(2007) 186–192www.elsevier.com/locate/enggeo

Engineering Geology 90

Error inflation in Probabilistic Seismic Hazard Analysis

Jens-Uwe Klügel

NPP Goesgen Daeniken, 4658 Daeniken, Switzerland

Received 10 September 2006; received in revised form 20 December 2006; accepted 10 January 2007Available online 17 January 2007

Abstract

Based on a consistent interpretation of earthquake occurrence as a stochastic process I demonstrate that the mathematical modelof Probabilistic Seismic Hazard Analysis (PSHA) as it is in use today is inaccurate and leads to systematic errors in the calculationprocess. These mathematical errors may be regarded as an important contributor to the unrealistic results obtained by traditionalPSHA for low probabilities of exceedance in recent projects.© 2007 Elsevier B.V. All rights reserved.

Keywords: Probabilistic Seismic Hazard Analysis; Error; Ground motion prediction

1. Introduction

Recent applications of the traditional PSHA meth-odology based on the SSHAC-procedures (SSHAC,1997) have shown surprising results, which appear to beincompatible with simple plausibility checks (Klügel,2005a). Tentative explanations include the introductionof the ergodic assumption (Anderson et al., 2000;Klügel, 2005b) and inappropriate treatment of depen-dencies between random parameters in logic trees(Klügel, 2005a). Here, a further, albeit related, funda-mental contributing cause to the problems with largescale PSHA-projects is proposed. We submit that thecalculation of conditional probabilities of exceedance ofground motion by approximating the multivariateprobability distribution of exceedance by a univariatedistribution (SSHAC, 1997) is incorrect. Personaldiscussions with experts involved in the PEGASOS pro-ject, a SSHAC level 4 (SSHAC, 1997) project (Klügel,2005a; Zuidema, 2006) suggest that key practitioners of

E-mail address: jkluegel@kkg.ch.

0013-7952/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.enggeo.2007.01.003

the PSHA-methodology may be unaware of the impli-cations of this simplification.

2. Some basic considerations on stochastic processes,measurements and epistemic uncertainty

Earthquake occurrence as well as the occurrence ofthe associated ground motion in a specified location inthe theory of probability is described as a stochasticprocess. Therefore, we will shortly repeat some basicconsiderations on stochastic processes. Furthermore, Iwill provide a link to the observation of the realizationsof a stochastic process by measurements and theassociated observation errors and their propagation.

2.1. Stochastic (point) processes

A stochastic process is a consecutive set of randomquantities defined on some known state space, Ω, in-dexed so that the order is known: {ω[t]: t∈T}. Astochastic process must also be defined with respect to astate space, Ω, which identifies the range of possible

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values of ω (Gill, 2002). Accordingly, a (one-dimen-sional) stochastic process is a function X(ω,t), such thatX:Ω×T→R(Rinne, 2003), with R the space of realnumbers. A stochastic process is a point process if thestate space Ω is countable. Stochastic processes can bemultidimensional and complex.

Under specific conditions a particular stochasticprocess is determined by specifying the joint probabilitydistributions of the various random variables (consis-tency theorem of Kolmogoroff, (Rinne, 2003)).

In many stochastic process models, a point processarises as a component of a more complex model. Fre-quently, the point process is the component that carriesthe information about the locations in time or space ofobjects that may themselves have a stochastic structureand stochastic dependency relations. In the theory ofstochastic point processes many such models can besubsumed under the heading of marked point processes(Daley and Vere-Jones, 2002).

