Comparison of Time Series and Random-Vibration Theory Site

Comparison of Time Series and Random-Vibration

Theory Site-Response Methods

by Albert R. Kottke and Ellen M. Rathje

Abstract The random-vibration theory (RVT) approach to equivalent-linear site-response analysis is often used to simulate site amplification, particularly when largenumbers of simulations are required for incorporation into probabilistic seismic-hazardanalysis. The fact that RVT site-response analysis does not require the specification ofinput-time series makes it an attractive alternative to other site-response methods.However, some studies have indicated that the site amplification predicted by RVTsite-response analysis systematically differs from that predicted by time-seriesapproaches. This study confirms that RVT site-response analysis predicts site amplifi-cation at the natural site frequencies asmuch as 20%–50% larger than time-series analy-sis, with the largest overprediction occurring for sites with smaller natural frequenciesand sites underlain by hard rock. The overprediction is caused by an increase in durationgenerated by the site response, which is not taken into account in the RVT calculation.Correcting for this change in duration brings the RVT results within 20% of the time-series results. A similar duration effect is observed for the RVT shear-strain calculationused to estimate the equivalent-linear strain-compatible soil properties. An alternativeto applying a duration correction to improve the agreement between RVT and time-series analysis is the modeling of shear-wave velocity variability. It is shown that intro-ducing shear-wave velocity variability throughMonte Carlo simulation brings the RVTresults consistently within �20% of the time-series results.

Introduction

Local soil conditions influence the characteristics ofearthquake ground shaking and these effects must be takeninto account when specifying ground-shaking levels forseismic design. These effects are often quantified via site-response analysis, which involves the propagation of earth-quake motions from the base rock through the overlying soillayers to the ground surface. Site-response analysis providessurface acceleration-time series, surface acceleration-response spectra, and/or spectral-amplification factors basedon the dynamic response of the local soil conditions.

In most cases, 1D site-response analysis is performed toassess the effect of soil conditions on ground shaking becausevertically propagating, horizontally polarized shear wavesdominate the earthquake ground-motion wave field. Themostcommon technique for site-response simulations is equiva-lent-linear analysis using time-series input motions. The mo-tion-to-motion variability between input-time series requiresthat simulations be performed using a suite of time series togenerate a stable estimate of the expected site response. A po-tential alternative is random-vibration theory (RVT), whichuses the same equivalent-linear wave-propagation approach,but allows for the calculation of the mean response with onlyone simulation. RVT-based site-response analyses are com-

monly used for site-response calculations that are incorpo-rated into probabilistic seismic-hazard analysis for nuclearpower plants (Regulatory Guide 1.208; Nuclear RegulatoryCommission [NRC], 2007) because in these cases, numeroussite-response simulations are performed to account for a suiteof scenario events and the variability in the site characteristics(i.e., depth to bedrock, shear-wave velocity profile, and non-linear properties).

Extensive validation of RVT within the context of sto-chastic simulations of ground motion has been performedwithin the seismological community; examples of such workinclude Hanks and McGuire (1981), Boore (1983), andMcGuire et al. (1984). Within the context of site-responseanalysis, fewer comparisons between RVT and time-seriesanalyses have been conducted. Electric Power ResearchInstitute (EPRI) (1993) compared RVT site-response predic-tions with recordings from a number of downhole arrays. Thestudy focused on comparisons between observed surfacemotions and the surface motions computed by various site-response methods, including an RVT method. The studyfound that the RVT method did just as well as the time-seriesmethods in predicting the surface motions. Rathje and Ozbey(2006) compared the results from site-specific site-response

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Bulletin of the Seismological Society of America, Vol. 103, No. 3, pp. 2111–2127, June 2013, doi: 10.1785/0120120254

analyses using time series and RVT methods. This study gen-erally showed a favorable comparison between RVT andtime-series analyses, but in several cases the RVT analysissystematically predicted larger amplification at the siteperiod. The study by Rathje and Ozbey (2006) was the firstin the literature to identify a potential overprediction of siteamplification by RVT, but the analyses were limited to only asingle, hypothetical soil site. This paper expands upon thework of Rathje and Ozbey (2006) to better elucidate thedifferences between RVT and time-series site-responseanalyses.

In this paper a comprehensive comparison of the timeseries (TS) and RVT-based site-response techniques is per-formed for a range of site conditions (i.e., shallow soil to verydeep soil) and over a range of input intensities that inducedifferent levels of nonlinearity. The site conditions and inputintensities used in the comparisons are selected to span therange of site conditions for which site-response analyses areperformed. The goals of these comparisons are to identifysystematic differences between TS and RVT site-responseanalyses for deterministic site conditions and to explain theobserved differences. As such, this paper focuses on compar-ing two different site-response simulation techniques ratherthan comparing site-response simulations with site-responseobservations.

Site-Response Methods

Equivalent-linear (EQL) site-response analysis (Schnabelet al., 1972) uses frequency domain-transfer functions tomodel the 1D, linear-elastic wave propagation through a lay-ered soil deposit. The method incorporates soil nonlinearitythrough the use of strain-compatible soil properties for eachsoil layer. The key to the EQL approach is the selection of soilproperties, shear-modulus (G), and damping (D), for each soillayer that are consistent with the level of shear strain inducedby the input rock motion. Development of strain-compatibleproperties requires an iterative approach in which the strainsare computed, the properties are revised based on the strains,and revised strains are computed based on the updated proper-ties. These iterations continue until the properties assigned toeach layer in the wave-propagation analysis are consistentwith the strains generated in each layer. The strain level usedto select the strain-compatible properties is not the peak time-domain shear strain, but rather an effective shear strain (γeff )that is typically taken as about 65% of the peak value.

