Spatial variation of seismic ground motions: An overview

Spatial variation of seismic ground motions: An overview

Aspasia ZervaDepartment of Civil and Architectural Engineering, Drexel University, Philadelphia PA 19104;[email protected]

Vassilios ZervasDeceased

This study addresses the topic of the spatial variation of seismic ground motions as evaluatedfrom data recorded at dense instrument arrays. It concentrates on the stochastic description ofthe spatial variation, and focuses on spatial coherency. The estimation of coherency from re-corded data and its interpretation are presented. Some empirical and semi-empirical coherencymodels are described, and their validity and limitations in terms of physical causes discussed.An alternative approach that views the spatial variation of seismic motions as deviations inamplitudes and phases of the recorded data around a coherent approximation of the seismicmotions is described. Simulation techniques for the generation of artificial spatially variableseismic ground motions are also presented and compared. The effect of coherency on the seis-mic response of extended structures is highlighted. This review article includes 133references.@DOI: 10.1115/1.1458013#

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1 INTRODUCTION

The termspatial variation of seismic ground motionsdenotesthe differences in amplitude and phase of seismic motiorecorded over extended areas. The spatial variation of smic ground motions has an important effect on the respoof lifelines such as bridges, pipelines, communication tranmission systems, etc. Because these structures extendlong distances parallel to the ground, their supports undedifferent motions during an earthquake. This differential mtion can increase the response of lifelines beyond thesponse expected if the input motions at the structures’ sports were assumed to be identical. The results of panalyzes reported in the literature indicate that the effectthe spatial variation of seismic ground motions on the rsponse of bridges cannot be neglected, and can be, in cadetrimental@1#. The incorporation of the spatial variation othe seismic motions in lifeline design response spectraalso been attempted@2–5#. It has been recently recognize@6# that the spatial variation of seismic ground motions chave a dramatic effect on the response of extended sttures. Presently, in studies performed by the California Dpartment of Transportation~Caltrans!, spatially variableground motions are used as input motions at the supportvarious bridges, such as the West Bay Bridge in San Frcisco and the Coronado Bridge in San Diego, California@7#.The modeling of spatial variability and its effects on thresponse of lifelines is a topic under renewed, active invtigation, eg,@8–10#.

Transmitted by Associate Editor DE Beskos

ASME Reprint No AMR328 $22.00Appl Mech Rev vol 55, no 3, May 2002 271

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The spatial variability in seismic ground motions can rsult from relative surface fault motion for recording statiolocated on either side of a causative fault; soil liquefactiolandslides; and from the general transmission of the wafrom the source through the different earth strata toground surface. This paper will concentrate on the latcause for the spatial variation of surface ground motions

The spatial variation of seismic ground motions hstarted being analyzed after the installation of dense insment arrays. Before the installation of these arrays, the stial variation of the motions was attributed to the apparepropagation of the waveforms on the ground surface, iewas considered, at least in engineering applications, thatdifference in the motions between two stations was causolely by a time delay in the arrival of the time history at thfurther away station, eg,@11–13#. The data recorded at densseismograph arrays have provided valuable informationadditional causes and more detailed descriptions for thetial variation of the motions.

One of the first arrays installed was the El Centro Diffeential array @14# that recorded the 1979 Imperial Valleearthquake; the array was linear and consisted of seventions with a total length of 312.6 m and minimum separatidistance of 7.6 m. The array, however, which has providedabundance of data for small and large magnitude eventshave been extensively studied by engineers and seismgists, is the SMART-1 array~Strong Motion ARray in Tai-wan!, located in Lotung, in the north-east corner of Taiw

© 2002 American Society of Mechanical Engineers

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272 Zerva and Zervas: Spatial variation of seismic ground motions Appl Mech Rev vol 55, no 3, May 2002

@15#. This 2D array, which started being operative in 198consisted of 37 force-balanced triaxial accelerometersranged on three concentric circles, the inner denoted bI ,the middle byM , and the outer byO ~Fig. 1!. Twelve equi-spaced stations, numbered 1-12, were located on eachand stationC00 was located at the center of the array. Spavariability studies based on the SMART-1 data startedpearing in the literature almost as early as the array recings became available~eg, @16,17#!. Two additional stationsE01 andE02, were added to the array in 1983, at distanof 2.8 and 4.8 km, respectively, south ofC00 @18#. The arraywas located in a recent alluvial valley except for stationE02that was located on a slate outcrop@18#. A smaller scale, 3D~ground surface and down-hole instrumentation! array, theLSST array, was constructed in 1985 within the SMARTarray close to stationM08 ~Fig. 1!. The LSST array consistof 15 free-surface triaxial accelerometers located along tharms at 120° intervals, and eight down-hole acceleromeat depths up to 47 m@18,19#. Additional arrays, permanenand temporary, have been and are being deployed arounworld for studies of the characteristics of seismic groumotions, eg, EPRI Parkfield, USGS Parkfield, HollistStanford, Coalinga, USGS ZAYA, Pinyon Flat@20#, Tarzana@21#, San Fernando Valley@22#, Sunnyvale@23#, all in Cali-fornia, USA; Chiba Experiment Station, Tokyo, Japan@24#;Nice, France@25#; Thessaloniki, Greece@26#; L’Aquila, Italy,@27#.

This paper presents an overview of the spatial variationseismic ground motions, its modeling, and the interpretaof the models. It starts in Section 2 with the descriptionthe stochastic approach for the evaluation of the spatial vability from recorded data. Section 3 interprets the stochadescriptor of the spatial variability, the coherency, in termsthe phase variability it represents. Section 4 presents som

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the available empirical and semi-empirical spatial variabilmodels and discusses their validity and limitations; the effof coherency on the seismic response of lifelines~bridges! ishighlighted. Section 5 describes a recent approach thatlyzes spatial variability without relying on coherency esmates, and identifies correlation patterns in the amplituand phase variation of recorded data around a common,herent component. Section 6 describes simulation techniqfor the generation of spatially varying seismic ground mtions; the characteristics of simulations generated fromferent coherency models and by means of different simution schemes are compared. Finally, Section 7 providebrief summary, conclusions, and recommendations.

2 STOCHASTIC DESCRIPTION OF THESPATIAL VARIATION OF SEISMIC GROUNDMOTIONS FROM RECORDED DATA

For illustration purposes, consider that seismic ground mtions ~accelerations!, a(rW j ,t), in one of the horizontal~N-Sor E-W! or vertical direction at a locationrW j5(xj ,yj ) on theground surface and at timet can be described by the sum oan infinite number of sinusoids as follows:

a~rW j ,t !5 (n51

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A~vn ,rW j ,t ! sin@vnt1kW ~vn ,t !•rW j

1f~vn ,rW j ,t !# (1)

wherevn is the ~discrete! frequency;kW (vn ,t) is the wave-number that specifies the average apparent propagatiothe waveforms on the ground surface; andA(vn ,rW j ,t) andf(vn ,rW j ,t) are the amplitude and phase shift, respectiveFor the seismic time history recorded at a different stationk,Eq. ~1! still holds, but the amplitudesA(vn ,rWk ,t) and phasesf(vn ,rWk ,t) would differ from those at stationj . The propa-gation term~kW (vn ,t)•rW j in Eq. ~1!! also incorporates stationdependent arrival time perturbations around the time decaused by the average apparent propagation of the moon the ground surface. The differences in the time histoof Eq. ~1! between stations constitute the spatial variationthe motions, and are attributed to source-rupture characttics, wave propagation through the earth strata, scatterand local site effects. The physical causes underlyingspatial variation of seismic motions are presented extensiin @28#, and are highlighted herein in Sections 4 and 5.

The procedure for the evaluation of the stochastic spavariation of seismic motions from recorded data considthat the motions are realizations of space-time random fieIn order to extract valuable information from the availablimited amount of data, such as the recorded time historiethe array stations during an earthquake, certain assumpneed to be made:• It is assumed that the random field is homogeneous

space, ie, all stochastic descriptors of the motions~jointprobability distribution functions! are functions of theseparation distance between stations, but independenabsolute location. This assumption implies that the fquency content~amplitude! of the seismic motions at dif-ferent recording stations does not vary significantly. Sin
Fig. 1 The SMART-1 array

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Appl Mech Rev vol 55, no 3, May 2002 Zerva and Zervas: Spatial variation of seismic ground motions 273

the majority of dense instrument arrays are locatedfairly uniform soil conditions, the assumption of homogneity is valid. Significant variation in the frequency conteof the motions can be expected if the stations are locatedifferent local soil conditions~eg, one on rock and theother on alluvium!.

• It is further assumed that the time histories recorded atarray stations are stationary random processes. Stationimplies that the probability functions do not depend onabsolute time, but are functions of time differences~ortime lag!; in this sense, the time histories have neithbeginning nor end. Although this assumption appears tounrealistic, this is not the case: Most characteristics of smic ground motions for engineering applications are evaated from the strong motion shear~S-! wave window, ie, asegment of the actual seismic time history. This stromotion segment from the actual time history can be viewas a segment of an infinite time history with uniform chaacteristics through time, ie, a stationary process. For ationary process, the amplitude and phase of the moti~Eq. ~1!! are not functions of time.

• It is also assumed that the stationary time histories atrecording stations are ergodic. A stationary process isgodic, if averages taken along any realization of the pcess are identical to the ensemble averages, ie, the inmation contained in each realization is sufficient for tfull description of the process. The assumption of ergodity is necessary: The evaluation of probabilistic modelsthe description of the spatially variable seismic groumotions would require, ideally, recordings at the samefrom many earthquakes with similar characteristics, so tan ensemble of data can be analyzed and averages oensemble can be evaluated. However, in reality, theronly one realization of the random field, ie, one setrecorded data at the array for an earthquake with specharacteristics.

It is obvious that reality does not fully conform to theassumptions, but actual data recorded at dense instruarrays during the strong motion S-wave window mayviewed as homogeneous, stationary, and ergodic in a limor weak sense. These necessary, but restrictive, assumpfor the modeling of the spatial variation of seismic groumotions are typically retracted in the simulation of such mtions for engineering applications.

The following stochastic descriptors characterize the rdom field.

2.1 Cross spectral density

The random field of seismic ground motions~accelerations!is best described by means of the cross spectral density odata recorded at two stations~locations! on the ground sur-face @29#: Let a(rW j ,t)5aj (t) and a(rWk ,t)5ak(t) be twotime series in the same direction recorded at locationsrW j andrWk on the ground surface, and letj jk5urW j2rWku indicate thestation separation distance. Let the duration of the strmotion S-wave window be 0<t<T, T5NDt, with N beingthe number of samples in the recorded time series forwindow, andDt the time step. It is iterated here that th

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window of the actual time history is assumed to be a segmof an infinite one with uniform characteristics through tim~stationarity assumption!. The cross covariance function othe motions between the two stations is defined as@30#:

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The cross covariance function is generally smoothed beit is further used as an estimator; the smoothed cross covance function is:

Rjk~t!5w~t!Rjk~t! (3)

where w(t) is the lag window, with propertiesw(t)5w(2t), w(t50)51 and*2`

1`w(t)51.It is customary—for stationary processes—to work in t

frequency rather than the time domain. The cross specdensity~or cross spectrum! of the process is defined as thFourier transform of the covariance function~Eq. ~2!!. Thesmoothed cross spectrum is evaluated by Fourier transfoing Eq. ~3!:

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with i 5A21 andv being the frequency~in rad/sec!. Alter-natively, the cross spectral estimates of Eq.~4! can be evalu-ated directly in the frequency domain as follows: LetAj (v)andAk(v) be the scaled discrete Fourier transforms oftime historiesaj (t) andak(t), respectively, defined as:

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The ~smoothed! cross spectrum of the series becomes:

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where the spectral window,W(v), is the Fourier transformof the lag windoww(t), Dv is the frequency step, (Dv5 2p/T), and* indicates complex conjugate. The presenof the smoothing windows in the time or frequency domareduces the variance in the estimates but, also, the resoluof the spectra.

