Bradley_EpistemicUncertaintiesNzPsha_BSSA_2012 (1).pdf

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Consideration and Propagation of Epistemic Uncertaintiesin New Zealand Probabilistic Seismic-Hazard Analysis

by Brendon A. Bradley, Mark W. Stirling, Graeme H. McVerry, and Matt Gerstenberger

Abstract This article presents results from the consideration of epistemic uncer-tainties in New Zealand ( NZ) probabilistic seismic-hazard analysis. Uncertainties inground-motion prediction are accounted for via multiple ground-motion predictionequations within the logic-tree framework. Uncertainties in the fault-based seismicityof the earthquake rupture forecast due to uncertainties in fault geometry, slip param-eters, and magnitude-scaling relationships are considered in a Monte Carlo simulationframework. Because of the present lack of fault-specific data quantifying uncertaintiesfor many faults in NZ, representative values based on judgement and available data for NZ and foreign faults were utilized. Uncertainties in the modelling of backgroundseismicity were not considered. The implications of the considered epistemic uncer-tainties in terms of earthquake magnitude frequency distributions and probabilistic

seismic-hazard analyses for two spectral acceleration ordinates, two soil classes,and two locations (Wellington and Christchurch) are examined. The results illustratethat, for the uncertainties considered, the variation in seismic hazard due to theadopted ground-motion prediction model is larger than that due to the uncertaintiesin the earthquake rupture forecast. Of the earthquake rupture forecast uncertaintiesconsidered, the magnitude-geometry scaling relationships was the most significant,followed by fault rupture length. Hence, the obtained results provide useful guidanceon which modelling issues are the most critical in the reliability of seismic-hazardanalyses for locations in NZ.

IntroductionThe location of New Zealand ( NZ) astride the boundary

of the Australian and Pacific plates makes it a country of highseismicity. For the purposes of mitigation, seismic hazard inNew Zealand is routinely computed on the basis of a prob-abilistic seismic-hazard analysis ( PSHA ). The two basic in-gredients of a PSHA are (i) an earthquake rupture forecast (ERF ), quantifying the location and likelihood of all possibleearthquake ruptures that may occur; and (ii) a ground-motionprediction equation ( GMPE ), quantifying the ground motionshaking at a specific location due to the occurrence of anearthquake rupture.

Because of the complexity of earthquake rupture, wavepropagation, and local site effects, ERFs and GMPE s aretypically probabilistic in that they incorporate uncertaintiesin the parameters they attempt to quantify. In PSHA , it iscommon to distinguish between two main types of uncertain-ties. The first is uncertainties that, for the given modelsadopted, are deemed to be purely random and unpredictableand are referred to as aleatory variability. The second is that which arises due to limited knowledge of the phenomena being predicted and is referred to as epistemic uncertainty.An example of the aleatory variability is the variability in

ground-motion amplitudes at a given distance from anearthquake predicted using a GMPE , while an example of epistemic uncertainty is the assessment of which GMPE ismost appropriate to be used for a particular problem under consideration.

The benefit of making the distinction between aleatoryvariability and epistemic uncertainties is that, in principle,epistemic uncertainties can be reduced with improvedknowledge (both empirical and theoretical), while aleatoryvariability is assumed to be purely random. Clearly, sucha distinction is somewhat idealistic in that some of theobserved aleatory variability could be due to systematiceffects (hence, strictly being a source of epistemic uncer-tainty), resulting from, for example, the simplified natureof GMPE s. Despite this idealization, the separation of alea-tory variability and epistemic uncertainties is still important in assessing performance over time (e.g., Der Kiureghian andDitlevsen, 2008 ).

The consideration of epistemic uncertainties in PSHA via logic trees was first proposed by Kulkarni et al. (1984) .Logic trees allow the consideration of alternative modelsand the values of their parameters to be used in PSHA , with

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Bulletin of the Seismological Society of America, Vol. 102, No. 4, pp. 1554 1568, August 2012, doi: 10.1785/0120110257


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each alternative possibility given a degree of belief such that the sum of the possibilities adds to 1.0. Although, there isrelatively little literature on the logic-tree concept (as a result of the field of PSHA being largely practice driven), Bommer et al. (2005) and Bommer and Scherbaum (2008) provideuseful overviews and potentially adverse consequences.Since their inception, logic trees have become commonly

employed in PSHA to attempt to account for epistemic un-certainties. However, at present, nationwide seismic-hazardanalyses for New Zealand ( Stirling et al. , 2011 , 2002) do not explicitly account for epistemic uncertainties, instead usingonly preferred values of parameters in the ERF and a sin-gle GMPE .

This article presents a summary of the results obtainedfrom consideration of epistemic uncertainties in nationwidePSHA in NZ. First, the methodology by which epistemic un-certainties are considered is discussed. Second, the influenceof epistemic uncertainties on the nationwide magnitude frequency distribution of fault-based seismicity and seismic-

hazard analyses for various intensity measures and locationsare examined. The observations are discussed with referenceto the presently employed PSHA methodology, as well asthe significance of epistemic uncertainties obtained in rela-tion to other studies. Finally, the limitations of the study areaddressed. This article is intended to provide an overview of this study, and more elaborate discussion of the methodologyand presentation of results can be found in Bradleyet al. (2011) .

