIG. 4. Easy pulse fits for each data set, — = fit, - - = data. (a) near f ault, nf11; (b) LA10%/50yr, la03; (c) LA 2%/50yr, la30; (d) SE 10%/50yr, se14; (e) SE 2%/50yr, se23

be due to the need to constrain the parameter estimates to sensible values. For example, the number of pulsecycles,Nc, was constrained to a minimum value of 0.5. The nonlinear least squares procedure for a few ofthe records attempted to drive the number of cycles below this value, however, in order to compensate forthis, the pulse amplitude value,Vp, would grow arbitrarily large, leading to nonsensical results. Changesin parameter values throughout the convergence process were carefully examined for consistency beforechoosing a final result. Examples of records with “medium” pulse-fitting difficulty from each of the groundmotion suites are shown in Figure 5.

A record labeled as “difficult” to fit was typically one for which the least squares method would havetrouble converging or would converge to different results based on different initial guesses with no cleardistinction between pulse options. In some cases, particularly in the Seattle records, it was so difficultto isolate a distinct pulse that it was decided that no discernible pulse was present in the record. In suchcases, the pulse velocity would be assigned a value of zero. Several other records contained what appearedto be multiple cycle pulses where convergence was difficult due either to the variability of results fromdifferent initial guesses or simply the inability of the converged result to capture the pulse behavior. Inmost cases, either convergence (or at least general stability in parameters) was achieved that would capturemost of any pulse-like characteristic of the record based on choices thataimed to fit long-period and largeramplitude trends towards the beginning of the record. The greatest problem with the difficult pulse fits wasthat the pulse parameters were outliers; the method would frequently sacrifice capturing the high amplitude


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−20

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20

40

VE

LOC

, cm

/s (a) nf18

0 5 10 15 20 25−10−5051015

TIME, s

DIS

PL,

cm

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02040

VE

LOC

, cm

/s (b) la07

0 5 10 15 20 25 30 35 40−20

−10

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20

TIME, s

DIS

PL,

cm

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VE

LOC

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/s (c) la34

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TIME, s

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PL,

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VE

LOC

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TIME, s

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0204060

VE

LOC

, cm

/s (e) se26

0 5 10 15 20 25 30 35

−20−10

01020

TIME, s

DIS

PL,

cm

FIG. 5. Medium pulse fits for each data set, — = fit, - - = data. (a) near fault, nf18; (b) LA10%/50yr, la07; (c) LA 2%/50yr, la34; (d) SE 10%/50yr, se13; (e) SE 2%/50yr, se26

oscillations for longer period trends. Examples of records with “difficult” tofit pulse to are shown in Figure6. The numbers of “easy,” “medium,” difficult,” and “no pulse” recordsfor each of the ground motion suites

TABLE 2. Pulse fit difficulty for five data sets

Data Set Near Fault LA 2% in 50 LA 10% in 50 SE 2% in 50 SE 10% in 50

Easy 21 9 10 8 5Medium 12 8 8 5 6Difficult 7 3 2 0 1No Pulse 0 0 0 7 8

are shown in Table 2.To find Vs, the peak of the stochastic part of the velocity record, the acceleration time history was

bandpass filtered between 0.1 and 20 Hz and integrated once. The velocitypulse fit was then subtractedfrom the overall velocity record to leave only the stochastic part of the velocity record. The peak of thestochastic velocity record,Vs, was then determined and recorded.

To fit the envelope the instantaneous acceleration amplitude,e(ti), was computed from the original


0 5 10 15 20−60−40−20

0204060

VE

LOC

, cm

/s (a) nf20

0 5 10 15 20−20

−10

0

10

20

TIME, s

DIS

PL,

cm

0 2 4 6 8 10 12 14 16

−20−10

01020

VE

LOC

, cm

/s (b) la19

0 2 4 6 8 10 12 14 16−6−4−20246

TIME, s

DIS

PL,

cm

0 5 10 15 20 25

−50

0

50

VE

LOC

, cm

/s (c) la31

0 5 10 15 20 25

−20−10

01020

TIME, s

DIS

PL,

cm

0 5 10 15 20 25−30−20−10

010203040

VE

LOC

, cm

/s (d) se11

0 5 10 15 20 25

−5

0

5

10

TIME, s

DIS

PL,

cm

0 5 10 15 20 25 30 35 40 45

−100−50

050

100V

ELO

C, c

m/s (e) se36

0 5 10 15 20 25 30 35 40 45

−20−10

01020

TIME, s

DIS

PL,

cm

FIG. 6. Difficult pulse fits for each data set, — = fit, - - = data. (a) nea r fault, nf20; (b) LA10%/50yr, la19; (c) LA 2%/50yr, la31; (d) SE 10%/50yr, se11; (e) SE 2%/50yr, se36

acceleration record,a(ti), and its Hilbert transform,a(ti),

e(ti) =√

a(ti)2 + a(ti)2 . (7)