2.2 Power spectral density

The power spectral densities of the motions~ie, j [k in Eqs.~4! and ~6!! are estimated from the analysis of the datacorded at each station and are commonly referred to aspointestimates of the motions:

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It is obvious that the Fourier spectra of the motions atvarious stations will not be identical. However, the assumtion of spatial homogeneity in the random field implies ththe power spectrum of the motions is station independen

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Once the power spectra of the motions at the stationinterest have been evaluated, a parametric form is fittethe estimates, generally through a regression scheme.most commonly used parametric forms for the power sptral density are the Kanai-Tajimi spectrum@31,32#, or its ex-tension developed by Clough and Penzien@33#. The physicalbasis of the Kanai-Tajimi spectrum is that it passes a wprocess through a soil filter; the following expression for tpower spectral density of ground accelerations results:

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in which S+ is the amplitude of the white process bedroexcitation, andvg and zg are the frequency and dampincoefficient of the soil filter. A deficiency of Eq.~8! is that thespectrum produces infinite variances for the ground veloand displacement. For any stationary process, the pospectral densities of velocity,Sv(v), and displacementSd(v) are related to that of acceleration through:

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v4 Sa~v! (9)

It is apparent from Eqs.~8! and ~9! that the velocity anddisplacement spectra of the Kanai-Tajimi spectrum aredefined asv→0. Clough and Penzien@33# passed the KanaiTajimi spectrum~Eq. ~8!! through an additional soil filterwith parametersv f andz f , and describe the power spectruof ground accelerations as:

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which yields finite variances for velocities and displacments. For the simulation of seismic motions at firm sconditions, the soil filter parameters assume the valuesvg

55p rad/sec,v f50.1vg , andzg5z f50.6 @34#; the valuesof the parameters for other soil conditions can be found,in @4,35#.

The aforementioned spectra model only the effect oflocal soil conditions, since the bedrock excitation~S+ in Eqs.~8! and ~10!! is a white process. However, seismic groumotions are the result of the rupture at the fault andtransmission of waves through the media from the faultthe ground surface. Alternatively, seismological spectraincorporate these effects can be used instead of Eqs.~8! and~10! for simulation purposes. For example, Joyner and Bo@36# presented the following description for the stochasseismic ground motion spectrum:

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S~v!5CF SF~v!AF~v!DF~v!IF ~v! (11)

in which CF represents a scaling factor which is a functiof the radiation pattern, the free surface effect, and theterial density and shear wave velocity in the near souregion;SF(v) is the source factor that depends on the mment magnitude and the rupture characteristics;AF(v) isthe amplification factor described either by a frequencypendent transfer function as, eg, the filters in the KanTajimi or Clough-Penzien spectra~Eqs.~8! and ~10!!, or, interms of the site impedanceA(r0v0)/(r rv r), wherer0 andv0 are the density and shear wave velocity in the souregion andr r and v r the corresponding quantities near threcording station;DF(v) is the diminution factor, that accounts for the attenuation of the waveforms; andIF (v) is afilter used to shape the resulting spectrum so that it repsents the seismic ground motion quantity of interest.

2.3 Coherency

The coherency of the seismic motions is obtained fromsmoothed cross spectrum of the motions between thestationsj andk, normalized with respect to the corresponing power spectra as, eg,@19,37#:

g jk~v!5Sjk~v!

ASj j ~v!Skk~v!(12)

where the subscriptn in the frequency has been dropped fconvenience.g jk(v) is a complex number; the square of thabsolute value of the coherency, the coherence:

ug jk~v!u25uSjk~v!u2

Sj j ~v!Skk~v!(13)

is a real number assuming values 0<ug jk(v)u2<1. Coher-ency is commonly written as:

g jk~v!5ug jk~v!u exp@ iu jk~v!# (14)

with

u jk~v!5tan21S I@Sjk~v!#

R@Sjk~v!# D (15)

being the phase spectrum;R andI in Eq. ~15! indicate thereal and imaginary part, respectively.ug jk(v)u is commonlyreferred to as the lagged coherency, andR@g jk(v)# as theunlagged coherency.

Once coherency estimates are obtained from the recotime histories, a functional form~model! is fitted through thecoherency data points by means of regression analysesisting coherency models are extensively discussed in Sec4; Section 3 presents the physical interpretation of cohency.

3 INTERPRETATION OF COHERENCY

The complex coherency of Eq.~14! is alternatively expressedas:

g~j,v!5ug~j,v!u exp@ iu~j,v!# (16)

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where the subscriptsj and k have been dropped; Equatio~16! then indicates the variation of coherency for any seration distancej between two locations on the ground suface.

3.1 Wave passage effect

The complex term in the above equation, exp@iu(j,v)#, de-scribes the wave passage effect, ie, the delay in the arrivthe waveforms at the further away station caused bypropagation of the waveforms. Consider that the ground mtions consist of a unit amplitude, monochromatic randwave propagating with velocitycW on the ground surfacealong a line connecting the stations. The coherency expsion for this type of motion would be@38#:

gwp~j,v!5expF2 iv~cW•jW !

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For a single type of wave dominating the window analyzas is most commonly the case for the strong motion S-wwindow used in spatial variability evaluations, the considation that the waves propagate with constant velocity onground surface is a valid one@37#. Furthermore, since bodwaves are non-dispersive, except in highly attenuated methey have the same velocity over a wide range of frequen@39#. Thus, in Eq.~16!,

u~j,v!52v j

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where nowc indicates the apparent propagation velocitythe motions along the line connecting the stations. Abrahsonet al @19# introduced a correction factor in Eq.~16!, thatis controlled by the constant apparent propagation velocitthe motions at lower frequencies, but allows random vaability in the velocity at high frequencies.

3.2 Lagged coherency

The lagged coherency,ug(j,v)u, is a measure of ‘‘similar-ity’’ in the seismic motions, and indicates the degreewhich the data recorded at the two stations are relatedmeans of a linear transfer function. If, for example, one pcess can be obtained by means of linear transformation oother and in the absence of noise@40#, coherency is equal toone; for uncorrelated processes, coherency becomes zeis expected that at low frequencies and short separationtances the motions will be similar and, therefore, coherewill tend to unity asv→0 andj→0. On the other hand, alarge frequencies and long station separation distancesmotions will become uncorrelated, and coherency will teto zero. The value of the coherency in between these extrcases will decay with frequency and station separationtance. This observation has been validated from the analof recorded data, and the functional forms describinglagged coherency at any site and any event are exponefunctions decaying with separation distance and frequen

A significant characteristic of the lagged coherency, tdoes not become apparent directly from Eq.~12!, is that it isonly minimally affected by the amplitude variability betweethe motions at the two stations. Spudich@41# presents a

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simple example to illustrate this: Consider that the time htory at stationk is a multiple of that at stationj , ie, ak(t)5naj (t); the substitution of this expression in Eqs.~6!, ~7!,and ~12! would yield exactly the same coherency for avalue ofn. Even when the motions at the various stationsnot multiples of each other, it can be shown that the absovalue of the coherency is not sensitive to amplitude vations. Thus, lagged coherency describes the phase variabin the data in addition to the phase difference introducedthe wave passage effect~Eq. ~18!!.

Abrahamson @42# presented the relation between thlagged coherency and the random phase variability, whicsummarized in the following: Letf j (v) and fk(v) be the~Fourier! phases at two stationsj and k on the ground sur-face, after the wave passage effects have been removedrelation betweenf j (v) andfk(v) can be expressed as:

f j~v!2fk~v!5b jk~v!p e jk~v! (19)

in which e jk(v) are random numbers uniformly distributebetween@21,11#, andb jk(v) is a deterministic function ofthe frequencyv and assumes values between 0 andb jk(v) indicates the fraction of random phase variability btween@2p,1p# ~from the productpe jk(v)! that is presentin the Fourier phase difference of the time histories. Fexample, ifb jk(v)50, there is no phase difference betwethe stations, and the phases at the two stations are idenand fully deterministic. In the other extreme case, ie, whb jk(v)51, the phase difference between stations is copletely random. Based on Eq.~19!, and neglecting the amplitude variability of the data, Abrahamson@42# noted thatthe mean value of the lagged coherency can be expresse

E@ ug jk~v!u#5sin~b jk~v!p!

b jk~v!p(20)

It is easy to verify from Eq.~20! that when the lagged coherency tends to one,b jk(v) is a small number, ie, only asmall fraction of randomness appears in the phase differebetween the motions at the two stations; as coherencycreases,b jk(v) increases, and, for zero coherency,b jk(v)51.

The following peculiarity of the coherency estimate neeto be mentioned: Consider that the smoothing window isused in the evaluation of the cross covariance function nalternatively, in the cross spectrum of the series~Eqs.~3! and~6!!. Substitution of Eqs.~6! and ~7!, which becomeSjk(v)5Aj* (v)Ak(v), Sj j (v)5uAj (v)u2, and Skk(v)5uAk(v)u2, into Eq.~12! yields the identityug(j,v)u[1 forany frequencyv and station pair (j , k) regardless of the truecoherency of the data. The information about the differenin the phases of the motions at the stations is introducethe estimate through the smoothing process, which contthe statistical properties of the coherency as well as its relution. As an example, Fig. 2 presents the lagged cohereevaluated from data recorded at stationsI06 andI12 ~sepa-ration distance of 400 m! and stationsC00 andM06 ~sepa-ration distance of 1000 m! during the strong motion S-wavwindow of motions recorded in the N-S direction of Eventat the SMART-1 array. For the evaluation of the lagged c

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herencies in the figure, 3-, 7-, 11-, 15-, and 19-point Haming windows~ie, M51, 3, 5, 7, and 9, respectively, in Eq~6!! are used. It can be seen from the figure that the mnear-neighbor frequencies are used in the process, the hthe loss of resolution in the estimates. Abrahamsonet al @19#note that the choice of the smoothing window shoulddirected not only from the statistical properties of the cohency, but also from the purpose for which it is derived. If tcoherency estimate is to be used in structural analysis,time windows less than approximately 2000 samples andstructural damping coefficient 5% of critical, an 11-poiHamming window is suggested@19#. Recently, Zerva andBeck @43# have proposed an alternative approach for bypaing the strict requirement of coherency smoothing if pametric models of the coherency are to be evaluated: by fit

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parametric expressions to unsmoothed cross spectralmates, parametric models of lagged coherency are recovindirectly.

