Seismic hazard analysis for alternative measures of ground motionintensity employing stochastic simulation methods
F. JalayerUniversity of Naples “Federico II”, Naples, Italy
P. FranchinUniversity of Rome “La Sapienza”, Rome, Italy
ABSTRACT: The wide application of spectral acceleration as a ground motion intensity measure (IM) is in partdue to the fact that various empirical (attenuation) relationships, that relate spectral acceleration to ground motionsource and path parameters, are already available and tabulated. In the recent years, various scalar and vectorIM’s have been proposed and shown to be more suitable than spectral acceleration; however, their wider use ishindered by limited availability of site-specific empirical attenuation relations for these variables. Subset Simu-lation is an advanced simulation method which is particularly efficient for calculating small failure probabilities.The (target) failure region in the Subset Simulation is regarded as the last in a sequence of nested intermediatefailure regions in which, a Markov Chain Monte Carlo algorithm is used to sample from the original probabilitydistribution conditioned on the previous intermediate failure region in the sequence. This study demonstratesthat Subset Simulation, based on a stochastic model for ground motion, can be effectively employed in order todevelop hazard curves for alternative scalar and vector IM’s. Two example applications are illustrated in whichhazard curves are derived for a scalar structure-specific IM that includes the effect of both higher modes andinelastic response and a vector IM consisting of spectral acceleration and spectral shape for a California site.The algorithm is efficient and relatively straight-forward, however, some care should be taken in defining thesequence of nested failure regions which guide the simulation procedure into regions with small exceedanceprobabilities.
1 INTRODUCTION
The state of the practice in probability-based seis-mic performance assessment of structures is toadopt an intensity measure (IM) in order to repre-sent the uncertainty in the future ground motion.This involves a probabilistic seismic hazard analy-sis (PSHA, McGuire, 1995) that relies on empiricalattenuation relations in order to obtain the (mean andstandard deviation of) IM as a function of parameterssuch as magnitude and distance.
Spectral acceleration at the first-mode period,denoted by Sa(T1), is a common choice for a measureof earthquake ground motion intensity. Its wide-spreaduse is in part facilitated by availability of tabulatedempirical attenuation relations and regional hazardcurves. However, studies (Shome and Cornell 1999,Luco and Cornell 2006, Baker and Cornell, 2006)demonstrate that first-mode spectral acceleration isnot always sufficient as a single parameter for relayingthe ground-motion source characteristics.
For example, for high-rise long-period structures,the effect of higher modes becomes important.Anothercase involves near-source ground motions, where stud-ies (Alavi and Krawinkler 2000) have demonstratedthat, even for an SDOF structure, Sa(T1) may not besufficient and the effect of inelastic response needs tobe taken into account.
Considerable research effort has been focused onfinding ways to employ more suitable intensity mea-sures in the scalar or vector form (Luco and Cornell2006, Baker and Cornell 2005 and Cordova et al.,2002); however, these efforts have been restrained bythe practical need to base the results on the PSHA forspectral acceleration, for which site-specific empiricalattenuation relations already exist.
A vector IM which consist of Sa(T1) and Sa(T )at a period T other than the first-mode one couldrepresent a better, more informative IM compared toSa(T1) alone. The ratio Sa(T )/Sa(T1), which is knownas the spectral shape, is found to be a parameter that
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directly affects structural response (Shome and Cor-nell 1999, Baker and Cornell 2006). Depending onthe structure, the period T could be either shorterthan T1 (e.g., second mode period) to reflect thehigher mode effect, or longer, to reflect the softeningeffect in the regime of significant non-linear behav-ior. The application of this vector IM, however, wouldrequire calculation of the joint probability distributionof Sa(T1) and Sa(T ) or at the very least would ask forestimation of the correlation between the two spectralvalues.
Subset Simulation (Au and Beck 2001) is anadvanced simulation procedure which is efficient forcalculating the small probabilities corresponding tostrong earthquake occurrence. Moreover, it is robustwith respect to the number of uncertain parameters inthe problem and it is applicable to any type of structuralmodel and loading.