The envelope function, equation (3), was fit to the instantaneous acceleration amplitudes normalized to apeak value of 1. The envelope fit function was divided by

√2 so that the approximation would pass through

the normalized data. In fitting the envelope function, the residuals were weighted by the square root of thedata such that the higher amplitude values of the normalized acceleration wouldbe more significant,

χ2(τ0, τ1, τ2, τ3) =1

2

m∑

i=1

[

e(ti)/max(e(ti))− e(ti; τ0, τ1, τ2, τ3)/√2

1/√

e(ti)

]2

. (8)

It was found that this weighting method resulted in envelope fits that reachedtheir maximum value veryclose to the time of the peak ground acceleration (within the first half of the ground motion record), whichwas considered preferable to envelopes that had maxima toward the end ofthe record. Subtracting the pulseacceleration prior to fitting the envelope did not noticeably affect envelopeparameters. Initial guesses forthe four envelope parameters were chosen by visual inspection. Despitea variety of initial guesses with longvalues ofτ2, the least squares method consistently removed the plateau region indicating that this parametercould be excluded from the envelope function. Examples of envelope functions from each of the groundmotion suites are shown in Figure 7.


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0.1

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0.3

0.4

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0.8

0.9

1

TIME, s

EN

VE

LOP

E

(a) nf02

0 2 4 6 8 10 12 140

0.1

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TIME, s

EN

VE

LOP

E

(b) la16

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0.1

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1

TIME, s

EN

VE

LOP

E

(c) la34

0 5 10 15 20 250

0.1

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EN

VE

LOP

E

(d) se13

0 10 20 30 40 500

0.1

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0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

TIME, s

EN

VE

LOP

E

(e) se39

FIG. 7. Example envelope fits for each data set, — = fit, - - = data. (a) near fault, nf02; (b) LA10%/50yr, la16; (c) LA 2%/50yr, la34; (d) SE 10%/50yr, se13; (e) SE 2%/50yr, se39

The power spectral density of the difference between each acceleration record and its pulse was esti-mated using Welch’s averaged periodogram method and normalized to a maximumof 1. The normalizedspectra were then averaged over all records for each of the five datasets to obtain an averaged PSD for eachof the five sets. Each of the five average PSD’s were then fit with equation(2). In fitting power spectrathe residuals were weighted by the square root ofS(fi) such that the higher amplitude values of the powerspectrum were considered more significant.

χ2(fg, ζg) =1

2

m∑

i=1

[

S(fi)− S(fi; fg, ζg)

1/√

S(fi)

]2

(9)

This method was applied consistently to each of the five average PSD functions and the resulting parametervalues for the ground damping,ζg, and ground frequency,fg, were recorded along with their associatedstandard errors. The averaged PSD’s were easily fit by the Levenberg-Marquardt method; parameter esti-mates converged within a few iterations. The power spectral density fits foreach of the ground motion suitesare shown in Figure 8.

PARAMETER DISTRIBUTIONSThe systematic fitting of waveform expressions to the SAC ground motion data sets provide a statistical

sample of the characteristics of each suite. The near fault set provides 40 sample measurements of ten