The statistics of the coherency estimateug jk(v)u, as ob-tained form the recorded data~Eq. ~12!!, are not simple. Itsvariance depends on its value: asug jk(v)u decreases, its variance increases@44#. On the other hand, tanh21@ugjk(v)u# isapproximately normally distributed with constant varianinversely proportional to the duration of the strong motiwindow and the bandwidth of the smoothing spectral wdow @30#. This transformation is sometimes used~eg,@29,44#!, before any parametric form of coherency is fittedthe empirical data. However, the statistical propertiestanh21@ugjk(v)u# assume that the spectrum of the processapproximately constant over the frequency bandwidth ofwindow; if this is not the case, then an additional sourcebias is introduced in the estimate@44#. Coherency functionsas estimates obtained from limited, finite segments of dare biased estimators: Bias is expected at low frequendue to the sensitivity characteristics of seismometers andsmall intensity of the low frequency components of tground motions@45–47#. Bias is introduced at higher frequencies and long separation distances due to the usenite length series that may produce finite values for the emate when the true coherency is zero@29#. Additionaluncertainty is introduced by inaccuracies in recorder schronization~eg, @45,47,48#!, and by imperfect eliminationof time lags caused by wave passage effects@47#. The timelags caused by the wave passage effect appear asdetermin-istic (j/c) in Eq. ~18!. However, the wave passage effealso incorporates random, station dependent time delay fltuations around these deterministic delays that affect theherency and should be given proper consideration.

3.3 Arrival time perturbations

The apparent propagation velocity of the seismic motioacross the seismograph array can be estimated by measignal processing techniques, such as the conventimethod~CV! @49,50#, the high resolution method~HR! @50#,or the multiple signal characterization method~MUSIC!@51,52#. All techniques evaluate, in different forms, thfrequency-wavenumber~F-K! spectrum of the motions, iethe triple Fourier transform of the recorded time historieand identify the average propagation velocity of the motiofrom the peak of the spectrum. An evaluation of the advtages and disadvantages of the techniques can be foundin @53#.

Seismic ground motions, however, incorporate randtime delay fluctuations around the wave passage delay,are particular for each recording station. These arrival tiperturbations are caused by the upward traveling ofwaves through horizontal variations of the geologic structunderneath the array@41# and, also, due to deviations of thpropagation pattern of the waves from that of plane wapropagation@54#. They can be so significant that the apparepropagation pattern of the wavefronts between sets oftions can be very different from the estimated overall pattof constant apparent propagation throughout the array@55#.

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Fig. 2 Comparison of lagged coherency evaluated from recordata at separation distances of 400m and 1000m for diffesmoothing windows

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oh

ist

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i

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Whereas the wave passage effects control the complex enential term of the coherency in Eq.~16!, the arrival timeperturbations affect its absolute value, namely the laggedherency. An approach that partially eliminates these effefrom coherency estimates is the alignment of the data wrespect to a reference station. In this process, the crossrelation of the motions~the normalized cross covariancfunction of Eq.~3!! relative to the reference station is evalated. The time corresponding to the highest correlation pvides the delay in the arrival of the waves at the variostations relative to their arrival at the reference one. Oncemotions are aligned, they become invariant to the referestation selection, but the value of the time delay requiredalignment is relative, ie, it is affected by the choice of treference station. More refined approaches for the estimaof the arrival time perturbations have been proposed bygeophysical community for the analysis of data collectedclosely spaced sensors and used for exploration purp@41#. For example, Rothman@56,57# proposed an approacthat views the problem as nonlinear inversion, estimatescross correlations of the data, but, instead of picking thpeak, the correlation functions are transformed to probabdistributions; random numbers are generated from the dibutions in an iterative scheme that minimizes the objecfunction, until convergence to the correct time shiftsachieved.

It should be noted, however, that if the motions at tarray stations are aligned with respect to a reference stabefore their lagged coherency is estimated, then speciatention should be placed in separately analyzing and estiming these delays due to their significance in the spatial vation of the motions@41,54,55,58#. Boissieres and Vanmarcke@54# modeled these fluctuations from SMART-1 data. Thconsidered an extension of the closure property@37#, inwhich ‘‘closure’’ is checked by relating the lags of all tripleof stations taken two by two. Their model considers thattime lag between two stations is given by:

Dt jk5Dt jkwp1Dt jk

r (21)

in which Dt jkwp5j jk /c is the time lag between the two sta

tions, j andk, due to the average propagation of the wavand Dt jk

r represents the random fluctuations. From thanalysis of the SMART-1 data they concluded thatDt jk

r is anormally distributed random variable with zero mean astandard deviation equal to 2.73102215.4131025j jk , inwhich j jk is measured in m.

4 SPATIAL VARIABILITY MODELS

A mathematical description for the coherency was first intduced in earthquake engineering by Novak and Hin@34,59# in 1979. The expression, based on wind engineerwas:

ug~j,v!u5expF2kS vj

VsD nG (22)

where,k andn are constants andVs is an appropriate sheawave velocity. It should also be noted that Novak and Hin@34,59# presented the first stochastic analysis of a lifel

xpo-

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tiontheatses

theeirlitytri-

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ey

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rdyne

system~buried pipeline! subjected to seismic motions expriencing loss of coherence; until that time only the propation of the motions was considered in deterministic analyof lifelines eg,@11,12,60#, with the exemption of the work ofBogdanoff et al @61#, who considered random earthquaktype excitations and Sandi@13#, who presented a stochastanalysis of lifeline response to non-synchronous seismictions. After the installation of strong motion arrays and,particular, the SMART-1 array, the stochastic descriptionthe seismic motions and the stochastic response analyslifelines have been extensively investigated by researche

The following two subsections present some of the exing empirical and semi-empirical models for the spatvariation of the seismic ground motions; it is noted, howevthat the list of models presented is not exhaustive.

4.1 Empirical coherency models

Because of i! the variability in seismic data recorded at diferent sites and during different events; ii! the differences inthe numerical processing of the data used by various invtigators; and iii! the different functional forms used in thregression fitting of a function through data with large scter, there is a multitude of spatial variability expressionsthe literature. The procedures for the removal of the wapassage effects in lagged coherency estimates also vsome procedures evaluate coherency directly from the dsome remove the apparent propagation effects first andevaluate the coherency, while others align the data beforecoherency is estimated. Some of the developed expressfor the description of the coherency of the seismic groumotions at the SMART-1 array are presented in the folloing:

due to Loh@62#:

ug~j,v!u5exp~2a~v!uju! (23)

with a(v) being a function ofv determined from the data oEvent 5;

due to Loh and Yeh@63#:

ug~j,v!u5expS 2avuju2pcD (24)

with parameters determined from Events 39 and 40;due to Loh and Lin@64# for the description of 1D~isotro-pic! coherency estimates:

ug~j,v!u5exp~2aj2!

ug~j,v!u5exp~~2a2bv2!uju! (25)

ug~j,v!u5exp~~2a2bv!ujuc!

and due to Loh and Lin@64#, again, but for the descriptionof directionally dependent coherency estimates:

ug~j,v!u5exp~~2a12b1v2!uj cosuu!exp~~2a2

2b2v2!uj sinuu! (26)

with u indicating the angle between the direction of propgation of the waves and the station separation, and the

ts

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r

a

l

r

-e

v

a

h

ringdatais-

20

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ofthe

esve-a-

s of

y is

al-

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dia,fre-


maining parameters depending on SMART-1 dadue to Haoet al @65# and Oliveiraet al @66#:

ug~j l ,j t ,v!u5exp~2b1uj l u2b2uj tu!

3expF2~a1~v!Auj l u1a2~v!Auj tu!

3S v

2p D 2G (27)

with a1(v) and a2(v) given by functions of the form:a i(v)52p ai /v1biv/(2p)1ci ; j l andj t in the equationare projected separation distances along and normal todirection of propagation of the motions, and the parameand functions in the equation are obtained through regresanalyses of the data.

Most lagged coherency estimates assume that the ranfield is isotropic, in addition to being homogeneous; isotroimplies that the rotation of the random field on the grousurface will not affect the joint probability density functionAs a consequence of this assumption, most derived expsions for the lagged coherency are functions of separadistance only (j jk5urW j2rWku) and not direction (jW jk5rW j

2rWk). It can also be observed from Eq.~16! that the onlydirectionally dependent term in the coherency is the oneresenting the wave passage effect~Eq. ~17!!. The last twoexpressions, however, are based on the description ofrandom field as anisotropic and account for the directiodependence of the spatial variation of the motions. LohLin @64# indicate that, since seismic ground motions are pdependent, the spatial variability will be direction dependeas different directions represent different paths; their evation of isocoherence maps for two events recorded atSMART-1 array suggested that the field is not axisymmetIndependent analyses of additional SMART-1 data suppthis observation: Abrahamsonet al @19# also observed directional dependence of coherency and presented a possiblplanation for this effect, namely that scattering in the forwadirection tends to be in phase with the incident wawhereas scattering to the side tends to loose phase@67#. Ra-madan and Novak@47#, in order to preserve the simpler representation of the field as isotropic, model its weak anisropy characteristics by defining the separation distancej5j l1m tj t , in which m t is a separation reduction factor taccount for the directional variability in the data. They cosidered the two expressions~Eqs.~26! and ~27!! and the setof data from which they were obtained and concluded tthis simple separation distance transformation with variam t preserves the isotropy of the field@47#.

Due to Abrahamsonet al @68#:

tanh21@ ug~j,v!u#5~2.5420.012j!

3H exp@~20.11520.00084j!v#

1v20.878

3 J 10.35 (28)

ta;

theersion

dompynd.res-tion

ep-

thenalnd

athnt,ua-theic.ort

ex-rde,

-ot-s

on-

atble

from nonlinear regression analysis of data recorded du15 earthquakes; the above expression was derived fromrecorded at the LSST array and is valid for separation dtances less than 100 m.

Due to Harichandran and Vanmarcke@37# and Harichan-dran @29,55#:

ug~j,v!u5A expS 22 B ujuan~v! D1~12A!expS 2

2 B ujun~v! D

(29)

n~v!5kF11S v

2p f 0D bG21/2

; B5~12A1aA!

ug~j,v!u5A expS 22uju~12A!

a k D3F11S v

2p f 0D bG1/2

1~12A! (30)

with the parameters determined from the data of Event@37# and Events 20 and 24@29#; Harichandran@29# intro-duced the second coherency expression~Eq. ~30!! to accountfor the difference in the behavior of the coherency at lonseparation distances and higher frequencies.

The dependence of the functional form of the coherenon shorter and longer separation distances was noted bylast two coherency models. In particular, Abrahamsonet al@19# investigated the same events recorded at the SMARarray ~separation distances of 200–4000 m! and the LSSTarray ~separation distances of 0–100 m!. They noted thatcoherency values extrapolated at shorter separation distafrom the SMART-1 data tended to overestimate the trueherency values obtained from the LSST data. Rieplet al @69#made a similar observation from their analyses of an extsive set of weak motion data recorded at the EUROSEIEST site in northern Greece: the loss of coherency with dtance for their data was marked by a ‘‘cross over’’ distanthat distinguished coherency for shorter~8–100 m! andlonger ~100–5500 m! separation distances.

Another point worth noticing regarding the behaviorcoherency at shorter and longer separation distances isfollowing: Abrahamsonet al @68#, Schneideret al @20#, andVernonet al @70# from analyses of data at close by distanc(,100 m) observe that coherency is independent of walength@20#, and decays faster with frequency than with sepration distance. On the other hand, independent studiedata at longer separation distances (.100 m) by Novak andhis coworkers@47,71#, and Tokso¨z et al @72# observe that thedecay of coherency with separation distance and frequencthe same: Novak@71# and Ramadan and Novak@47# ob-served that if coherency is plotted as function of a normized separation distance with wavelength (j/l), with l5Vs /(2pv) being the wavelength andVs an appropriateshear wave velocity, then the coherency plots at variousquencies collapse onto the same curve; they used datathe 1979 Imperial Valley earthquake and Events 20 and 4the SMART-1 array. Similarly, Tokso¨z et al @72#, from theiranalysis of data recorded at three arrays in FennoscanFinland, noted that the curves of coherency decay with

oi

l

n

h

t

o

e

hh

ar

w

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er-

on00,and


quency were very similar if separation distance was scawith wavelength. The observed coherency dependencewavelength implies that coherency is a function of the pruct (j v), ie, that the decay with frequency and separatdistance is the same. It appears then that different faccontrol the loss of coherency in the data at shorter and lonseparation distances.

