Seismic Design Loads from Site-Specific and Aggregate

Seismic Design Loads from Site-Specific and

Aggregate Hazard Analyses

by Praveen K. Malhotra

Abstract A significant limitation of site-specific probabilistic seismic hazardanalysis (known as PSHA) is usually overlooked. The seismic design loads derivedfrom PSHA can only be expected to control the risk at individual locations (site-specific risk); they cannot be expected to control the risk at multiple locations simul-taneously affected by an earthquake (aggregate risk). This article presents a method ofcalculating the seismic design loads for controlling both the site-specific and the ag-gregate risks.

Introduction

Buildings and other structures are designed for seismicloads in order to keep the risk to life and property at a toler-able level. There are two kinds of risk: (1) site specific and(2) aggregate. The site-specific risk is from the perspective ofa building owner. The aggregate risk is from the perspectiveof an entire community. The site-specific risk can be quan-tified as the annual probability of a loss exceeding a certainamount from a given building. The aggregate risk can bequantified as the annual probability of a loss exceeding amuch greater amount from multiple buildings in the region.The site-specific and aggregate risks have been called indi-vidual and societal risks by May (2001).

The site-specific and the aggregate risks are not neces-sarily correlated. For example, if a region is affected by onlysmall but frequent earthquakes, the probability of significantloss at a location is high due to the frequent occurrence ofstrong shaking at the location, but the probability of a sig-nificant aggregate loss is low because each small earthquakewill affect only a few locations. On the other hand, if thesame region is affected by only large but rare earthquakes,the probability of significant loss at a location is low due tothe rare occurrence of strong shaking at the location, but theprobability of a significant aggregate loss is high because thelarge earthquake will affect many locations simultaneously.This is further aggravated by the fact that the low frequencyof earthquakes in the region decreases risk awareness,making risk mitigation less popular, thus further increasingthe aggregate risk in the region.

Seismic design should be aimed at reducing both thesite-specific and the aggregate risks. Because of the highlyrandom nature of earthquake ground motions and uncertainconstruction quality, it is not possible to eliminate the seismicrisk entirely, but it is possible to reduce it to a tolerable level.Ignoring the uncertainty in construction quality, one can saythat the site-specific risk is proportional to the chance of ex-

ceeding the seismic design load at a given location; the ag-gregate risk is proportional to the chance of exceeding theseismic design load at many locations simultaneously. There-fore, the seismic design load for controlling both the site-specific and the aggregate risks should satisfy the followingtwo criteria:

1. Site-specific criterion: the seismic design load at each sitein the region should not be exceeded more than λ timesper year, where λ depends on the risk tolerance of theindividual.

2. Aggregate criterion: the seismic design load should notbe simultaneously exceeded over an area ≥Ae more thanλ times per year, where Ae and λ depend on the risk tol-erance of the community.

The first criterion can be satisfied by performing a site-specific probabilistic seismic hazard analysis (PSHA) (e.g.,Cornell, 1971; Reiter, 1991; McGuire, 2004) of each site inthe region and selecting design ground motions with an ex-ceedance rate of λ per year. This is the current approach forobtaining the design ground motions (hence loads). The sec-ond criterion can be satisfied by performing an aggregatehazard analysis of the region as described in this article.The seismic design load should be the higher of the loadsobtained from the two criteria. The idea of aggregate hazardhas been discussed in the past: Rhoades and McVerry (2001)estimated the joint hazard at two sites; Leonard and Steinberg(2002) discussed the need for calculating the hazard atmultiple sites simultaneously affected by an earthquake;Wesson and Perkins (2001), Smith (2003), Bazzurro andLuco (2005), and Crowley and Bommer (2006) discussedthe use of hazard at multiple locations to estimate the aggre-gate (or portfolio) loss. This article focuses on design groundmotions for controlling the aggregate loss.

1849

Bulletin of the Seismological Society of America, Vol. 98, No. 4, pp. 1849–1862, August 2008, doi: 10.1785/0120070241

Outline and Scope

First, a quick overview of the site-specific PSHA is pre-sented. Next, the aggregate hazard analysis is discussed andrelated to the site-specific hazard analysis. Key param-eters that control the aggregate hazard are identified througha parametric study. Finally, the seismic design loads thatsatisfy both the site-specific and the aggregate criteria arecalculated.

Problem Statement

Figure 1a shows a 400 × 400-km region with only twoseismic sources (faults). Source A produces onlyMw 6 earth-quakes with a recurrence interval (RI) of RIA � 50 yr or anoccurrence rate of 1=50 � 0:02=yr. Source B produces onlyMw 7.5 earthquakes with an RI of RIB � 450 yr or an occur-rence rate of 1=450 � 0:0022=yr. The lengths of thesesources are in accordance with the Wells and Coppersmith(1994) relationship. It is assumed that the occurrence ofan earthquake on source A or source B has no effect onthe future occurrence of earthquakes on these two sources(time-independent assumption). For the sake of simplicity,all sites in the region are assumed to be on firm rock (averageshear wave velocity of 760 m=sec in the top 30 m; BuildingSeismic Safety Council [BSSC], 1998). Earthquakes onsource B are only 1=9 as frequent as earthquakes on sourceA, but they are much bigger in size and hence affect a muchlarger area compared to earthquakes on source A. Figure 1bshows 6400 grid points at a uniform spacing of 5 × 5 km.The seismic design loads satisfying both the site-specific

and the aggregate criteria need to be calculated at all gridpoints.

