Lin Et Al (2013) Exact CS, BSSA

Conditional Spectrum Computation Incorporating Multiple Causal

Earthquakes and Ground-Motion Prediction Models

by Ting Lin, Stephen C. Harmsen, Jack W. Baker, and Nicolas Luco

Abstract The conditional spectrum (CS) is a target spectrum (with conditionalmean and conditional standard deviation) that links seismic hazard information withground-motion selection for nonlinear dynamic analysis. Probabilistic seismic hazardanalysis (PSHA) estimates the ground-motion hazard by incorporating the aleatoryuncertainties in all earthquake scenarios and resulting ground motions, as well as theepistemic uncertainties in ground-motion prediction models (GMPMs) and seismicsource models. Typical CS calculations to date are produced for a single earthquakescenario using a single GMPM, but more precise use requires consideration of at leastmultiple causal earthquakes and multiple GMPMs that are often considered in a PSHAcomputation. This paper presents the mathematics underlying these more precise CScalculations. Despite requiring more effort to compute than approximate calculationsusing a single causal earthquake and GMPM, the proposed approach produces an exactoutput that has a theoretical basis. To demonstrate the results of this approach andcompare the exact and approximate calculations, several example calculations are per-formed for real sites in the western United States. The results also provide some in-sights regarding the circumstances under which approximate results are likely toclosely match more exact results. To facilitate these more precise calculations for realapplications, the exact CS calculations can now be performed for real sites in theUnited States using new deaggregation features in the U.S. Geological Survey hazardmapping tools. Details regarding this implementation are discussed in this paper.

Introduction

Ground-motion selection for structural and geotechnicalsystem analysis is often associated with a target responsespectrum that is derived from probabilistic seismic hazardanalysis (PSHA) results. The conditional spectrum (CS) isone such target spectrum that estimates the distribution (withmean and standard deviation) of the response spectrum, con-ditioned on the occurrence of a target spectral accelerationvalue at the period of interest. As this CS concept is consideredfor practical use, several common approximations need to befurther explored. Typical CS calculations to date are producedfor a single earthquake ground-motion scenario (i.e., magni-tude, distance, and ground-motion intensity of interest) andcomputed using a single ground-motion prediction model(GMPM, previously known as an attenuation relation, aground-motion prediction equation, a ground-motion model,or a ground-motion relation). The scenario is generally deter-mined from PSHA deaggregation, but PSHA deaggregationcalculations for real sites often show that multiple earthquakescenarios contribute to the occurrence of a given ground-motion intensity. Additionally, modern PSHA calculationsare performed with multiple GMPMs using a logic tree thatalso includes seismic source models. Incorporating those fea-

tures is thus necessary to compute a CS that is fully consistentwith the PSHA calculations upon which it is based.

This paper presents the methodology for performing re-fined CS computations that precisely incorporate the aleatoryuncertainties (which are inherently random) in earthquakeevents with all possible magnitudes and distances, as wellas the epistemic uncertainties (which are due to limitedknowledge) from multiple GMPMs and seismic sourcemodels. Three approximate calculation approaches and theexact calculation approach are presented, with increasinglevels of complexity and accuracy. To demonstrate, severalexample calculations are performed for representative siteswith different surrounding seismic sources: Stanford innorthernCalifornia, Bissell in southern California, and Seattlein the Pacific Northwest. The results evaluate the exact andapproximate calculations, and analyze factors that contributeto the differences in accuracy. Note that while the exact ap-proach is more cumbersome, it does not need to be computedby the user, because these exact CS calculations have beenimplemented in the U.S. Geological Survey (USGS) seismichazard mapping tools, and could be incorporated into otherPSHA software as well. Details regarding this new tool

1103

Bulletin of the Seismological Society of America, Vol. 103, No. 2A, pp. 11031116, April 2013, doi: 10.1785/0120110293

and issues related to implementation of these concepts areprovided.

Basic Conditional Spectrum Computation

A wide variety of techniques has been developed in thepast to select ground-motion inputs for nonlinear dynamicanalysis (Haselton et al. 2009; Katsanos et al. 2010). Onecommon approach involves selecting ground motions withresponse spectra that match the target spectrum (Watson-Lamprey and Abrahamson, 2006; Beyer and Bommer, 2007;American Society of Civil Engineers, 2010; Applied Technol-ogy Council, 2011). The conditional mean spectrum (CMS) isone such spectrum that incorporates correlation across periodsto estimate the expected Sa values at all periods Ti [SaTi]given the target Sa value at the period of interest T [SaT](Baker and Cornell, 2006; Somerville and Hamburger, 2009;Abrahamson and Al Atik, 2010; Baker, 2011; Gulerce andAbrahamson, 2011; Somerville and Thio, 2011).

The basic CMS computation procedure is as follows.First, we obtain the target spectral acceleration at periodT, SaT from PSHA and its associated mean causal earth-quake magnitude (M), distance (R), and other parameters (),from deaggregation. Next, we use a GMPM to obtain the log-arithmic mean and standard deviation of Sa at all periods Ti,denoted as ln SaM;R; ; Ti and ln SaM; ; Ti. For anySaTi value, we compute the Ti, the number of standarddeviations by which ln SaTi differs from the mean spectralordinate predicted by a given GMPM, ln SaM;R; ; Ti, at Ti

Ti ln SaTi ln SaM;R; ; Ti

ln SaM; ; Ti: (1)

The target T, for the target SaT value, can also becomputed using equation (1).We can then compute the condi-tional mean spectral acceleration at other periods Ti,ln SaTij ln SaT, using the correlation coefficient betweenpairs of values at two periods, Ti; T, hereafterreferred to as Ti; T (Baker and Jayaram, 2008):

ln SaTij ln SaT ln SaM;R; ; Ti Ti; TlnSaM; ; TiT: (2)

The spectrum defined by ln SaTij ln SaT in equation (2)has been termed the CMS, because it specifies the meanvalues of ln SaTi, the exponentials of which are equivalentto the median values of SaTi if it is lognormally distrib-uted, conditional on the value of ln SaT (Baker, 2011).

Similarly, we can compute ln SaTij ln SaT, the condi-tional standard deviation of spectral acceleration at periodTi, conditioned on the value of Sa at T:

lnSaTij ln SaT ln SaM; ; Ti1 2Ti; T

q: (3)

The conditional standard deviation ln SaTij ln SaTfrom equation (3), when combined with the conditional mean

value ln SaTij ln SaT from equation (2), specifies a com-plete distribution of logarithmic spectral acceleration valuesat all periods (where the distribution at a given period isGaussian, as justified by Jayaram and Baker, 2008). We termthe resulting spectrum distribution as a CS, to be distin-guished from the CMS that does not consider the variabilityspecified by equation (3). It is noteworthy that as an exten-sion of the CMS approach, Bradley (2010) proposed a gen-eralized conditional intensity measure (GCIM) approach thatconsiders the complete distribution of conditional intensitymeasures other than Sa.