Adopted Epistemic Uncertainty Methodology

Consideration of epistemic uncertainties in a hazardassessment serves two primary objectives. First, it providesan assessment of the significance of various model uncertain-ties of the estimates of the particular hazard considered andsubsequently can be used to provide confidence estimates.Second, the results can be used to understand the sensitivityof the results to specific inputs and hence which sources of modelling uncertainty are critical in precisely quantifying thehazard. In this study, emphasis was placed primarily on thelatter of these two objectives. The primary reason for this isthat no attempt was made to consider epistemic uncertaintiesin the inputs for a PSHA in an exhaustive manner; and, there-fore, the results obtained cannot be considered to represent the full distribution of uncertainty in the hazard estimates.Second, the study was conducted in the context of continuingimprovement of seismic-hazard estimation in NZ and there-fore ascertaining how the precision of such hazard estimatescan be most efficiently improved with limited resources. Assuch, and in light of the fundamental nature of several of theassumptions used in PSHA (as elaborated by Field, 2007 ), theconsideration of epistemic uncertainties is kept relativelysimple. The adopted methodology is separately explainedfor the GMPE and ERF components of PSHA .

Epistemic Uncertainty in NZ GMPE

Epistemic uncertainty in ground-motion prediction re-sults from (i) uncertainties in the values of input parametersof a GMPE and (ii) uncertainty regarding the appropriatenessof one or more prospective GMPE s. Uncertainties in GMPEparameter inputs relating to fault and wave-propagation de-tails are incorporated within the ERF. Uncertainties in GMPEparameter inputs relating to local site effects are not consid-ered herein; the primary reason for this omission is that theconventional development of GMPE s does not remove theepistemic uncertainty of such parameters ( Gehl et al. , 2011 ;Moss, 2011 ), which would result in double-counting of suchuncertainty. Uncertainty regarding the appropriateness of prospective GMPE s is considered via the conventional useof logic trees as discussed subsequently in this article.

The McVerry et al. (2006) model, developed based onpre-1995 NZ strong-motion data, is the only NZ-specificGMPE developed to predict both peak ground accleration(PGA ) and spectral acceleration ( SA) and consequently is the

only GMPE considered in conventional NZ PSHA (Stirlinget al. , 2011 ), referred to herein as the 2010 New Zealandnational seismic hazard model ( NSHM ). Hence, the present consideration of GMPE uncertainty requires the considera-tion of foreign GMPE s. Bradley (2010) and Bradley et al.(2011) examined the applicability of various foreign GMPE s,as well as the McVerry et al. (2006) model, based on ob-served strong-motion data in NZ up to 2009. These studiesconsidered four GMPE s for active shallow crustal events andthree for subduction slab and subduction interface events. Acomparison of the magnitude and source-to-site distancescaling of the considered GMPE s is given in Figure 1. For brevity, specific details of these studies are omitted here,

other than the recommended logic tree and associatedweights to account for GMPE uncertainty provided in Table 1.In particular, it is noted that the Atkinson and Boore (2003)model was assigned zero weight due to its poor comparisonwith recorded data ( Bradley, 2010 ). Te considered GMPE s of McVerry et al. (2006) , Zhao et al. (2006) , Boore and Atkin-son (2008) , Chiou et al. (2010) , and Atkinson and Boore(2003) henceforth are abbreviated as McV06 , Z06, BA08 ,C10 , and AB03 , respectively.

Epistemic Uncertainty in NZ ERF

The current methodology adopted for PSHA in NZ, as

implemented by Stirling et al. (2002, 2011) , is based on a combination of fault-based sources and distributed seismicityas depicted in Figure 2. The fault-source model uses thedimensions and slip rates of mapped fault sources to developmagnitude frequency estimates for characteristic earth-quakes, and the spatial distribution of historical seismicity isused to develop magnitude frequency estimates for the back-ground seismicity model. Because the fault-based seismicityis assumed to be characteristic in nature, it is assumed that events less than this characteristic magnitude are modeled bythe distributed background sources.

Consideration and Propagation of Epistemic Uncertainties in New Zealand PSHA 1555


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Only epistemic uncertainties in the fault-based compo-nent of the ERF are considered in this study, which is con-sistent with similar studies conducted elsewhere (e.g.,Working Group on California Earthquake Probabilities[WGCEP], 2003 ). That is, no uncertainties in backgroundseismicity are considered. The implications of this are dis-cussed in light of the results obtained subsequently. Further-more, given underlying fundamental assumptions (e.g., fault segmentation and time-independence), the consideration of uncertainties in the fault-based component of the ERF is lim-ited to the consideration of uncertainties in the parametersused to specify the characteristic distribution of each fault source rather than to consider different magnitude frequencydistributions for fault sources. Hence, the consideration andpropagation of such uncertainties first requires an overviewof the methodology by which the characteristic magnitude-frequency distribution of modeled faults is determined.

Deterministic Calculation of Source Magnitudes and Ratesof Occurrence. The seismic hazard for characteristicruptures, as modeled in the fault-model of Stirling et al.