Earthquake occurrence in PSHA is frequentlydescribed as a homogenous stationary Poisson processor a Poisson cluster process. State (e.g. size) parameters,ω, are for example earthquake magnitude or intensity. Inaddition, state parameters may include seismic sourcecharacteristics like focal depth, fault style, rake andothers. Observations of earthquakes in specific locationsin terms of ground motion characteristics representanother, dependent stochastic process which can beregarded as a marked point process with the locations ofthe ground motion observations (e. g. a network)representing the associated ground process (Daley andVere-Jones, 2002). Therefore, the earthquake effectsdescribed in a specific location by ground motion char-acteristics represent realizations of a stochastic processtriggered by the (primary) stochastic process of earth-quake occurrence. In engineering applications we areinterested in the following characteristics of the stochas-tic process of earthquake occurrence:

• The mean function:

lt ¼ EXfX ðx; tÞg ð1Þ

The mean function of a stochastic process assignsto each t∈T the expected value of all realizations ofX(ω,t) at the corresponding t. The calculated meanvalues are also called the ensemble mean values(Rinne, 2003). The mean function of a stochasticprocess is a non-random function.

• The variance function:

r2t ¼ EX ½X ðx; tÞ−lt�2n o

¼ EX X 2ðx; tÞ� �−l2t ð2Þ

shows for each t∈T the scatter (or the variability) of therandom variable Xt around its expected value at thecorresponding t. The square root

rt ¼ffiffiffiffiffir2t

qð3Þ

represents the standard deviation of the stochasticprocess. Another mean of a stochastic process can bederived by averaging along the time sequence:

lx ¼ ETfX ðx; tÞg ð4ÞIt is also worth mentioning that knowledge of the

moments of the stochastic process is frequently insuf-ficient for a complete description of the process. Themain reason for this is that the random variables X(ω,ti)may depend on each other. This dependency can beexpressed by the correlation function:

qðti; tjÞ ¼ EXfX ðx; tiÞX ðx; tjÞg−lðtiÞlðtjÞrðtiÞrðtjÞ ð5Þ

The question of dependency between the randomvariables in a stochastic process is important with res-pect to error propagation in transformations. Note thatattenuation equations are frequently developed by re-gression analysis in logarithmic scale, whereas for ap-plications the absolute values of acceleration are ofinterest. Therefore, a transformation is needed. Withoutconsideration of dependency between the random vari-ables involved error propagation is not transformationinvariant (Brandt, 1999).

In additional, it is necessary to remember that inseismic hazard analysis multiple seismic sources have tobe considered. Earthquake occurrences from each of thesesources follow an individual source-specific stochasticprocess characterized by different joint probabilitydistributions.

2.2. Conditional probability of exceedance

The core of traditional PSHA (SSHAC, 1997) is thecalculation of the probability of exceedance of aspecified ground motion level at a given site. Thesolution to this problem is decomposed into thecalculation of the frequency of earthquakes with somedamaging potential and the calculation of the condi-tional probability of exceedance for each of thesecontributing earthquakes (summed over all potentiallycontributing sources) that a specified ground motionlevel is exceeded. Because ground motion occurrence ata certain site is the result of a stochastic process, a largeamount of random variables is involved. Large scale

188 J.-U. Klügel / Engineering Geology 90 (2007) 186–192

probabilistic models (e.g. using logic trees) model theseismic hazard as a random function of multiple randominput variables. Therefore, it is beneficial to give ageneral expression of the calculation of the conditionalprobability of exceedance of value z by the randomvariable Xn conditioned to the other contributing randomvariables for the multivariate case (Fisz, 1978):

FðXnNzjX1;X2; N Xi; N Xn−1Þ¼ 1−

R z−l f ðX1;X2; N :Xi N :XnÞdXnRl−l f ðX1;X2; N Xi N XnÞdXn

ð6Þ

Here f denotes the probability density function of then-dimensional random variable (X1, X2,…, Xi,…, Xn).The Xi represent the random variables of our probabi-listic model. The denominator represents the (n−1)-dimensional marginal distribution of the variables de-fining the conditions. Eq. (6) illustrates that for thecalculation of conditional probabilities knowledge of thejoint probability distribution of the various random vari-ables of the stochastic process is required.