The input earthquake parameters required for the pre-vious CS calculations are those required by the GMPM andcan include magnitude (M), distance (R), and other param-eters such as rupture mechanism and site conditions (). Inthis paper, we will sometimes use M=R in the text as anabbreviation for M=R= (although will be included inequations where appropriate). To implement the CS in prac-tice, we need to determine which M=R and GMPM to use.Common approximations to compute the CS include usingthe mean magnitude and distance from deaggregation, alongwith a single GMPM (Baker, 2011). As explained more in thenext section, these approximations need to be evaluated forthe practical implementation of CS as a target spectrum forselecting ground motions.

Deaggregation

Computing CS at real sites requires us to consider thefollowing two factors: First, deaggregation will producemultiple causal earthquake magnitude and distance valuesfor a given SaT amplitude, as illustrated in the USGSdeaggregation plots in Figure 1, where the height of eachcolumn represents the percentage contribution from eachM=R combination. Second, PSHA calculations use multipleGMPMs and seismic source models to compute seismic haz-ard at a site through a logic tree (Kramer, 1996; McGuire,2004; Scherbaum et al., 2005; Bommer and Scherbaum,2008; Petersen et al., 2008). A refined CS computation,therefore, needs to consider multiple causal earthquakes andGMPMs, if not multiple seismic source models.

Probability seismic hazard analysis accounts for thealeatory uncertainties in earthquake events by combining thefrequencies of occurrence of all earthquake scenarios withdifferent magnitudes and distances with predictions of result-ing ground-motion intensity, in order to compute seismichazard at a site (Kramer, 1996; McGuire, 2004). Probabilityseismic hazard analysis also incorporates the epistemic un-certainties in the seismic source models and ground-motionpredictions by considering multiple models in a logic tree.For instance, the USGS utilizes three GMPMs (Boore andAtkinson, 2008; Campbell and Bozorgnia, 2008; Chiouand Youngs, 2008) for crustal seismic sources in the westernUnited States (Petersen et al., 2008). Traditional PSHAdeaggregation (McGuire, 1995; Bazzurro and Cornell, 1999;Harmsen, 2001), however, only reports distributions of

1104 T. Lin, S. C. Harmsen, J. W. Baker, and N. Luco

causal M=R= values given an Sa amplitude. The McGuire(1995) deaggregation is conditional on Sa that equals a targetvalue, termed Sa occurrence, while the Bazzurro and Cornell(1999) deaggregation is conditional on Sa that exceeds a tar-get value, termed Sa exceedance. Depending on the Sa val-ues of interest, either deaggregation approach can be used.Such deaggregation can be extended to include distributionsof the logic-tree branches, such as GMPMs, that contribute topredictions of Sa occurrence (or exceedance).

Just as the deaggregation of magnitude and distanceidentifies the relative contribution of each earthquake sce-nario to Sa occurrence (or exceedance), the deaggregationof GMPMs tells us the probability that the occurrence (orexceedance) of that Sa level is predicted by a specificGMPM. Note that the GMPM deaggregation weights differfrom the logic-tree weights; in decision analysis (Benjaminand Cornell, 1970), the logic-tree weights are equivalent toprior weights, whereas the deaggregation weights may be

interpreted as posterior weights given the occurrence (orexceedance) of the ground-motion amplitude of interest.Additional details on GMPM deaggregation are provided inLin and Baker (2011). For the purpose of response spectrumpredictions, the key elements are the GMPMs and their inputearthquake parameters (e.g., M, R, ). Hence, the focus ofdeaggregation here will be on these two. The other portionsof the logic tree (e.g., recurrence type, rates, maximum mag-nitude) do not influence spectrum predictions and so can begrouped for the purpose of these calculations. The USGS hasrecently begun providing GMPM deaggregation outputs (seeData and Resources). These outputs facilitate the exactcalculations of CS described next.

Conditional Spectrum Calculation Approaches

We now consider several options for computing CS thatincorporate consideration of multiple causal earthquake

010

2030

4050

60

Closest Distance, Rcd

(km)

0

10

20

30

40

50

60

Closest D

istance,

Rcd

(km)

5.0

5.5

6.0

6.5

7.0

7.5

8.0Magnitude (Mw)

5.05.5

6.06.5

7.07.5

8.0

Magnitud

e (Mw)

510

1520

25%

Con

tribu

tion

to H

azar

d

-1 < 0

scenarios and multiple GMPMs, as well as multiple seismicsource models and their logic-tree branches. We introduceseveral approximate calculation approaches with increasingcomplexity but also with increasing accuracy, followed bythe exact calculation. Differences between the approachesare highlighted, and these approaches are evaluated laterto determine the accuracy of the approximate approaches.

Method 1: Approximate CS Using Mean M=R and aSingle GMPM

The most basic method for computing an approximateCS is to utilize a single earthquake scenario and single

GMPM, so that equations (2) and (3) can be used directly.In current practice, this is done by taking the mean valueof the causal magnitudes and distances from deaggregation,denoted here as M and R (Baker, 2011). Similarly, the meanvalue of other causal parameters () can be obtained or in-ferred from deaggregation. These mean values can then beused with a single GMPM (even though the underlying haz-ard analysis utilized several GMPMs). The resulting CS cal-culations are given in the following equations, utilizing thesubscript k to denote that the calculations are based on a sin-gle GMPM indexed by k. The equations are also denoted asbeing approximately equal to the true CS values, given thesimplifications made here.

ln SakTij ln SaT lnSak M; R; ; Ti Ti; Tln Sak M; ; TiT; (4)

and

ln SakTij ln SaT ln Sak M; ; Ti1 2Ti; T

q; (5)

where the mean and standard deviation predicted by GMPM kare denoted ln Sak and lnSak , and the CS computed forGMPM k is denoted ln SakTij ln SaT and lnSakTij ln SaT.Note that the correlation coefficient, , is assumed to beconstant for each GMPM; this could be revised if desired.

Method 2: Approximate CS Using Mean M=R andGMPMs with Logic-Tree Weights

We can refine method 1 by considering all GMPMs usedin the PSHA computation. The PSHA logic tree weights eachmodel (these weights can be equal or unequal), and herewe denote the weight for model k as plk, where the super-script l refers to logic tree. To obtain an approximate CSusing all of these GMPMs, we repeat the single-GMPM cal-culation (equations 4 and 5) to obtain ln SakTij lnSaT andln SakTij ln SaT for each GMPM. We then sum up the result-

ing mean spectra [ln SaTij ln SaT], weighted by the logic-tree weights, to get a mean spectrum

ln SaTij ln SaT Xk

plkln SakTij ln SaT: (6)

The computation of conditional standard deviations isslightly more complicated, because it not only accountsfor the mean of the standard deviations from the GMPMs,but it also includes the additional uncertainty introducedby the variation in mean predictions among the GMPMs.Formal probabilistic modeling (Ditlevsen, 1981) can be usedto show that the resulting conditional standard deviation is

ln SaTij ln SaT Xk

plk2lnSakTij ln SaT lnSakTij ln SaT ln SaTij ln SaT2r

: (7)

In equations (6) and (7) we no longer have a subscript k onthe resulting ln SaTij lnSaT and ln SaTij ln SaT, becausethe results are no longer specific to a single GMPM butincorporate multiple GMPMs.