Site-Specific Hazard Analysis

The site-specific hazard is computed at each site (gridpoint) independently of other sites in the region. A site, in-dicated by a star (�) in Figure 1b, is selected. If an Mw 6earthquake occurs on source A, the peak ground acceleration(PGA) at the site can assume any value due to the highly un-predictable nature of the rupture and the propagation of thereleased seismic energy. Esteva (1970) observed that thevariability in PGA can be expressed by a lognormal distribu-tion. This is the underlying assumption in all modern ground-motion prediction relations. According to the Boore and At-kinson (2007) relation, if an earthquake occurs on source A,the median (fiftieth percentile) value of the PGA at the site is0:0688g and the standard deviation (variability) of the naturallogarithm of PGA is σT � 0:565. The variability has twocomponents: (1) interevent and (2) intraevent (Brillingerand Preisler, 1984; Abrahamson and Youngs, 1992; Bommerand Crowley, 2006). The interevent variability is the sys-tematic difference in the ground motions throughout theregion produced by different earthquakes of the same mag-nitude and style of faulting. The intraevent variability is thedifference in ground motions at various sites of same soilclass and distance from the source. In the site-specific hazardanalysis, the inter- and intraevent variabilities are not sepa-rated, but in the aggregate hazard analysis, the inter- and in-traevent variabilities will be treated separately.

Source A

Source B

(a) (b)

Figure 1. A region affected by two seismic sources: (a) source A producesMw 6 earthquakes and source B producesMw 7.5 earthquakes;(b) 6400 grid points at a spacing of 5 km in each direction. The selected site, shown by a star (�), is 27.5 km from source A and 31.8 km fromsource B.

1850 P. K. Malhotra

Figure 2a shows the lognormal distribution of PGAconstructed from the median and the standard deviation ofln PGA at the site. Figure 2b is a plot of the area under theprobability density function (Fig. 2a) to the right of variousPGA values. It gives the exceedance probabilities of vari-ous PGAs; higher PGAs have lower exceedance probabili-ties. PGA � 0:1g has an exceedance probability of 0.256

(25.6%) if an earthquake occurs on source A. PGA � 0

has an exceedance probability of 1 (100%) if an earthquakeoccurs on source A. Because the occurrence rate of earth-quakes on source A is 0:02=yr, the exceedance rates of vari-ous PGAs are obtained by multiplying the curve in Figure 2bby 0:02=yr. This gives the PGA hazard curve for the sitedue to source A (Fig. 2c). It is a plot of the exceedance

0 0.1 0.2 0.3 0.40

4

8

12

Pro

babi

lity

Den

sity

(a)

0 0.1 0.2 0.3 0.40

0.5

1

0.256

Pro

babi

lity

of E

xcee

danc

e

(b)

0 0.1 0.2 0.3 0.40

0.01

0.02

0.00512/year

Peak Ground Acceleration, PGA (g)

Exc

eeda

nce

Rat

e, λ

(pe

r ye

ar)

(c)

Figure 2. Construction of PGA hazard curve for the site (shown by a star [�] in Fig. 1b) due to source A: (a) probability density functionof PGA at the site, (b) exceedance probability function obtained by integrating the probability density function, and (c) hazard curve obtainedby multiplying the exceedance probability function with the RI of earthquakes on source A.

Seismic Design Loads from Site-Specific and Aggregate Hazard Analyses 1851

rate λ versus PGA. PGA � 0:1g has an exceedance rate of0:00512=yr due to earthquakes on source A.

Following the same steps as for source A, the PGA haz-ard curve for the site due to source B is also constructed.Assuming that the earthquakes on source A and source B

occur independently of each other, the exceedance rates ofvarious PGAs due to source A and source B are added toobtain the overall PGA hazard curve for the site (Fig. 3).For example, PGA � 0:1g has an exceedance rate of0:00512=yr due to source A and 0:00169=yr due to sourceB. Therefore, the overall exceedance rate of PGA � 0:1g

is 0:00512� 0:00169 � 0:0068=yr. By definition, the returnperiod (RP) (of exceedance) is the reciprocal of the rate ofexceedance, i.e.,

RP � 1=λ: (1)

Therefore, the RP of exceedance of PGA � 0:1g isRP � 1=0:0068 � 147 yr. From Figure 3, the 500-yrRP PGA for the site is 0:17g. In a similar manner, thePGA hazard curves for all 6400 grid points shown in Fig-ure 1b are constructed. In engineering analyses, the spectralaccelerations at various natural periods are also useful, but inthis article, the results shown are only for PGA, the spectralacceleration at zero natural period.

Figure 4a displays the 500-yr RP PGAs read from thePGA hazard curves for various grid points in the region.For comparison, the higher of the median values of PGAsdue to source A and source B are displayed in Figure 4b.The 500-yr RP PGAs near source A are much higher thanthe median PGAs because source A is highly active; its RIis much shorter than 500 yr. The 500-yr PGAs near sourceB are much lower than the median PGAs because sourceB is less active; its RI is only slightly shorter than 500 yr.Despite its much smaller size, source A has a significant in-fluence on the 500-yr RP site-specific PGAs in the region.