This work shows how the subset simulation schemecan be efficiently employed, together with a stochasticground motion model (Atkinson and Silva 2000) andthe results of site-specific PSHA for Sa(T1), in order togenerate ground motions and to calculate the hazardfor alternative intensity measure, such as the vector[Sa(T1), Sa(T )]. Marginal distributions for each ofthe components are computed from the obtained jointprobability distribution and compared to the respec-tive PSHA results. Moreover, the procedure is usedto calculate the hazard curve for a scalar structure-specific intensity measure IM1I ,2E that is proposed byLuco and Cornell (2006) which takes into account theeffect of the first two modes of vibration and the effectof inelastic response.
PROPOSAL
A standard probabilistic seismic hazard analysisinvolves calculation of the probability of exceeding agiven level of the adopted IM in a given time intervalor calculation of the mean annual rate of exceedingsuch level due to all possible seismic events capa-ble of producing a ground motion of intensity greaterthan the specified level at the site. In order to achievethis result, PSHA requires the integration of the con-tributions from all influential seismogenetic zones(Cornell, 1968 and McGuire, 1995). A PSHA proce-dure relies on attenuation relations in order to obtainthe (mean and standard deviation of) IM as a func-tion of parameters such as magnitude and distance,whereas, magnitude and distance are sampled basedon the geometry of the surrounding seismic zone, themagnitude scaling laws, and the marginal probabilitydistribution for moment magnitude.
Alternatively, the above integration can be carriedout by means of simulation. When using simulation,instead of using an attenuation relation, the groundmotion time-history can be directly generated based
on semi-empirical stochastic ground motion modelswhich provide Fourier amplitude spectrum for a givenmagnitude and distance. This in turn can be used asthe transfer function for a linear SDOF system in thefrequency domain which takes Gaussian white noise asthe input and produces the ground motion time-historyas the output. The magnitude and distance could besimulated similar to a standard PSHA. Given the verylow probability levels of interest, use of an efficientsimulation scheme is mandatory in order to make suchan approach feasible. This work employs the SubsetSimulation (Au and beck, 2001, 2003) coupled with astochastic ground motion model (Atkinson and Silva,2000) in order to calculate the joint complementaryCDF and the corresponding hazard surface for boththe vector IM consisting of Sa(T1) and Sa(T ) and thescalar intensity measure IM1I ,2E .
1.1 The Subset Simulation Procedure
Given a n-dimensional vector θ of uncertain param-eters, with joint density f (θ), and a failure domainF ⊂ Rn in the space of θ, the probability of failure canbe written as:
The right-most term in the above expression showsthat P(F) can be evaluated as the expectation of thefailure indicator function IF (θ) = 1 if θ ∈ F and zerootherwise. This is the starting point of Monte Carlosimulation methods.
Subset simulation method achieves its great effi-ciency in computing estimates of very low P(F)’s bybreaking up the problem into a sequence of smallerones. If the failure domain F can be decomposed ina ordered sequence of nested failure regions F1 ⊃F2 ⊃ . . . ⊃ Fm = F such that Fk = ⋂k
i=1 Fi, P(F)can be correspondingly expressed as the product ofa sequence of (much larger) conditional probabilitiesaccording to:
Central to this simulation method are two aspects:1) the availability of an algorithm to simulate sam-ples based (in an asymptotic manner) on the originalprobability distribution conditioned on being in anintermediate failure region (the Metropolis-Hastingsalgorithm) 2) the choice of the intermediate failureregions. As it regards the latter, consideration of theshape of the failure domain helps in the choice of thenested sequence.
The failure region for a system modeled as ns seri-ally connected sub-systems, with the set of componentindices of the j-th subsystem denoted by Ij , can be
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written as union and the intersection of componentfailure regions:
where Fi(θ) is the failure domain for the ith componentwith its demand and capacity denoted by Di(θ) andcapacities Ci(θ), respectively.