Although the use of logic-tree weights is not rigorouslycorrect, we introduce it here as a convenient approximationbecause these equations do not require GMPM deaggregationoutputs, which are not currently available from many PSHAsoftware tools.

Method 3: Approximate CS Using GMPM-SpecificMean M=R and GMPMs with Deaggregation Weights

In this section, we further refine the CS calculations bytaking advantage of GMPM deaggregation if it is available(e.g., as it is from the new USGS tools). First, GMPM deag-gregation will provide separate M=R deaggregation for eachprediction model. Here we will use the meanM and R valuesfor each model, denoted as Mk and Rk for GMPM k. Usingthese values, means and standard deviations of the CS can becomputed for GMPM k as follows:

lnSakTij ln SaT ln Sak Mk; Rk; k; Ti Ti; Tln Sak Mk; k; TikT; (8)

and

ln SakTij lnSaT ln Sak Mk; k; Ti1 2Ti; T

q:

(9)

Note that the GMPM-specific Mk and Rk (in equations 8 and9) are different from the M and R with respect to all GMPMs(in equations 4 and 5). As an intermediate step (e.g., betweenR and Rk), the concept of conditional deaggregation givenGMPM can also be extended to compute Mk, Rkj Mk, andkj Mk, Rk in a cascading or Rosenblatt-distribution manner.


The second GMPM deaggregation output used in thismethod is the probability that GMPM k predicted occurrence(or exceedance) of the Sa. These deaggregation probabilitiesare denoted as pdk , where the superscript d refers to deaggre-gation, and they are generally not equal to the PSHA logic-tree weights that have been denoted as plk. Utilizing theseweights, along with the GMPM-specific ln SakTij ln SaTand ln SakTij ln SaT from equations 8 and 9, a compositeCS can be computed:

lnSaTij ln SaT Xk

pdkln SakTij ln SaT; (10)

and

ln SaTij ln SaT Xk

pdk 2ln SakTij lnSaT ln SakTij lnSaT ln SaTij ln SaT2r

: (11)

As with method 2, the mean spectrum is the mean (overGMPMs) of the GMPM-specific means, except that herewe have utilized the more appropriate deaggregation weightspdk . The standard deviation of the spectrum again containscontributions from the individual GMPM conditional stan-dard deviations, plus the uncertainty from the variations inmean spectra across GMPMs. While this method incorporates

more exact information than methods 1 or 2, it is stillapproximate because it utilizes only the mean earthquakescenario for a given GMPM. Method 4 will resolve that finalapproximation.

Method 4: Exact CS Using Multiple CausalEarthquake M=R and GMPMs with DeaggregationWeights

With this final method we now account exactly for,when we compute the CS, the contribution that each earth-quake magnitude/distance and each GMPMmakes to the seis-mic hazard. For each causal earthquake combination Mj=Rjand GMPM k, we can obtain a corresponding mean and stan-dard deviation of the CS, denoted ln Saj;kTij ln SaT andln Saj;kTij ln SaT, as

lnSaj;kTij lnSaT lnSakMj;Rj; j; Ti T; TijT lnSakMj; j; Ti; (12)

and

ln Saj;kTij ln SaT ln SakMj; j; Ti1 2Ti; T

q:

(13)

A PSHA deaggregation that includes GMPM deaggrega-tion will provide the weights, pdj;k, that indicate the contri-bution of each Mj=Rj and GMPMk to occurrence (orexceedance) of the Sa of interest. Note that here we areconsidering the contributions of individual Mj=Rj ratherthan approximating all contributing earthquake scenarios bysimply a mean M and R, as we did in equations (8) and (9).The exact CS incorporating multiple M=R and GMPMscan then be evaluated by combining these individual

lnSaj;kTij ln SaT and ln Saj;kTij lnSaT with their corre-sponding weights, pdj;k

ln SaTij ln SaT Xk

Xj

pdj;kln Saj;kTij ln SaT; (14)

and

ln SaTij ln SaT Xk

Xj

pdj;k2ln Saj;kTij ln SaT ln Saj;kTij ln SaT ln SaTij ln SaT2s

: (15)

Equations (14) and (15) are similar to those in method 3,except that there is now a second set of summations in theequations to account for the effect of multiple Mj=Rj valuesinstead of the single-mean values Mk and Rk from method 3.These equations provide an exact answer to the question, whatis the mean and standard deviation of the response spectraassociated with ground motions having a target SaT,when that SaT could potentially result from multipleearthquake scenarios, and the Sa predictions come from alogic tree with multiple GMPMs? It requires more effort tocompute than the approximate approaches commonly usedtoday, and it requires detailed deaggregation information in-cluding hazard contributions of GMPMs and other parameters,, that may not be available.

Alternatively, an exact CS can be computed directlyfrom the earthquake parameters and GMPMs that were usedin PSHA computation to aggregate the hazard, as described inthe next section. Although treatments of aleatory and episte-mic uncertainties are often separated in PSHA, a single seis-mic hazard curve is typically derived with the considerationof aleatory uncertainties from multiple causal earthquakesand epistemic uncertainties from multiple GMPMs and

Spectrum Computation Incorporating Multiple Causal Earthquakes and Ground-Motion Prediction Models 1107

seismic source models (Petersen et al., 2008). Here we take asimilar approach to compute a single CS that combines thesealeatory and epistemic uncertainties.

Aggregation Approach to Method 4 (Exact CS)

Although the GMPM deaggregations used in methods 3and 4 are now available from the USGS, it would be imprac-tical to provide deaggregations with respect to all of the otherbranches of the PSHA logic tree (e.g., for alternative moment-area equations in California and for body wave to momentmagnitude equations in the central and eastern UnitedStates). Likewise, it would be cumbersome to provide deag-gregations with respect to all of the other GMPM inputparameters besides M and R (e.g., rupture mechanism andhanging/footwall indicators), which were previously denotedas . To account for these other branches and parameters inmethod 4 without additional deaggregation results, a CS canbe calculated during the PSHA computation (using equa-tions 12 and 13) for each and every earthquake sourceand logic-tree branch. These numerous CS results can thenbe combined using equations (14) and (15), but now with jrepresenting all the earthquake sources and k representing allthe logic-tree branches. In this case, the deaggregationweights in the equations are simply taken from the PSHAcontribution (mean annual exceedance frequency) for eachearthquake source and logic-tree branch, normalized bythe total aggregated hazard.