2.3. Measurements and epistemic uncertainty

In seismic hazard analysis we are interested in themean function (ensemble mean) of the stochastic pro-cess describing ground motion characteristics at specificlocations. Assuming stationarity of earthquake occur-rence (e.g. model of an homogeneous stationary Poissonprocess) empirical attenuation equations must be lookedat as an estimate of the mean function of this markedpoint process (or of an envelope of the mean functionsof the marked point processes in case of a regionalmodel covering several seismic sources). They expressthe dependence of the ground motion characteristic ofinterest on a selection (or ideally a complete set) of stateparameters of the triggering stochastic process of earth-quake occurrence and the members of the associatedground process (location). Therefore, a series of mea-surements of the realisations of the stochastic process ofground motion occurrence and the subsequent data anal-ysis (e.g. by regression) result in an estimate of the meanfunction of the stochastic process and the associatedvariability (Eq. (3)). The mean function provides theexpected value of the random function which should notbe mixed with a median. Similarly, our measurementsprovide an estimate of the variability of the stochasticprocess. Unfortunately (as in many other engineeringapplications), we only have a limited set of observedrealizations of the stochastic process available. Further-more (this is especially true for seismic hazard analysis),the data of single realisation paths are limited to a few

anchor points (or time intervals) t in T and to a limitedamount of locations. Therefore, ergodic assumptionsmust be made to compensate the lack of knowledge.Furthermore, we do not have sufficient knowledge con-cerning the real dependency of ground motion in dif-ferent locations on the state variables of the stochasticprocess (we may not even know all random variablesinvolved in our stochastic process). Therefore, our modelsare incomplete and affected by “lack of knowledge”uncertainties usually denoted as “epistemic uncertainty”.It is important to acknowledge these limitations of ourmodels because the consequence is that we are not able toprovide a bias-free estimate of the postulated true inherentvariability of ground motion. We are not even able toprove that inherent variability exists. Based on considera-tions related to quantum mechanics we assume that suchan “inherent” component exists (Woo, 1999). We alwaysmeasure total uncertainty, a combination of epistemic un-certainty and inherent variability (frequently denoted asaleatory variability). Note that our whole discussion isbased on the additional strong assumption of stationarityof the stochastic process of earthquake occurrence (or theinvariance of ground motion attenuation with time). Theconclusion is that the total uncertainty to be considered ina probabilistic model as well as the distinction of differenttypes of uncertainty (reducible epistemic, irreducible epis-temic or inherent (aleatory) variability) depends on ourmodel.

2.4. Summary

Based on the interpretation of earthquake occurrenceas a multivariate stochastic process with ground motionobservations considered as a dependent marked sto-chastic point process, we have arrived at the followingconclusions:

1. The calculation of the conditional probability of ex-ceedance of a specified ground motion level in Prob-abilistic Seismic Hazard Analysis is a multivariatestatistical problem. Its calculation requires the know-ledge of the joint probability distribution of all randomvariables involved.

2. Due to our limited knowledge we are not able toprovide a complete description of the stochastic pro-cess parameters involved. Therefore, we must rely onmodels associated with a corresponding amount ofepistemic uncertainty.

3. In our measurements of ground motion we observethe total uncertainty of ground motion characteristics.This total uncertainty represents a combination ofepistemic uncertainty and postulated inherent

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variability of the stochastic process. The inherentvariability includes not only the variability ofattenuation but also the variability in state parameterssuch as seismic source parameters ( e. g. faultingstyle, fault rupture velocity etc.) and the character-istics of the travel path of seismic waves.

4. Our model of total uncertainty depends on the modelwhich we use (model of stochastic process, model ofstate parameters explicitly considered in the model).We cannot even prove the existence of aleatoryvariability in the sense of an objective, inherent pro-perty of earthquake occurrence, besides some generaldiscussions based on considerations related toquantum mechanics (which are not yet applied inseismic hazard analysis). With respect to measure-ments our estimate of uncertainty also contains themeasurement errors of the state parameters used inthe model (e.g. magnitude, distance).