EQL wave-propagation analysis requires specification ofan input rock motion. This input acceleration-time series isconverted to the frequency domain using the Fast FourierTransform (FFT). The resulting Fourier amplitude spectrum(FAS) is multiplied by the frequency domain-transfer func-tion that represents wave propagation to the ground surface,and the FAS at the surface is converted to an acceleration-time series using the inverse FFT. Commonly a suite of threeto seven input motions, selected and scaled to match a targetresponse spectrum, is specified for use in this analysis

because motion-to-motion variability induces different levelsof nonlinearity and different responses.

To avoid selection and scaling of input acceleration-timehistories and the computation time required to performsite-response analyses for multiple ground motions, the RVTapproach can be used. A full description of the RVTapproachcan be found elsewhere (e.g., Silva et al. 1997; Boore, 2003;Rathje and Ozbey, 2006), but a summary is provided here. Inan RVT site-response analysis, the input motion is character-ized solely by the amplitude of the FAS and the ground-motion duration (Dgm). Because only the amplitudes ofthe FAS are known, without the associated phase angles,the specified FAS cannot be directly converted into anacceleration-time series. However, the FAS can be used tocalculate peak time-domain parameters of motion, such aspeak ground acceleration (PGA), through the use of extremevalue statistics. In this approach, first the root mean square(rms) acceleration (arms) is computed from the FAS and therms duration (Drms) using Parseval’s theorem, as follows:

arms ��2

Drms

Z ∞0

jA�f�j2dfs

��m0

Drms

r; (1)

where A�f� is the Fourier amplitude at frequency f,m0 is thezero-th moment of the FAS, and Drms is taken as the ground-motion duration (Dgm) when using RVT to compute PGA.The kth spectral moment of the FAS is defined by

mk � 2

Z ∞0

�2πf�kjA�f�j2df: (2)

To compute the expected maximum acceleration in the timedomain (amax, which represents PGA), arms is multiplied by apeak factor (pf � amax=arms). Cartwright and Longuet-Higgins (1956) derived an expression for the expected valueof the peak factor for bandlimited white noise as

pf � amax

arms�

��2

p Z ∞0

1 − �1 − ξ exp�−z2��Nedz; (3)

where ξ is the bandwidth and is defined as

ξ ��m2

2

m0m4

s: (4)

The bandwidth varies from 0 to 1 and quantifies whether themotion is narrow- or broadband, with values approaching 1.0for a narrowband signal. In equation (3), Ne is the number ofextrema and is defined as

Ne �Dgm

π

��m4

m2

r: (5)

Ne increases as the duration of the motion increases and asthe signal becomes more broadband (i.e., smaller ξ).

To compute spectral acceleration rather than PGA usingthe RVT approach, the FAS is first multiplied by the transferfunction of a single-degree-of-freedom oscillator before

2112 A. R. Kottke and E. M. Rathje

equations (1) through (5) are applied. Additionally, Drms

must be modified from Dgm, particularly for longer-periodoscillators, because the oscillator response causes the motionto extend beyond the duration of the original motion. Booreand Joyner (1984) first proposed this oscillator correction as

Drms � Dgm �Do

�γn

γn � α

�; (6)

where γ � Dgm=To, To is the oscillator natural period, Do isthe oscillator duration given by To=2πζ, and ζ is the fractionof critical damping of the oscillator. Based on numerical sim-ulations, Boore and Joyner (1984) proposed n � 3 andα � 1=3. Liu and Pezeshk (1999) developed different valuesof n and α, whereas more recently Boore and Thompson(2012) developed an improved model forDrms. The oscillatorcorrection of Liu and Pezeshk (1999) is used in this studybecause it is adequate for the earthquake scenarios of engi-neering interest (Boore and Thompson, 2012).

Given that the input motion for RVT is specified by theamplitude of an FAS with no ability to reconstruct a timeseries, the RVT approach can only be applied to EQL site-response analysis in which the site response is modeledthrough frequency domain-transfer functions. The RVT ap-proach cannot be applied to nonlinear analysis. For EQLanalysis, the input rock FAS is multiplied by the amplitudeof the site-transfer function, and the surface FAS is convertedto a surface-response spectrum using the RVT methodologyoutlined above. The EQL soil properties are determined

through iterations that incorporate RVT estimates of the peakshear strain. The RVT response spectrum represents themean-response spectrum at the surface and thus providesin one analysis the same information as the mean from a suiteof input motions from a TS analysis.

Analyses Performed

EQL site-response analyses using TS-input motions andusing the RVT method are conducted for a range of site con-ditions. The sites are based on measured shear-wave velocityprofiles from western North America (Sylmar County Hos-pital) and central and eastern North America (Calvert Cliffs),such that different fundamental site frequencies and regionalcharacteristics can be investigated. Additionally, eight hypo-thetical site profiles are analyzed to better understand the ob-served trends. The shear-wave velocity profiles of the sitesare described below. For each site, nonlinear-modulus reduc-tion and damping curves are assigned based on soil type(plasticity index) and confining pressure using the empiricalmodel of Darendeli (2001). The ground motions used in theanalyses correspond to a magnitude 6.5 earthquake at a dis-tance of 20 km, as described below.

Site Profiles

The Sylmar County Hospital parking lot (SCH) site islocated in the San Fernando Valley of southern Californiaand consists of 90 m of alluvium over rock (Fig. 1).

Figure 1. Shear-wave velocity (VS) profiles and theoretical site-transfer functions for the Sylmar County Hospital parking lot and CalvertCliffs sites.