4 5 6 7 8

10

-2

10-1

Moment magnitude, M w

Moment magnitude, M w

, S A ( T ) ( g )

n o i t a r e l e c c a l a r t c e p

S

Active shallow crustalPGA

McV06Z06BA08C10

R rup

=50km

10km

100

101

102

10-2

10-1

100

Source-to-site distance, R rup (km)

, S A ( T ) ( g )


S

Active shallow crustalPGA

McV06Z06BA08C10

M w=5.5

M w=7.5

4 5 6 7 8

10-2

10-1 ,

S A ( T ) ( g )


S

Subduction interfacePGA

McV06Z06AB03

R rup

=120km

R rup =50km

101

10210

-3

10-2

10-1

Source-to-site distance, R rup (km)

, S A ( T ) ( g )


S

Subduction interfacePGA

McV06Z06AB03 M w=5.5

M w=7.5

10-2

10-1

1000

0.1

0.2

0.3

0.4

Period, T (s)

I n t e r - e v e n

t S t d D e v ,

McV06Z06BA08C10

M w = 7

M w = 5

10-2

10-1

1000.2

0.3

0.4

0.5

0.6

0.7

Period, T (s)

, v e

D d t S t n e v e - a r t n I

McV06Z06BA08C10

M w = 5

M w = 7

(a) (b)

(c) (d)

(e) (f)

Figure 1. Variation in PGA amplitudes predicted by the various GMPEs considered in this study: (a,b) PGA scaling with magnitude andsource-to-site distance for active shallow crustal earthquakes; (c,d) PGA scaling with magnitude and source-to-site distance for subductioninterface earthquakes; and (e,f) interevent and intraevent standard deviation dependence on vibration period. Figures for subduction slabevents are omitted and can be found in Bradley (2010) . (McV06, McVerry et al. , 2006; Z06, Zhao et al. , 2006; BA08, Boore and Atkinson,2008 ; C10, Chiou et al. , 2010; AB03, Atkinson and Boore, 2003 .) The color version of this figure is available only in the electronic edition.

Table 1Logic-Tree Weights Used for Various GMPEs for

Different Tectonic Environments ( Bradley et al. , 2011)Model Abbreviated Name Weight

Active Shallow Crustal SourcesMcVerry et al. (2006) McV06 0.2Zhao et al. (2006) Z06 0.2Boore and Atkinson (2008) BA08 0.28Chiou et al. (2010) C10 0.32

Subduction Slab and Interface SourcesMcVerry et al. (2006) McV06 0.4Zhao et al. (2006) Z06 0.6Atkinson and Boore (2003) AB03 0.0

1556 B. A. Bradley, M. W. Stirling, G. H. McVerry, and M. Gerstenberger


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(2002 , 2011) , is governed by the magnitude M w and recur-rence interval of the characteristic rupture. The fault isdescribed by several geometrical parameters (length, L; topof rupture extent, D top ; bottom of rupture extent, Dbottom ;dip, ) and deformation parameters (slip rate, _s; couplingcoefficient, c). Based on the fault geometry, a magnitude-scaling relation is used to compute the magnitude of the rup-ture. For New Zealand faults, four different magnitude-scaling relations are used, depending on the classification of the fault. For crustal plate-boundary faults, the relationshipof Hanks and Bakun (2002) is utilized:

Median: M w 3 :394

3log10 A; and

Sigma: M w 0 :22 : (1)

For normal faults in volcanic and rift environments, therelation of Villamor et al. (2001) is utilized:

Median: M w 3 :394

3log10 A; and

Sigma: M w 0 :195 : (2)

For all other crustal faults, the relationship of Berrymanet al. (2001) is utilized:

Median: M w 4 :192

3log10 W

4

3log10 L; and

Sigma: M w 0 :18 : (3)where W is the fault width.

For subduction interface events, the relationship of Strasser et al. (2010) is utilized:

Median: M w 4 :441 0 :846 log10 A; and

Sigma: M w 0 :286 : (4)No scaling relation is given for subduction slab events

because these are modeled as point sources in the back-ground seismicity model. The median of equations (1)(3)are used in computing the characteristic magnitudes of crus-tal sources in the NSHM (i.e., Stirling et al. , 2002, 2011 ), but the standard deviations are not considered. For subductioninterface events, Stirling et al. (2011) used assumed valuesof average displacement and computed magnitudes from thedefinition of moment magnitude (equation 6), however, thescaling relation of equation (4) will be used herein.

For the given moment magnitude estimated from a mag-nitude-scaling relation, the seismic moment M 0 (in units of Nm) can be computed from ( Hanks and Kanamori, 1979 )

log10 M 0 9 :05 1 :5 M w

: (5)

The total seismic moment rate for the fault can becomputed as follows ( Brune, 1968 ):

_M 0 A_sc; (6)

where is the shear rigidity of the fault interface; _s is theaverage slip rate over the area of the fault surface, and c isthe coupling coefficient. Based on the assumption that all themoment rate accumulating on the modeled fault surface isreleased in characteristic events (recalling the assumption

Figure 2. (a) Modeled characteristic fault sources and (b) seismotectonic background seismicity regions in the 2010 New Zealand Na-

tional Seismic Hazard Model (NSHM; figure modified from Stirling et al. , 2011 ). The locations of Christchurch and Wellington, which areconsidered as case study locations in this article, are also annotated. (KMF, Kapiti-Manawatu faults; North Is, North Island; MFS, Marlbor-ough fault system). The color version of this figure is available only in the electronic edition.



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that noncharacteristic events are modeled as backgroundsources), the mean occurrence rate can be computed from

_M 0

M 0: (7)

Consideration of Epistemic Uncertainties in the Fault Model. The values of the characteristic magnitude and fre-quency of modeled faults in the NSHM are uncertain as a result of uncertainties in fault geometry and fault deforma-tion and of uncertainty in empirical magnitude geometry-scaling relationships (i.e., equations 14). To account for such uncertainties, the characteristic magnitude and fre-quency of modeled faults used in PSHA were obtained basedon the Monte Carlo simulation procedure given in Figure 3.

