influence of spatial correlation of strong ground motion on uncertainty in earthquake loss...

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2011; 40:993–1009Published online 22 November 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/eqe.1074

Influence of spatial correlation of strong ground motion onuncertainty in earthquake loss estimation

Vladimir Sokolov∗,† and Friedemann Wenzel

Geophysical Institute, Karlsruhe Institute of Technology (KIT ), Hertzstr. 16, Karlsruhe 76187, Germany

SUMMARY

In addition to the mean values of possible loss during an earthquake, parameters of the probabilitydistribution function for the loss to a portfolio (e.g. fractiles and standard deviation) are very important.Recent studies have shown that the proper treatment of ground-motion variability and, particularly, thecorrelation of ground motion are essential for the estimation of the seismic hazard, damage and lossfor distributed portfolios. In this study, we compared the effects of variations in the between-earthquakecorrelation and in the site-to-site correlation on seismic loss and damage estimations for the extendedobjects (hypothetical portfolio) and critical elements (e.g. bridges) of a network. A scenario earthquakeapproach and a portfolio containing a set of hypothetical building and bridges were used for the purpose.We showed that the relative influences of the types of correlation on characteristics of loss distributionand the probability of damage are not equal. In some cases, when the median values of loss distributionor the probability that at least one critical element of a lifeline will be damaged are considered andwhen the spatial correlation of ground motion is used, the possible variations in the between-earthquakecorrelation may be neglected. The shape of the site-to-site correlation function (i.e. the rate of decrease ofthe coefficient of spatial correlation with separation distance) seems also to be important when modellingspatially correlated ground-motion fields. Copyright � 2010 John Wiley & Sons, Ltd.

Received 23 March 2010; Revised 7 September 2010; Accepted 8 September 2010

KEY WORDS: strong ground motion; spatial correlation; loss uncertainty

1. INTRODUCTION

A key element of seismic hazard assessment (SHA) is the consideration of uncertainties, which areclassified as epistemic and aleatory. The epistemic uncertainty reflects the incomplete knowledge ofthe nature of all inputs to the assessment, the variability of the interpretation of available data, andthe limitations of the technique applied for the analysis. Epistemic uncertainty can be incorporatedinto SHA using the logic tree method (as in [1]). At present, an approach based on copula analysisis under examination [2, 3].

Aleatory uncertainty is related to the inevitable unpredictability regarding the nature of ground-motion parameters. In other words, the aleatory uncertainty describes the disagreement betweenobservations and predictive models that is due to the absence of a physical explanation or due tovariables that are not included in the predictive equations. Additional explanatory variables needto be added to the model to represent repeatable (as opposed to random) influences on the groundmotion. Thus, the aleatory component of uncertainty may also reflect epistemic modelling uncer-tainty regarding the factors controlling the ground-motion component that have not been included

∗Correspondence to: Vladimir Sokolov, Geophysical Institute, Karlsruhe Institute of Technology (KIT), Hertzstr. 16,Karlsruhe 76187, Germany.

†E-mail: [email protected]

Copyright � 2010 John Wiley & Sons, Ltd.

994 V. SOKOLOV AND F. WENZEL

in ground-motion models (as in [4]). Aleatory uncertainty is mainly quantified in SHA through theuse of the standard deviation of the data’s scatter about the ground motion-prediction equations.It has become common practice to separate the total aleatory variability into two independentcomponents [4–8], namely, the earthquake-to-earthquake (between-earthquake) variability and thesite-to-site (within-earthquake) variability.

The between-earthquake variability results from event-specific factors that have not been includedin the predictive model. The within-earthquake variability reflects the fact that earthquake groundmotion for a given event at different sites must vary to some extent. The within-earthquakevariability is determined mostly by peculiarities in the propagation path and local site conditions,and there have been attempts to separate the within-earthquake variability into its path-to-path andsite-to-site components [9].

The ground-motion parameter Y at n locations during m earthquakes is represented by

logYi, j = f (ei ,si, j )+�i +εi, j , i =1, 2, . . . ,m, j =1,2, . . . ,n, (1)

where ei denotes variables that are properties of the earthquake source, si, j are the properties ofsite location j during earthquake i , and f is a suitable function that describes the dependenceof the mean value of ground-motion parameter log Yi, j on the magnitude, distance, local siteconditions, etc. (i.e. log Yi, j = f (ei ,si, j )). The random variables �i and εi, j represent the between-earthquake and within-earthquake components of variability (independent and normally distributedwith variances �2

� and �2ε), respectively. The value of �i is common to all sites during a particular

earthquake i , and the value of εi. j depends on the site. Assuming the independence of the tworandom terms, the total aleatory variance �2

T is given by �2T =�2

�+�2ε .