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Frequency, Hz

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(a) Near Fault

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rum

(e) SE10in50

FIG. 8. Power spectra fits for each data set, — = fit, - - = data. (a) ne ar fault; (b) LA 10%/50yr;(c) LA 2%/50yr; (d) SE 10%/50yr; (e) SE 2%/50yr

of the model parameters and the others provide 20 sample measurements of tenof the model parameters.The remaining two parameters, ground frequency,fg, and ground damping,ζg, are obtained from spectraaveraged over each set, thereby giving one value per set. The means and variances of these statistical samplesare provided in Table 3. In this table the mean values for ground frequency, fg, and ground damping,ζg,were obtained via the nonlinear least squares fitting of the averaged power spectral density functions for eachdata set. The reported standard deviations for these two parameters are simply the standard parameter errorsas determined by the fitting procedure. Although the parameter describing theinitial period of no groundmotion,τ0, was required for fitting the envelopes, it simply represents an arbitrary amount of measurementtime before the strong ground motion begins. As such, it has no effect in computing structural responses.Therefore, the distributions of values forτ0 were not explored further. Since this time period is removedfrom models of simulated ground motions, the arrival in time of the pulse,Tpk, was also adjusted in orderto account for this change, by subtractingτ0 from Tpk.

The parameter estimates show that the 2% in 50 years Los Angeles data set has the largest velocity am-plitudes, the near fault data set has the second largest amplitude parameters. The mean velocity amplitudesfor the Los Angeles 2% in 50 years records are a little more than twice the mean velocity amplitudes for theLos Angeles 10% in 50 years records. The mean velocity amplitudes for the Seattle 2% in 50 years recordsare about three times larger than the mean velocity amplitudes for the Seattle 10% in50 years records.For the near-fault data set the pulse amplitudes are 60 percent larger than the amplitude of the stochastic


TABLE 3. Fit statistics and offsets for five data sets

suite parameterVp Tp Nc Tpk φ Vs τ1 τ2 τ3 fg ζg

cm/s s - s rad cm/s s s s Hz -

nrfaultmean 90.83 1.78 0.88 5.04 2.72 57.04 4.02 0.48 4.08 0.76 1.41

std.dev. 74.10 1.06 0.66 3.23 1.64 33.06 3.66 0.75 2.31 0.08 0.14

la10in50mean 45.28 2.05 0.80 4.97 2.71 34.83 3.77 0.14 4.48 0.78 1.76

std.dev. 22.81 1.25 0.82 3.32 1.42 16.77 2.69 0.14 3.10 0.08 0.14

la2in50mean 99.99 1.92 0.64 4.81 3.43 74.88 4.46 0.53 3.81 0.73 1.36

std.dev. 47.62 0.94 0.47 3.49 1.47 31.72 3.64 0.77 1.94 0.07 0.14

se10in50mean 11.60 1.95 0.84 6.11 3.26 22.22 12.25 0.14 10.40 1.33 0.82

std.dev. 11.97 0.65 0.30 2.47 1.50 11.91 13.04 0.10 8.28 0.140.08

se2in50mean 34.13 2.02 1.17 6.61 3.73 63.76 15.66 0.13 10.62 1.15 0.81

std.dev. 36.68 1.05 0.59 3.37 1.34 26.12 14.63 0.14 7.24 0.140.08offset 0 0.8 0.5 0 −π 10 1 0 1 0 0

content. For the two Los Angeles data sets the pulse amplitudes are 33 percent larger than the amplitudeof the stochastic content. In the two Seattle data sets the pulse amplitudes are onlyhalf of the stochasticcontent amplitudes. The pulses identified from the Seattle data set are therefore relatively inconsequential.The coefficient of variation (c.o.v.) in the pulse velocity is approximately 70% for near-fault records andapproximately 50% for the other Los Angeles records. The c.o.v. for the stochastic velocity amplitude isabout 50% for all data sets.

Mean pulse periods ranged from 2.58 s to 2.85 s across the five data sets.The c.o.v. for the pulse periodis about 60% for all data sets. Based on these waveforms, pulse period statistics are relatively insensitive tothe hazard level or geographical location.

The central frequency of the stochastic content of Seattle ground motions(≈ 1.20 Hz) is higher thanthat of Los Angeles ground motions (≈ 0.75 Hz), and the duration of Seattle ground motions is about threetimes those of Los Angeles. In all five data sets the bandwidth (ζfg) of the stochastic content is about 1.0/s.