Specific Values of Fault-Based Uncertainties Used in thePresent Study. Given that epistemic uncertainties are a re-sult of a lack of knowledge (both theoretical and empirical),those faults that have had less attention devoted to them willbe modeled with parameters/relationships that have a greater

epistemic uncertainty. Therefore, ideally one would haveestimates for the magnitude of the parameter/relationshipuncertainties that are fault specific. Unfortunately, this isnot the case for the faults in the NZ seismic-hazard model,with the majority of such data not presently cataloged. As a result, the approach taken here is to make use of availablefault-specific data and judgement in the absence of such data to assign uncertainties to each of the parameters. A key con-sideration in the subsequent analyses conducted is a sensi-

tivity study to assess the importance of each parameter uncertainty in the overall picture of seismic hazard, thereforeindicating which parameters deserve more rigorous estima-tion in future.

In determining the magnitude of uncertainties to assignto faults without specific parameter uncertainty estimates,use was made of both NZ-specific and foreign data, primarily

from the Working Group on California Earthquake Probabil-ities (WGCEP, 2003 ). The following information from WGCEP was obtained ( Bradley et al. , 2011 ): (i) coefficientsof variation ( COV) for fault length range from 0.06 to 0.28,with a mean of 0.15; (ii) fault width COVs range from 0.08 to0.12, with a mean of 0.10; (iii) fault slip rate COVs rangefrom 0.06 to 0.34, with a mean of 0.18. In addition to theparameter uncertainty magnitudes in the WGCEP study,various estimates are also available for well-studied and/or recently studied NZ faults. Examples of such estimatesinclude the following slip rate COVs: 0.14, Wellington fault (Van Dissen, 2010 ); 0.10, Ostler and Irishman Creek faults(Amos et al. , 2007); 0.16, Ohariu fault ( Heron et al. , 1998);0.1, Hope fault ( Langridge and Berryman, 2005 ); 0.15,Porters Pass fault ( Howard et al. , 2005); 0.16, Blue Mountainfault (Pace et al. , 2005); 0.15, Ohariu fault ( Litchfield et al. ,2006); 0.25, Taupo rift faults ( Villamor and Berryman,2006); 0.13, Wairau fault ( Zachariasen et al. , 2006); 0.08,Alpine fault ( Zachariasen et al. , 2006); 0.08, Wairarapa fault (Van Dissen and Berryman, 1996 ); and 0.1, Hikurangi sub-duction zone sources ( Wallace et al. , 2009); among others.On the basis of these and the WGCEP values, a COV of 0.20was assigned to slip-rate uncertainties for general faults.Estimates for fault length, depth, and dip uncertainties for NZ faults are few and far between; and, therefore, use was

largely made of the WGCEP (2003) study in assigningvalues for these uncertainties.Table 2 provides the values of parameter uncertainties to

assign to faults without fault-specific data, as well as the as-sumed distribution. Mean values of parameters for individualfaults are documented in Stirling et al. (2011) . For use in thelater Monte Carlo simulations, these parameter distributionswere truncated at two standard deviations from the mean.Truncated normal distributions were used for representing

Table 2Uncertainties Assigned to Fault Parameters in the Absence of Fault-Specific Data

Parameter Uncertainty Assumed Distribution

Length of fault plane, L COV 0 :15 * Truncated normalTop of rupture extent, D top

1 12p km Bounded uniform

Bottom of rupture extent, D bottom 1 km Truncated normalFault dip, 5 Truncated normalFault slip rate, _s COV 0 :20 Truncated normalCoupling coefficient, c COV 0 :15 [interface sources only] Truncated normalMagnitude-scaling relationship Fault-type specific [i.e., equations (1)(4)] Truncated normal

*COV, coefficient of variation; equal to the standard deviation divided by the mean.For a uniform distribution, a standard deviation of 1 =

12p refers to maximum and minimum values that are

0.5 units above and below the mean value.If the minimum depth was negative, then it was set to zero and the distribution function renormalized.

for i=1:nsimulation for j=1:nfaults

1. Generate a random set of geometrical fault parameters ( )and deformation parameters (

2. Using the appropriate magnitude scaling relation (Equations (1)(4)), deter-mine the mean and standard deviation of the characteristic magnitude.

3. From the mean and standard deviation of the characteristic magnitude, gener-ate a randomly realized magnitude, (for fault j and realization i).

4. For the generated magnitude get the associated seismic moment (Equa-tion (5) and (6)), and determine the mean annual rate of occurrence,(Equation (7)).

end end

).

Figure 3. Monte Carlo procedure for fault-based epistemicuncertainty consideration.



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large magnitudes, where the parameter uncertainties result ina smoother variation in frequency compared with the stepchanges that result in the case of no uncertainties. It is alsoworth recalling that, in this study, magnitudes for subduction

interface events (which dominate the magnitude frequencydistribution around M w 8) were estimated using the scalingrelation of Strasser et al. (2010) , compared with the assumedaverage displacements utilized by Stirling et al. (2011) . At small magnitudes, the consideration of epistemic uncertain-ties leads to a higher exceedance rate. The reason for thisdifference is that if the simulated characteristic magnitude issmaller than the mean value, then the corresponding occur-rence rate will be higher. Comparison of the size of the uncer-tainty in the magnitude frequency distribution due to eachof the parameter uncertainties illustrates that magnitude-scaling-relation uncertainty is the most significant, followedby fault-length uncertainty. Thisobservation is seen explicitlyin Figure 5a , which illustrates the lognormal standard devia-tion of the exceedance frequency as a function of magnitudefor various parameter uncertainties. Figure 5b presents theuncertainty in the magnitude frequency distribution whenmultiple uncertainties are considered. Specifically, all theparameter uncertainties related to fault geometry (i.e., fault length, rupture top, rupture depth,fault dip), fault deformation(i.e., slip rate and coupling coefficient for interface sources)were considered simultaneously. Figure 5b illustrates that,despite the grouping of uncertainties, magnitude-scalingrelation uncertainty remains the dominant uncertainty in thenationwide fault-based magnitude frequency distribution.