An advantage of this aggregation approach to method 4is that the other GMPM input parameters no longer need tobe inferred in calculating a CS. This is because we can firstcalculate a CS for each and every earthquake source (not tomention logic-tree branch) using the same correspondingGMPM input parameters used in the PSHA computation.Based on the examples for three sites presented next, anynumerical differences between the aggregation approach andmethod 4 are not expected to be significant for the CMS (ormean CS) at a site. Also based on the examples, however, thestandard deviation of a CS (via equation 15) using the resultsfor every earthquake source and logic-tree branch would beexpected to yield differences, although the practical signifi-cance of the differences is not known. The standard deviationfrom the aggregation approach would capture all of the un-certainties considered in the PSHA computation.

An implementation of this aggregation approach to cal-culating an exact CS is now provided as part of USGS onlinehazard tools. Details and limitations of this implementationare discussed in the final section, Conditional SpectrumCalculation Tools from USGS.

Example Calculations for Three Sites

To demonstrate the numerical results that are nowavailable using the previous equations and the USGS onlinehazard tools, this section provides a set of example calcula-tions to determine whether using the exact method 4 pro-

vides results that have practical differences from thoseobtained using the simpler approximate methods 1 to 3.

To evaluate the accuracy of the approximate methods,CS for three locations are computed using the methods de-scribed previously. Target spectral accelerations are obtainedfor Sa0:2s with 10% probability of exceedance in 50 years(i.e., a return period of 475 years), and for Sa1s with 2%probability of exceedance in 50 years (i.e., a return period of2475 years).

Description of Example Sites and GMPMs

We consider three locations in the western United Stateswith relatively high hazard but differing surrounding seismicsources: Stanford, Bissell, and Seattle. Deaggregation resultsfor these three sites are shown in Figure 1. Ground-motionhazard at Stanford, located in northern California, is domi-nated by a single, shallow crustal earthquake source, theSan Andreas fault zone. Ground-motion hazard at Bissell,located in southern California, has contributions from multi-ple earthquake sources, but they are all shallow crustalsources. Ground-motion hazard at Seattle has contributionsfrom multiple earthquake sources of different types: shallowcrustal, subduction zone interface, and intraplate (each ofwhich has its own set of GMPMs). All three sites are assumedto have a time-averaged shear-wave velocity in the top30 meters of the soil (VS30) of 760 m=s.

The hazard calculations and deaggregation results allcome from the USGS models and hazard mapping tools. TheUSGS model (Petersen et al., 2008) assigns equal logic-treeweights to GMPMs near Stanford and Bissell, but unequallogic-tree weights to GMPMs near Seattle. The USGS modeluses three GMPMs for the crustal sources near Stanford,Bissell, and Seattle (Boore and Atkinson, 2008; Campbelland Bozorgnia, 2008; Chiou and Youngs, 2008), threeGMPMs for the subduction zone interface sources (Youngset al., 1997;Atkinson andBoore, 2003; Zhao et al., 2006), andthree GMPMs for the intraplate sources near Seattle (Youngset al., 1997; Atkinson and Boore, 2003, which provides mod-els that can be used in both the Global and Cascadia regions).The Youngs et al. (1997), Atkinson and Boore (2003), andZhao et al. (2006) models are developed for both subductionzone interface and intraplate sources. The GMPMs vary intheir required input variables. The mean ln Sa predictionsdepend onM, R, and other source and site characteristics ()such as depth to top of rupture, faulting mechanism, andhanging-wall effect. Some of the GMPMs predict standarddeviations of ln Sa that depend only on the period of interest(Boore and Atkinson, 2008), while others are magnitudedependent (Chiou and Youngs, 2008).

Deaggregation Information

The targets Sa0:2s with 10% probability of exceed-ance in 50 years and Sa1s with 2% probability of exceed-ance in 50 years are obtained from PSHA for the threeexample sites considered. The associated causal earthquake


magnitudes and distances are obtained from USGS deaggre-gation (see Data and Resources), and their deaggregationplots are shown in Figure 1. Other parameters, , that areassociated with each causal earthquake M=R combinationcan be obtained directly from the parameters that were usedfor PSHA computation (i.e., via the aggregation approach tomethod 4), from deaggregation outputs if available, orinferred from the characteristics of contributing earthquakesources. With the target SaT and ln SaM;R; ; T andln SaM; ; T predictions for each causal M=R=, we canthen back calculate T using equation (1). The logic-treeweights for each GMPM (plk) are obtained from the USGSdocumentation (Petersen et al., 2008), and the deaggregationweights (pdk and p

dj;k) are obtained from the USGS deaggre-

gation tools (see Data and Resources).

Conditional Spectra Results

Conditional spectra results can be computed for eachexample case using the approximate or exact methods de-scribed previously. Depending on the level of approximation,the CS for a GMPM k can be computed using method 1 (equa-tions 4 and 5) for approximate CS with mean M=R ( M andR), method 3 (equations 8 and 9) for approximate CS withGMPM-specific meanM=R ( Mk and Rk), or method 4 (equa-tions 12 and 13) for all contributing CS with GMPM-specificcausal earthquake M=R (Mj and Rj). The composite CS canthen be computed with the corresponding weights usingmethod 2 (equations 6 and 7) for approximate CS withGMPM logic-tree weights (plk), method 3 (equations 10 and11) for approximate CS with GMPM deaggregation weights(pdk), or method 4 (equations 14 and 15) for exact CS withdeaggregation weights associated with each causal earth-quake and GMPM (pdj;k). The resulting CS obtained fromthese four methods are plotted in Figure 2. Also plotted inthe figure are the CMS (but not conditional standard devia-tions) resulting from the aggregation approach to method 4,which are nearly identical to the method 4 results despite theinclusion of additional logic-tree branches and earthquakesources. These CMS are obtained from the USGS calculationtools described in the final section.

We can make several observations from the results ofFigure 2. For the Stanford and Bissell sites, CS computedusing method 1 are very similar to results from methods 2and 3, but they differ more for Seattle. This is because theGMPMs used in Seattle, some of which are for subductionzone events, result in more varied predictions; using a meanM= R that represents a variety of earthquake sources with aGMPM appropriate for a single-source type could be antici-pated to produce these varied predictions. For Seattle, meth-ods 2 and 3, which more carefully address the contributionsof multiple GMPMs, do a better job of approximating theexact results from method 4.

For all three sites, the approximate methods 2 and 3work better for conditional mean estimation than for condi-tional standard deviation estimation compared with the exact

method 4, because method 4 produces higher conditionalstandard deviations in every case. A closer examination ofequation (15) reveals two components of contribution tothe exact conditional standard deviations: first, a contributionfrom ln Sa, that is, variance in ln Sa for a given Mj=Rj andGMPMk; second, a contribution from ln Sa, due to variationin Mj=Rj and GMPMk. In other words, the total variancecomes from the expectation of the variance, which is the firstterm, and the variance of the expectation, which is the secondterm (Ditlevsen, 1981). The individual contributions fromthese two terms are plotted in Figure 3ac. Methods 2and 3 very closely approximate the contribution from ln Safor the Stanford and Bissell sites, although the match is not asgood for Seattle. Method 4 also includes the contributionfrom ln Sa, which is, however, not well captured by meth-ods 2 and 3. Figure 3df shows the ln Sa predicted for eachMj=Rj and GMPMk, and it is the weighted variance of thesespectra that creates the contribution to the overall standarddeviation from lnSa. Thus, we see that this variance ofexpectations contribution may not be negligible; the approxi-mate methods do not capture this well, because they consideronly mean M=R values and thus cannot identify the contri-bution to uncertainty from multiple M=R contributions toground-motion hazard. Similarly, we anticipate that the deag-gregation approach to the exact method 4 underestimates theconditional standard deviation, relative to the aggregationapproach that accounts for every logic-tree branch and earth-quake source used in the PSHA computation. We do not yetknow whether this underestimation is practically significant.