Comparison of Time Series and RVT Site-Response Methods 2113

Characteristic of southern California soil sites, there is not asignificant impedance contrast as the profile encounters ashear-wave velocity typical of soft rock (∼760 m=s). The sitecharacteristics used in this study are based on measurementsmade by Gibbs et al. (1999). The theoretical linear-elastictransfer function for the SCH site, assuming 1% dampingin all soil layers, is shown in Figure 1 and indicates a first-mode natural frequency of 1.7 Hz with an amplitude ofabout 2.5.

The Calvert Cliffs (CC) site is located in Maryland, onthe coast of the Chesapeake Bay. The site consists of alter-nating layers of sand and clay/silt to a depth of over 750 m(Fig. 1). The shear-wave velocity within the profile rangesfrom 400 to 900 m=s, while the rock underlying the soil col-umn is late Paleozoic with a shear-wave velocity of almost3000 m=s. The site characteristics used in the analyses arebased on the site investigation reported by UniStar Nuclear(2007). The first-mode natural frequency is 0.25 Hz, and thetheoretical linear elastic-transfer function for 1% damping inthe soil layers shows an amplitude of almost 5.0 at this fre-quency (Fig. 1). This large amplitude is a result of the largeimpedance contrast at the soil/rock interface.

Eight additional hypothetical shear-wave velocity pro-files are created to better understand how the first-mode natu-ral frequency and rock-impedance ratio influence the results.

Each hypothetical site is composed of a soil layer with a con-stant velocity underlain by a rock half-space. The shear-wavevelocity and thickness of the soil layer are varied, as well asthe shear-wave velocity of the rock half-space, as summa-rized in Table 1. The rock shear-wave velocities are selectedto be representative of western North America (WNA) andcentral and eastern North America (CENA). The first-modenatural frequency of the sites ranges from 0.32 Hz for thedeepest site to 10 Hz for the shallowest site. The theoreticalsite-transfer functions for a select group of the hypotheticalsites are shown in Figure 2. For sites with a bedrock shear-wave velocity of 1000 m=s (Fig. 2a), the amplification at thefirst-mode frequency is approximately 3 and variations insoil thickness only shift the modal peaks. As the shear-wavevelocity is increased to 3000 m=s (Fig. 2b), the amplificationat the first-mode frequency increases to approximately 8.

Input Ground Motions

Input motions for the site-response analyses in this studyare generated by stochastic simulation (e.g., Boore, 2003).The stochastic method first generates an FAS based on a seis-mological model that incorporates source, path, and siteeffects. This FAS can be used to generate input-time series forsite-response analysis by scaling the FAS of a windowed timeseries of white noise or the FAS can be used directly as inputinto RVT site-response analysis. Because the time series gen-erated by the stochastic method are consistent with a speci-fied FAS and because RVT site-response methods require aninput FAS, the stochastic method provides a good approachto generate input motions for comparisons of RVT and TSsite-response analyses. However, a single stochasticallysimulated time series does not match the FAS exactly andtherefore a suite of time series are required to match, onaverage, the FAS.

Table 1Characteristics of the Hypothetical Sites used in the

Site-Response Analyses

SoilThickness (m)

VS, Soil(m=s)

SiteFrequency (Hz)

VS, Rock(m=s)

10 400 10 1000, 300032 400 3.2 1000, 3000100 400 1.0 1000, 3000320 400 0.32 1000, 3000

Figure 2. Theoretical site-transfer functions for selected hypothetical velocity profiles.


The input motions for this study are based on a hypo-thetical Mw 6.5 earthquake at a distance (R) of 20 km inWNA. The stochastic-input motions for this event are com-puted for a point source using the program SMSIM (Boore,2005) and model parameters typical for shallow crustalevents in WNA, as given by Boore (2005). The key modelparameters are presented in Table 2. The computed durationof the event for the RVT calculation (Dgm) is 6.80 s. The re-sulting RVT-generated response spectrum and FAS for theseismological model are shown in Figure 3a,b. One hundredtime series are generated from the seismological FAS and theresulting response spectra and FAS are shown in Figure 3c,d.The FAS and response spectra agree favorably between RVTand time series, demonstrating the consistency in the input-motion specifications for RVT using stochastic simulations.All 100 simulated time series were used in the TS site-response analyses.

The level of variability in a suite of input-time series caninfluence the mean predictions of site response, such that thisissue should be considered when comparing with RVTresults. The logarithmic standard deviation of the responsespectra (σln Sa) of the 100 time series is about 0.40 at0.20 Hz and decreases to 0.15 at 100 Hz. To better under-stand the influence of the variability in the suite of input-timeseries on the results from TS site-response analyses, the sto-chastic time series are modified to reduce their variabilityusing two techniques. In the first technique, the amplitudesof each FAS are modified to match the seismological FAS.This technique is referred to as Fourier amplitude matchingor FA matched. The FA matched motions are time series thatagree very closely with the seismological FAS (Fig. 3f) andhave some reduced variability in the response spectra par-ticularly at longer periods (Fig. 3e). While the resultingmotions may no longer have realistic characteristics in thetime domain, these motions are used in this study to modelminimal variability in the frequency domain. In the secondtechnique, the response spectra of the 100 simulated timeseries are spectrally matched (SA matched) to the RVT re-sponse spectrum using RSPMATCH (Hancock, 2006). Thespectral matching is performed from 0.2 to 30 Hz, which re-sults in the TS response spectra agreeing well over this fre-quency range but not as well outside of this range (Fig. 3g).While the response spectra show little variability for the

spectrally matched motions, the FAS still contain variability(Fig. 3h).