Figure 6 presents the uncertainty in the magnitude frequency distribution due to the consideration of all fault parameter uncertainties. As with the previous figures, theNSHM and mean curves are similar and show the aforemen-tioned variation at larger magnitudes. Based on the time-independent nature of the ERF (i.e., a Poisson temporaldistribution), the 50-year exceedance probability of an event with magnitude greater than 7 is essentially 100%. TheNSHM 50-year probability of a magnitude 7.5 or greater event is 79%, while this study results in a mean probabilityor 84% and a 68% confidence interval (i.e., that between the

16th and 84th percentiles) of 73% 88%. For a magnitude 8or greater event, the NSHM probability is 35%, while themean probability is 25% with a 68% confidence intervalof 1231% for this study. As a result of the fault parameter

uncertainties not correlating between different faults, a magnitude frequency distribution for all faults has a smaller uncertainty than that for a smaller subset of faults. Hence, if the seismic hazard at a single site is dominated by a smallsubset of seismic sources, such uncertainties will be morepronounced.

Figures 46 illustrate that the magnitude-scaling relationuncertainty tends to be the dominant uncertainty for modeledfaults. Magnitude-scaling relation uncertainty was also trea-ted on a fault-specific basis through the use of equations (1)(4). Hence, this suggests that the approximate uncertaintyvalues that are prescribed for the other fault parameters

(in cases where fault-specific parameter uncertainties were

5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Magnitude, M w Magnitude, M w

S i g m a ,

l n

S i g m a ,

l n

M w ScalingLengthRup BottomRup TopSlip RateDipCoup Coef

5 6 7 8 90

0.2

0.4

0.6

0.8

1

1.2M w ScalingGeometryDeformationAll

(a) (b)

Figure 5. Lognormal standard deviation in the exceedance rate of various magnitudes considering various fault parameter uncertainties:(a) individual parameter uncertainties and (b) parameter uncertainties by group. (See text for details.) The color version of this figure isavailable only in the electronic edition.

A n n u a

l e x c e e

d a n c e

f r e q u e n c y ,

M

w

5 6 7 8 910 -4

10-3

10 -2

10 -1

10 0

Magnitude, M w

NSHM

Mean

16 th, 84 th

Considering all fault uncertainties

Figure 6. Uncertainty in the nationwide fault-based seismicitymagnitude frequency distribution, considering all fault parameter uncertainties. Values are included for the 16th and 84th percentiles.The color version of this figure is available only in the electronicedition.



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unavailable) are likely to have a second-order effect on thesize of the total uncertainty.

Seismic-Hazard Analysis Case-Study SitesConsidered

In order to examine the importance of various epistemic

uncertainties for a range of situations, seismic-hazardanalyses for two different sites, soil classes, and intensitymeasures are presented. Generic locations in Wellington andChristchurch are selected, both because they are NZs largest two cities and because at the 475-year return period, Well-ington s hazard is dominated by fault sources, while in con-trast, background sources provide a significant contributionfor Christchurch s hazard. Hazard analyses are performed at these two generic locations for both site class B (weak rock)and site class D (deep soil) site conditions (NZS 1170.5,Standards New Zealand, 2004 ). Finally, hazard analyses areperformed for both PGA and SA(2.0), in order to generallydepict the magnitude of epistemic uncertainties at short and

long vibration periods. A minimum magnitude of M w 5.0was considered in calculations.

Uncertainty in Seismic Hazard for Wellington

Figure 7 illustrates seismic-hazard curves for PGA andsite class D at a generic location in central Wellington(latitude 41 :2889 , longitude 174.7772) for an exposureperiod of 50 years. In Figure 7, and similar subsequent figures, 50 simulations for each different GMPE logic-treecombination (i.e., Table 1) were used to sample the consid-ered ERF epistemic uncertainties (i.e., Fig. 3). These indivi-dual simulations are shown in thin lines, with the mean of

this ERF uncertainty (but for a single GMPE ) shown in thicker lines. From the observation that the individual simulationstend to be in distinct groups for each GMPE combination,it is apparent that the GMPE uncertainty tends to be larger than that due to ERF uncertainty. The importance of GMPEuncertainty for different tectonic source types can be seen byexamining the differences in the seismic-hazard curves dueto different GMPE combinations in Figure 7. For example,

Figure 7a illustrates the difference between the hazard usingthe Z06 subduction GMPE but different crustal GMPEs, whileFigure 7b illustrates the difference in the hazard using theMcV06 subduction GMPE but different crustal GMPE s. BothFigure 7a and 7b clearly illustrate that the Z06 crustal GMPEleads to a significantly larger hazard than that using other crustal GMPEs, which give similar hazards and are consistent

with the median and sigma GMPE scaling in Figure 1.While the seismic hazards obtained using the C10 , BA08 ,and McV06 GMPE s in Figure 7a and 7b are similar relative toeach other, they differ due to the differences in the subduc-tion zone GMPE used. Figure 8 explicitly illustrates the sig-nificance of subduction zone GMPE uncertainty for PGA andsite class D conditions in Wellington. Both Figure 8a and 8billustrate that the Z06 GMPE tends to result in a larger hazardcompared with the McV06 GMPE .