Figure 4 shows the target CS computed using methods 2and 4 for all three sites and both target Sa values: Sa0:2swith 10% probability of exceedance in 50 years and Sa1swith 2% probability of exceedance in 50 years. Method 2 ischosen as the reference approximate method here because ithas an appealing combination of incorporating multipleGMPMs but not requiring any GMPM deaggregation informa-tion. For all three sites, the mean estimates of the approxi-mate and exact methods are in close agreement, while theapproximate standard deviations underestimate the exact re-sult; this effect is most pronounced at periods far from theconditioning period (i.e., 0.2 s or 1 s).

The method 2 approximation appears to work best forsites with a single earthquake source (e.g., Stanford),followed by sites with multiple earthquakes sources of thesame type (e.g., Bissell) and sites with multiple differingearthquake source types (e.g., Seattle). This is because thereare several contributing factors to the accuracy of theapproximation: (1) the input causal earthquake parameters,(2) the GMPMs used, (3) the GMPM deaggregation weights.

First, the importance of considering multiple causalM=R values depends upon howmanyM=R values contributesignificantly to the hazard. In cases where all contributions tohazard come from a narrow range of magnitudes and distan-ces (e.g., Stanford), the mean M=R is representative of themost important individual contributing M=R, so computa-tions based only on the mean M=R are very precise.


However, in cases where hazard contributions come from abroader range of magnitudes and distances (especially in thecase of Seattle), the meanM=R deviates from any individualcontributingM=R, so computations based on the meanM=Ronly may result in a slight shift in conditional mean estimatesin addition to reduced conditional standard deviations.Similarly, for the common situation of low- or moderate-seismicity sites in the central and eastern United States, forexample, those that are located several hundred kilometersfrom the New Madrid seismic region where the low-frequency deaggregation can be strongly bimodal, computa-tions based on the mean M=R only may not be precise.Furthermore, variations in other parameters besides M andR, such as depth to the top of rupture for different source

types, can also affect the accuracy of the approximation be-cause they contribute to the Sa prediction.

Second, the similarity of the GMPMs affects approxima-tions in their treatment. For cases where the ground-motionpredictions for a given M=R vary significantly betweenmodels (e.g., Seattle, where the subduction zone and crustalprediction models vary significantly), approximate treatmentof GMPMs is less effective, and so a CS using a single GMPM(or using approximate weights on multiple models) may pro-duce inaccurate results. Similar inaccuracies can be expectedin the central and eastern United States. On the other hand,the three GMPMs used at Stanford and Bissell all tend toproduce similar predictions, so for those cases the choiceof the GMPM used to compute CS may not be as critical.

0.05 0.2 1 3103

102

101

100

101

Sa(T

) [g]

T [s]

1: Mean M/R, single GMPM2: Mean M/R, logictree weights3: Mean Mk/Rk, deagg weights4: Exact (USGS)

0.05 0.2 1 3103

102

101

100

101

Sa(T

) [g]

T [s]


0.05 0.2 1 30

0.2

0.4

0.6

0.8

1

1.2

lnSa

(T)

T [s]

1: Mean M/R, single GMPM2: Mean M/R, logictree weights3: Mean Mk/Rk, deagg weights4: Exact

0.05 0.2 1 30

0.2

0.4

0.6

0.8

1

1.2

lnSa

(T)T [s]


0.05 0.2 1 30

0.2

0.4

0.6

0.8

1

1.2

lnSa

(T)

T [s]


0.05 0.2 1 3103

102

101

100

101

Sa(T

) [g]

T [s]


(a)

(b)

(c)

(d)

(e)

(f)

Figure 2. CMS at Stanford, Bissell, and Seattle (a, b, and c, respectively) and conditional standard deviation spectra at Stanford, Bissell,and Seattle (d, e, and f, respectively) using methods 1 to 4 for Sa0:2s with 10% probability of exceedance in 50 years. The color version ofthis figure is available only in the electronic edition.


Third, as the GMPM deaggregation weights differ morefrom the GMPM logic-tree weights, approximate treatment ofthose weights works less effectively. For instance, in the (nottoo uncommon) case where one GMPM strongly dominatesthe deaggregation, the approximation that assumes GMPMlogic-tree weights is expected to deviate more substantiallyfrom the computation that utilizes GMPM deaggregationweights. The GMPM deaggregation weights vary with theperiod of interest (T), the target SaT amplitude of inter-est, and the location, so it is difficult to develop simple rulesfor when the approximations work well. But if the GMPMpredictions are similar to each other for theM=R values con-tributing significantly to hazard, then the deaggregation

weights are often similar to the logic-tree weights, and thepredictions are also in good agreement (as noted in the pre-vious paragraph), so the approximations are generally goodin those cases.

Impact of Approximations on Ground-MotionSelection

The importance of approximations in the CS computa-tions will depend upon how the results affect any engineeringdecisions that may be made. The most common use fora CS is as a target response spectrum for ground-motion se-lection and scaling (Baker and Cornell, 2006; Baker, 2011).

(a)

(b)

(c)

(d)

(e)

(f)

0.05 0.2 1 30

0.2

0.4

0.6

0.8

1

1.2

lnSa

(T)

T [s]

2: Mean M/R, logictree weights3: Mean Mk/Rk, deagg weights4: ExactContribution from Contribution from

0.05 0.2 1 30

0.2

0.4

0.6

0.8

1

1.2

lnSa

(T)

T [s]


0.05 0.2 1 30

0.2

0.4

0.6

0.8

1

1.2

lnSa

(T)

T [s]


0.05 0.2 1 3103

102

101

100

101

Sa(T

) [g]

T [s]

Mj/Rj, GMPMk2: Mean M/R, logictree weights3: Mean Mk/Rk, deagg weights4: Exact

0.05 0.2 1 3103

102

101

100

101

Sa(T

) [g]

T [s]


0.05 0.2 1 3103

102

101

100

101

Sa(T

) [g]

T [s]


Figure 3. Conditional standard deviation spectra with contribution from ln Sa (variance in ln Sa for a given Mj=Rj and GMPMk) andln Sa (due to variation inMj=Rj and GMPMk) at Stanford, Bissell, and Seattle (a, b, and c, respectively) and CMS for each consideredMj=Rjand GMPMk at Stanford, Bissell, and Seattle (d, e, and f, respectively) using methods 2 to 4 for Sa0:2s with 10% probability of exceedancein 50 years. The color version of this figure is available only in the electronic edition.