If the seismic design of buildings throughout the regionis based on the 500-yr RP site-specific accelerations, the riskat each location will remain at a tolerable level because thedesign acceleration at each location will only be exceeded0.002 times per year. However, during a future earthquakeon source B, the design accelerations will be exceeded atnumerous locations because the 500-yr RP site-specific ac-celerations surrounding source B are much smaller thanthe median accelerations (Fig. 4a,b); this may pose a signif-icant aggregate risk to the region. In order to control the

0 0.1 0.2 0.3 0.410

−5

10−4

10−3

10−2

Source A

B A & B

500 years

0.17

Peak Ground Acceleration, PGA (g)

Exc

eeda

nce

Rat

e, λ

(pe

r ye

ar)

0.00512 + 0.00169 = 0.0068/year (RP = 147 years)

0 0.1 0.2 0.3 0.4

102

103

104

105

Ret

urn

Per

iod,

RP

(ye

ars)

Figure 3. Construction of the overall PGA hazard curve for thesite shown by a star (�) in Figure 1b.

0.1 to 0.2 g> 0.2 g

500−year PGA

0.1 to 0.2 g> 0.2 g

Median PGA(a) (b)

Figure 4. Comparison of (a) the 500-yr RP site-specific PGAs with (b) the higher of the median PGAs due to earthquakes on source A andsource B.

1852 P. K. Malhotra

aggregate risk, the design accelerations should satisfy theaggregate criterion, i.e., the design accelerations should notbe simultaneously exceeded over an area ≥Ae at more than acertain rate.

Aggregate Hazard Analysis

So far, the hazard at each site has been computed inde-pendently of other sites in the region, or, the spatial distribu-tion of the ground accelerations during an earthquake hasbeen ignored. In the aggregate hazard analysis, the spatialdistribution of the ground accelerations is also consideredbecause of the need to know the effect of an earthquakeon many sites simultaneously. The purpose of aggregatehazard analysis is to obtain the design accelerations that willbe simultaneously exceeded over a certain minimum area at acertain rate.

The aggregate hazard analysis is performed simulta-neously for all sites in the region. The following approachis used. First, numerous ground-motion scenarios are simu-lated for earthquakes on source A and source B. Then, thedesign accelerations are systematically raised throughoutthe region until they are high enough to satisfy the aggregatecriterion. Again, it is assumed that the occurrence of an earth-quake on source A or source B has no effect on the futureoccurrence of earthquakes on these two sources (time-inde-pendent assumption).

Simulation of Ground-Motion Scenarios

The distance of each of the 6400 grid points (Fig. 1b)from source A is computed, and the mean and the standarddeviation of the natural logarithm of PGA (ln PGA) are cal-culated from the Boore and Atkinson (2007) predictionrelationship. As already mentioned, the total variability inln PGA has two components: (1) interevent (earthquake-to-earthquake) variability τ and (2) intraevent (site-to-site)variability σ. Because these two components are assumedto be independent of each other, they are related to the overallvariability by the following relation:

σ2T � τ 2 � σ2: (2)

The interevent variability exists because two earthquakes ofthe same size and style of faulting do not necessarily producethe same average ground accelerations throughout the re-gion. The intraevent variability exists because sites of thesame soil class and distance from the source experience dif-ferent ground motions due to differences in the stratigraphyand the influence of deeper geological structures.

It is assumed that the interevent variability produces ran-dom but fully correlated ground accelerations at all 6400 gridpoints. It is also assumed that the intraevent variability pro-duces random and uncorrelated ground accelerations at all ofthe grid points. With these two assumptions, the covariancematrix of ground accelerations at 6400 grid points is

C �τ2 τ 2 … τ 2

τ2 τ 2 … τ 2

..

. ... . .

. ...

τ2 τ 2 � � � τ 2

2664

3775�

σ2 0 … 0

0 σ2 … 0

..

. ... . .

. ...

0 0 � � � σ2

2664

3775: (3)

Other models with nonzero intraevent correlation (e.g., Wangand Takada, 2005) could also be used instead of the modelused in this study. It follows from equation (3) that the spatialcorrelation between ground accelerations at any two gridpoints is (Brillinger and Preisler, 1984; Wesson and Perkins,2001)

ρ � τ 2

τ 2 � σ2�

�τσT

�2

: (4)

According to the Boore and Atkinson (2007) prediction re-lationship, the interevent variability of ln PGA is τ � 0:26,the intraevent variability is σ � 0:502, and the total variabil-ity is σT � 0:565; Campbell and Bozorgnia (2007) have pro-posed somewhat smaller values of the inter- and intraeventvariabilities. From equation (4), the spatial correlation be-tween the logarithm of ground accelerations at any two gridpoints is ρ � 0:212. It is assumed that within the 25-km2

area, represented by each grid point, the ground accelerationsare fully correlated.

Using the mean values of ln PGA at 6400 grid points, thecovariance matrix given by equation (3), and assuming thatthe ln PGAs are normally distributed (PGAs are lognormallydistributed), 45,000 sets of ln PGAs at 6400 grid points aregenerated using the Matlab (MathWorks, 2006) routinelhsnorm. The exponentials of the ln PGAs are the PGAs at6400 grid points. One of the 45,000 sets of simulated PGAsfor an earthquake on source A is displayed in Figure 5a.