The between-earthquake correlation of earthquake ground motion, or the similarity of ground-motion variability during different earthquakes at the same site, is determined by the relationbetween the components of variability (as in [10]):

�� = �2�

�2�+�2

ε

= �2�

�2T

. (2)

At the same time, two close sites may exhibit correlation of ground motion during an earthquakedue to the commonality of wave paths (within-earthquake site-to-site correlation), which dependson the site separation distance �. The function �ε(�) is usually represented as

�ε(�)= (exp(a�b)), (3)

where a and b are the region-dependent coefficients. The so-called ‘correlation distance’ RC maybe considered a characteristic of the correlation [11]. The correlation distance shows the site-to-sitedistance for which the correlation coefficient �ε(�) decreases up to 1/e=0.368. For earthquake iand site j , the total correlation in εi, j values is as follows (as in [12]):

�T (�)= �2�+�ε(�)�2

ε

�2T

=��+�ε(�)

(�2

ε

�2T

), (4)

where �ε(�)=�εi, j,1;εi, j,2(�) is the empirical correlation coefficient calculated for within-earthquake εi, j values separated by a distance �, �� is the between-earthquake correlationcoefficient (Equation (2)) and �ε(�) is the within-earthquake site-to-site (spatial) correlationcoefficient (Equation (3)).

The ground-motion correlation should be taken into account when estimating ground-motionparameters in a wide area, for example, in the assessment of seismic hazard and loss for widelylocated building assets (portfolio), spatially distributed systems (lifelines) and ShakeMap gener-ation. The rigorous methodology described by Wesson and Perkins [10], Rhoades and McVerry[13] and McVerry et al. [14] for assessing the joint hazard for scales of kilometres to tensof kilometres requires consideration of the between-earthquake and within-earthquake compo-nents of uncertainty. Several modern ground-motion attenuation equations (e.g. New Generation

Copyright � 2010 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2011; 40:993–1009DOI: 10.1002/eqe

INFLUENCE OF CORRELATION IN EARTHQUAKE LOSS ESTIMATION 995

Attenuation models, NGA) allow the recognition of the between-earthquake correlation, becausethe equations include specification of the between-earthquake and within-earthquake componentsof variability (as in [9, 15–18]).

The site-to-site correlation should be empirically evaluated for a given area. A dense observa-tion of records from numerous earthquakes is necessary for modelling the site-to-site correlationstructure; therefore, such correlations have not been extensively investigated. Boore et al. [19],Hok and Wald [20] and Lin et al. [21] considered one or a few particular earthquakes in California;Wang and Takada [11] separately analysed a few earthquakes in Japan and the Chi-Chi earthquakein Taiwan. Kawakami and Mogi [22] used records from several earthquakes in Japan (the Chibaarray 30 km east of Tokyo and the SIGNAL array in Tokyo) and Taiwan (SMART-1 array). Evansand Baker [23] used the NGA database [24], which is primarily based on Californian data and afew earthquakes in Taiwan (the Chi-Chi earthquake and large aftershocks). Goda and Hong [25]and Jayaram and Baker [26] considered the Chi-Chi earthquake and some Californian earthquakes,and Hong et al. [27] analysed only Californian data. Recently, Goda and Atkinson [28] used alarge database collected in Japan (records from the K-NET and KiK-net strong-motion networks)to study the spatial correlation for peak ground acceleration and pseudo-spectral acceleration atdifferent periods from 0.1 to 5.0 s. Sokolov et al. [29] analysed the characteristics of ground-motion correlation using the database accumulated in Taiwan. The results reported by these studiesreveal different rates of decay of the correlation with separation distance. It has been shown thatthe differences relate to the frequency content of ground motion [25, 30]. On the other hand, thedifferences may be caused by regional peculiarities [28, 29]. Table I summarizes the data used andthe obtained results of some of the studies; Figure 1 shows several models of spatial correlation.

As can be seen from the foregoing definitions, the random variability associated with estimationsof the earthquake ground motion contains both aleatory and epistemic components. Differentrealizations of the random ground-motion field (see Section 2) in terms of between-earthquakeand within-earthquake residuals and spatial correlation represent aleatory variability. The spatialcorrelation, in principle, depends on the chosen ground motion model, because the correlationdescribes the behaviour of residuals between the observations and predictions. Thus, differentmodels for between-earthquake and spatial correlation combined with the corresponding groundmotion models would represent a source of epistemic uncertainty.