It is possible to parameterize F with a scalar param-eter such that the sequence of failure regions can begenerated by varying a single parameter. For a failureregion that can be expressed according to the generalformat in (3) such a parameter is given by:
It is immediate to show that F = {θ : Y (θ) > 1}from which it follows that the sequence of failureregions can be generated as: Fk = {θ : Y (θ) > yk }with 0 < y1 < . . . < ym = 1. The choice of the inter-mediate yk -values usually results from a compromisebetween the number of nested domains (levels) and thenumber of simulations per level. A convenient choiceis to choose these thresholds adaptively by keepingthe magnitude of the conditional probabilities in (2)constant (e.g. equal to P(F1) = p0).
1.2 The stochastic ground motion model
In this work, a stochastic ground motion model pro-posed by Atkinson and Silva (2000) is used to obtainthe (mean) Fourier amplitude spectrum denoted byA(f ; M , r) for a given magnitudeMand source-to-sitedistance r. This mean amplitude spectrum is used asthe transfer function for a linear SDOF oscillator in thefrequency domain that takes as input a windowed time-series of zero-mean Gaussian uncertain variables andproduces ground motion time-history as output. TheFourier amplitude can be written as:
where AAS2000(f ; M , r) is the Fourier amplitude pro-posed in (Atkinson and Silva 2000) and εmodel isassumed to be a unit-median Lognormal uncertainvariable which takes into account — in the absenceof more specific information — the overall effectof uncertainty in ground motion parameters on thespectra predicted by the stochastic model.
1.3 Application of subset simulation to thedetermination of hazard for a vector IM
Determination of the hazard and/or complementaryCDF for an alternative scalar IM (one for which thereis no attenuation available) is a straightforward appli-cation of subset simulation as described in Section 1.1.
One can regard the problem as that of a system withone component of deterministic unit-capacity. In thiscase equation (4) reduces to Y = D(θ) = IM (θ). In gen-eral, the algorithm can be used in order to calculate thejoint probability distribution for a vector-valued IM,denoted by S(θ) = (S1(θ), . . . , Sn(θ)).
For a vector-valued IM, the sequence of nested fail-ure regions can be represented by a scalar variablesimilar to the one stated in Equation 4:
which parameterizes the nested failure boundarysequence Fk = {k : 1, . . . , m} with (monotonically)increasing scalar sequence yk = {k : 1, . . . , m} :
where P(Fk ) = P(Y > yk ) can be calculated by per-forming the Subset Simulation procedure for the scalarvariableY and the si values are shape factors that con-trol the aspect ratio of the failure boundary. In thisparticular case, the failure surface has the shape of ann-dimensional box. Finally, the results of the SubsetSimulation need to be post-processed in order to renderthe values of FS (s) for a desired mesh of s = (s1, . . . , sn)values.
Based on the sequence of failure regions, a setof mutually exclusive and collectively exhaustive(MECE) regions can be defined as (Figure 1):
Identification of a set of MECE regions defined asin (8) allows the determination of CCDF (GS) valuesby use of the total probability theorem:
Figure 1. Decomposition of the failure domain into mutu-ally exclusive and collectively exhaustive regions.
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The probability terms FS (s|Ai) and P(Ai) in (9)are easily evaluated: FS (s|Ai) can be calculated basedon simple statistics on the θ-samples that belong todomain Ai, (i.e. those for which yi < Y (θ) ≤ yi+1);P(Ai) is given by:
2 NUMERICAL EXAMPLES
2.1 Example 1: Sa(T1) and Sa(T2)
As the first example, the hazard curve for a vector IMconsisting of the spectral acceleration at periods T1and T2 is calculated using Subset Simulation. This isequivalent to an IM consisting of first-mode spectralacceleration and the spectral shape factor at anotherperiod. In order to determine the modeling error for thestochastic ground motion model, the marginal distri-butions for Sa(T1) and Sa(T2) calculated using SubsetSimulation are (roughly) fit to the results of PSHA.