3. Sources of error in traditional PSHA

Modern PSHA logic trees represent a probabilisticframework where at each branch of the tree the con-ditional probability of different (calculation) alternativesmust be estimated. It is a conditional probability becausethe value to be selected at the branching points in generaldepends on the preceding path through the logic tree. It isnot important whether the conditional probabilities dis-cussed are subjective probabilities estimated by expertsor whether they are derived from observed data. It is alsonot important whether the probability density distribu-tion function is represented in a discrete or in a conti-nuous form. But it is important to understand that theresult of each of the possible calculation paths processedthrough a logic tree represents one of the possible valuesof a multivariate probabilistic distribution where thenumber of variates corresponds to the number of randomvariables represented in the model (logic tree). A PSHAlogic tree represents one of the possible models of thestochastic process of earthquake occurrences triggeringthe dependent process of ground motion observations atdifferent locations. Comparing the logic tree methodol-ogy and the recommendations of how to treat uncertaintyin a PSHA-model (SSHAC, 1997) with the conclusionsfrom the discussion in Section 2, we are able to identifythe following sources of systematic error:

1. The standard error of empirical attenuation equations(or the difference between observation and of a physicalpredictive model) is interpreted as aleatory uncertaintyand not understood as an (limited by our knowledge)estimate of total variability of the stochastic pro-

cess of earthquake occurrence (SSHAC, 1997,Appendix F). Uncertainties in source parameters aremodeled additionally and independently in the sourcemodel (SSHAC, 1997). This leads to a systematicdouble counting of uncertainties. This double count-ing is inherently embedded in the methodology.

2. The dependence of the mathematical description ofuncertainty on the model used is ignored, leading to amathematical incorrect formulation of the hazardcalculation in the traditional PSHA-model.

3. The calculation of the conditional probability of ex-ceedance of a specified ground motion level is notcorrect, largely as a consequence of point 2.

I will discuss the issues related to the mathematicalmodel in more detail in Section 4.

4. Problems in the mathematical model of PSHA

4.1. Mathematical models in PSHA

What are themost frequently usedmathematicalmodelsin PSHA?

1. (SSHAC, 1997):

kðaÞ ¼XSi¼1

mi

ZR

ZMU V lna−gðm; rÞ

r

� �

� fRðrjmÞfMðmÞdrdm ð7Þ

2. (Abrahamson, 2006):

kðaÞ ¼XSi¼1

mi

Zm

Zr

ZefmiðmÞfRiðrjmÞ

� fePðlnðaÞNajm; r; eÞdedrdm ð8Þ

Here λ is the mean annual rate of events exceeding aground motion level a, νi is the rate of earthquakes inthe magnitude range of interest for source i, the fR andfM describe the frequency or conditional frequency dis-tributions for magnitude and distance.

Both models are based on an attenuation model ofthe following form:

lnðaÞ ¼ gðm; rÞ þ erln ð9Þ

σln is the standard deviation of the attenuation equation(we assume for simplicity reasons that we are dealingwith an empirical attenuation equation) and the param-eter ε defines the confidence intervals of the

190 J.-U. Klügel / Engineering Geology 90 (2007) 186–192

attenuation equation. In case of a measurement ε has aStudent's t-distribution with n−1 degrees of freedom.For large n, it is usually approximated by a normaldistribution with a zero mean and the standard deviationσln.

The SSHAC-model simply uses the complementarycumulative density function of a standard unit normalfor the calculation of the probability of exceedance. Thesecond model (in use in some of the standard computercodes for PSHA today) calculates the conditional prob-ability of exceedance as:Z e

−lfePðlnðaÞNajm; r; eÞde ð10Þ

Here P(ln(a)Na|m,r,ε) is a jump function attaining avalue of 1 if the logical condition in parentheses isfulfilled, otherwise it attains a value of 0. Therefore, thesecond model takes into account the truncation ofground motion acceleration at an upper limit. Otherwiseit is equivalent to the SSHAC-model.