Figure 3 presents the decay with frequency at separadistances of 100, 300, and 500 m of the coherency modeHarichandran and Vanmarcke@37# ~Eq. ~29!! and of Haoet al @65# ~Eq. ~27!!. The parameters of Harichandran aVanmarcke’s model areA50.736, a50.147, k55210 m,f o51.09 Hz, andb52.78; their values were determinefrom the data of Event 20 at the SMART-1 array. For tcoherency model of Haoet al @65# it is considered thatj l

5j, j t50, ie, the stations are located on a line alongepicentral direction; the parameters of the model areb1

52.2531024, a15106.631024, b150.26531024, andc1

520.99931024, which were determined from the dataEvent 30 at the SMART-1 array. In both expressions~Eqs.~27! and ~29!!, frequency (v) is measured in rad/sec anseparation distance (j) in m; however, in all figures, forconsistency with subsequent results, frequency is givenHz. The differences in the representation of coherencythe same site but different events is obvious from the figuFor comparison purposes, Fig. 3c presents the coherency dcay with frequency at the shorter separation distances~50and 100 m! of the LSST array as proposed by Abrahamset al @68#.

The expressions in Eqs.~23–30! represent the lagged coherency of motions recorded at the alluvial site of tSMART-1 array. It should be emphasized at this point tmotions recorded at rock sites also experience loss of coence that is attributed to the near surface rock weatheringcracking @73#. Cranswick @74#, Toksoz et al @72#, Menkeet al @75#, and Schneideret al @20# estimated coherency fromdata at rock sites and observed that it also decays expotially with frequency and station separation distance. Crswick @74# presented a thorough investigation of thesponse~spectral amplitudes and spatial variability! of rocksites, and explained the drop of coherence in terms ofrock properties and layer formation. Schneideret al @20#compared coherency estimates at alluvial and rock sites,concluded that, for the separation distances exami(,100 m), coherencies computed at rock sites were lothan those recorded on alluvium.

4.2 Semi-empirical models

Semi-empirical models for the spatial variation of the semic ground motions, ie, models for which their functionform is based on analytical considerations but their paraeter evaluation requires recorded data, have also been iduced.

Somervilleet al @76# proposed a model that attributes thspatial variation of the motions to the wave propagationfect, the finite source effect, the effect of scattering ofseismic waves as they propagate from the source to theand the local site effects. It has been shown, however, f

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data analyses@19# and seismological observations@41#, thatearthquake magnitude may not particularly influence cohency estimates. Abrahamsonet al @19#, from their analysis of15 events of small (M<5) and large (M>6) magnitudes

Fig. 3 Variation of empirical coherency models basedSMART-1 data with frequency at separation distances of 100, 3and 500 m, and on LSST data at separation distances of 50100 m.

c

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recorded at the LSST array, observed that there was nosistent trend indicating dependence of coherency ontended faults. Spudich@41# gives a possible explanation fothe reason why the source finiteness may not significaaffect coherency estimates: For large earthquakes of uneral rupture propagation, the waves radiating from the souoriginate from a spatially compact region that travels wthe rupture front, and, thus, at any time instant, a relativsmall fraction of the total rupture area radiates. Unilaterupture at the source constitutes the majority of earthqua@41#; Spudich@41# cautions, however, that bilateral ruptueffects on spatial ground motion coherency are yet unmsured. Der Kiureghian@58# developed a stochastic modelwhich the total spatial variation of the seismic motionscomposed of terms corresponding to wave passage effbedrock motion coherence effects, and site response cobution; in the model evaluation, Der Kiureghian@58# alsonoted that the attenuation of the waveforms does not afcoherency estimates.

Perhaps the most quoted coherency model was introduby Luco and Wong@77#, and is based on the analysis of shewaves propagating a distanceR through a random medium

pa-

ara-

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uchs of

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on-ex-rtly

ilat-rcethelyralkeseea-niscts,

ntri-

ect

cedar

ug~j,v!u5e2(hvj/vrm)25e2a2v2j2

h5mS R

r 0D 1/2

a5h

v rm(31)

wherev rm is an estimate for the elastic shear wave velocin the random medium,r 0 the scale length of random inhomogeneities along the path, andm2 a measure of the relativevariation of the elastic properties in the medium.a, the co-herency drop parameter, controls the exponential decathe function; the higher the value ofa, the higher the loss ofcoherence as separation distance and frequency increWith appropriate choices for the coherency drop paramethe model has been shown to fit the spatial variation ofcorded data, and has been used extensively by researchtheir evaluation of the seismic response analysis of lifelin~eg,@4,77–79#!. Figure 4a presents the exponential decaythe model with frequency at separation distances of 100, 3and 500 m for a median valuea52.531024 sec/m from theones suggested by Luco and Wong@77#. It is noted that thisanalytically based model produces coherencies equal tozero frequency for any separation distance, and tendzero—rather than to a finite value—as frequency and seration distance increase~cf. Fig. 4a with Figs. 3a andb, thatpresent empirical coherency models!. Luco and Wong’smodel also considers that the exponential decay with seption distance and frequency is the same.

Zerva and Harada@80# introduced a semi-empirical modefor the coherency that approximates the site topography bhorizontally extended layer with random characteristics ovlaying a half-space~bedrock!. The model includes the effectof wave passage with constant velocity on the ground sface, the loss of coherence due to scattering of the wavethey travel from the source to the site by means of Luco aWong’s @77# expression~Eq. ~31!!, and the local site effectsapproximated by vertical transmission of shear wavthrough a horizontal layer with random properties. The radom layer properties are evaluated from the spatial variaof the soil profile for the site under investigation. The resuing coherency for a soft random layer is shown in Fig. 4b;its decay with separation distance and frequency is identto that of Luco and Wong’s model~ie, the loss of coherencydue to the scattering of the waves in the bedrock!, except fora drop at the stochastic layer predominant frequency. Sdrops in coherency have been observed from analyserecorded data between two stations: Cranswick@74# noticesthat perturbations with small deviations in the layer charteristics will produce the greatest changes in the site respofunctions, and, since coherency is a measure of similaritythe motions, it will be low at the resonant frequencies. Athough such a drop in coherency~Fig. 4b! may seem insig-nificant in terms of the overall coherency decay, it contrseismic ground strains, which in turn control the seismicsponse of buried pipelines@80#. In a similar approach, Kanda@9#, using finite element modeling for a layered medium wirregular interfaces and random spatially variable incidmotions, analyzed coherency and amplitude variabilitythe free surface of the site.-
Fig. 4 Variation of semi-empirical coherency models with fr
quency at separation distances of 100, 300, and 500 m

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Zerva, Ang, and Wen@81,82# developed an analyticamodel for the estimation of the spatial coherency. The mois based on the assumption that the excitation at the socan be approximated by a stationary random process, wis transmitted to the ground surface by means of frequetransfer functions, different for each station. The frequentransfer functions are the Fourier transform of impulsesponse functions determined from an analytical wave progation scheme and a system identification technique intime domain. The resulting surface ground motions aretionary random processes specified by their power and cspectral densities, from which the spatial variation of tmotions is obtained. The approach was applied to the sitthe SMART-1 array for Event 5; the analytical coherencaptured the trend of the recorded data, even thoughwave propagation scheme did not consider local site effe

4.3 Effect of spatial variabilityin lifeline seismic response

The significance of the effect of the spatial variation of semic ground motions on the response of lifelines, and, in pticular, bridges, has been recognized since the mid 19@11,13,60,83#, and has been continuously investigated wincreased interest. Examples include: Abdel-Ghaffar@84,85#analyzed the response of suspension and cable-stbridges using random vibration theory; Harichandran aWang @86,87#, and Zerva@88,89# examined simple bridgeconfigurations in random vibration analysis using variospatial variability models; Harichandranet al @90# analyzedsuspension and arch bridges subjected to simulated spavariable motions; Montiet al @91# examined the response omulti-span bridges to simulated spatially variable excitions; recently, Deodatiset al @8# and Saxenaet al @1# inves-tigated the nonlinear response of highway bridges subjeto simulated spatially variable seismic ground motions.

The most definite outcome that can be drawn from all pstudies is that the use of identical motions as excitationthe structures supports will not always yield a conservaresponse; indeed—in cases—the response induced by idcal motions can be grossly unconservative@8#. Specific con-clusions regarding the effect of spatially variable motionsthe response of bridges compared to that induced by idenexcitations cannot be easily drawn: This effect dependsthe structural configuration and properties, as well as onground motion characterization, ie, the apparent propagavelocity of the motions, the soil conditions at the bridge suports, and the choice of the coherency model from the mtitude of models existing in the literature. The time delacaused by the apparent propagation velocity result in outphase motions at the structures’ supports. The consideraof variable soil conditions at the bridge supports affectamplitude variation of the motions, and produce generahigher response than if the soil conditions were assumebe identical@1,8#. In order to analyze the effect of the seletion of a coherency model on the seismic response oftended structures, Zerva@5,79# isolated the contribution othe different coherency models to the quasi-static andnamic response of linear, generic models of lifelines. It wshown @79# that the root-mean-square~rms! quasi-static re-

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sponse of lifelines is proportional to the rms differential dplacement between the supports, and the rms contributiothe excitation of individual modes is proportional to the dferential response spectrum. The differential response strum was defined as the rms response of a single-degrefreedom oscillator resting at two supports, which wesubjected to spatially variable seismic ground motionsnormalized Clough-Penzien spectrum~Eq. ~10!! and themodels of Harichandran and Vanmarcke@37# ~Fig. 3a! andLuco and Wong@77# ~Fig. 4a! were used in the comparisonno wave passage effects were included, so that the variabdue to the lagged coherency estimates could be clearlyserved. Figure 5 shows the comparison of the lifelinesponse to the two different models: it can be seen thatcases, the response can as much as double depending oexponential decay in the models. Thus, the selection oparticular coherency model in the seismic response evation of lifelines can have an important effect on the seismresistant design of the structures.

The question that arises, then, is which one of the cohency models available in the literature should be usedlifeline earthquake engineering. From one side, empiricalherency models are essentially event-, or mostly site-, scific. It was recognized, since the early studies of the spa

Fig. 5 Quasi-static and dynamic response of lifelines subjecteseismic motions experiencing different degree of exponential dein their lagged coherency

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variation from SMART-1 data, that there are large diffeences in the correlations between events@17#, so that thesame parametric expression may not describe the spvariability of different events at the same site@37#; this ob-servation can also be made from the comparison of theherency models in Figs. 3a andb. As such, empirical coherency models cannot be reliably extrapolated to different sand events. From the other side, semi-empirical modelsbased on certain simplifying assumptions that may notways capture reality. It is being examined recently whethegeneric model can reproduce the spatially variable naturseismic motions at different sites and various events. Sostudies@47,68# suggest that such models may be feasibwhereas others@20# suggest that they may not. The underling difficulty in establishing appropriate coherency exprsions for any site and any event is that coherency, as a pustatistical measure, is not related to physical parameters

An additional consideration for the description of sptially variable seismic ground motions is their amplituvariability. The amplitude variability has not been invesgated as widely as the phase variability: Abrahamsonet al@68# and Schneideret al @20# evaluated the variation of thamplitude spectra of the data by considering that the Fouamplitude spectrum is lognormally distributed; it was oserved that the variance of the difference between the lorithms of spectral amplitudes between stations increasedfrequency and tended to constant values at higher freqcies.