The earthquakes on source B are 1=9 as frequent as theearthquakes on source A because the RI of earthquakes onsource B is nine times the RI of earthquakes on source A.Therefore, only 45; 000=9 � 5000 sets of PGAs are gener-ated for source B. One of the 5000 sets of PGAs is displayedin Figure 5b. Next, the simulated ground-motion scenariosare validated.

Validation of Ground-Motion Scenarios

Figure 6a shows a histogram of 50,000 simulated PGAsat the grid point indicated by an asterisk in Figure 1b: 45,000PGAs for source A and 5000 PGAs for source B. Figure 6bshows the exceedance probabilities of the PGAs, computedfrom the 50,000 simulated values. The exceedance rate ofPGA is the exceedance probability divided by the RI of earth-quakes, i.e.,

λ � Ep

RI: (5)

By definition, the exceedance RP is the reciprocal of the ex-ceedance rate, i.e.,


0 0.1 0.2 0.3 0.40

400

800

1200

Num

ber

of T

imes

(a)

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

PGA (g)

Exc

eeda

nce

Pro

babi

lity,

Ε p

(b)

0.17

Figure 6. Distribution of 50,000 simulated values of PGA at the grid point shown by a star (�) in Figure 1b: (a) histogram of simulatedPGAs and (b) exceedance probability curve.

Source A Earthquake

0.1 to 0.2 g> 0.2 g

0.1 to 0.2 g> 0.2 g

Source B Earthquake(a) (b)

Figure 5. PGA contours for two possible earthquake scenarios for (a) source A and (b) source B.

1854 P. K. Malhotra

RP � 1

λ� RI

Ep

; (6)

where the RI of earthquakes (on any of the two sources) inthe region is given by

1

RI� 1

RIA� 1

RIB: (7)

Equation (7) is derived by simply adding the occurrence rates(reciprocal of RIs) of earthquakes on source A and source Bto obtain the overall occurrence rate of earthquakes in theregion. Substituting RIA � 50 yr and RIB � 450 yr in equa-tion (7) gives RI � 45 yr. Equation (6) is rewritten to expressthe exceedance probability in terms of the RI and the RP:

Ep � RI

RP: (8)

Substituting RI � 45 yr and RP � 500 yr in equation (8)gives Ep � 0:09. Therefore, the PGA with an exceedanceprobability of 0.09 in Figure 6b, i.e., PGA � 0:17g, is the500-yr RP PGA. This is the exact value obtained fromthe site-specific probabilistic hazard analysis described inthe previous section. Similarly, the 500-yr RP PGAs at all6400 grid points are calculated and displayed in Figure 7a.Figure 7b shows the 500-yr PGAs from the site-specific prob-abilistic analysis described in the previous section (Fig. 4a).The PGAs in Figure 7a,b are identical. Therefore, the simu-lated scenarios can also be used to perform the site-specifichazard analysis of the region using a statistical approach, but

the probabilistic approach described in the previous sectionwas much more efficient. This confirms that the simulatedscenarios have adequately captured the probability distribu-tion of ground accelerations at each location.

Next, the correlation between the ln PGA at the site(shown by a star [�] in Fig. 1b) and the ln PGAs at the re-maining 6399 sites are calculated from 45,000 scenarios forsource A. The same calculations are also performed using5000 scenarios for source B. Apart from some scatter, asso-ciated with the finite set of earthquake scenarios, the correla-tion coefficient was found to be 0.21 for both source A andsource B ground motions. This confirms that the scenarioshave adequately captured the spatial correlation of groundaccelerations.

Finally, to confirm the adequacy of the number ofsimulated scenarios the exceedance probability curves fromprobabilistic and statistical analyses are compared (resultsnot shown). The statistical values of PGA were within 1%of the probabilistic values of PGA for exceedance proba-bility Ep ≥ 0:0443. Therefore, 50,000 scenarios are suf-ficient to calculate the ground motions up to an RP ofRP � RI=Ep � 45=0:0443≈ 1000 yr.

Seismic Design Accelerations Satisfying theAggregate Criterion

First, the following two areas are defined: (1) Ai as anarea of interest. This is the total area of the region excludingareas with no current or future planned development. It re-presents the total value exposed to seismic risk in the region.(2) Ae as an area of exceedence. This is the net area over

0.1 to 0.2 g> 0.2 g

Statistical Analysis

0.1 to 0.2 g> 0.2 g

Probabilistic Analysis(a) (b)

Figure 7. Site-specific hazard map of 500-yr RP PGAs from (a) statistical analysis of 50,000 earthquake scenarios and (b) probabilisticanalysis.


which the design accelerations are allowed to be simulta-neously exceeded at a certain rate. This area is not necessa-rily contiguous, and it can be anywhere within the area ofinterest Ai.

The objective is to obtain the design accelerations (forthe entire region) that are simultaneously exceeded over anarea ≥Ae at a certain rate. The number of scenarios availablefor the analysis is NA � 45; 000 for source A andNB � 5000 for source B. The following steps are used toperform the analysis:

1. At the start, the design PGA for the entire region (all 6400grid points in Fig. 1b) is assumed to be a small value thatrequires no seismic design, say, PGA � 0:075g.