4.2. Discussion on the calculation of probability ofexceedance

It is obvious that the calculation of the conditionalprobability of exceedance in the SSHAC-model is in-correct. The probability of exceedance is calculated com-pletely independently of all other parameters of theprobabilistic model. Its expression is derived based onEq. (9) by multiplying it with the probability densityfunction of ε, performing integration and converting theexpression into a complementary probability (to obtain a“probability of exceedance”). Indeed, assuming inEq. (6) that the random variate Xn corresponds to εand that the error term E=εσ (with σ=const) iscompletely independent from all other random variablesused in the probabilistic model we can derive Eq. (10)(or the SSHAC-model for the untruncated case).Therefore, both PSHA-models use an unconditional(because independence is assumed) univariate approx-imation for the calculation of the conditional probabilityof exceedance. As the discussion in Section 2 has shownthe assumption of independence of the error term fromthe other model parameters is not valid. The dependenceof the error term on distance was observed inmeasurements (e.g. Boore et al., 2003; Lin et al.,2005). Uncertainty as an estimator of the variability ofthe stochastic process of earthquake occurrence dependson the model of the stochastic process used. This cannotbe compensated by regression techniques attempting toeliminate correlation effects between σln and the model

parameters, because lack of correlation is not proof forlack of dependency. Furthermore, the regression errorσln depends on all state variables of the stochasticprocess (including source parameters and travel pathcharacteristics). This means that the separation of theerror term from the model used in both PSHA-models(Eqs. (7) and (8)) is mathematically incorrect and bothmodels are invalid. The simplified expression for thecalculation of the probability of exceedance wasintroduced into the original risk analysis model ofCornell (1968) by McGuire (1978). It was assumed thatall uncertainty of the problem is concentrated in theuncertainty term of the attenuation equation and all otherparameters in the model are non-random. With respectto the model of a stochastic process discussed in Section2 this assumption is equivalent to the assumption thatthe term g(m,r) in our probabilistic model behaves likethe mean function of the stochastic process, which is anon-random function by definition. But in a probabilis-tic model that treats all input parameters of the analysisas random the term g(m,r) obtains a different meaning.It becomes a random function:

C ¼ fgðm; r;XiÞ ð11Þ

Here the Xi represent the random state variableswhich participate in our stochastic process in addition tom and r (large scale probabilistic models like the largelogic tree used in the PEGASOS project (Zuidema,2006) contain much more random variables than just mand r). These parameters may or may not be included inthe probabilistic model. The SSHAC-model replaces therequested random function Γ in the calculation effec-tively by the mean function of the underlying stochasticprocess. This mean function represents an estimator ofthe expected value of ground motion in dependence ofm and r, g(m,r). The random function Γ in our prob-abilistic model in general can attain values that differgreatly from the values calculated from the mean func-tion of the stochastic process. Furthermore, due to theconsideration of additional random variables besides mand r in a probabilistic model (logic tree) not consideredin the calculation of the conditional probability of exceed-ance, an additional error term is introduced. Therefore, thesimplified calculation of the probability of exceedanceintroduces a systematic error into the calculation process.

This is illustrated by the following discussion.In the probabilistic model (e. g. logic tree) ln (a) is

calculated in dependence of the random function Γ,which according to Eq. (11) may depend on much morerandom variables then just m and r (e.g. the sourcecharacteristics, travel path characteristics). Therefore,

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the effective calculation in the logic tree model is per-formed based on an equation of the following shape:

lnðaÞ ¼ Cþ erln ð12ÞThe random function Γ as used in the probabilistic

(logic tree) model can be represented by the sum:

C ¼ gðm; rÞ þ Dðm; r;XiÞ ð13ÞΔ(m,r,Xi) is a nontrivial random function. Perform-

ing the substitution of Eq. (13) into Eq. (12) and thenreplacing ln (a) in Eq. (9) we obtain:

Dðm; r;XiÞ ¼ 0 ð14ÞThis is obviously incorrect in terms of howΔ(m,r,Xi)

was introduced. Accordingly, we obtained a statementof the same logical value as “x equals 0 for any x” as theresult of the simplifications made in the traditionalPSHA model.

It is worth to mention that in large scale probabilisticmodels even the mean rate of earthquakes νi above alower magnitude threshold becomes a random variablebecause the maximum magnitude used for hazard trun-cation is described by an uncertainty distribution. There-fore νi cannot be separated from the integration of f (m)in Eqs. (7) and (8).