Approximations to the CS might have an influence on groundmotions selected from a database (because the selectedground motions match the target response spectrum), andthat could affect nonlinear dynamic analysis results. On theother hand, the finite size of recorded ground-motion data-bases means that minor changes in the spectrum target maynot result in substantially different ground motions beingselected; in that case, the approximations would not havean appreciable impact on structural analysis results.

To illustrate, the target CS mean and variance are com-puted using both the exact (method 4) and approximate(method 2) approaches at the Bissell site for the Sa0:2samplitude with 10% probability of exceedance in 50 years.Ground motions can be selected to match these targetspectrum mean and variance using the procedure of Jayaram

et al. (2011), which assumes a Gaussian distribution. Ingeneral, the conditional logarithmic Sa distribution is notGaussian when multiple causal earthquakes and/or multipleGMPMs are considered, and hence the mean and variancealone may not describe the entire distribution; they still pro-vide useful insights, however. Alternatively, Bradley (2010)considers multiple causal earthquake sources and the com-plete distribution of the conditional ground-motion intensitymeasure. The target response spectra and correspondingground motions selected via Jayaram et al. (2011) are shownin Figures 4b and 5. The means of the CS using both methodsare in close agreement, but the standard deviation of the CSusing method 4 is higher than that using method 2, especiallyat periods farther away from the conditioning period of 0.2 s.Consequently, the spectra of the ground motions selected

(a)

(b)

(c)

(d)

(e)

(f)

0.05 0.2 1 3103

102

101

100

101

Sa(T

) [g]

T [s]

4: Exact 4: Exact +/ 22: Mean M/R, logictree weights 2: Mean M/R, logictree weights +/ 2

0.05 0.2 1 3103

102

101

100

101

Sa(T

) [g]

T [s]


0.05 0.2 1 3103

102

101

100

101

Sa(T

) [g]

T [s]



0.05 0.2T [s]

103

102

101

100

101

Sa(T

) [g]


103

102

101

100

101

Sa(T

) [g]

0.05 0.2T [s]

103

102

101

100

101

Sa(T

) [g]

0.05 0.2

1 3

1 3

1 3T [s]


Figure 4. CS computed using methods 2 and 4 for Sa0:2s with 10% probability of exceedance in 50 years at Stanford, Bissell, andSeattle (a, b, and c, respectively) and Sa1s with 2% probability of exceedance in 50 years at Stanford, Bissell, and Seattle (d, e, and f,respectively). The color version of this figure is available only in the electronic edition.


using method 4 are expected to show a similar mean but ahigher standard deviation than those using method 2. Thelarger exact CS standard deviation results in selecting a fewmore ground motions with high spectra at periods other than0.2 s, and this can result in a slightly increased probability ofobserving large structural responses or collapses (as seen inJayaram et al., 2011). Note that the spectra of the selectedground-motion sets, however, do not differ substantially inFigure 5, relative to variations between ground-motion spec-tra within a set. Structural analyses to date suggest that incases such as this, the exact method results in similar medianbut increased dispersion in structural response estimates.

The difference in structural response can be larger,however, if methods 2 and 4 result in substantial differencesin both the mean and the variance of the CS, such as thoseillustrated in Figure 4c,f for the Seattle site. In such cases, inaddition to a slight increase in structural response dispersion,the median of the structural response may shift as well. Toreflect contributions from different earthquake sources, indi-vidual CS can be constructed for each earthquake source, andseparate sets of ground motions selected to match both thetarget spectra as well as other characteristics of each source(Goda and Atkinson, 2011). These CS for different earth-quake sources are available as intermediate steps to computethe exact CS described previously and are provided by theUSGS as part of the CMS feature that is described in the nextsection. As the difference between approximate and exactspectra increases, more refined target spectra will have in-creasing benefits for ground-motion selection and structuralresponse assessment.

Conditional Spectrum Calculation Tools from USGS

The USGS National Seismic Hazard Mapping Projectwebsite now provides an option for CMS, as part of the2008 Interactive Deaggregations web tool (see Data andResources). The spectra are consistent with the 2008 nationalseismic hazard maps for the continental United States and arecomputed via the Aggregation Approach to Method 4 (ExactCS) section. Described next are the inputs to and outputs

from the tool, issues with the current implementation, andfuture features. One such future feature is providing thestandard deviations of CS in addition to the means.

Description of Calculation Tool

As shown in Figure 6, the USGS 2008 InteractiveDeaggregation web tool provides CMS for a user-specified:

location (address or latitude and longitude) anywhere inthe continental United States;

mean exceedance probabilities of 1%, 2%, 5%, 10%, 20%,or 50% in 30, 50, 75, 100, or 200 years;

spectral acceleration period of 0 s (corresponding to peakground acceleration), 0.1 s, 0.2 s, 0.3 s, 0.5 s, 1 s, or 2 s, oradditionally 3 s, 4 s, or 5 s for locations in the westernUnited States (west of 115 longitude);

VS30 of soil between 180 and 1300 m=s for locations in thewestern United States, of 760 or 2000 m=s in the centraland eastern United States (east of 100 longitude), or of760 m=s for locations in between.As output, the tool currently provides graphs, tables, and

text files for four different types of CMS, all calculatedaccording to the Aggregation Approach to Method 4 (ExactCS) section. The four different types are:

an overall CMS that accounts for all of the earthquakesources and logic-tree branches considered in the USGSPSHA computation (with a few exceptions described in thenext section);

a CMS for each GMPM that accounts for all of the earth-quake sources and logic-tree branches related to the par-ticular GMPM;

a CMS for each of several (currently seven) M=R= binsthat contribute most to the total aggregated hazard, ac-counting for all of the earthquake sources and logic-treebranches within that bin;

a CMS for each M=R= bin and GMPM (currently only intext output), accounting for all of the earthquake sourcesand logic-tree branches within the bin that are related to theparticular GMPM.

(a) (b)

4: Exact 4: Exact +/ 2Selected ground motions

103

102

101

100

101

Sa(T

) [g]

0.05 0.2 1 3T [s]

2: Mean M/R, logictree weights 2: Mean M/R, logictree weights +/ 2Selected ground motions

103

102

101

100

101

Sa(T

) [g]

0.05 0.2 1 3T [s]

Figure 5. Response spectra of ground motions selected match CS obtained using (a) method 4 and (b) method 2 at Bissell site forSa0:2s with 10% probability of exceedance in 50 years. The color version of this figure is available only in the electronic edition.


The CMS for each GMPM demonstrates the effect ofusing only a single GMPM. As mentioned previously, thiseffect can be particularly significant in cases when more thanone earthquake source type (e.g., subduction zone, shallowcrustal) contributes significantly to the total aggregatedhazard. In such cases, the CMS for two or more M=R= binsand/or GMPMs, each corresponding to a different sourcetype, may be more useful than a single overall CMS.