It becomes then apparent, from the aforementionedcussion, that the description of the spatial variation ofseismic motions ought to incorporate both the amplitudethe phase variation in the data. It should also be tied to phcal parameters, so that it can be reliably extrapolated tosite. An alternative approach@53,92# that deals directly withtime histories rather than coherency estimates, recognizqualitatively—physical patterns in the spatial variabilitythe data, and analyzes simultaneously the amplitudephase variability in the motions is presented in the followin

5 CORRELATION PATTERNS IN THEAMPLITUDE AND PHASE VARIATIONOF THE MOTIONS

The approach@92# models the seismic ground motionssuperpositions of sinusoidal functions described by their aplitude, frequency, wavenumber, and phase. For each eand direction~horizontal or vertical! analyzed, it identifies acoherent, common component in the seismic motions oextended areas. The common component represents a cent wavetrain that propagates with constant velocity onground surface and approximates to a satisfactory degreactual motions. The spatial variation of the seismic motiois determined from the differences between the recordedand the coherent estimates of the motions. The methodowas applied to data recorded at the SMART-1 dense insment array in Lotung, Taiwan, and is presently appliedadditional recorded data. For illustration purposes, the strmotion S-wave window~7.0–12.12 sec actual time in th

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records with a time step of 0.01 sec! in the N-S direction ofEvent 5 at the SMART-1 array is presented herein.

Identification of propagation characteristics of themotions—Signal processing techniques, eg,@49,53#, are ini-tially applied to the data recorded at all stations in orderidentify the propagation characteristics of the motions. Tapplication of the conventional method@50# with slownessstacking@39# to the data identified the slowness of the domnant broad-band waves in the window assW5(0.1 sec/km,20.2 sec/km), ie, the waves impinge the array at an azimof 153° with an apparent propagation velocity of 4.5 km/sThe~horizontal! wavenumber,kW 5$kx ,ky%, of the motions isrelated to the slowness throughkW 5v sW, with v indicatingfrequency.

Identification of amplitude and phase of the signals—Theseismic ground motions are then described byM sinusoidsand expressed as:

a~rW,t !5 (m51

M

Am sin~kW m•rW1vmt1fm! (32)

in which rW indicates location on the ground surface andt istime. Each sinusoidal component is identified by its~dis-crete! frequency and wavenumber (vm ,kW m); Am andfm areits amplitude and phase shift, respectively. It is noted thatnoise component is superimposed to the ground motion emate of Eq.~32!. The subscriptm in the parameters of Eq~32! indicates frequency dependence, and the number onusoids,M , used in the approach depends on the cut-frequency, above which the sinusoids do not contribute snificantly to the seismic motions. The amplitudes,Am , andphases,fm , of the sinusoidal components can then be demined from the system of equations resulting from the leasquares minimization of the error function between thecorded time historiesa(rW,t) and the approximate onesa(rW,t)~Eq. ~32!! with respect to the unknownsAm andfm @53#:

E5(l 51

L

(n51

N

@a~rW l ,tn!2a~rW l ,tn!#2 (33)

evaluated at discrete locations~stations! l and timesn. AnynumberL of stations—ranging from one to the total numbof recording stations—can be used for the evaluation ofsignal amplitudes and phases. WhenL.1 in Eq. ~33!, theidentified amplitudes and phases represent the commonnal characteristics at the number of stations considered; wL51, the amplitudes and phases correspond to the motat the particular station analyzed.

Reconstruction of seismic motions—Five stations (L55)are used in Eq.~33! for the identification of their commonamplitudes and phases; the stations areC00, I03, I06, I09and I12 ~Fig. 1!. Once the common characteristics are idetified, they are substituted in Eq.~32!, and an estimate of themotions~reconstructedmotions! at the stations consideredobtained. The comparison of the recorded motions withreconstructed ones is presented in Fig. 6; no noise~random!component is added to the reconstructed signals. Sinceplitudes and phases at each frequency are identical fo

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s ter


five stations considered, the reconstructed motions reprea coherent waveform that propagates with constant veloon the ground surface. Figure 6 indicates that the recstructed motions reproduce, to a very satisfactory degreeactual ones and, although they consist only of the broad-bcoherent body wave signal~Eq. ~32!!, they can describe thmajor characteristics of the data. The details in the acmotions, that are not matched by the reconstructed ones,stitute the spatially variable nature of the motions, after~deterministic! wave passage effects have been removed

Variation of amplitudes and phases—When only one sta-tion at a time~L51 in Eq. ~33!! is used in the evaluation oamplitudes and phases at different frequencies for thatticular station, the reconstructed motion is indistinguishafrom the recorded one. This does not necessarily meanthe analyzed time histories are composed only of the idefied broad-band waves, but rather that the sinusoidal futions of Eq.~32! can match the sinusoidally varying seismtime histories, ie, Eq.~33! becomes essentially compatiblea Fourier transform. The comparison of the results atindividual stations with the common ones provides insiginto the causes for the spatial variation of the motions.

Figure 7 presents the amplitude and phase variation osinusoidal components of the motions with frequency;continuous, wider line in these figures, as well as in the ssequent ones, indicates the common signal characterisnamely the contribution of the identified body wave to tmotions at all five stations, whereas the thinner, dashed lrepresent the corresponding amplitudes and phases whestation at a time is considered in Eq.~33!. In the lower fre-quency range (,1.5 Hz), amplitudes and phases identifiat the individual stations essentially coincide with thosethe common component. In the frequency range of 1.5Hz, the common amplitude represents the average of theamplification and phases start deviating from the commphase. It is noted that phases were restricted in the ra@0,2p), and, therefore, jumps of approximately 2p do not

Fig. 6 Comparison of recorded and reconstructed strong S-wmotions in the N-S direction at the center and inner ring stationthe array for Event 5

sentcityon-the

and

ualcon-he

ar-blethatnti-nc-icotheht

theheub-tics,e

nesone

dof–4siteonnge

indicate drastic variations in their values. At higher frequecies, the common amplitude becomes lower than the oidentified at the stations and phases vary randomly.

Alignment of time histories—Part of the variabilitiesaround the common component in Fig. 7 are due to thethat the time history approximation~Eq. ~32!! does not allowfor the random arrival time perturbations at the array stati~Section 3.3!. Their effect can be noted in the comparisonthe recorded and reconstructed motions in Fig. 6, when,the reconstructed motion atI12 arrives later than the recorded one. In the present approach, these arrival timeturbations are partially eliminated through the alignmentthe seismic motions with respect toC00 ~Fig. 1!. As ex-pected, the arrival time delays evaluated from the alignmprocess between individual sets of stations exhibited adom behavior around the delays determined from the apent propagation of the motions identified from the constslowness of the broad-band waves. However, the ave

aveofFig. 7 Amplitude and phase variation of the motions at the cenand inner ring stations

,rdaf

ssn

he

theta-n

in-

ilar,ereest

kmer-in thevesee-ex-ncepli-rageithles


pattern of wave propagation across the array was consiswith the one identified from the signal processing techniq@92#.

Common components in aligned motions—For the identi-fication of the amplitude and phase variation of the alignmotions, the error function~Eq. ~33!! is used again in theminimization scheme but, in the sinusoidal approximationthe motions~Eq. ~32!!, the termkW m•rW is set equal to zerosince the aligned motions arrive simultaneously at all arstations. Figure 8 presents the variation of the amplituand phases determined from the application of the lesquares minimization scheme to the aligned motions atinner ring stations~I03, I06, andI09, andI12! and the cen-ter stationC00, together with the amplitude and phase vartion identified using the aligned motions recorded at onetion at a time. The variability range of amplitudes and phaaround the common component is reduced in the aligdata results compared to those of the actual motions~Figs. 7and 8!, indicating the significance of the consideration of tspatial arrival time perturbations in the approach. Figur

tentue

ed

of

ayesst-

our

ia-ta-esed

e9

presents the results of the application of the approach toaligned seismic motions recorded at four middle ring stions,M03, M06, M09, andM12, and the reference statioC00.

The common amplitude and phase identified from thener and the middle ring station data~Figs. 8 and 9!, showntogether for comparison purposes in Fig. 10, are very simparticularly considering the facts that separate analyses wperformed for the two sets of stations, and that the longseparation distance for the middle ring stations is 2whereas that of the inner ring ones is 400 m. Their diffences are an expected consequence of the larger scatterdata at the further away stations due to attenuation of waand more significant variations in site topography. The agrment of the common amplitudes and phases over antended area of radius of 1 km strongly suggests the existeof the coherent component in the data. The common amtude can be viewed as a mean value representing the aveamplification of the motions at the site and is associated wthe common phase variability with frequency, that resemb

the
Fig. 8 Amplitude and phase variation of the aligned motions atcenter and inner ring stations
theFig. 9 Amplitude and phase variation of the aligned motions atcenter and middle ring stations

ad

oie-

the; the

li-lsoym-nscythanonandve-ingnd

cor-nd


random distribution between@0,2p). The spatial variabilityof the motions, in addition to their propagation effectsready considered, results from deviations in both amplituand phases at the individual stations around the commcomponents~Figs. 8 and 9!, which are described in terms odifferential amplitudes and phases in Figs. 11 and 12.

Differential amplitude and phase variability in alignedmotions—The normalized differential amplitudes~Figs. 11and 12! are obtained by subtracting at each frequencycommon amplitude from the amplitudes identified at thedividual stations and dividing by the common amplitudThe normalized differential amplitudes are cut off at a mamum value of 7.5; their actual values, which are not imptant for the subsequent analysis, can be significantly hbecause the common component can assume low valucertain frequencies~Figs. 8 and 9!. Furthermore, the normalized differential amplitudes, from their definition, cannot asume values lower than (21). The differential phases~Figs.

ai

l-eson

f

thein-e.xi-r-

gh,s at

s-

11 and 12! are obtained by subtracting at each frequencycommon phase from the phases identified at each stationdifferential phases~Figs. 11 and 12! are allowed to vary be-tween @2p,1p) rather than between@0,2p), as was thecase in Figs. 7–10.