Comparison of Time Series and RVTSite-Response Results

Time series and RVT site-response analyses are per-formed using the program Strata and used to calculate theamplification factor (AF � Sa;soil=Sa;rock, where Sa is spec-tral acceleration). The TS and RVT approaches to EQLsite-response analysis are compared using the median ampli-fication factor predicted from a suite of time series (AFTS)and the AF values predicted by RVT (AFRVT). AFTS is cal-culated as the mean value in logarithmic space (i.e., thegeometric mean). In earthquake-engineering practice, themedian value is commonly used, whereas RVT predictsthe mean value. The comparisons in this study will focus onthe median of the time series because little difference wasobserved between the mean and median values of AF com-puted from the suite of simulated time series. The compari-son between the RVT and TS results are quantified by theratio of the RVT amplification factor to the median TS am-plification factor, defined as

αAF � AFRVTAFTS

: (7)

The agreement between the TS and RVT analyses is quanti-fied using αAF versus frequency, as well as the minimum αAF

and the maximum αAF over all frequencies.

Linear-Elastic Comparisons

Time series and RVT site-response analyses are first per-formed for linear-elastic (LE) soil conditions. The TS andRVT amplification factors for the SCH site (Fig. 4) are verysimilar, with the RVT results falling within the range of thetime series. The differences between the TS and RVT resultsare better shown through the AFRVT=AFTS ratio (αAF). αAF

for SCH varies between 0.95 and 1.1 with the maximumoccurring at the site frequency of 1.7 Hz (Fig. 4). The TSand RVT amplification factors for the CC site (Fig. 4) againshow the RVT results falling within the range of the timeseries, but now there are significant differences between themedian TS and RVT results, particularly at the site frequency.αAF for CC varies from 0.8 to 1.3 with the maximum of about1.3 occurring at the first-, second-, and third-mode naturalfrequencies (Fig. 4).

The comparisons using the SCH and CC sites show thatthe agreement between the RVT and TS methods dependson the site characteristics. To better understand the influenceof the site characteristics on the differences between the RVTand TS results, LE analyses are performed using the hypo-thetical sites. The associated αAF are shown in Figure 5.These results show that the maximum αAF always occurs ata site frequency and that it increases as the site frequency

Table 2Selected Input Parameters from Boore (2005) used by

SMSIM for the Calculation of the FAS shown inFigure 3

Parameter Value

Source spectrum Point source Brune ω-squaredStress drop, Δσ (bar) 80Site diminution, κ (s) 0.03Density of crust, ρ (g=cm3) 2.8VS of crust, β (km=s) 3.6Dgm Atkinson and Boore (2006)Crustal Amplification Boore (2005)


decreases (i.e., soil thickness, Hsoil, increases) and as VS;rock

increases. For example, as the site frequency decreases from10 Hz (Hsoil � 10 m) to 0.32 Hz (Hsoil � 316 m), the maxi-mum αAF increases from 1.05 to 1.2 for VS;rock � 1000 m=sand increases from 1.15 to 1.5 for VS;rock � 3000 m=s. Theinfluence of VS;rock on the results indicates that the height ofthe transfer function is also important, with large VS;rock

associated with larger peaks in the transfer function (Figs. 1and 2). These results demonstrate that the amplification pre-dicted by RVT may be as much as 20%–50% larger than TS,which is significant. The potential under prediction is lesssevere, but still notable at 10% to 20%.

One potential cause for the differences between RVTandTS analysis is the modification of the duration of the time

Figure 3. Acceleration-response spectra and Fourier amplitude spectra for (a and b) RVT-input motion, (c and d) stochastically simulatedtime-series motions, (e and f) simulated motions matched to the FAS, and (g and h) simulated motions spectrally matched to the RVT-responsespectrum. The thin gray lines represent all time series generated. The thick gray line represents a single time series.


series due to the response of the site, which is ignored in thecurrent implementation of RVT site response (i.e., the rockDgm is used as the basis for Drms in the RVT calculation ofthe surface response spectrum). As previously noted, the in-crease in duration due to the single-degree-of-freedom oscil-lator response must be included in the calculation of RVTresponse spectra. Thus it follows that the increase in duration

due to the soil-column response should be taken into ac-count. For example, Figure 6 shows an input acceleration-time history and the computed surface-time history for theHsoil � 316 m site with VS;rock � 3000 m=s. The durationof the input motion, as defined based on D5–75 (i.e., thetime interval between the occurrence of 5% and 75% ofthe Arias intensity of the acceleration-time history), is equal

Figure 4. Amplification factors for TS and RVT analyses and αAF for the SCH and CC sites.

Figure 5. αAF for the eight hypothetical sites listed in Table 1.


to 3.77 s. This duration is increased to 4.98 s, a 32% increase,at the ground surface due to the site response. The durationeffect is even more pronounced when considering the re-sponse of an oscillator subjected to these motions (Fig. 6).For an oscillator with a natural frequency of 1.0 Hz, theduration increases from 3.95 to 6.46 s, a 64% increase.An increase in duration for a given FAS results in a reductionin arms (equation 1), which then translates into smaller valuesof PGA and spectral acceleration. Ignoring these increases induration generates RVT-computed spectral accelerations thatare larger than the TS values.