Current Implementation Issues

While the web tool implementation described previouslyaccounts for practically all of the earthquake sources andlogic-tree branches considered in computing the USGS 2008national seismic hazard maps, there are some exceptions.The least significant of these is that CMS calculations arenot carried out for earthquake sources that do not contributeappreciably to the total aggregated hazard (i.e., those withmean annual exceedance frequency less than 106). Also forthe sake of limiting computation time, CMS are not calcu-lated for the two USGS logic-tree branches that quantify addi-tional epistemic uncertainty among the GMPMs for shallowcrustal earthquakes in the western United States (see Petersenet al., 2008 for details). The logic-tree branches correspond-ing to the GMPMs themselves are fully accounted for,however. In the central and eastern United States, the USGSlogic-tree branch for temporal clustering of New Madridseismic zone earthquakes is not yet incorporated into the2008 Interactive Deaggregations web tool, and hence is notaccounted for in the CMS calculations. The numerical im-pacts of these exceptions on the calculated CMS have notyet been quantified, but they are anticipated to be relativelyinsignificant.

In being consistent with the 2008 national seismichazard maps, the USGS web tool utilizes several differentGMPMs for shallow crustal, subduction zone, and stablecontinental earthquake sources. However, for the correlationcoefficients needed to calculate CS, the tool currently only

uses the Baker and Jayaram (2008) model, which was devel-oped with ground-motion data exclusively from shallowcrustal earthquakes. A recent study of subduction zoneground motions from Japan suggests that this correlationmodel is also a reasonable representation for subduction zoneearthquake sources (Jayaram et al., 2011). For stablecontinental earthquake sources, little data either confirm orcontradict this model. In general, studies of correlation mod-els have shown them to be relatively insensitive to the par-ticular GMPM, earthquake magnitude, distance, and rupturemechanism (Baker, 2005; Jayaram et al., 2011). Thus, thenumerical impact of using the single correlation model isanticipated to be relatively insignificant. Additional and/orupdated correlation models corresponding to different earth-quake sources and/or GMPMs could be incorporated into theweb tool as they become available.

Future Features

As mentioned previously, in the future the USGS webtool will provide CS, not just CMS. The conditional standarddeviations will likewise be calculated according to the aggre-gation approach to computing an exact CS. Whereas thecurrent implementation for the conditional means has beenseen to match results from the deaggregation approach ofmethod 4, the standard deviations of a CS calculated bythe web tool are anticipated to be higher and more inclusiveof all the uncertainties accounted for in the USGS nationalseismic hazard maps.

Lastly, the current web tool calculates CMS with weightsthat are for exceedance of the Sa value specified by a user(via a selected mean exceedance probability). A future toolwill provide weights for occurrence of an Sa value. In orderto do so, the tool will optionally allow a user to specify an Savalue of interest, for example, a risk-targeted maximumconsidered earthquake (MCER) ground-motion value fromthe American Society of Civil Engineers (2010).

Figure 6. Interface for USGS online interactive deaggregations, including options to request GMPM deaggregation and CMS computation(reprinted with permission from USGS; see Data and Resources). The color version of this figure is available only in the electronic edition.


Conclusions

Approximate and exact computations of CS wereproposed and used for example calculations for Stanford,Bissell, and Seattle. Exact CS mean and standard deviationcalculations can incorporate multiple GMPMs and causalearthquakeM=R= combinations, as well as multiple seismicsource models and their logic-tree branches. Varying levelsof approximations were also considered, that replaced multi-ple M=R combinations with simply the mean M=R fromdeaggregation, and either considered only a single GMPMor performed an approximate weighting of several GMPMs.These approximations are potentially appealing because oftheir ease of computation and because they do not requiredeaggregation of GMPM weights, a result that is not yetwidely available in conventional PSHA software.

The approximate CS calculations appear to be more ac-curate for conditional mean estimation than for conditionalstandard deviation estimation. The exact conditional stan-dard deviation is always higher than approximate resultsbecause of the additional contribution from the variance inmean logarithmic spectral accelerations due to variation incausal earthquakes and GMPMs. The input causal earthquakeparameters and the GMPMs used along with the correspond-ing weights affect the accuracy of the approximation in CScomputation. Exact calculation methods may be needed forlocations with hazard contributions from multiple earthquakesources, where errors from approximations are higher as aresult of multiple contributing earthquake magnitudes anddistances, and variation in predictions from the inputGMPMs.

The exact CS calculations require extension of tradi-tional PSHA deaggregation, which considers only magnitude,distance, and , to deaggregation of GMPMs. This additionaldeaggregation output is now available as part of the hazardresults provided by the USGS. Further, the USGS nowprovides CMS results using the exact calculation aggregationapproach described here. These new calculation tools shouldbe useful in facilitating hazard-consistent ground-motionselection for nonlinear dynamic analysis and will allowfor exact spectra to be computed without requiring cumber-some calculations by users.

Data and Resources

Deaggregation and CMS data used in this paper canbe obtained from the USGS National Seismic HazardMapping Project website (http://earthquake.usgs.gov/hazards, last accessed June 2012), as part of the 2008 Inter-active Deaggregations web tool at http://geohazards.usgs.gov/deaggint/2008/ (last accessed June 2012).

Acknowledgments

The authors thank Brendon Bradley, Gabriel Toro, Ned Field, EduardoMiranda, and Ivan Wong for their helpful reviews of the paper. Also, thanksto Eric Martinez for his assistance on the USGS implementation. This work

was supported by the USGS external grants program, under Awards07HQAG0129 and 08HQAG0115. Any opinions, findings, conclusions,or recommendations presented in this material are those of the authors,and do not necessarily reflect those of the funding agency.

References

Abrahamson, N. A., and L. Al Atik (2010). Scenario spectra for designground motions and risk calculation, in 9th U.S. National and 10thCanadian Conference on Earthquake Engineering, Toronto, Canada,2529 July 2010.

American Society of Civil Engineers (ASCE) (2010). Minimum DesignLoads for Buildings and Other Structures, ASCE/SEI 710. AmericanSociety of Civil Engineers/Structural Engineering Institute, Reston,Virginia, 608 pp.

Applied Technology Council (ATC) (2011). Guidelines for seismic perfor-mance assessment of buildings, ATC-58 100% draft, Technical report,Applied Technology Council, Redwood City, California.

Atkinson, G. M., and D. M. Boore (2003). Empirical ground-motionrelations for subduction-zone earthquakes and their application toCascadia and other regions, Bull. Seismol. Soc. Am. 93, no. 4,17031729.

Baker, J. W. (2005). Vector-valued ground motion intensity measures forprobabilistic seismic demand analysis, Ph. D. Thesis, StanfordUniversity, Stanford, California.

Baker, J. W. (2011). Conditional mean spectrum: Tool for ground-motionselection, J. Struct. Eng. 137, no. 3, 322331.

Baker, J. W., and C. A. Cornell (2006). Spectral shape, epsilon and recordselection, Earthquake Eng. Struct. Dynam. 35, no. 9, 10771095.