Envelope functions, drawn by eye, containing the amptude and the phase differential variability range are ashown in the figures. The phase envelope functions are smetric with respect to the zero axis. The envelope functiofor the amplitudes are symmetric only in the lower frequenrange, as their values are restricted in the range greater(21). Isolated peaks within the dominant site amplificatifrequency range are excluded from both the amplitudephase envelope functions. The trend of the positive enlopes of both amplitudes and phases is very similar, implythat the amplitude and phase variability of the data aroutheir respective common component characteristics arerelated. Since amplitude variability is easier visualized a

tne

Fig. 10 Comparison of the common component amplitudephase identified from the analysis of the inner and middle rstation aligned data

ndng

Fig. 11 Differential amplitude and phase variability with respecto the common component of the center and inner ring statioaligned motions; envelope functions containing the variability ardrawn by eye

yn

a

e

m

a

t

ndin-

ave-oreifi-ies,avend,-loci-

ig-sitenti-dif-

t a-


attributed to physical causes than phase variability, the vation of both amplitudes and phases can be—qualitativelassociated with physical parameters: In the low frequerange, the envelope functions are at close distance to theaxis, that increases consistently for both amplitudesphases as separation distance between stations increase~in-ner vs middle ring results!; this is attributed to the longwavelength of the contributing waves at low frequencithat do not ‘‘see’’ the site irregularities, particularly for thinner ring stations. As frequency increases within the donant site amplification frequency range, the distance ofamplitude and phase envelope functions from the zeroincreases gradually with a slower rate for the inner ring stions than the middle ring ones. In this range, the commamplitudes reproduce the average of the site amplifica~Figs. 8 and 9! implying that the motions are controlled bthe signal that is modified in amplitude and phase astraverses the horizontal variations of the layers underne

theo-

n-ri-are

litymonlessreaareis-ta,m-ari-nc-aeenpeve-hisnd

ged

isver,, the-ve-mesthe

ncere-onsy bens,to

ta

ria-—cy

zeronds

s,e

i-thexista-oniony

itath

the array. The increase in the variabilities of amplitudes aphases around the common component as frequencycreases may also be associated with the decreasing wlength of the signals at increasing frequencies, which is mobvious for the middle ring results, and to the more signcant contribution of scattered energy. At higher frequencpast the dominant site amplification frequency range, wcomponents in addition to the broad-band signal, amainly, scattered energy~noise! dominate the motions. Because these wave components propagate at different veties, phases at the individual stations~Figs. 8 and 9! deviatesignificantly from the common phase, and the common snal amplitude no longer represents the average of theamplification and becomes lower than the amplitudes idefied at the individual stations. Consequently, the phaseferences vary randomly between@2p,1p), ie, the differ-ential phase envelopes are parallel to the zero axis adistance equal top, and the normalized differential amplitudes assume high values. Noise is also the cause ofisolated peaks in the dominant frequency range of the mtions; this variability, however, may not be of significant cosequence for the modeling or the simulation of spatially vaable seismic ground motions: it occurs when amplitudeslow within the dominant frequency range of the motions.

Differential phase variability and spatial coherency—Theenvelope functions in Figs. 11 and 12 contain the variabirange of amplitudes and phases with respect to the comcomponent for motions recorded at separation distancesthan the maximum station separation distance for the aconsidered. Thus, the envelope functions, as upper limits,functions of frequency and maximum station separation dtance, ie, 400 m and 2 km for the inner and middle ring darespectively. Constrained within the envelope functions, aplitudes and phases vary randomly, suggesting that the vability can be described by the product of the envelope fution and a random number uniformly distributed withinspecific range. Differential amplitudes and phases betwstations also vary randomly within the bounds of envelofunctions that have trends very similar to those of the enlope functions with respect to the common component; tobservation can also be made, indirectly, from Figs. 11 a12. It is recalled from Eqs.~19! and ~20! that differences inphases between stations are directly related to the lagcoherency through envelope functions,b jk(v)p, containingtheir random variability@42#. Thus, the differential phasevariability identified by means of this methodologyequivalent to conventional coherency estimates. Howethe shape of the envelope functions, and, consequentlyassociated coherency~Eq. ~20!! can now be related to physical parameters: In the low frequency range, where the enlope functions are close to the zero axis, coherency assuvalues close to one; this behavior is a consequence oflong signal wavelength at low frequencies and the distabetween stations. Within the dominant site amplification fquency range of the motions, where the envelope functiincrease with frequency, coherency decreases; this maattributed to the decreasing wavelength of the motiowhich is more prominent for the further away stations, and

ctionre

Fig. 12 Differential amplitude and phase variability with respeto the common component of the center and middle ring staaligned motions; envelope functions containing the variabilitydrawn by eye

t

eirah

d

o

ot

o

n

-

n

eg

ih

thefirstundncy; thede-ech-gi-

ingad-d asto

ne.theari-tionans forey

nlyels

e-ted

pro-

nsukam-

tralrgem-ts

FFTfor

ionsari-

etrall-u-ofAshehe

ith


the more significant contribution of scattered energy. Inhigher frequency range, where phases vary randomlytween @2p,1p) and noise dominates, coherency assumzero values. It is noted that coherency evaluated fromcorded data~Eq. ~12!! can never reach zero in the highfrequency range, although the true coherency of the mottends to zero@29,47#. The qualitative association of coheency with physical parameters introduced in this approsets the bases for the possible physical modeling of coency.

6 SIMULATIONS OF SPATIALLYVARIABLE GROUND MOTIONS

The spatially variable seismic ground motion modelsscribed in Sections 2, 3, and 4 can be used directly as inmotions at the supports of lifelines in random vibratianalyses; this approach, however, is applicable to relativsimple structural models and for the linear response ofstructures. For complex structural lifeline systems, andtheir nonlinear seismic response evaluation, only the deministic solution is known or can be evaluated with sufficieaccuracy. In this case, the generation of artificial spatiavariable seismic ground motions is required; these motiare then used as input motions at the structures’ suppordeterministic seismic response analyses. Thus, Monte Csimulations become the most versatile tool for such prlems.

An extensive list of publications addressing the topicsimulations of random processes and fields has appearethe literature. The following list is by no means exhaustivOne of the techniques that has been widely applied in eneering problems is the spectral representation met@93,94#. Simulations based on AR~auto-regressive!, MA~moving-average! and ARMA ~auto-regressive-movingaverage! techniques have also been used extensively~eg,@95–99#!. Simulations by means of the local average subvision method@100#, the turning bands method@101#, andbased on wavelet transforms@102,103# have been generatedSimulations of non-stationary time series by means of pcesses modulated by time varying functions~eg, @104–106#!have also been reported. More specifically, spatially variaseismic ground motions have been simulated by meanswide variety of techniques. Examples include, but arelimited to: covariance matrix decomposition@65,107#; spec-tral representation method@78,79,108#; envelope functionscontaining random phase variability@42#; coherency functionapproximation by a Fourier series@109,110#; ARMA ap-proximation @111#; FFT @112# and hybrid DFT and digitalfiltering approach@113# for non-stationary random processeand conditional simulations@114,115# and interpolation@116#.

The appropriate simulation technique for the particuproblem at hand depends on the characteristics of the plem itself @117#. The main objectives are that the charactistics of the simulated motions match those of the tarfield, and that the computational cost for the simulationsnot excessive. An additional consideration in the simulatof spatially variable ground motions is which one of t

hebe-esre-r

ons-cher-

e-putnelytheforter-ntllyns

s inarlob-

ofd ine:gi-

hod

di-

.ro-

bleof aot

s;

larrob-r-etis

one

extensive list of coherency models should be used inanalysis. In order to illustrate these considerations, thesubsection herein simulates spatially variable seismic gromotions based on two commonly used spatial coheremodels by means of the spectral representation methodsimilarities and differences in the resulting motions arescribed. The second subsection presents two simulation tniques that are commonly used in lifeline earthquake enneering; the differences in the simulated motions resultfrom the use of the particular simulation scheme aredressed. In both subsections, the motions are simulatestationary, ie, no deterministic modulation function is usedtransform the simulated time history to a non-stationary oThe reason for this is to preserve the characteristics ofcoherency model and the simulation scheme in the compson. For the same reason, the contribution of the propagaof the motions in the simulated ones is only illustrated inexample in Section 6.1. Section 6.3 presents approachethe modification of the simulated time histories, so that threproduce more closely actual seismic ground motions.

6.1 Selection of coherency model

Simulations resulting from two coherency models commoused in lifeline earthquake engineering, namely the modof Harichandran and Vanmarcke@37# ~Eq. ~29!! and of Lucoand Wong@77# ~Eq. ~31!! are compared. The spectral reprsentation method is used for the generation of the simulatime histories.

The concept of representations of Gaussian randomcesses was introduced by Rice in 1944@93# ~reprinted in@118#!, but the use of the approach in generating simulatioof random processes and fields originates from Shinoz@94,119,120#. The methodology has been presented in a coprehensive article by Shinozuka@121#, and was more re-cently elaborated upon in Shinozuka and Deodatis@122,123#and references therein. In its initial formulation, the specrepresentation method dealt with the summation of lanumbers of weighted trigonometric functions, which is, coputationally, not efficient. Yang@124# introduced the FasFourier Transform~FFT! technique in simulating envelopeof random processes, and Shinozuka@125# extended the ap-proach to random processes and fields. The use of thedramatically reduces the computational requirementssimulations. Improvements on the approach and evaluatof its properties and capabilities have been reported by vous researchers over the last few years~eg, @78,126–129#!.Mignolet and Harish@129# compared the performance of thspectral representation algorithm, the randomized specrepresentation scheme@120# and the random frequencies agorithm @119#, and concluded that, irrespectively of comptational effort, the latter performs generally better in termsfirst and second order distributions than the other two.computational efficiency is a significant consideration, tsimulation scheme described in the following utilizes tspectral representation method at fixed frequencies.

Consider a homogeneous space-time random field wzero mean, space-time covariance functionR(j, t), j beingthe separation distance, andt being the time lag, and

v

f

hF

m

of.

n-

f

and

y asalue,cay.gh-

r-onon-

are,

andngless at

ong

ased

hedel

ander ofis--cy-and

r-lly

dieso-tionw

s-t of

. The


frequency-wavenumber~F-K! spectrumS(k, v), in which kindicates the wavenumber andv indicates frequency. Thefrequency-wavenumber spectrum and the space-time coance function are Wiener-Khintchine transformation paand possess the same symmetries@38#. The F-K spectrum isobtained through the Fourier transform of the cross specdensity of the motions. Simulations of spatially variable semic ground motions can then be generated by means ospectral representation method@78,121#:

f ~x,t !5& (j 50

J21

(n50

N21

~@2 S~k j ,vn!Dk Dv#1/2

3cos~k j x1vnt1f jn(1)!

1@2 S~k j ,2vn!Dk Dv#1/2

3cos~k j x2vnt1f jn(2)!! (34)

in which f jn(1) andf jn

(2) are two sets of independent randophase angles uniformly distributed between (0, 2p#, k j5( j1 1

2)Dk and v j5(n1 12)Dv are the discrete wavenumbe

and frequency, respectively, andDk and Dv are the wave-number and frequency steps. Simulations based on Eq.~34!can be obtained if there exists an upper cut-off wavenumku5J3Dk and an upper cut-off frequencyvu5N3Dv,above which the contribution of the F-K spectrum to tsimulations is insignificant for practical purposes. The Falgorithm can then be introduced in Eq.~34! to improve thecomputational efficiency of the method. In order to accomodate the fact that the FFT algorithms start atk050 andv050, rather thank05 1

2 Dk and v05 12 Dv, as is required

in the present case, Eq.~34! is rewritten as@78#:

f rs5& RFeipr /Meips/LH (j 50

M21

(n50

L21

~@2 S~k j ,vn!DkDv#1/2

3eif jn(1)

!ei2pr j /M ei2psn/LJ G1& R Feipr /Me2 ips/LH (

j 50

M21

(n50

L21

~@2 S~k j ,2vn!