2.1.1 Fine-tuning of stochastic-model variabilityterm
In the absence of error estimations specific to thestochastic ground motion model, a single parame-ter denoted by εmodel is adopted to model the overalleffect of uncertainties in the ground motion source andpath parameters. εmodel is assumed to be a unit-medianlognormal variable whose variance is determined bymatching the spectral acceleration hazard estimationsprovided by subset simulation with those provided bya standard PSHA. Figures 2-a and 2-b show the spec-tral acceleration hazard curves λSa(T )(x) obtained forT1 = 0.8s and T2 ≈ 2T1 = 1.5s and the correspondingvalue of ε.
It can be observed that σln εmodel = 0.50 andσln εmodel = 0.45 achieve a good match with PSHAresults at T = 0.80s and T = 1.50, respectively.
2.1.2 Calculating the joint complementary CDF forSa(T1) and Sa(T2)
The Subset Simulation procedure is employed tocalculate the joint complementary CDF for Sa(T1)and Sa(T2), where the failure boundary sequence isparameterized by :
The Subset Simulation is performed at 6 succes-sively increasing level, each with the same conditionalprobability of failure of 10%. P(Y > y) at the highest
Figure 2. Spectral acceleration seismic hazards from PSHA(IM-based) and Subset Simulation (probabilistic representa-tion of ground motion) for a) T1 = 0.80s and b) T2 = 1.50s.
level, which is equal to the product of the condi-tional failure probabilities at all levels, is equal to10−6. At each level 500 simulations are performed.In order to calculate exceedance probabilities as smallas 10−6 using standard Monte Carlo simulation, oneneeds to perform on the order of 4 million (4 × 106)simulations to get a coefficient of variation in the esti-mate for the failure probability equal to 50%; whileSubset Simulation has been carried out by perform-ing 500 + (500 − 50) × 5 = 2750 analyses, giving acoefficient of variation for the lowest level of 13%that increases with each intermediate level to approxi-mately 66% at the highest level. This demonstrates theefficiency of Subset Simulation for calculating verysmall failure probabilities. It is observed, however, thatthis CoV refers to the estimate of the distribution ofthe scalar parameter Y and the estimate of the jointdistribution of the vector IM could be characterizedby a different CoV.
Figure 3 illustrates the joint complementary CDFsurface for Sa(T1) and Sa(T2) that is calculatedusingεmodel = 0.45 following the procedure outlined
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Figure 3. Joint complementary distribution function ofSa(T1) and Sa(T2).
Figure 4. Joint PDF of Sa(T1) and Sa(T2) (the contour valuesare in logarithmic scale).
in Section 1.3 with . This procedure also provides thejoint PDF whose contours are plotted in Figure 4.
The significant correlation between the two spectralvalues can be observed in the contour lines of Figure 4.For the sake of presentation, the contour values areshown in the logarithmic scale.
2.1.3 Comparing the marginal distributionsIn order to benchmark the above procedure for calcu-lating the joint distribution for the two variables, themarginal distributions obtained by integration of thejoint distribution are compared against those obtainedby performing separate “scalar” Subset Simulation foreach variables. Figures 5-a and 5-b demonstrate theresults for the two periods. The marginal distributionsare plotted in dashed lines and the solid lines representthe results of the Subset Simulation carried out for eachof the variables individually. The results seem to havea good agreement. However, it should be noted that thechoice of the failure region parameterY is fundamentalin achieving a good agreement. More specifically, the
ratio of the factors s1 and s2 (i.e., the aspect ratio of thefailure boundary) in Equation 6 should be consistentwith the correlation observed between the two vari-ables. This is partly because the algorithm performsthe simulations and moves forward conditioned on thesequence of failure regions. The results of PSHA foreach variable are not shown in the figures; neverthe-less, as observed in Figures 2-a and 2-b, the hazardcurves for each variable were reasonably matched withthe corresponding PSHA results.