2. For each of the NA scenarios for source A, the total num-ber of grid points, where the assumed value of the designPGA is exceeded, is calculated. Remembering that eachgrid point represents 25 km2, the total area, over whichthe assumed value of the design PGA is exceeded, is cal-culated for each scenario. Then the total number ofsource A scenarios for which the exceedance area is≥Ae is calculated; let this number be nA. Similarly, thetotal number of source B scenarios for which the excee-dance area is ≥Ae is calculated; let this number be nB.Then, the exceedance rate of the assumed design accel-erations over an area ≥Ae is

λ � nANA

λA � nBNB

λB; (9)

where λA and λB are the occurrence rates of earthquakeson source A and source B.

3. The grid point that appears in most of the scenarios forwhich the exceedance area is ≥Ae is identified; this gridpoint has the highest contribution to the aggregate hazardin the region. The design PGA for this grid point is raisedby a small amount (say, 0:001g). (To increase the effi-ciency of the analysis, the design PGAs are simulta-neously raised for the top 5% of the most frequentlyappearing grid points.)

4. Again, the total number of source A scenarios for whichthe exceedance area is ≥Ae is calculated; this is the newnA. Also, the total number of source B scenarios forwhich the exceedance area is ≥Ae is calculated; this isthe new nB. Substituting new nA and nB in equation (9)gives the new exceedance rate. The reciprocal of the newexceedance rate is the new RP (equation 1).

5. The design accelerations are continuously raised by themethod described in Step 3 and the RP is recalculated bythe method described in Step 4 until the design accelera-tions have been obtained for all the desired RP, say, upto 500 yr.

Figure 8 displays the design PGAs from the aggregatehazard analysis. The PGAs in Figure 8a,b are exceeded overan area Ae ≥ 4000 km2 with RPs of 250 and 500 yr, respec-tively. The first and the second terms on the right-hand sidein equation (9) are the contributions of source A and source Bto the overall exceedance rate. The relative values of theseterms depend on the overall exceedance rate (or RP). Figure 9

RP = 250 years

0.1 to 0.2 g> 0.2 g

0.1 to 0.2 g> 0.2 g

RP = 500 years(a) (b)

Figure 8. Accelerations from aggregate hazard analysis for two different RPs: (a) RP � 250 yr and (b) RP � 500 yr, and exceedance areaAe � 4000 km2.

1856 P. K. Malhotra

shows the percentage contributions of source A and source Bto the overall exceedance rates corresponding to differentRPs. This is similar to the deaggregation of the site-specificprobabilistic hazard (e.g., McGuire, 1995; Cramer and Peter-sen. 1996; Harmsen and Frankel, 2001). Figure 9 shows thatsource B’s contribution increases with increase in the RP.

A comparison between Figures 8b and 7b shows asignificant difference between the aggregate and the site-specific hazards. Whereas the site-specific PGAs at 500-yrRP are controlled by the more active seismic source (sourceA), the aggregate PGAs are controlled by the bigger seismicsource (source B).

Although the method of analysis of aggregate hazardhas been illustrated with only two seismic sources, both ofwhich follow a characteristic magnitude-recurrence model,

the method has general applicability with multiple sources,some of which may follow a Gutenberg–Richter magnitude-recurrence model. The only change will be to include scenar-ios with different magnitude-recurrence pairs. To improvethe efficiency of the analysis, only those scenarios that areexpected to contribute significantly to the aggregate hazardcan be considered. Earthquakes in remote areas and/or thosewith small magnitudes do not contribute significantly to theaggregate hazard, hence these earthquakes need not be con-sidered in the aggregate hazard analysis.

Effect of Exceedance Area Ae

The exceedance area Ae (along with the exceedance rateλ) represents the aggregate risk tolerance of the community;the lower the risk tolerance, the smaller the area over whichthe design accelerations can be allowed to be exceeded at acertain rate. Figure 10 displays the 500-yr RP PGAs for twodifferent exceedance areas Ae � 2000 and 8000 km2. As ex-pected, an increase in the exceedance area decreases the de-sign PGAs for the region. Figure 11 shows the percentagecontribution of source A to the overall exceedance rate fortwo different values of the exceedance area. An increase inthe exceedance area decreases the contribution of the smallersource (source A), because the smaller source is less likely tocause high accelerations over a large area.

Effect of Area of Interest Ai

The area of interest represents the total value exposed toearthquakes in the region. So far, the entire 400 × 400-kmarea has been assumed to be uniformly developed. Next, only

100 200 300 400 5000

20

40

60

80

100

Source A

Source B

Return Period, RP (years)

Con

trib

utio

n (%

)

Figure 9. Contributions of source A and source B to the excee-dance rate of aggregate hazard at different RPs.

Ae = 2000 km2

0.1 to 0.2 g> 0.2 g

0.1 to 0.2 g> 0.2 g

Ae = 8000 km2(a) (b)

Figure 10. Effect of changing exceedance area Ae on the aggregate hazard for 500-yr RP.


half of the area, shown by shaded squares in Figure 12, isassumed to be developed. Because there is no aggregate riskfrom the undeveloped area, it is excluded from the area ofinterest Ai. Now the area of interest is only 80; 000 km2

(3200 grid points). Aweight of zero is assigned to grid pointsoutside of the area of interest and a weight of 1 is assigned togrid points within the area of interest. The aggregate hazardanalysis is performed as before, except that the area repre-sented by each grid point is multiplied by its respectiveweight before calculating the exceedance area Ae. The aggre-gate hazard is still calculated at all 6400 grid points, in casean isolated structure needs to be built outside of the area ofinterest.