To further investigate the modification in duration due tothe site response, significant durations are computed forboth the input-time series (Dinput

5–75) and surface-time series(Dsurface

5–75 ). Significant durations are computed for input- andsurface-time series, as well as the oscillator responses at eachof the oscillator frequencies for the input- and surface-timeseries. The ratio ofDsurface

5–75 toDinput5–75 is plotted versus oscillator

frequency in Figure 7 for the surface motions computed fromanalysis of the SCH, CC, and eight hypothetical sites. The datashow that for sites with first-mode natural frequencies greaterthan or equal to about 3 Hz (SCH, Hsoil � 10 m and 32 m)there is little change in the durations, with the duration ratiosfalling between 0.98 and 1.05. However, as the natural fre-quency of a site decreases, differences between the input-motion and surface-motion durations increase and thesechanges are most prevalent at the natural frequencies of thesite. For example, when Hsoil � 316 m (fsite � 0:32 Hz),

the ratio of Dsurface5–75 to Dinput

5–75 at the natural frequency of thesite is 1.2 for VS;rock of 1000 m=s and 1.6 for VS;rock of3000 m=s (Fig. 7f). The change in duration is also influencedby the height of the transfer function, with the duration modi-fiedmorewhen the bedrock shear-wavevelocity is larger. Thismore significant increase in duration is caused by strongermultireflections in the soil column for larger VS;rock. Theshapes of the curves in Figure 7 for duration are very similarto those in Figures 4 and 5 for amplification factor, suggestingthat the overprediction in the RVT-amplification factor at asite’s natural frequencies is related to the changes in theground-motion duration due to site response.

The surface-response spectra and AF computed by RVTcan be corrected to account for the change in duration due tothe site response. Equation (1) indicates that arms is inverselyproportional to the square root of Drms. Because the com-puted AF is directly proportional to arms, the AF computedby RVT can be corrected by dividing by the square root of theobserved duration ratio (Dsurface

5–75 =Dinput5–75). The amplification

factors computed by RVT are corrected using this methodol-ogy and the αAF computed from the corrected RVT results areshown in Figure 8. For the sites with high natural frequencies(SCH, Hsoil � 10 m and 32 m), the difference between thecorrected and uncorrected αAF are insignificant because ofthe relatively small changes to the duration (Fig. 7). How-ever, for the sites with smaller natural frequencies (CC,Hsoil � 100 m and 316 m), the duration correction reducesthe largest αAF such that no value is above about 1.2.

Figure 6. The influence of site response and oscillator response on ground-motion duration. Acceleration-time histories represent aTS-input motion, the computed surface motion for the Hsoil � 316 m site for the TS-input motion, and the computed 1-Hz oscillator(5% damping) responses for the input and surface motions.


Summarizing the results for all analyses performed, thecomputed values of minimum αAF and maximum αAF areplotted versus site frequency in Figure 9. Before the durationcorrection, the minimum αAF ranges from 0.82 to 0.98 withsmaller values (more underprediction) occurring at smallersite frequencies. The duration correction has little effect onthe minimum αAF. For the maximum αAF, the uncorrectedvalues range from 1.05 to 1.5 with larger values (more over-prediction) occurring at smaller site frequencies. The durationcorrection reduces the maximum αAF to below about 1.2for all analyses performed and the sites with VS;rock �1000 m=s generally displaymaximumαAF values around 1.1.

It is clear from the preceding analyses that site responseinfluences ground-motion duration and that ignoring thischange in duration in RVT analysis results in significantdifferences between RVT and TS estimates of site amplifica-tion. The duration correction for RVT explored here is basedon computed values of D5–75 of the input and surface mo-tions from the TS analysis. However, the duration correctionsdeveloped by others to deal with duration changes due tooscillator response (e.g., Boore and Joyner, 1984; Liu andPezeshk, 1999; Boore and Thompson, 2012) are baseddirectly on the spectral amplitudes computed by RVT andTS simulations. An alternative to the approach explored hereis to use the site-amplification results from the RVT and TSsite-response analyses (e.g., Fig. 5) to directly compute the

duration correction required to make the RVT and TS predic-tions match.

Influence of Input-Motion Variabilityon RVT–TS Comparisons

The influence of the variability in the input motions onthe amplification factors computed by TS analysis is ex-plored because the RVT site-response analysis uses a smoothFAS as input while the TS analysis uses multiple-time seriesthat have motion-to-motion variability and nonsmooth FAS.Site-response simulations are conducted for all sites using LEsoil conditions and the simulated, FA-matched andSA-matched time series (Fig. 3). The amplification factorscomputed for the hypothetical site with a thickness of316 m and VS;rock � 3000 m=s are shown in Figure 10 forthe three sets of input motions. Both the FA-matched andSA-matched motions generate moderately larger median-amplification factors than the original TS at the site frequen-cies, with the FA-matched motions providing the largestmedian response. The median-spectral ratio at the site fre-quency (0.32 Hz) is 4.1 for the original time series, 4.2 forthe SA-matched motions, and 4.5 for the FA matched. Thechange in the median-amplification factor is caused by thematched motions generating almost no “very low” amplifi-cation factors (e.g., many motions have AF < 3:0 for the TSmotions at the site frequency, whereas almost no motions fall

Figure 7. The ratio of the significant duration of the oscillator response of the surface motion to the significant duration of the oscillatorresponse of the input motion.


Figure 8. αAF computed with the uncorrected RVT results and duration-corrected RVT results for each site analyzed.

Figure 9. The maximum and minimum αAF as a function of site natural frequency for the uncorrected and duration-corrected RVT results.


below 3.0 for the FA- and SA-matched motions, Fig. 10), butmaintaining larger predictions in amplification factor (e.g.,above 5.0 in Fig. 10).