Comparison of Figures 7 and 8 illustrates that bothactive shallow crustal and subduction zone sources contri-bute significantly to the hazard, and its epistemic uncertainty,for the given conditions. This observation is consistent withthe knowledge that Wellington is located close to several ac-tive shallow crustal faults (most notably the Wellington fault)and to the Hikurangi subduction zone, for which the interfacepasses approximately 20 km underneath Wellington ( Stirlinget al. , 2011 ).

In interpretation of the observed epistemic uncertaintiesfor Wellington, it is worth bearing in mind that the seismichazard at this location is generally dominated by fault-basedseismicity rather than background sources. Hence, theneglect of background seismicity epistemic uncertainties islikely inconsequential for Wellington.

Uncertainty in Seismic Hazard for Christchurch

Figure 9 illustrates the seismic-hazard curves for PGAand site class D at a generic location in central Christchurch(latitude 43 :5300 , longitude 172.6203) for an exposureperiod of 50 years. Because Christchurch is subject tonegligible seismic hazard from subduction zone sources, onlyresults for the four different crustal GMPE s being consideredin this study are explicitly annotated. An obvious observation

10 -2 10 -1 10 010 -3

10 -2

10 -1

10 0

Peak ground acceleration, PGA ( g)10 -2 10 -1 10 0

Peak ground acceleration, PGA ( g)

WellingtonSite Class D

C10Z06McV06BA08

Effect of crustal GMPEAll results use the Z06subduction GMPE

Mean of realizations

with a single GMPE

Individualrealizations


C10McV06Z06BA08

Mean of realizationswith a single GMPE

Effect of crustal GMPEAll results use the McV06subduction GMPE


5 0

- y e a r e x c e e

d a n c e

p r o

b a

b i l i t

y

10 -3

10 -2

10 -1

10 0

5 0

- y e a r e x c e e

d a n c e

p r o

b a

b i l i t

y(a) (b)

Figure 7. Seismic-hazard curves for site class D PGA in Wellington: (a) using the Z06 subduction GMPE and various crustalGMPEs; and (b) using the McV06 subduction GMPE and various crustal GMPEs (which tend to be clustered together and appear as a thicker gray line here). The color version of this figure is available only in the electronic edition.



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in the nature of epistemic uncertainty in the seismic-hazardcurves in Figure 9, relative to that for Wellington (i.e., Figs. 7

and 8), is the reduction in the seismic-hazard uncertainty dueto ERF uncertainty. This observation is the result of twofactors. First, unlike Wellington, which is a high seismicregion dominated by a handful of fault-based seismic sources,Christchurch is a regionof relatively lowerseismicity, andsig-nificant contributions to the seismic hazard are made by bothnumerous fault-based sources and background seismicity asdepicted in Figure 10. As no epistemicuncertainties were con-sidered in the background seismicity model, there is conse-quently less uncertainty in the total seismic hazard due toboth fault andbackground seismicity. Thesecondfactor in the

reduction in ERF uncertainty for Christchurch is a result of thethere being numerous fault-based sources that significantly

contribute to the seismic hazard. As previously noted withrespect to the uncertainty in the nationwide fault-basedmagnitude frequency distribution (i.e., Fig. 6), because fault-source uncertainty is not correlated between different faults, a larger number of fault sources providing a substantial contri-bution to theseismic hazard will consequently lead to a reduc-tion in the magnitude of the epistemic uncertainty in theseismic hazard. The effect of removing the backgroundsources for the epistemic uncertainty in the Christchurchseismic hazard is explored in the following section.

Christchurch Hazard with the Removal

of Background SeismicityFigure 11 illustrates the seismic hazard computed with

and without the consideration of background seismicitysources. Comparison with Figure 9, in which fault and back-ground source are considered, illustrates that the backgroundsources are significant, with the 10% in 50-year exceedanceprobabilitybeingapproximately 0 :2 0 :24 g when backgroundsources are considered (i.e., Fig. 9) and 0 :12 0 :19 g whenbackground sources are neglected (i.e., Fig. 11). In additiona,Figure 11 shows that, even in the fault-based seismicityonly case, the ERF component of the total epistemic uncer-tainty is insignificant compared with the GMPE uncertainty

(i.e., the variation in the seismic hazard considering ERFuncertainty, but only a single GMPE is relatively small).Interestingly, the McV06 GMPE results in the largest seismichazard for exceedance probabilities greater than 10% in50 years when background sources are considered but resultsin the smallest seismic hazard when background sources areneglected. This occurs because the McV06 model is known tosignificantly overpredict ground motions produced by smallmagnitude ( M w < 6 ) events (e.g., Fig. 1a ), which are the pre-dominant background sources for the PGA hazard at the 10%in 50-year exceedance probability (e.g., Fig. 10a ).

5 0 - y e a r e x c e e

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Z06McV06


Effect of subduction GMPEAll results use the C10crustal GMPE



McV06

Z06



Effect of subduction GMPEAll results use the McV06crustal GMPE

Figure 8. Seismic-hazard curves of peak ground acceleration for site class D in Wellington: (a) using the Z06 subduction GMPE andvarious crustal GMPEs; and (b) using the McV06 subduction GMPE and various crustal GMPEs. Individual realizations are shown in thinlines, while thicker lines represent the mean hazard of the 50 realizations of fault-based seismicity for each GMPE combination (which tendto be clustered together and appear as a thicker gray line here). The color version of this figure is available only in the electronic edition.