Figure 13 compares the 500-yr RP PGAs for two differ-ent values of the area of interest Ai. As expected, the aggre-gate hazard decreases with decrease in the area of interest Ai.Figure 14 compares the contributions of source A to the ag-gregate hazard for two different values of the area of interest.A decrease in the area of interest Ai amounts to an apparentincrease in the exceedance area Ae; therefore, source A’s con-tribution to the aggregate hazard decreases with decrease inthe area of interest. In a risk-based approach, different areasin a region can be assigned weights proportional to the ex-posed value/population density and vulnerabilities of exist-ing structures.

The design accelerations from the aggregate hazardanalysis decrease with an increase in the exceedance areaAe or a decrease in the area of interest Ai. Next, the aggre-gate hazard is calculated for two different sets of Ae andAi, but the same ratio Ae=Ai � 0:025 (2.5%). Figure 15ashows the 500-yr RP PGAs for Ae � 4000 km2 and Ai �160; 000 km2. Figure 15b shows the 500-yr PGAs for Ae �400 km2 and Ai � 16; 000 km2. In both cases, the 500-yrPGAs are nearly the same. This implies that the ratio Ae=Ai

is more important in the aggregate hazard than the Ae and Ai

themselves.

Seismic Design Loads Satisfying Both Site-Specificand Aggregate Criteria

As already mentioned, the seismic design loads shouldsatisfy both the site-specific and the aggregate criteria. As-sume that only 2.5% of the region (Ai � 4000 km2) is de-veloped (shaded area in Fig. 16). The next step is to establishthe exceedance rate and the exceedance area based on therisk tolerance of the individuals and the community. A sys-tematic analysis of the risk tolerance is outside of the scopeof this article, but others (e.g., Bommer and Pinho, 2006)have discussed the need for such an analysis for establishingthe exceedance rate (RP) of seismic design loads. TheHurricane Katrina Review Panel (American Society of CivilEngineers [ASCE], 2007) emphasized the need to communi-cate the risk and to decide the acceptable (tolerable) risk. Letus say that the following criteria reflect the risk tolerance ofthe individuals and the community:

1. Site-specific criterion: the seismic design load at each sitein the region should not be exceeded more than 0.002times per year.

2. Aggregate criterion: the seismic design load over an area≥100 km2 should not be simultaneously exceeded morethan 0.002 times per year.

Figure 17a shows the PGAs satisfying the site-specificcriterion, and Figure 17b shows the PGAs satisfying the ag-gregate criterion. Figure 18a displays the higher of the PGAsfrom Figure 17a,b; therefore, the PGAs displayed in Fig-ure 18a satisfy both the site-specific and the aggregatecriteria. Figure 18b shows the controlling criterion for eachgrid point. As expected, the site-specific criterion controls

100 200 300 400 5000

20

40

60

80

100

Ae = 2000 km2

Ae = 8000 km2


Con

trib

utio

n of

Sou

rce

A (

%)

Figure 11. Source A’s contribution to the aggregate hazard fortwo different exceedance areas.

Figure 12. Area of interest, Ai � 80; 000 km2, shown byshaded squares.

1858 P. K. Malhotra

the design accelerations near the more active seismic source(source A) and the aggregate criterion controls the design ac-celerations near the bigger seismic source (source B).

U.S. Building Code Map

In an effort to unify the seismic design of structuresthroughout the U.S., the Building Seismic Safety Council([BSSC], 1998) requested the California Geological Survey(CGS) and the United States Geological Survey (USGS) togenerate the 475-yr RP ground motions for the entire country.The CGS and USGS calculated the 475-yr RP ground motionsfrom the site-specific PSHA. These ground motions did notcapture the effects of rare but large earthquakes capable of

producing significant aggregate losses in the New MadridSeismic Zone, the Pacific Northwest, and the Wasatch re-gions (Earthquake Engineering Research Institute [EERI],2006, p. 51). The BSSC then requested the CGS/USGS togenerate the 2475-yr RP ground motions, which capturedthe effects of rare but large earthquakes. BSSC decided toinclude the 2475-yr ground motions in the 2000 InternationalBuilding Code (IBC) (International Code Council [ICC],2000) but require the design of buildings for only 2=3 of the2475-yr ground motions. The 2=3 of 2475-yr RP groundmotions were significantly higher than the prevailing designground motions in parts of the western U.S. The BSSC (1998)then truncated the 2475-yr RP ground motions by the deter-ministic limit to obtain the so-called maximum consideredearthquake ground motions (Leyendecker et al., 2000; EERI,2006, p. 51–52).

As a result of these adjustments (change of RP from 475to 2475 yr, deterministic limit, and 2=3 factor), the risk to lifeand property from code-designed buildings varies consider-ably from location to location (Malhotra, 2005, 2006). If theintent of the building code is only to control the site-specificrisk, the unequal risk to code-designed buildings throughoutthe U.S. cannot be justified. If, on the other hand, the intentof the code is also to control the aggregate risk, the site-specific hazard analysis alone cannot be the basis of seismicdesign loads in the building code. The confusion caused bythe seismic design loads in the building code has been thesource of a debate among researchers (e.g., Stein, 2005;Wang et al., 2005). In the interest of stakeholders, the seismicdesign loads in building codes should be based on a trans-

100 200 300 400 5000

20

40

60

80

100

160000 km2

Ai = 80000 km2


Con

trib

utio

n of

Sou

rce

A (

%)

Figure 14. Source A’s contribution to the aggregate hazard fortwo different values of the area of interest.