The knowledge of the loss distribution about the mean (e.g. the variance or standard deviation)is very important for decision making and mitigation activities. For example, primary insurers areconcerned with the central part of the distribution (mean and median values), while re-insurers dealmostly with the tail of the distribution. Several studies analysed the influence of ground-motionuncertainty and correlation on characteristics of loss distribution. Specific (scenario) earthquakeswere considered by Lee et al. [33], Lee and Kiremidjian [34], Molas et al. [35], Goda and Atkinson[28] and Crowley et al. [36]. Multiple earthquakes and loss probability curves were analysed byMcVerry et al. [14], Wesson and Perkins [10], Park et al. [12], and Goda and Hong [37]. Bommerand Crowley [38] and Crowley and Bommer [39] discussed two approaches to calculating lossprobability curves: one based on independent probabilistic SHA and the other involving MonteCarlo simulation based on the seismicity model.

Loss estimates were performed using the techniques based on different ground-motion char-acteristics (namely, Modified Mercalli intensities [14, 35], peak ground acceleration [10, 28] andspectral acceleration or spectral displacement [12, 33, 34, 36, 37, 39]) and using various descriptionsof loss (i.e. the mean damage ratio [10, 28, 33, 34, 36, 39] and monetary loss [12, 14, 25, 37]). Allof these studies considered the influence of between-earthquake and within-earthquake componentsaccepting a single model of the between-earthquake correlation or analysing, in addition to themodel, two extreme cases (no correlation of ground motion, �� =0, and perfect correlation, �� =1).Different models of site-to-site correlation were used by Park et al. [12], Goda and Atkinson [28],Molas et al. [35] and Crowley et al. [36].

The general findings obtained in the works mentioned above may be summarized as follows.The higher the spatial correlation and between-earthquake variability, the larger the variation inlosses to a portfolio and the higher the probability of extreme loss values. For the case of a



Table I. Parameters of site-to-site within-earthquake correlation functions �ε(�) reported in the literature.

CorrelationReference Function distances, km Data and models used

Wang and Takada [11] exp(�/�) 27.3 The 1999 Chi-Chi(Taiwan) earthquake; PGV;ground motion model developedfor Japan by Midorikawaand Ohtake [31]

Goda and Hong [25] exp(a�b) 28.7 The 1999 Chi-Chi(Taiwan) earthquake; PGA;NGA ground motion modeldeveloped by Boore andAtkinson [32]

Sokolov et al. [29] exp(a�b) 18.2 The 1999 Chi-Chi(Taiwan) earthquake; PGA;ground motion modeldeveloped in the study

Sokolov et al. [29] exp(a�b) 34.1 The 2003 (MW 6.8, Taiwan)earthquake; PGA; ground motionmodel developed in the study

Sokolov et al. [29] exp(a�b) 4.0–23.5 Records from 13 (MW>6.0)earthquakes in Taiwan; PGA;ground motion model developedin the study; region-dependentestimations

Goda and Hong [25] exp(a�b) <2.0(≈1.7) Records from six largeearthquakes in California; PGA; groundmotion model developed in the study

Hong et al. [27] exp(a�b) <2.0(≈1.4) Records from 39 earthquakesin California; PGA; ground motionmodel developed in the study

Goda and Atkinson [28] �exp(a�b)−�+1.0 19.0–29.0 Records from 155 earthquakesthat occurred in Japan; PGA;ground motion model developedin the study; earthquakedepth-dependent estimations

single (scenario) earthquake, within-earthquake variability increases the possibility of obtainingextreme motion at one of the multiple locations compared with the single-site probability. Between-earthquake variability increases the possibility of obtaining extreme motions at all consideredlocations simultaneously.

Usually, within-earthquake standard deviation is found to be larger than that for between-earthquake standard deviation, although the relation is a period-dependent quantity [40].Atkinson [41] and Morikawa et al. [42] showed that the total standard deviation of ground-motionprediction may be reduced when using a single site-specific model or a region-specific correctionfactor, which is determined by grouping ground-motion data at specific stations of a densestrong-motion array. Tsai et al. [9] showed that a reduction of the total standard deviationcould be achieved if the path effect could be specified. This reduction, obviously, is related tothe within-earthquake component, and Morikawa et al. [42] obtained almost similar values forbetween-earthquake and within-earthquake standard deviations after the correction. Thus, thebetween-earthquake correlation �� may vary, at a minimum, between 0.06 [40] and 0.5 [42].