2.2 Example 2: Luco’s IM
As a second example, the subset simulation methodis used to simulate stochastic ground motion in orderto calculate the hazard curve for a scalar IM whichis a functional of both elastic and inelastic spectralvalues. Luco and Cornell (2006) have proposed ascalar structure-specific intensity measure denoted byIM1I ,2E that takes into account not only the ground-motion frequency content around the first two modal
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Figure 6. The mean annual frequency of exceeding IM1I,2E
periods but also, to some extent, the inelastic structuralbehavior. IM1I ,2E can be calculated as:
where PF1 and PF2 are the modal participation factorsfor maximum inter-story drift corresponding to thefirst two modes of vibration, Sd (T1, ξ1) and Sd (T2, ξ2)are the spectral displacements with periods T1 and T2and damping ratios ξ1 and ξ2 corresponding to thefirst two modes, and SI
d(T1, ξ1, dy) is the spectral dis-
placement of an elastic-perfectly plastic oscillator withperiod T1, damping ratio ξ1 and yield displacement dy.IM1I ,2E is decidedly more sufficient than Sa(T1) in pre-dicting the maximum inter-story drift ratio response ofmoment-resisting frames.
The Subset Simulation is performed at 3 succes-sively increasing level, each with the same conditionalprobability of failure of 10%.At each level 500 simula-tions are performed. Figure 6 shows the correspondinghazard curve.
3 CONCLUSIONS
An advanced simulation method known as Subset Sim-ulation is applied to generate ground motions froma stochastic ground motion model and to calculatethe hazard values and exceedance probabilities foralternative scalar and vector intensity measures (IM).This is while the state of the practice is to perform aprobabilistic seismic hazard analysis (PSHA) in orderto calculate the hazard curve for an IM. However,PSHA relies on the availability of empirical (attenua-tion) relations between the considered IM and groundmotion parameters such as magnitude and distance.
This work demonstrates the efficiency of the SubsetSimulation in calculating the hazard for those IM’s for
which the PSHA results cannot be readily obtainedfor lack of the corresponding attenuation relation-ships. Magnitude and distance are simulated from ajoint distribution that is obtained by post-processing(deaggregation) the results of a PSHA for spectralacceleration.
As a first example, the joint hazard surface for avector IM consisting of spectral acceleration at two dif-ferent periods is calculated. Since the spectral shape isfound to be a parameter that directly affects structuralresponse, it is important to have the hazard informa-tion for this type of IM. The resulting marginal hazardcurves for the two spectral acceleration values demon-strated a good match with the available hazard curvesfor each of the spectral acceleration values. However,the final results and their accuracy is sensitive to the(pre-determined) shape of the nested failure regionsequence used in the algorithm. In comparison to thecomputation effort needed for a standard Monte Carlo,Subset Simulation proves to be an efficient meansfor calculating the hazard information (based on thePSHA results available for spectral acceleration) for avector-valued IM.
As the second example, the proposed procedureis used to calculate the mean annual frequencies ofexceedance for a scalar intensity measure IM1I,2E in theform of generalized maximum inter-story drift ratiotaking into account the participation of the first twomodes and the inelastic behavior in the structure tosome extent. IM1I,2E is observed to be more suffi-cient than spectral acceleration in representing groundmotion intensity; thus, the availability of hazard infor-mation for it would facilitate it use as an intensitymeasure.
ACKNOWLEDGEMENTS
This work was partially supported by the LESSLOSSProject funded by the European Commission – underAward Number GOCE-CT-2003-505488.This supportis gratefully acknowledged. Any opinions, findingsand conclusions or recommendations expressed in thismaterial are those of the authors and do not necessar-ily reflect those of the funding body. The first authoralso acknowledges the support from the Masters inEarthquake Engineering and Engineering SeismologyProgram (MEEES) by the European Union.
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