Baker, J. W., and N. Jayaram (2008). Correlation of spectral accelerationvalues from NGA ground motion models, Earthquake Spectra 24,no. 1, 299317.

Bazzurro, P., and C. A. Cornell (1999). Disaggregation of seismic hazard,Bull. Seismol. Soc. Am. 89, no. 2, 501520.

Benjamin, J. R., and C. A. Cornell (1970). Probability, Statistics, andDecision for Civil Engineers, McGraw-Hill, New York.

Beyer, K., and J. J. Bommer (2007). Selection and scaling of real accelero-grams for bi-directional loading: A review of current practice and codeprovisions, J. Earthquake Eng. 11, 1345.

Bommer, J. J., and F. Scherbaum (2008). The use and misuse of logic trees inprobabilistic seismic hazard analysis, Earthquake Spectra 24, no. 4,9971009.

Boore, D. M., and G. M. Atkinson (2008). Ground-motion predictionequations for the average horizontal component of PGA, PGV, and5%-damped PSA at spectral periods between 0.01 s and 10.0 s,Earthquake Spectra 24, no. 1, 99138.

Bradley, B. A. (2010). A generalized conditional intensity measure approachand holistic ground-motion selection, Earthquake Eng. Struct. Dynam.39, no. 12, 13211342.

Campbell, K. W., and Y. Bozorgnia (2008). NGA ground motion model forthe geometric mean horizontal component of PGA, PGV, PGD and 5%damped linear elastic response spectra for periods ranging from 0.01 to10 s, Earthquake Spectra 24, no. 1, 139171.

Chiou, B. S.-J., and R. R. Youngs (2008). An NGA model for the averagehorizontal component of peak ground motion and response spectra,Earthquake Spectra 24, no. 1, 173215.

Ditlevsen, O. (1981). Uncertainty Modeling: With Applications to Multidi-mensional Civil Engineering Systems, McGraw-Hill, New York.

Goda, K., and G. M. Atkinson (2011). Seismic performance of wood-framehouses in south-western British Columbia, Earthquake Eng. Struct.Dynam. 40, no. 8, 903924.

Glerce, Z., and N. A. Abrahamson (2011). Site-specific design spectra forvertical ground motion, Earthquake Spectra 27, no. 4, 10231047.

Harmsen, S. C. (2001). Mean and modal in the deaggregation of probabi-listic ground motion, Bull. Seismol. Soc. Am. 91, no. 6, 15371552.

Haselton, C., J. Baker, Y. Bozorgnia, C. Goulet, E. Kalkan, N. Luco,T. Shantz, N. Shome, J. Stewart, P. Tothong, J. Watson-Lamprey,


and F. Zareian (2009). Evaluation of ground motion selection andmodification methods: Predicting median interstory drift responseof buildings, Technical Report 2009/01, Pacific EarthquakeEngineering Research Center, Berkeley, California.

Jayaram, N., and J. W. Baker (2008). Statistical tests of the jointdistribution of spectral acceleration values, Bull. Seismol. Soc. Am.98, no. 5, 22312243.

Jayaram, N., J. W. Baker, H. Okano, H. Ishida, M. W. McCann, andY. Mihara (2011). Correlation of response spectral values in Japaneseground motions, Earthquakes and Structures 2, no. 4, 357376.

Jayaram, N., T. Lin, and J. W. Baker (2011). A computationally efficientground-motion selection algorithm for matching a target responsespectrum mean and variance, Earthquake Spectra 27, no. 3, 797815.

Katsanos, E. I., A. G. Sextos, and G. D. Manolis (2010). Selection ofearthquake ground motion records: A state-of-the-art review from astructural engineering perspective, Soil Dynam. Earthquake Eng.30, no. 4, 157169.

Kramer, S. L. (1996). Geotechnical Earthquake Engineering, Prentice-HallInternational Series in Civil Engineering and Engineering Mechanics.Prentice-Hall, Upper Saddle River, New Jersey.

Lin, T., and J. W. Baker (2011). Probabilistic seismic hazard deaggregationof ground motion prediction models, in 5th International Conferenceon Earthquake Geotechnical Engineering, Santiago, Chile, 1013 January 2011.

McGuire, R. K. (1995). Probabilistic seismic hazard analysis and designearthquakes: Closing the loop, Bull. Seismol. Soc. Am. 85, no. 5,12751284.

McGuire, R. K. (2004). Seismic Hazard and Risk Analysis, EarthquakeEngineering Research Institute, Oakland, California.

Petersen, M. D., A. D. Frankel, S. C. Harmsen, C. S. Mueller, K. M. Haller,R. L. Wheeler, R. L. Wesson, Y. Zeng, O. S, D. M. Perkins, N. Luco,E. H. Field, C. J. Wills, and K. S. Rukstales (2008). Documentation forthe 2008 update of the United States national seismic hazard maps,U.S. Geol. Surv. Open-File Rept. 20081128.

Scherbaum, F., J. J. Bommer, H. Bungum, F. Cotton, and N. A. Abrahamson(2005). Composite ground-motion models and logic trees: Methodol-ogy, sensitivities, and uncertainties, Bull. Seismol. Soc. Am. 95, no. 5,15751593.

Somerville, P. G., and R. O. Hamburger (2009). Development of groundmotion time histories for design, in Seismic Mitigation for MuseumCollection: Papers from the J. Paul Getty MuseumNational Museumof Western Art (Tokyo) Jointly Sponsored Symposium, Tokyo, Japan.

Somerville, P. G., and H. K. Thio (2011). Development of groundmotion time histories for seismic design, in Proc. of the 9th PacificConference on Earthquake Engineering, Auckland, New Zealand,1416 April 2011.

Watson-Lamprey, J., and N. A. Abrahamson (2006). Selection of groundmotion time series and limits on scaling, Soil Dynam. EarthquakeEng. 26, no. 5, 477482.

Youngs, R. R., S.-J. Chiou, W. Silva, and J. Humphrey (1997). Strongground motion attenuation relationships for subduction zone earth-quakes, Seismol. Res. Lett. 68, no. 1, 5873.

Zhao, J. X., J. Zhang, A. Asano, Y. Ohno, T. Oouchi, T. Takahashi,H. Ogawa, K. Irikura, H. K. Thio, P. G. Somerville, Y. Fukushima,and Y. Fukushima (2006). Attenuation relations of strong groundmotion in Japan using site classification based on predominant period,Bull. Seismol. Soc. Am. 96, no. 3, 898913.

Department of Civil and Environmental EngineeringStanford UniversityStanford, California [email protected]@stanford.edu

(T.L., J.W.B.)

United States Geological SurveyDenver Federal CenterMS 966, Box 25046Denver, Colorado [email protected]@usgs.gov

(S.C.H., N.L.)

Manuscript received 10 October 2011


Lin Et Al (2013) Exact CS, BSSA

Documents

Transcript of Lin Et Al (2013) Exact CS, BSSA