Ai = 160000 km2

0.1 to 0.2 g> 0.2 g

0.1 to 0.2 g> 0.2 g

Ai = 80000 km2(a) (b)

Figure 13. Effect of the area of interest Ai on the 500-yr RP PGAs from the aggregate hazard analysis.


parent analysis of the site-specific and aggregate hazardswithout the need for arbitrary adjustments.

The ongoing concentration of life and property in urbancenters across the world makes it even more important to takethe aggregate risk seriously and to base the seismic design

loads on a systematic analysis of the site-specific and aggre-gate hazards. As a start, the aggregate hazard analysis shouldbe performed for urban centers with high concentration oflife and property. In remote areas, the chance of a huge ag-gregate loss is low; therefore, the aggregate hazard is lesslikely to control the design loads in those areas.

Conclusions

1. There are two types of seismic risk: (1) site-specific riskat each location and (2) aggregate risk to the whole re-gion. Reducing the site-specific risk to a tolerable leveldoes not automatically reduce the aggregate risk to a tol-erable level.

2. The seismic design loads for controlling both the site-specific and the aggregate risks should satisfy the follow-ing criteria:

i. Site-specific criterion: the seismic design load at eachsite in the region should not be exceeded more than λtimes per year.

ii. Aggregate criterion: the seismic design load over anarea ≥Ae (contiguous or separated) should not be si-multaneously exceeded more than λ times per year.

The site-specific criterion is satisfied by performing asite-specific hazard analysis (conventional PSHA) ofevery site in the region. The aggregate criterion is satis-fied by performing an aggregate hazard analysis as de-scribed in this article. Site-specific hazard analysis is aspecial case of the aggregate hazard analysis in which

Figure 16. Area of interest, Ai � 4000 km2, shown by shadedsquares.

Ae = 4000 km2 & A

i = 160000 km2

0.1 to 0.2 g> 0.2 g

0.1 to 0.2 g> 0.2 g

Ae = 400 km2 & A

i = 16000 km2(a) (b)

Figure 15. Effect of keeping the ratio Ae=Ai fixed on the 500-yr RP PGAs from the aggregate hazard analysis.

1860 P. K. Malhotra

the area of interest and the exceedance area both reduce toa point.

3. The site-specific hazard is controlled by frequently occur-ring earthquakes even if they affect only a few locationsat a time, and the aggregate hazard is controlled by largeearthquakes that affect many locations simultaneously.

4. The results of site-specific PHSAwere arbitrarily adjustedto obtain the seismic design loads in the IBC. At leastpartly, these adjustments were needed to overcome thelimitation of the site-specific PSHA. A systematic analy-sis of the aggregate hazard should eliminate the need forarbitrary adjustments.

Design PGA Map

0.1 to 0.2 g> 0.2 g

Aggregate

Site−specific

Controlling Criteria(a) (b)

Figure 18. Combining site-specific and aggregate maps: (a) design accelerations satisfying both site-specific and aggregate criteria;(b) controlling criterion for design accelerations at different sites.

Site−specific PGA Map

0.1 to 0.2 g> 0.2 g

0.1 to 0.2 g> 0.2 g

Aggregate PGA Map(a) (b)

Figure 17. Design accelerations satisfying the (a) site-specific criterion and (b) aggregate criteria.


5. The ongoing concentration of life and property in urbancenters across the world makes it even more important forbuilding codes to focus on controlling the aggregate risk.

Acknowledgments

Helen Crowley, Julian Bommer, and an anonymous reviewer providedseveral helpful suggestions to improve the presentation of this article. ChrisCramer drew attention to some relevant publications and commented on thepresentation of the site-specific hazard analysis. Francesco Tamanini andHosam Ali provided an internal review of the article.

References

Abrahamson, N. A., and R. R. Youngs (1992). A stable algorithm for regres-sion analyses using the random effects model, Bull. Seismol. Soc. Am.82, no. 1, 505–510.

American Society of Civil Engineers (ASCE) (2007). The New Orleans hur-ricane protection system: what went wrong and why, a report by theAmerican Society of Civil Engineers Hurricane Katrina ExternalReview Panel, ASCE, Reston, VA.

Bazzurro, R., and N. Luco (2005). Accounting for uncertainty and correla-tion in earthquake loss estimation, Proc. ICOSSAR 2005, Safety andReliability of Engineering Systems and Structures.

Bommer, J. J., and H. Crowley (2006). The influence of ground-motionvariability in earthquake loss modeling, Bull. Earthq. Eng. 4, no. 3,231–248.

Bommer, J. J., and R. Pinho (2006). Adapting earthquake actions in Euro-code 8 for performance-based seismic design, Earthq. Eng. Struct.Dyn. 35, 39–55.

Boore, D. M., and G. M. Atkinson (2007). Boore-Atkinson NGA groundmotion relations for the geometric mean horizontal component of peakand spectral ground motion parameters, PEER Report 2007/01, PacificEarthquake Engineering Research Center, University of California,Berkeley.