3Dk Dv#1/2eif jn(2)

!ei2pr j /M e2 i2psn/LJ G (35)

in which, f rs5 f (xr ,ts); xr5rDx; Dx5 2p/MDk ; r50,..,M21; ts5rDt; Dt5 2p/LDv ; s50,..,L21; andRindicates the real part. The FFT is applied to the two exprsions in the braces in Eq.~35!. M andL are powers of 2, andought to satisfy the inequalitiesM>2J andL>2N, so thataliasing effects can be avoided; ie, forJ< j ,M and N<n,L the value of the F-K spectrum in Eq.~35! is zero. Thefollowing characteristics are inherent in the simulations:!they are asymptotically Gaussian asJ,N→` due to the cen-tral limit theorem; ii! they are periodic with periodTo

54p/Dv and wavelengthLo54p/Dk; iii ! they are er-godic, at least up to second moment, over an infinite tiand distance domain or over the period and wavelengththe simulation; and iv! asJ,N→`, the ensemble mean, co

ari-irs

tralis-the

m

r

ber

eT

m-

es-

i

eof

-

variance function and frequency-wavenumber spectrumthe simulations are identical to those of the random field

Seismic ground motion simulations~displacements! basedon the spatial variability models of Harichandran and Vamarcke@37# ~Eq. ~29!! and that of Luco and Wong@77# ~Eq.~31!! are generated by means of Eq.~35!. The parameters othe Harichandran and Vanmarcke model wereA50.736, a50.147,k55210m, f o51.09 Hz, andb52.78. Two differ-ent values for the coherency drop parameter of the LucoWong model were used: a low value,a5231024 sec/m,that reproduces slow exponential decay in the coherencfrequency and separation distance increase, and a high va51023 sec/m, that represents a sharp exponential deThe power spectral density considered was the ClouPenzien spectrum~Eqs. ~9! and ~10!! with parameters:So

51 cm2/sec3, vg55p rad/sec, v f50.1vg , and zg5z f

50.6, ie, firm soil conditions. The simulations were peformed at a sequence of six locations along a straight linethe ground surface with separation distance between csecutive stations of 100 m; the resulting time historiespresented in Figs. 13a, b, andc. For comparison purposesthe same number of wavenumber and frequency points~JandN!, the same discrete wavenumbers and frequencies,the same seeds for the generation of the random phase awere used in the simulations. As a result, the simulationx50 look similar for all cases~Figs. 13a, b, andc!. Thereare, however, significant differences asx increases: Figure3b suggests that simulations based on the Luco and Wmodel with low coherency drop,a5231024 sec/m, are es-sentially unchanged at all locations, whereas the ones bon the other two models~Figs. 13a andc! vary with distance.It is also noted that the variability in the time histories of tmotions for the high coherency drop Luco and Wong mo(a51023 sec/m) in Fig. 13c is dominated by high fre-quency components, whereas that of the HarichandranVanmarcke model~Fig. 13a! by lower frequency ones. Thesdifferences in the time histories are caused by the behaviothe models in the low frequency range that controls the dplacements~0–2 Hz!. The model of Harichandran and Vanmarcke is only partially correlated even at zero frequen~Fig. 3a! yielding low frequency variation in the time histories between the stations. On the other hand, LucoWong’s model is fully correlated asv→0 for any value ofthe coherency drop parameter~Eq. ~31!!. For low values ofa~Eq. ~31! and Fig. 4a!, it results in high values for the coheency in the lower frequency range, thus yielding essentiaidentical displacement simulations~Fig. 13b!. For higher val-ues ofa ~Eq. ~31!!, it decreases rapidly with frequency anseparation distance, and the variability in the time historshown in Fig. 13c is caused by higher frequency compnents. It needs to be mentioned, however, that acceleratime histories based on Luco and Wong’s model for lovalues ofa show variability in space: acceleration time hitories are controlled by a higher frequency range than thadisplacements; in this higher frequency range~Fig. 4a!, themodel decreases with frequency and separation distancecomparison of the simulated time histories in Figs. 13a–c

,

u

nt

he-

onge-

ifi-hes atper-

ns—

u-or


indicates that the exponential deay of the coherency moaffects the characteristics of the ground motions, whichturn, affect the seismic response of lifelines.

It should also noted from Figs. 13a–c that the seismicground motions do not propagate on the ground surfaIndeed, it can be shown@78#, that for quadrant-symmetricspace time fields ~ie, S(k,v)5S(2k,v)5S(k,2v)5S(2k,2v), @38#!, such as the random fields resultinfrom both the Harichandran and Vanmarcke and the Land Wong models, Eq.~34! can be rewritten as:

f ~x,t !52& (j 50

J21

(n50

N21

@2 S~k j ,vn!Dk Dv#1/2

3cosS k j x1f jn

(1)1f jn(2)

2 D cosS vnt1f jn

(1)2f jn(2)

2 D(36)

which represents the superposition of standing waves,reflects correctly the characteristics of the lagged cohereIf the apparent propagation of the motions is included in

delin

ce.

gco

andcy.

he

description of the random field, then it is reflected in tsimulated motions. Figure 13d presents displacement simulations based on Luco and Wong’s model fora5231024 sec/m and an apparent propagation velocityc5500 m/sec~Eqs. ~17! and ~18!!. It can be observed fromthe figure that the seismic ground motions propagate althe x direction with the specified apparent propagation vlocity.

6.2 Selection of simulation scheme

Another important consideration in the generation of artcial spatially variable ground motions is the selection of tsimulation scheme, so that the resulting seismic motionall locations on the ground surface possess the same proties. To this effect, Katafygiotiset al @108# and Zerva andKatafygiotis @107# compared commonly used simulatioschemes in lifeline earthquake engineering. Their resultfor two techniques—are summarized in the following:

Let vu be a cut-off frequency, above which the contribtion of the spectrum to the simulations is insignificant f

ichandran
Fig. 13 Comparison of simulated time histories generated by means of the spectral representation method and the models of Harand Vanmarcke~1986! and Luco and Wong~1986!

o

t

u

-

ms

e

orderrst

therim-esethecon-en

Pac-

mly

ec-

s

r aghpli-ner,sat-sly

e

ionto-thisis-

en-n-

ca-nt

se-ble-

eti-


practical purposes. The interval@0,vu# is then divided intoNequal parts, each having lengthDv5vu /N. Let vk be somepoint in the interval@(k21)Dv, kDv# chosen in such a waythat for anyk>1, vk212vk5Dv.

The Cholesky decomposition with random amplitudes„CDRA… method—The random field at a pointxW on theground surface is simulated as the sum of trigonometerms with random space-dependent coefficients:

f ~xW ,t !5ADv(k51

N

@hk1~xW !cos~vkt !1hk2~xW !sin~vkt !#

(37)

wherehk j(xW ), j 51, 2, is a sequence of independent randfields. Equation~37! is, essentially, the classical spectral reresentation simulation scheme: For the simulation of randprocesses, ie,xW is fixed,hk1(xW ) andhk2(xW ) are independenrandom variables having Gaussian distribution@118#, or, thetrigonometric functions of Eq.~37! are combined into asingle sinusoidal function with a random phase, that hasform distribution over the interval@0, 2p) @94#.

If simulations atM locations on the ground surface arebe generated,hk1(xi

W ) andhk2(xiW ), i 51, . . . ,M , ~Eq. ~37!!,

are written in vector form as:

nk j5@hk j~xW1!, . . . . . . .,hk j~xW M !#T,

1<k<N, j 51, 2 (38)

The elements of the autocovariance matrixRnk jnk jof the ran-

dom vectornk j are assigned the form:

@Rnk jnk j# l ,m5E@hk j~xW l !hk j~xWm!#5S~vk!ug~j lm ,vk!u

(39)

wherej lm is the distance between the two points (xW l ,xWm),S(vk) is the power spectral density of accelerations, veloties, or displacements, andug(j lm ,vk)u is the lagged coherency. If Rnk jnk j

5CkTCk is the Cholesky decomposition of th

autocovariance matrix of the random vectornk j , thennk j canbe simulated as:

nk j5CkQj (40)

where the random vectorQj consists ofM independent com-ponents having normal distribution and unit variance. Tgeneratednk j are substituted into Eq.~37! and the simulatedtime histories are obtained. The characteristics of the silated field relative to the target random field are discusextensively in Katafygiotiset al @108#. It is noted herein, asit will be elaborated upon later, that the simulated motioare sums of trigonometric functions with random amplitudwhich, for stationl and frequencyk, take the form:

Akl5ADv~hk12 ~xW l !1hk2

2 ~xW l !! (41)

The Hao-Oliveira-Penzien„HOP… method—The simulatedtime histories according to this method@65# are superposi-tions of sinusoidals, each of which has a deterministic aplitude and a random phase, and corresponds to thequency vk in the partitioned frequency domain@0,vu#.Consider that simulations need to be generated at a sequof locations. For each of these locations, the number of

tric

mp-om

ni-

to

ci-

e

he

u-ed

nss,

m-fre-

encesi-

nusoids that needs to be superimposed depends on theof the particular location in the sequence. Hence, for the filocation, only one sinusoid is used per frequency, forsecond location in the sequence, two sinusoids are supeposed for each frequency, and so on. The amplitudes of thsinusoids convey the coherency relation of each point insequence with all the previous ones. Each sinusoid alsotains a random phase that is uniformly distributed betwe@0, 2p).

Let f i(t), i 51, . . . ,M denote the simulations ofM ran-dom processes corresponding toM prescribed locationsxW i ,i 51, . . . ,M , of the random field. According to the HOmethod, the simulation of these processes is achievedcording to:

f i~ t !5 (m51

i

(k51

N

Aim~vk!cos~vkt1fmk!, i 51, . . . ,M

(42)

where Aim(vk) are the amplitudes andfmk the randomphases, which are independent random variables unifordistributed between@0, 2p). The amplitudes at frequencyvk

are calculated by Cholesky decomposition of the cross sptral density matrix,S(vk), where

@S~vk!# im5S~vk!ug~j im ,v!u (43)

asS(vk)5L (vk)LT(vk)S(vk), 0,k,N; in this decompo-

sition, L (vk) is the lower triangular matrix. The amplitudein Eq. ~42! become

Aim~vk!5 l im~vk!A2 S~vk!Dv (44)

with l im(vk) being the elements ofL (vk) and depending onthe coherency. It is noted that the HOP method allows fosequential simulation at the locations of interest. Althouthe phases for all locations are chosen randomly, the amtudes for each station are chosen in a deterministic manso as to ensure that the currently generated time historyisfies the necessary coherency relations with all previougenerated time histories.

It is also noted that the ergodicity properties of the timhistories at the various locations as simulated by Eq.~42!differ: whereas the time history simulated at the first locatis ergodic over the period of the simulations, the time hisries at the subsequent locations are not. A way to remedydeficiency is by utilizing a location-dependent frequency dcretization that amounts to double-indexing of the frequcies @130#: The frequency domain is discretized at frequeciesvk1 , k51, . . . ,N, for the simulation at the first locationand the first term of the summation for all subsequent lotions ~Eq. ~42!!. The domain is then discretized at differefrequenciesvk2 , k51, . . . ,N, in the second term of thesummation for the simulation at the second and all subquent stations, and so on. Hence, frequencies are douindexed, ie,vkm , m51, . . . ,i , in Eq. ~42!. With a properselection ofvkm , m51, . . . ,i , the simulated motions areergodic over the longest period that results from the discrzation @130#.

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n 1get


Comparison of simulation techniques—For the simulationof ground accelerations based on the two techniques,Clough-Penzien spectrum~Eq. ~10!! and the coherency expression of Luco and Wong~Eq. ~31!! are used again; theparameters in this case are:So51 m2/sec3, vg

55p rad/sec,v f50.1vg , zg5z f50.6, ie, same firm soilconditions as in the previous subsection, anda52.531024 sec/m.