10 -2 10 -1 10 010 -3

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ChristchurchSite Class D

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d a n c e

p r o

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y

Figure 9. Seismic-hazard curves of peak ground accelerationfor site class D in Christchurch. Individual realizations are shownin thin lines, while thicker lines represent the mean hazard of the 50realizations of fault-based seismicity for each GMPE combination(see inset) (which tend to be clustered together and appear as a thicker gray line here). The color version of this figure is availableonly in the electronic edition.



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Magnitude of Epistemic Uncertainties in Termsof Design Seismic Demands

In order to understand the salient differences in episte-mic uncertainties for NZ PSHA , Figures 12 and 13 illustratethe seismic-hazard curves for the eight different cases con-sidered with explicit indication of the mean, median, and 5thand 95th percentiles, while Table 4 provides these values for the 10% and 2% exceedance probabilities in 50 years. Themagnitude of epistemic uncertainties is clearly significant indetermining the ground-motion hazard level for a given ex-ceedance probability or vice versa. For example, while themean hazard for the 10% and 2% exceedance probabilitiesin 50 years for PGA and site class B in Wellington are 0 :48 g

and 0 :93 g, respectively, the 90% confidence intervals are0 :41 g; 0 :57 g and 0 :76 g; 1 :19 g . That is, the 90% confi-dence interval for the 10% in 50-year exceedance probabilityis almost 0 :2 g and for the 2% in 50-year exceedance prob-ability is slightly over 0 :4 g. On the other hand, for SA(2.0)and site class D in Christchurch, the mean SA(2.0) hazard for the 10% and 2% exceedance probabilities are 0 :16 g and0 :28 g, respectively, while the 90% confidence intervalranges on these mean values are 0 :11 g; 0 :21 g and 0 :21 g;0 :35 g , respectively. That is, the range of the 90% confidenceinterval is 0 :1 g and 0 :14 g for these two exceedance probabil-ities, respectively. Table 4 also illustrates the values from theNew Zealand loadings standard NZS 1170.5 ( Standards NewZealand, 2004 ), for the considered cases. For the most part,the values are in agreement, although the mean values ob-

tained in this study are generally larger than the NZS1170.5 values for Wellington and smaller than the corre-sponding values for Christchurch.

Table 5 provides another interpretation of the data inTable 4 based on the ratio of the difference between the 95%and 5% fractiles and mean ground-motion intensity for a given exceedance probability. From Table 5, the averageratio is approximately 0.49, but such ratios vary significantlyfrom 0.20 0.85. There is relatively little difference betweenthe size of the uncertainty for Wellington and Christchurch,with average ratios of 0.46 and 0.52, respectively. Again, it isnoted that the magnitude of the uncertainty for Christchurchin particular is artificially low due to the neglection of back-ground seismicity uncertainty, which will be most noticeablefor PGA , in which frequent small magnitude events are most important (i.e., Fig. 10). The size of the ratios in Table 5generally increases with reducing exceedance probabilityfor Wellington but are essentially constant for Christchurch.Finally, it is noted that the ratios are not systematically larger for site class D cases than for site class B cases, which maybe assumed a priori because of the different manner in whichGMPE s account for local site response, something whichshould be relatively insignificant for soft rock (class B) sites.

Figure 10. Seismic-hazard deaggregation for Christchurch (site class D) for 2% in 50-year exceedance probability: (a) PGA and (b) SA(2.0), considering both fault and background seismicity sources. The color version of this figure is available only in the electronic edition.

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ChristchurchSite Class DNo BackgroundSources

C10

BA08

Z06

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0.1 0.2 0.30.05

0.1

0.15

Figure 11. Seismic-hazard curves of peak ground accelerationfor site class D in Christchurch as a result of only fault-based seis-micity (i.e., neglecting background sources). Individual realizationsare shown in thin lines, while thicker lines represent the mean ha-zard of the 50 realizations of fault-based seismicity for each GMPEcombination (see inset) (which tend to be clustered together andappear as a thicker gray line here). The color version of this figureis available only in the electronic edition.



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10-2

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Figure 12. Comparison of seismic-hazard analyses with epistemic uncertainty and those of the conventional NZ NSHM for the four cases considered in Wellington. The color version of this figure is available only in the electronic edition.

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Figure 13. Comparison of seismic-hazard analyses with epistemic uncertainty and those of the conventional NZ NSHM for the 4 casesconsidered in Christchurch. The color version of this figure is available only in the electronic edition.



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Comparison of Preferred Hazard and the Hazardwith Explicit Epistemic Uncertainties

A key benefit of the consideration of epistemic uncer-tainties is developing an understanding of the impact of certain assumptions on outcomes of a seismic-hazard analy-sis. Previous sections in this article have demonstrated that,of the considered uncertainties, GMPE epistemic uncertaintyis the largest source of uncertainty on PSHA results. There-fore, it is insightful to scrutinize how the present NZ seismic-hazard analyses using the McV06 model compare with theresults presented here (considering multiple GMPE s). Inaddition to the previously discussed results in Figures 12and 13, which are based on seismic-hazard analyses withepistemic uncertainties, the single hazard curve obtainedusing the NSHM methodology (that is, neglecting epistemic

uncertainties in ERF parameters and using the McV06 GMPE )is also depicted. This is referred to as the NSHM Hazardin Figures 12 and 13. For the PGA hazard in Wellington(for both site classes B and D), the NSHM hazard is approxi-mately equal to the 5% fractile of the hazard analysisconsidering epistemic uncertainties for the 10% and 2% in

50-year exceedance probabilities. For the SA (2.0) hazard inWellington (for both site classes B and D), the NSHM hazardis approximately the 95% fractile for the 10% in 50-year exceedance probability but is approximately the 5% fractilefor the 2% in 50-year exceedance probability. For Christch-urch, with the exception of PGA on site class D, the NSHMhazard is approximately equal to the 95% fractile of thehazard considering epistemic uncertainties. As GMPE uncer-tainty produces the largest variation in the considered seis-mic-hazard analyses, the observation of the NSHM hazard (ascompared with the mean and the 5th and 95th percentilesfrom the seismic hazard considering epistemic uncertainties)primarily results from the use of only the McV06 GMPE in theNSHM hazard calculations.