The αAF are recomputed using the amplification factorsfrom TS analyses with the different input motions along withRVT-amplification factors corrected for duration changes.The results are plotted in Figure 11. Using the FA-matchedor SA-matched motions only reduces the αAF moderately andthe effect is only noticeable for the sites with lower naturalfrequencies (i.e., CC, hypothetical with Hsoil � 100 and316 m). Thus, it appears that variability in the TS-input mo-tions does not significantly change the RVT–TS comparison,such that the minimum and maximum values of αAF stillrange from 0.9 to 1.2.

Influence of Soil Nonlinearity on RVT–TSComparisons

Equivalent-linear analyses are performed to investigatehow the differences between RVT and TS analyses changewith input intensity and induced strain level. Equivalent-linear RVT and TS site-response analyses are performed withthe original input motions shown in Figure 3 (medianPGA � 0:17g) and the same input motions scaled by 2.35to generate a median PGA of 0:4g. The αAF for the EQLanalyses are plotted in Figure 12 for the SCH and CC sitesalong with the LE results. The RVT results used in the αAF

calculation for Figure 12 are not duration corrected becausethe previously developed duration corrections are only theo-retically applicable to LE conditions. The induced shearstrains are also shown in Figure 12. For the SCH site(Fig. 12a), as the intensity of the input motion increases,the site frequency decreases due to softening of the strain-compatible properties, such that the peak in αAF shifts toa lower frequency. Additionally, the αAF at the site frequencyincreases from 1.06 under LE conditions to 1.11 for an inputPGA of 0:40g while the αAF at 100 Hz remains close to unityfor all analyses. The induced strains from TS and RVT analy-

ses (Fig. 12c) for the SCH site are very similar. For the CC site(Fig. 12b), there is little change in the site frequency withincreased input intensity due to the large depth of the sitepredominantly controlling the site frequency. The largestchanges in αAF for the CC site under different input inten-sities occur at frequencies greater than about 3 Hz, where theαAF is close to 1 under LE conditions and decreases to 0.85–0.95 at an input PGA of 0:40g. This result appears to becaused by the RVT analysis predicting somewhat largerstrains (20%–30% larger) than the TS analysis for this site(Fig. 12d). While the levels of damping associated with themoderately larger strains are not significantly greater, whenintegrated over a very deep site they result in the smaller am-plification factors at higher frequencies from RVT analysis.

Duration is a potential cause in the different peak strainspredicted by RVT and TS analyses. Extreme value statistics(equations 1–5) are used to compute the peak strains for RVTEQL analysis. First an FAS for shear strain is computed bymultiplying the input FAS for acceleration by a transfer func-tion that relates shear strain in a layer to input acceleration.The FAS for shear strain is then used to compute the rmsshear strain (γrms) using equation (1) and the appropriatepeak factor using equations (2)–(5). These terms togetherprovide the estimate of the peak shear strain. However, in allof these calculations the specified input-motion duration isused, which assumes that the shear-strain time history hasthe same duration characteristics as the input acceleration-time history.

To investigate the duration characteristics of the strain-time histories, the results from the TS analyses are consid-ered. Figure 13 shows an input acceleration-time historyand the associated shear-strain time history computed at adepth of 700 m within the CC site. The duration andfrequency-content characteristics of the shear-strain timehistory are very different from the input acceleration-timehistory. The shear-strain time history contains more low-frequency motion and its duration (D5–75) is 50% larger than

Figure 10. The influence of the input-motion variability on the amplification factors computed by TS analysis for the hypothetical sitewith Hsoil � 316 m and VS;rock � 3000 m=s.


the acceleration-time history (5.61 s versus 3.73 s). The dif-ference in frequency content for the shear-strain time historyis taken into account in RVT through the shear-strain transferfunction, but the duration change is not. It was also consid-ered whether the duration of the input velocity-time historyrelates better to the duration of the shear-strain time history,but the duration of the velocity-time history was similar tothe duration of the input acceleration-time history.

The input-motion duration (Dinput5–75) and shear-strain time

history durations (Dstrain5–75 ) are investigated for different depths

within the SCH and CC using the average across all motions.For the SCH site, where the TS and RVT shear strains are al-most indistinguishable, the average ratio of Dstrain

5–75 to Dinput5–75 is

very close to 1.0. However, the average ratio of Dstrain5–75 to

Dinput5–75 for the CC site ranges from 1.6 at a depth of 700 m to

1.15 at a depth of 25 m. The larger durations for strain areassociated with the depths where RVT overpredicts strain.

Because the current RVT implementation uses Dinput5–75 in the

calculation of γrms (equation 1), which is smaller than theactual Dstrain

5–75 , γrms is overestimated as well as the peak shearstrain. Using the same approach as was applied to the am-plification factors, the duration ratio can be used to correctRVT shear strain. The correction factor associated with a du-ration ratio of 1.6 is 0.79, which, when applied to the RVTshear strains, makes them very similar to the TS shear strains.

Influence of Site-Property Variations on RVT–TSComparisons

RVT site-response analyses are commonly used forseismic-hazard studies for nuclear power plants, and thesestudies require that site-property variability be taken into ac-count via Monte Carlo simulation. Introducing shear-wavevelocity variability reduces the peaks in the average site-amplification transfer function and because these peaks are

Figure 11. The influence of the input-motion variability on αAF. αAF computed from the median AF from TS analyses with differentsuites of input motions and the duration-corrected AF from RVT analysis.


generating the RVT overpredictions of site amplification, it isimportant to assess how site-property variability influencesthe comparison between RVT and TS site-response analyses.