The averages of the unsmoothed power spectral dens~PSD! and cross spectral densities~CSD! of 100 realizationsat a sequence of three locations along a straight line wevaluated by means of the two techniques; the distancetween consecutive stations was 500 m. The mean PSDCSD functions obtained from the simulations are compawith the target ones in Figs. 14 and 15. In particular, Fig.presents the PSD at the two first stations and the CSD

the

ities

erebe-andred14be-

tween the 1st and 2nd stations, and the 2nd and 3rd stat~500 m apart! evaluated by the CDRA method. Figure 1presents the corresponding results for the simulations geated by means of the HOP method. The CDRA method~Fig.14! reproduces well the prescribed spectral and cross spedensity functions as random functions with mean value eqto the corresponding target spectra and non-zero variaover the entire frequency domain. However, the HOP methhas the interesting inherent feature that it produces ‘‘quadeterministic’’ spectra, meaning that, for some frequenrange, the spectral values it produces are random numwith mean equal to the target values, while, for another raof frequencies, they are deterministic~zero variance! andequal to the target values. In particular, the PSD at statiogenerated by the HOP method fully coincides with the tarPSD; the PSD at station 2~and, similarly, at station 3!, gen-

Fig. 14 Comparison of power~PSD! and cross~CSD! spectralestimates generated by the CDRA method with target ones

Fig. 15 Comparison of power~PSD! and cross~CSD! spectralestimates generated by the HOP method with target ones

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erated by HOP, produces non-deterministic spectra onlthe frequency range from 0.3–3.0 Hz. This behavior isrectly related to the variability in the amplitudes of the simlations generated by the HOP method:

Figures 16 and 17 present the mean values of the amtudes of the 100 realizations at the first two stations as gerated by the CDRA method~Fig. 16! and the HOP method~Fig. 17!. In all subfigures, the target amplitude of thClough-Penzien spectrum, evaluated asA(v)5A2 S(v)Dv, is also presented. It is also noted that, whethe amplitudes behave as random variables, the mean amtude is lower than the target one. This is because the amtudes are random variables with non-zero variance; eg,the CDRA method, they result from the square root ofPSD function~Eqs. ~37! and ~41!!, which follows ax2 dis-tribution with two degrees of freedom. For the CDRmethod, the mean values of the amplitudes at both statio

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As 1

and 2 ~and, also, station 3! behave consistently as randovariables. This is not the case for the mean amplitudes gerated by the HOP method: The amplitudes of all simulatiat station 1 coincide with the target amplitudes. For statio~and, similarly, station 3!, the range where the amplitudeexhibit random characteristics is limited in the frequenrange 0.3–3.0 Hz; outside this range, they are determinand identical to the target amplitudes. This may be explaias follows@107#: The time series at station 1~first simulationin the sequence,i 51 in Eq. ~42!! is composed of only onecosine term at each frequencyvk with random phase anddeterministic amplitude, given by: A11(vk)5 l 11(vk)A2 S(vk)Dv5A2 S(vk)Dv ~Eq. ~44!!, as thevalue of the coherency of a time history with itself is equalone, resulting inl 11(vk)51. The time series at station~second simulation in the sequence,i 52 in Eq.~42!! is com-posed of two cosines at each frequencyvk ,

with amplitudes

OP
Fig. 16 Comparison of mean amplitudes generated by the CDmethod with target ones
RAFig. 17 Comparison of mean amplitudes generated by the Hmethod with target ones

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A21(vk)5 l 21(vk)A2 S(vk)Dv5g21(vk)A2 S(vk)Dv andA22(vk)5 l 22(vk)A2 S(vk)Dv5A12g21

2 (vk)A2 S(vk)Dv~Eq. ~44!!. Above 3 Hz, and for the given values of thcoherency drop parametera and the separation distancejcoherency becomes very low, so thatl 21(vk)5g21(vk)→0.The amplitude of the first cosine term,A21(vk) in Eq. ~42!,tends to zero, and the series is comprised, essentially, ofone cosine term at every frequency with amplitudeA22(vk).A2 S(vk)Dv. Consequently, the spectral densities andamplitudes of the generated motions~Figs. 15 and 17! be-come identical to the target ones, as was the case for sta1. Thus, the amplitudes of the simulated motions exhibit dferent properties depending on the order in which theysimulated. The amplitude variability in the motions genated by the HOP method, as well as their ergodicity propties discussed earlier, suggest that seismic ground motsimulated by this technique do not possess the same chteristics at all locations on the ground surface.

It should be noted that the behavior of the amplitude vaability in the motions simulated by the HOP method~Fig. 17!is, also, not consistent with the amplitude variability of atual recorded data, which vary randomly around the commcomponent amplitude in the dominant frequency range ofmotions ~Figs. 8 and 9!. On the other hand, the CDRAmethod, as well as the spectral representation method fosimulation of random fields described in the previous stion, produce simulated motions with characteristics content with those of the recorded data@108#; an extensive dis-cussion on the differential amplitude and phase variabiresulting from the simulated motions evaluated by meanthese techniques is presented in@107# and @108#. It is there-fore cautioned that the characteristics of the simulatscheme should be carefully evaluated before any techniquused in the seismic response evaluation of lifelines.

6.3 Simulations of spatially variable seismicground motions in engineering practice

The approaches described in the previous two subsecform the basis for the simulation of spatially variable seismground motions. The simulated motions, if properly evaated, should reproduce the characteristics of the randomfrom which they were generated; as a consequence, slated motions are stationary and homogeneous. These perties of the random field, although necessary for the evation of spatial coherency, are not desirable in the evaluaof seismic motions to be used in engineering applicationsinput motions at the supports of a lifeline. The followinmodifications can then be made, so that the simulated thistories exhibit spatially variable characteristics, butalso compatible with actual seismic ground motions:

Introduction of non-stationarity in simulatedmotions—Stationary time histories have neither beginninor end~Section 2!. In order to introduce finite duration inthe generated motions, the time histories simulated by anthe aforementioned techniques~eg, Eq.~34!! are multipliedby modulating~envelope! functions,a(t), namely,

f ~x,t !5a~ t ! f ~x,t ! (45)

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tionif-arer-er-ionsrac-

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one is

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where the modulating function can be given by, eg@61#:

a~ t !5a1t exp~2a2t !; t>0 (46)

The parametersa1 and a2 in the above equation can bselected so that the function’s maximum is equal to 1 aoccurs at the desired time during the strong motion parthe motions. Different modulating functions introduce eplicitly the duration of the seismic motions: a model devoped by Jenningset al @131# increases sharply at the beginning of the time history, remains constant for the durationthe strong motion, and decays exponentially thereafter. Omore sophisticated level, the modulating function can alsofrequency dependent@132#. The generation of non-stationarspatially variable simulated time histories has been reporamong others, in@7,109,112,132#.

Introduction of non-homogeneity in simulatedmotions—As has already been indicated~Section 2!, the ran-dom field of seismic data recorded at dense instrument aris considered as homogeneous, ie, the characteristics omotions, and, subsequently, the coherency, are indepenof absolute location and functions of separation distaonly. Since the majority of the dense instrument arrayslocated on fairly uniform soil conditions, the assumptionhomogeneity is valid. However, the supports of an extenstructure, such as a bridge, can be located in differentconditions that are characterized by different frequency ctent. In this case, the point estimates of the motions becofunctions of absolute location as well. The variations in tfrequency content of the point estimates of the motions ha very significant effect on the seismic response of extenstructures@1,8#. In order to accommodate the necessityhaving this fact reflected in the characteristics of the simlated input motions at the bridge supports, the simulatioare generated as random vector processes rather than rafields @132#; this consideration allows the power spectrdensities of the motions to vary at different locations.

Compatibility with seismic codes—As indicated in Sec-tion 2.2, the point estimates of the motions can be describy engineering models, such as the Kanai-Tajimi@31,32# orthe Clough-Penzien spectrum@33#, or seismological ones@36#. In many cases, however, simulated seismic ground mtions ought to be compatible with appropriate response sptra specified in design codes@133#. The simulation of spa-tially variable, response spectrum compatible seismic gromotions requires, generally, an iterative scheme, in whichpower spectral density of the simulated motions is upgraso as to match, to the degree possible, the prescribed seresponse spectra, eg,@1,8,132#.

Wave passage effects—The wave passage effects canintroduced in the simulated time histories by either imposa time delay compatible with the average apparent propation velocity after the motions have been generated,@7,132#, or by incorporating the wave passage term of tspatial variability~Eq. ~17!! in the description of the randomfield before the simulations are generated, eg,@78#. It should

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be noted, however, that the arrival time perturbations dcussed in Section 3.3 are not being considered in the silation of seismic ground motions.

It needs to be emphasized at this point that the coherefunctions used in all applications are not modified; theyused as they were developed from the recorded data, naas stationary and homogeneous.

7 SUMMARY AND CONCLUSIONS

The spatial variation of seismic ground motions denotesdifferences in the seismic time histories at various location the ground surface. Its significance for the seismicsponse of extended structures has been recognized sincearly 1960s; at the time, however, it was attributed onlythe wave passage effect. The modeling of the spatial variaity, and, particularly, the coherency, initiated with the anasis of the first recorded data that became available fromSMART-1 array in Lotung, Taiwan.

The spatial variability of seismic ground motions, as uderstood from analyses of data recorded at dense instruarrays, was described in this paper. The stochastic estimaof the spatial variability by means of the coherency, andinterpretation of the coherency in terms of the phase variaity of the data was presented. Empirical and semi-empirmodels for the coherency, their similarities, and, mostly, dferences, as well as their effect on the seismic responslifelines were briefly described. An alternative approachthe investigation of the spatial variability of seismic groumotions, that views spatial variability as deviations of amptudes and phases at individual stations around a coheapproximation of the seismic motions was described. Simlations of spatially variable seismic ground motions toused in the seismic response of lifelines were presentedvariability in the simulated motions resulting from the usea particular spatial coherency model as well as from theof the simulation scheme itself have been illustrated. Aproaches for the modification of the simulated seismic tihistories, so that they become compatible with actual onwere also highlighted.

The investigation of the spatial variability and its effecon lifelines has recently attracted renewed interest, thatbe partially attributed to the lessons learned during the dastating earthquakes of the last decade@6#. Unresolved issuesinvolving spatial variability include its physical modelinand its use in simplified design recommendations for alltended structures. Present basic and applied research otopic is moving in this direction.

ACKNOWLEDGMENTS

AZ was supported by USA NSF grants CMS-9725567 aCMS-9870509; this support is gratefully acknowledged.

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Aspasia Zervareceived her Diploma in Civil Engineering from the Aristoteleion University, Thsaloniki, Greece, in 1980, her MSc in Theoretical and Applied Mechanics, and her PhD inEngineering in 1982 and 1985, respectively, both from the University of Illinois at UrbaChampaign. Her research interests span the areas of earthquake engineering, engineering sogy, signal processing and probabilistic engineering mechanics. She taught at the City UniverNew York before joining Drexel University in 1989, where she presently serves as ProfessorCivil and Architectural Engineering Department and Courtesy Professor in the Electrical and Cputer Engineering Department.

Vassilios Zervasreceived his Diploma in Civil Engineering from the Metsoveion National TechnUniversity, Athens, Greece, in 1943. He founded a civil engineering consulting firm in ThessaGreece, and served as its president for over four decades. His professional expertise spanareas of seismic resistant design and construction of a wide variety of structures, including staand bridges.

Spatial variation of seismic ground motions: An overview

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