Comparison of Epistemic Uncertainty Magnitude for NZ with that of the San Francisco Bay Area, USA

One of the fundamental difficulties with assessing epis-temic uncertainties in seismic-hazard analyses is that whilethe ultimate aim is to represent the uncertainty in the seismic-hazard estimate, most often the consideration of epistemic

0 0.2 0.4 0.6 0.810

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Using 1997version GMPEs

Figure 14. Comparison of the magnitude of epistemic uncertainty in seismic hazard (dispersion) as a function of exceedance probabilityfor Wellington and Christchurch compared with that of the SFBA from the PSHA conducted by Bradley (2009) : (a) considering ERFuncertainties only; and (b) considering both ERF and GMPE uncertainty. The color version of this figure is available only in the electronicedition.

Table 4Summary of Mean, 5th, and 9th Percentile Hazard Values for 10% and 2%

Exceedance Probabilities (PE) in 50 Years

PE 10% in 50 years PE 2% in 50 years

IM Site Class Mean [5%,95%] Code* Mean [5%,95%] Code*

WellingtonPGA Rock, B 0.48 [0.41,0.57] 0.40 0.93 [0.76,1.19] 0.72

Soft/deep soil , D 0.53 [0.44,0.72] 0 .45 1.01 [0.72,1.50] 0 .80SA(2.0) Rock, B 0.21 [0.16,0.26] 0.23 0.47 [0.39,0.63] 0.42

Soft/deep soil , D 0.41 [0.35,0.47] 0 .48 0.92 [0.78,1.11] 0 .86ChristchurchPGA Rock, B 0.16 [0.12,0.21] 0.22 0.29 [0.22,0.35] 0.40

Soft/deep soil , D 0.22 [0.20,0.24] 0 .25 0.36 [0.32,0.41] 0 .44SA(2.0) Rock, B 0.07 [0.05,0.11] 0.13 0.13 [0.09,0.19] 0.23

Soft/deep soil , D 0.16 [0.11,0.21] 0 .26 0.28 [0.21,0.35] 0 .47

*NZS 1170.5 ( Standards New Zealand, 2004 ).



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were not available; therefore, maintenance of the fault data-base that is used to develop the fault-based component of theNZ ERF should devote further attention to the cataloguing of epistemic uncertainties in raw data.

Epistemic uncertainties were not considered in theparameters (including their spatial correlation) that definethe Gutenberg Richter distribution for background seismi-

city sources. For regions in which the background seismicityrepresents a significant component of the seismic hazard(e.g., Christchurch), clearly this leads to a systematic under-estimation of the epistemic uncertainty in PSHA results. Amethodology for consideration of uncertainties in back-ground source is therefore clearly warranted in order toassess the magnitude of such uncertainties.

Finally, this study focused on the uncertainties of theparameters of the fault-based component of the ERF but not on the ERF methodology itself. Field (2007) noted that thelevel of complexity in treating time dependence in the modelin WGCEP (2003) was inconsistent with other fundamentalassumptions such as fault segmentation. A data-driven meth-odology, with fewer ideologically driven assumptions istherefore a high priority for future seismicity models that in-tegrate both fault-based and background sources. The samecomment is applicable to the manner in which alternativeGMPE s are considered in PSHA .

Conclusions

This study presented the results of considering epistemicuncertainties in probabilistic seismic-hazard analyses for locations in New Zealand. The methodology accounted for uncertainties in ground-motion prediction viathe useof multi-pleground-motion prediction equations in a logic tree. Uncer-tainties in the characteristic rupture magnitude and recurrencerate of the fault-based component of the seismicity due touncertainties in fault deformation, geometry, and empiricalmagnitude-scaling relationships were considered via MonteCarlo simulation. Uncertainties in background seismicitywere not considered. Because of the present lack of fault-specific data quantifying uncertainties for many faults in NZ,representative values based on judgement and the data avail-able for NZ and foreign faults were utilized where required.

Probabilistic seismic-hazard analyses were conducted for two vibration periods of spectral acceleration [ PGA andSA(2.0)] for site class B (soft rock) and D (soft/deepsoil) con-

ditions in Wellington and Christchurch. The obtained resultsillustrated that, of the uncertainties considered, those due toground-motion prediction produced the largest variation inseismic hazard. Of the earthquake rupture forecast uncertain-ties considered, that due to the magnitude-scaling relation-ships was the most significant, followed by rupture length.

Data and Resources

All data used in this paper are from the publishedsources listed in the references.

Acknowledgments

Financial support for this study was provided by the New ZealandEarthquake Commission under award EQC 10/593. Constructive commentsfrom two anonymous reviewers improved the article and are greatlyappreciated.

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Department of Civil and Natural Resources EngineeringUniversity of CanterburyPrivate Bag 4800Christchurch, New [email protected]

(B.A.B.)

GNS ScienceP.O. Box 30368Lower Hutt 5044, New Zealand

(M.W.S., G.H.M., M.G.)

Manuscript received 11 September 2011

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