Cumulative distributions for each of the nine ground motion parameters for thefive data sets are shownin Figures 9 through 13. The associated standard errors for every parameter estimate are displayed as hor-izontal error bars. The standard error bars shown are truncated bythe parameter value bounds provided inTable 1. Several parameters have physically-motivated minimum values as shown in the last row of Table 3.These offsets are subtracted from each of the sample values prior to the estimation of the mean and varianceof each distribution. These offset values must then be added to the randomly generated parameter valueprior to the ground motion simulation. For example, the number of cycles in a pulse, Nc, for records inthe “la10in50” set, can be considered as 0.5 plus a lognormal random variable with a mean of 0.8 and astandard deviation of 0.8. This procedure serves the purpose of generating a better fit of the cumulative dis-tribution function as well as ensuring that physically meaningful parameter minimums are enforced duringsimulations. Empirical cumulative distributions of the parameter estimates match lognormal and Gammadistribution functions. The difference between the distributions is indiscernible in their upper tails.

SUMMARY AND CONCLUSIONSThe generation of synthetic ground motions in this study is accomplished via the superposition of a

coherent velocity pulse with a stochastic acceleration record corresponding to a specific power spectraldensity function and multiplied by a time modulating envelope function. Estimation of twelve groundmotion waveform parameters was accomplished via a systematic nonlinear leastsquares analysis for 120of the SAC ground motion records. Five parameters describe the pulse: pulse amplitude,Vp, pulse period,Tp, number of cycles,Nc, pulse arrival time,Tpk, and pulse phase,φ. One parameter describes the peak ofthe stochastic content of the velocity,Vs. Four parameters describe the envelope: the time before groundmotion begins,τ0, the period of increasing amplitude,τ1, a plateau region of strong ground motion,τ2, anda period of slow amplitude decay,τ3. And two parameters describe the power spectral density function: theground frequency,fg, and the ground damping,ζg. The ground frequency and ground damping parameters


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FIG. 9. Parameter estimates, standard errors, and cumulative dist ributions of ground mo-tion parameter estimates for the SAC “nrfault” set. ⊙=parameter estimates; ⊢⊣=standardparameter errors; •=Gamma distribution; —=lognormal distribution. (a) Vp; (b) Tp; (c) Nc;(d) Tpk; (e) φ, (f) Vs; (g) τ1; (h) τ2; (i) τ3

were fit for the average power spectral density for each data set rather than for individual records becausepower spectra computed from short-duration transient waveforms havea high variance. Sample means,variances, and distribution functions were calculated for each of the other ten parameters for each data setand parameter distributions are represented well by lognormal and Gamma distribution functions.

The pulse period statistics (Tp ≈ 2.7s ± 60%), the bandwidth statistics (ζgfg ≈ 1.0/s ± 10%) andthe coefficient of variation of the peak stochastic velocity (50%) are similar across the five suites of groundmotion analyzed in this study. The central frequency of the stochastic ground motionfg is about 1.20 Hzin the Seattle records and about 0.75 Hz for the Los Angeles records. The duration of the Seattle records isabout three times longer than the Los Angeles records and velocity pulses are relatively inconsequential inthe Seattle records, which are generally far-field records from subduction-zone ruptures.

This analysis provides a quantitative measure of the variability among groundmotions though a param-eterized ground motion waveform model. A companion study (Gavin and Dickinson 2010) uses this infor-mation to investigate correlations between parameters, to identify parameters having a significant impacton structural response, and to develop response-spectrum-compatiblemodels for bi-axial synthetic groundmotions representative of the original data sets. Provided that ground motion parameters can be identifiedor scaled for a sufficiently large number of hazard levels, this simulation procedure enables high resolutionfragility analyses.

ACKNOWLEDGEMENTSThis material is based upon work supported by the the Civilian Research and Development Foundation


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(i)

FIG. 10. Parameter estimates, standard errors, and cumulative dis tributions of ground mo-tion parameter estimates for the SAC “la10in50” set. ⊙=parameter estimates; ⊢⊣=standardparameter errors; •=Gamma distribution; —=lognormal distribution. (a) Vp; (b) Tp; (c) Nc;(d) Tpk; (e) φ, (f) Vs; (g) τ1; (h) τ2; (i) τ3

for the Independent States of the Former Soviet Union (CRDF) under Award No. MG1-2319-CH-02 andby the National Science Foundation under Grant No. NSF-CMMI-0704959 (NEES Research), and GrantNo. NSF-CMS-0402490 (NEES Operations). Any opinions, findings, and conclusions or recommendationsexpressed in this material are those of the authors and do not necessarilyreflect the views of the NationalScience Foundatioon.

The authors express sincere thanks to the reviewers, whose perceptive comments were instrumental inrevising this manuscript.

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