Brillinger, D. R., and H. K. Preisler (1984). An exploratory analysis of theJoyner-Boore attenuation data, Bull. Seismol. Soc. Am. 74, no. 4,1441–1450.

Building Seismic Safety Council (BSSC) (1998). 1997 Edition NEHRP rec-ommended provisions for seismic regulations for new buildings andother structures, FEMA 302/303, Part 1 (Provisions) and Part 2 (Com-mentary), developed for the Federal Emergency Management Agency,Washington, D.C.

Campbell, K. W., and Y. Bozorgnia (2007). Campbell-Bozorgnia NGAground motion relations for the geometric mean horizontal componentof peak and spectral ground motion parameters, PEER Report 2007/02, Pacific Earthquake Engineering Research Center, University ofCalifornia, Berkeley.

Cornell, C. A. (1971). Probabilistic analysis of damage to structures underseismic loads, in Dynamic Waves in Civil Engineering, D. A. Howells,I. P. Haigh and C. Taylor (Editors), Wiley, New York, 473–488.

Cramer, C. H., and M. D. Petersen (1996). Predominant seismic source dis-tance and magnitude maps for Los Angeles, Orange and VenturaCounties, California, Bull. Seismol. Soc. Am. 86, no. 5, 1645–1649.

Crowley, H., and J. J. Bommer (2006). Modelling seismic hazard in earth-quake loss models with spatially distributed exposure, Bull. Earthq.Eng. 4, no. 3, 249–273.

Earthquake Engineering Research Institute (EERI) (2006). Connections:Joseph P. Nicoletti. EERI Oral History Series, Interviewer: StanleyScott. Earthquake Engineering Research Institute, Oakland, CA.

Esteva, L. (1970). Seismic risk and seismic design decisions, in SeismicDesign of Nuclear Power Plants, R. J. Hansen (Editor), MIT Press,Cambridge, MA.

Harmsen, S., and A. Frankel (2001). Geographic deaggregation of seismichazard in the United States, Bull. Seismol. Soc. Am. 91, no. 1, 13–26.

International Code Council (ICC) (2000). International Building Code (IBC)2000, Building Officials and Code Administrators International, Inc.,Country Club Hills, IL; International Conference of Building Officials,Whittier, CA; Southern Building Code Congress International, Inc.,Birmingham, AL.

Leonard, G., and D. M. Steinberg (2002). Seismic hazard assessment: simul-taneous effect of earthquakes at close and distant sites, Earthq. Spectra18, no. 4, 615–629.

Leyendecker, E. V., R. J. Hunt, A. D. Frankel, and K. S. Rukstales (2000).Development of maximum considered earthquake ground motionmaps, Earthq. Spectra 16, no. 1, 557–578.

Malhotra, P. K. (2005). Return period of design ground motions, Seism. Res.Lett. 76, no. 6, 693–699.

Malhotra, P. K. (2006). Seismic risk and design loads, Earthq. Spectra 22,no. 1, 115–128.

MathWorks (2006). MATLAB routine for generating latin-hypercubesample with normal distribution, lhsnorm The MathWorks, Inc.,Natick, MA.

May, P. (2001). Social perspectives about earthquake performance: thefallacy of “acceptable risk”, Earthq. Spectra 17, no. 4, 725–737.

McGuire, R. K. (1995). Probabilistic seismic hazard analysis and designearthquakes: Closing the loop, Bull. Seismol. Soc. Am. 85, no. 5,1275–1284.

McGuire, R. K. (2004). Seismic Hazard and Risk Analysis, EarthquakeEngineering Research Institute, Oakland, CA, 221 pp.

Reiter, L. (1991). Earthquake hazard analysis—issues and insights,Columbia University Press, New York, 254 pp.

Rhoades, D. A., and G. H. McVerry (2001). Joint hazard of earthquake shak-ing at two or more locations, Earthq. Spectra 17, no. 4, 697–710.

Smith, W. D. (2003). Earthquake hazard and risk assessment in New Zealandby Monte Carlo methods, Seism. Res. Lett. 74, no. 3, 298–304.

Stein, S. (2005). Comment on “How can seismic hazard in the New MadridSeismic Zone be similar to that in California?” by Arthur Frankel,Seism. Res. Lett. 76, no. 3, 364–365.

Wang, M., and T. Takada (2005). Macrospatial correlation model of seismicground motions, Earthq. Spectra 21, no. 4, 1137–1156.

Wang, Z., B. Shi, and J. D. Kiefer (2005). Comment on “How can seismichazard in the New Madrid Seismic Zone be similar to that in Califor-nia?” by Arthur Frankel, Seism. Res. Lett. 76, no. 4, 466–471.

Wells, D. L., and K. J. Coppersmith (1994). New empirical relationshipsamong magnitude, rupture length, rupture width, rupture area, andsurface displacement, Bull. Seismol. Soc. Am. 84, no. 4, 974–1002.

Wesson, R. L., and D. M. Perkins (2001). Spatial correlation of probabilisticearthquake ground motion and loss, Bull. Seismol. Soc. Am. 91, no. 6,1498–1515.

Factory Mutual Insurance Company (FM Global)1151 Boston-Providence TurnpikeNorwood, Massachusetts [email protected]

Manuscript received 20 September 2007

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Seismic Design Loads from Site-Specific and Aggregate