5. Conclusions

The analysis performed demonstrates a fundamentalsystematic error in the mathematical model of traditionalPSHA. The question for practitioners is: how large isthis systematic error in PSHA? It is difficult to make aquantitative judgement, because the contributing effectsof several mathematical errors have to be regarded.Furthermore, the total error depends on the probabilisticmodel used to develop the seismic hazard (this modeldefines the true, multivariate conditional probability den-sity function to be used in the analysis). Some generalqualitative conclusions can be made based on Cheby-shev's inequality:

PrðjX−ljzerÞV 1e2

ð15Þ

Here X is a random variable with the mean μ. Theerror is small for slight perturbations of ground motionaccelerations around the regression mean of the atten-uation equation. This case corresponds approximately tothe case of high probabilities of exceedance of a PSHA-model. For low probabilities of exceedance we cannotsimplify the calculation of the conditional probability ofexceedance of a specified ground motion level and we

have to consider the dependency of uncertainty on themodel used. Therefore, the error of the results of aPSHA is expected to be high at low probabilities ofexceedance.

Eq. (15) can also be used to refute a frequently usedargument to justify the increase of seismic hazard insome recent projects by observations of increasinglyhigher accelerations in ground motion recordings ob-tained by the availability of denser networks. Eq. (15)(as well as Markov's inequality, Rinne, 2003) showsthat the observation of high acceleration time histories isstatistically compatible with a significantly lower meanseismic hazard.

The convoluted effect of the errors discussed can beestimated by comparing the total uncertainty of groundmotion characteristics as observed in measurementswith the results of PSHA for a fixed probability of ex-ceedance (corresponds for the discussed PSHA-proce-dures to a fixed magnitude and fixed distance). Usingthe model of an untruncated lognormal distribution (themodel of a lognormal distribution is widely acceptedamong seismologists) we can estimate the total uncer-tainty associated with the results of a PSHA and com-pare with the observed values. For low probabilities ofexceedance the mean uniform hazard spectra (UHS) forspectral accelerations resulting from PSHA intersect withthe 84%-quantile of the hazard distribution (BECHTEL/SAIC, 2004; Zuidema, 2006) .This means that the uncer-tainty σln≥2.0. The observed total uncertainty amountsto only σlm,ob≈0.65–0.7 or even less. This means that themathematically incorrect PSHA-procedures numericallylead to a tripling of the uncertainty in comparison toobservations. A full correction of the problem in a shortperiod of time seems to be unfeasible. A careful analysismust be made for each of the potentially affected projects.A principal and meaningful alternative for scientists andengineers consists in the use of deterministic scenario-based methods for seismic risk analysis. In (Klügel et al.,2006) it is shown that these methods can be expanded foruse in probabilistic risk analysis. A key advantage of thisexpansion of classical deterministic methods for use inrisk analysis consists in the separation of temporal andspatial characteristics of the stochastic process of earth-quake occurrence. In countrieswhere the discussed PSHA-methods are prescribed by regulation it is recommended toconstrain the results by a comparison to the results of adeterministic seismic hazard analysis.

Another interesting observation is that the use ofintensities instead of accelerations in the probabilisticmodel reduces the error effect, because intensities area more robust, composite parameter (it combinesinformation on spectral acceleration and magnitude).

192 J.-U. Klügel / Engineering Geology 90 (2007) 186–192

Furthermore, anyone understands that intensities cannotbe higher than XII, while the same understanding does notapply for accelerations, which have been extrapolated tomore than 10 g such as for the Yucca Mountain project,although meanwhile corrected (BECHTEL/SAIC, 2004).At the final stage of the analysis, intensities, rounded off tothe closest integer, can then be transformed into PGAor assigned to a spectrum reflecting the site conditions(Mohammadioun, 1986).

Acknowledgments

The author thanks Giuliano Panza for the very in-spiring discussions about the complexity of mathemat-ical problems in geophysics and their potential pitfalls aswell as many other colleagues concerned with the prob-lem and providing useful comments. The author alsothanks anonymous reviewers for their comments, whichcontributed significantly to improving the paper.

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