RVT site-response analyses are performed with theshear-wave velocity profiles varied through Monte Carlo

simulation, as incorporated in the program Strata (Kottkeand Rathje, 2008). The Monte Carlo simulations simulatevelocity profiles using the Toro (1995) model, which corre-lates the velocities in adjacent layers through an interlayer-correlation coefficient that varies with layer thickness and

Figure 12. (a, b) The influence of input motion intensity on αAF. (c, d) Peak shear-strain profiles from RVT and TS analyses and an inputPGA of 0:4g.

Figure 13. The input motion acceleration-time history and computed shear-strain time history at a depth of 700 m for the CC site.


depth. The thicknesses of the layers in the baseline-velocityprofiles (i.e., Fig. 1; Table 1) are maintained and velocityprofiles are generated for three different levels of variabilityin the shear-wave velocity (σlnVS

� 0:10, 0.20, and 0.30), asshown in Figure 14 for the SCH and CC sites. One hundredvelocity-profile realizations are generated for each σlnVS

andeach velocity profile is used to compute the LE site responseusing RVT and TS analysis. No duration corrections are ap-plied to the RVT results.

The influence of velocity variations (σlnVS� 0:20) on

the computed RVT and TS median amplification factors forthe SCH and CC sites is shown in Figure 15. The medianamplification factors are reduced when velocity variationsare included because the velocity variations essentially movethe locations of the transfer-function peaks, resulting insmaller average transfer-function peaks at the naturalfrequencies. However, the effect of velocity variations ismore significant for the RVT results, because the peaks in the

Figure 14. Shear-wave velocity profiles generated by Monte Carlo simulation for the SCH and CC sites for three different levels ofshear-wave velocity variability (σlnVS

).

Figure 15. The influence of shear-wave velocity variability on the site amplification predicted by RVT and TS site-response analysis.


transfer function have a larger effect on the predicted RVTamplification. For the CC site, the overprediction by RVTat the first-mode frequency is reduced from about 30% to15% when velocity randomizations with σlnVS

� 0:20 areused. Therefore, modeling velocity variations in the site-response analysis provides RVT results that are more similarto the TS results.

To evaluate the influence of site variability on the RVT–TS comparison, linear-elastic RVT analyses with site-property variability are computed for the hypothetical sitesas well as the SCH and CC sites. The minimum and maxi-mum αAF are plotted versus site natural frequency for allsites in Figure 16. The αAF values are shown for differentlevels of shear-wave velocity variation (i.e., σlnVS

). The datain Figure 16a show a moderate increase in the minimum αAF

towards 1.0 for σlnVSgreater than zero, but a 10%–15%

underprediction by RVT is still observed. The maximumαAF is influenced more by site variability than the minimumαAF (Fig. 16b). Using σlnVS

greater than zero results in a con-sistent improvement (i.e., reduction) in the maximum αAF

such that almost all values are below 1.2 for σlnVS≥ 0:2

(Fig. 16b). The results in Figure 16 show that when shear-wave velocity variability is included in site-response analy-ses, RVT analysis provides median-site amplification resultsmore similar to TS analysis. The agreement improves withincreasing levels of variability.

Including variability in the shear-wave velocity profilein RVT site-response analyses provides results that are in bet-ter agreement with the median predictions from TS analysis,but it does not completely resolve the differences betweenRVT and TS analyses. Differences as large as 10% to 20%are still observed. These levels of difference start to approacha level that is acceptable in earthquake engineering. How-ever, one must recognize that varying the shear-wave veloc-ity profile introduces variability in the RVT amplificationfactors, and this variability may be larger than one observes

in a suite of TS analyses that incorporates shear-wave veloc-ity variability. If one is considering only the median-site am-plification, then this difference in variability is not importantand using Monte Carlo simulations to improve the perfor-mance of RVT site-response analysis may be a viable option.However, the increase in variability in the RVT amplificationfactors will influence a soil-specific seismic-hazard curvethat incorporates site-specific site amplification and its vari-ability. This issue is explored in Pehlivan et al. (2012).

Discussion

The results from this study indicate that changes inground-motion duration due to site response, which arenot taken into account in RVT site-response analysis, can leadto RVT analysis overestimating site amplification relative toTS analysis. The overprediction occurs at the natural frequen-cies of a site. The RVT predictions can be improved to agreebetter with TS analysis through a duration correction. How-ever, this duration correction is frequency dependent and sitespecific (Fig. 7). One approach to developing the frequency-dependent correction factors for a site would be to perform asmall suite of linear-elastic TS and RVT analyses and use theresults to develop the required frequency-dependent correc-tion factors. While this approach eliminates one of the mainadvantages of RVT site-response analysis (i.e., no need toselect input motions), the input motions could be generatedquickly via the stochastic method and used to develop thecorrections factors. Alternatively, because the frequencydependence and amplitude of the correction factors are re-lated to the transfer function for the site, an empirical rela-tionship could be developed that predicts the correctionfactors as a function of the linear-elastic transfer function fora site. This empirical relationship could be developed fromcorrection factors computed for a large range of sites withdifferent characteristics. This approach would avoid the need

Figure 16. The influence of shear-wave velocity variability on the minimum and maximum αAF. αAF is computed using uncorrectedRVT results.


for users to perform TS analyses to correct their RVT analy-ses, but the correction factors would be less accurate thanthose developed on a site-specific basis.

Another required improvement to RVT site responserelates to the duration used in the shear-strain calculation.Using the input-motion duration in the RVT calculation ofshear strain can overestimate the induced shear strains, lead-ing to overdamping of the high-frequency components ofmotions for very deep sites and/or large strains. An adjust-ment to the input-motion duration that makes it representa-tive of a shear-strain time history can improve the RVTshear-strain calculation. Again, this adjustment is site spe-cific and could be estimated from TS analyses. Alternatively,an empirical relationship could be developed that predicts theadjustment factor based on site characteristics.

The corrections proposed here are most necessary forRVT site-response analyses that use deterministic soil proper-ties. When RVT site-response analyses are used within aMonte Carlo framework, in which the shear-wave velocityprofile is statistically varied and the site amplification is aver-aged over the shear-wave velocity realizations, the differ-ences between RVT and TS analyses are more modest(Fig. 16). While correction factors would further improvethe comparison between RVT and TS analyses when siteproperty variability is modeled, these correction factors maynot be as critical as for analyses with deterministic soilproperties.

Conclusions

The RVT approach to site-response analysis has beenused in seismic-hazard studies since the 1990s. It is an attrac-tive alternative to site-response analyses using TS inputmotions because a single RVT analysis can potentially pro-vide the mean site response without the need for a suite ofinput-time histories. RVT site-response analysis is commonlyused in seismic-hazard studies for nuclear power plants be-cause these studies often include numerous site-responsesimulations that account for variability in the site character-istics. Recent work by Rathje and Ozbey (2006) identifiedthe potential for RVT to overpredict site amplification relativeto TS analyses. The present article extends the work of Rathjeand Ozbey (2006) to better elucidate the differences betweenRVT and TS site-response analyses for a range of siteconditions.

The site-response results presented in this paper indicatethat RVT consistently overpredicts the site amplification atthe fundamental frequencies of a site as compared withthe median value from a suite of input-time series. This over-prediction is modest (i.e., less than 10%) for sites with natu-ral frequencies greater than about 1 Hz and with a soft rock(i.e., VS;rock ∼ 1000 m=s) half-space, but the overpredictioncan be as large as 50% for sites with low natural frequencies(i.e., 0.2–0.3 Hz) and a hard rock (i.e., VS;rock ∼ 3000 m=s)half-space. There is some underprediction of site amplifica-tion at frequencies between the fundamental modes, but the

underprediction is relatively modest (5%–15%). The over-predicted response by RVT is principally caused by a changein the duration of the ground motion and associated oscillatorresponse generated by the site response. Ignoring the in-crease in duration due to site response causes the arms accel-eration to be overestimated by RVT and thus the peakacceleration and spectral acceleration are overestimated aswell. Correcting the RVT spectral accelerations for thisduration increase using the observed TS durations bringsthe RVT estimates within 10% of the TS values for siteswith VS;rock ∼ 1000 m=s and within 20% for sites withVS;rock ∼ 3000 m=s.

RVT site-response analysis also tends to overpredictshear strains relative to TS analysis predominantly for deepsites (i.e., smaller natural frequencies) underlain by hardrock. Even a modest overestimation of shear strain in equiv-alent-linear site-response analysis can result in overdampingof high-frequency spectral accelerations when the largerstrains extend over significant depths. This overestimationagain is driven by changes in duration. The duration ofthe shear-strain time history is increased by the site responseand this increase currently is not taken into account by RVTanalysis.

When shear-wave velocity variability is incorporatedinto the site-response analysis via Monte Carlo simulation,the overprediction of site amplification by RVT is reducedfrom as much as 30%–50% to about 10%–20%. Theimproved agreement between RVT and TS analyses observedwhen shear-wave velocity variability is modeled is due to thevariability reducing the amplitudes of the peaks in the aver-age transfer function, which is the main cause for the RVToverprediction in site response. Nonetheless, there are stillconsistent differences between the RVT and TS results.

This paper focused on identifying systematic differencesbetween RVT and TS site-response analyses and explainingthese differences. An important finding is that changes to thedurations of ground shaking, oscillator response, and shearstrains due to site response influence the differences betweenRVT and TS results. Corrections for these durations wereestimated from the TS analyses and improved the agreementbetween RVT and TS analyses. A similar approach could beused in practice to improve RVT site-response predictions orempirical relationships could be developed that predict thecorrection factors based on the characteristics of a site.

Data and Resources

The time series and random-vibration theory site-response simulations performed in this study were conductedusing Strata, which can be obtained from http://nees.org/resources/strata (last accessed April 2012). The stochastictime series used in the analysis were created using the latestversion of SMSIM by Boore (2003) and can be obtainedfrom http://daveboore.com/software_online.html (last ac-cessed January 2012). Time-domain spectral matching ofthe input motions was performed using the program


http://nees.org/resources/strata



http://daveboore.com/software_online.html

RSPMATCH, which is described in Hancock (2006) andavailable from the authors. Examination of the results andcreation of the figures was done using Python (http://python.org, last accessed December 2012) in conjunction with thenumpy (http://www.numpy.org/, last accessed December2012), scipy (http://www.scipy.org), and matplotlib (http://matplotlib.sourceforge.net/, last accessed December 2012)libraries.

Acknowledgments

Financial support was provided by the Nuclear Regulatory Commis-sion under grant NRC-04-07-122. This support is gratefully acknowledged.We thank David Boore, Frank Scherbaum, Steven Day, and an anonymousreviewer for thoughtful comments and suggestions that improved themanuscript.

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Pacific Earthquake Engineering Research (PEER) Center325 Davis HallUniversity of CaliforniaBerkeley, CA [email protected]

(A.R.K.)

University of Texas at AustinDepartment of Civil, Architectural, and Environmental Engineering301 E. Dean Keeton St, Stop C1700Austin, Texas [email protected]

(E.M.R.)

Manuscript received 10 August 2012


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Comparison of Time Series and Random-Vibration Theory Site