Sokolov et al. [29] analysed the strong-motion database collected by the TSMIP network inTaiwan, which includes about 4650 records from 66 shallow earthquakes (ML>4.5, focal depth<30km) occurring between 1993 and 2004. They showed that the within-earthquake componentof uncertainty in Taiwan seems to be a magnitude-dependent quantity. When using the generalized(site-independent) ground motion-prediction model, the uncertainty of ground motion-prediction



Figure 1. Site-to-site intra-event (spatial) correlation functions �ε(�) estimated for particular earthquakes.(a) The Chi-Chi earthquake, 1—Wang and Takada [11]; 2—Goda and Hong [25]; 3—Sokolov et al. [29].(b) The data from moderate-to-large earthquakes (MW >6.0) in Taiwan [29], estimations for particularstrong-motion arrays in Taiwan. CHY array, thick sediments; TCU array, stiff soils in extended hilly areas;TAP array, triangular asymmetric alluvium-filled Taipei basin; ILA array, Quaternary alluvial Ilan basin.

may be reduced by the application of corresponding, empirically derived area- or site-dependentcorrection factors. The analysis of the site-to-site correlation in Taiwan suggests that the correlationstructure is highly dependent on the local geology and on peculiarities of the propagation path(azimuth-dependent attenuation) (Figure 1). The application of region- or site-dependent correctionreduces the site-to-site correlation, especially at large distances. Thus, a single, generalized modelof site-to-site correlation may not be adequate for the whole Taiwan territory or for large areas.

As can be seen from the foregoing short review, the proper treatment of ground-motion corre-lation is essential for the estimation of seismic hazard, damage and loss for distributed portfolios.If the information about correlation models is not available, it is necessary either to obtain upperand lower estimates by assuming the perfect correlation (�� =1.0;�ε(�)=1.0) and uncorrelatedground motion (�� =0.0;�ε(�)=0.0) or to make some assumption about the correlation. However,the parameters of ground-motion correlation may vary over wide ranges, and it is still necessaryto analyse the relative influence of the variation on estimations of aggregated loss for a portfolio(widely located constructions of several types) and on characteristics of joint hazard and damage.

In this study, we compared the effects of variations in the between-earthquake correlation and inthe spatial (site-to-site) correlation on the seismic loss and damage estimations for extended objects(hypothetical portfolio) and critical elements (e.g. bridges) of a network. Based on the results ofour previous work [29], the correlation distances vary from 0 to 30 km, and the between-earthquakecorrelation varies from 0.09 to 0.5, which is also within the reported values. A single event, aso-called ‘scenario’ earthquake, was used as the source of seismic influence. A set of hypotheticalbuildings representing a real building stock was constructed based on typical buildings in theTaiwan area.

2. APPLICATION OF GROUND-MOTION CORRELATIONTO GROUND-MOTION MODELLING

Let us consider the random variables (normally distributed with a zero mean and standard deviationsof �� and �ε) at two sites, x =�+εx and y =�+εy . The joint probability density function (PDF)



may be obtained using so-called copula functions (as in [43]). A copula is a function that combinesunivariate distributions to obtain a joint distribution with a particular dependence structure. In otherwords, a copula provides a means to address dependence structures that vary over locations. Forthe case of normal univariate margins, the dependence structure among the n margins is describedby the Gaussian (or normal) copula Cnormal, i.e. the copula of the multivariate normal distribution:

Cnormal(u1, . . . ,un)=�(�−1(u1), . . . ,�−1(un)), (5)

where � is the standard multivariate normal distribution function with linear correlation matrix� (the correlation matrix describes the correlation among n variables) and �−1 is the inverse ofthe standard univariate normal distribution function. For the considered case of random variablesat two sites, the joint PDF follows a bivariate normal distribution with a zero mean and standarddeviation of �T .

In ground motion models, the parameter of motion Y generated by earthquake i with magnitudeM at a site j at distance R is estimated as a lognormally distributed random variable as lnYij =N (ln Yij,�2), where ln Yij = f (M, R,site). In addition to the median value of ground motion Yij,we need to generate the standard normal variates (errors) of �i and εi, j . The dependence structureamong the variates is described by the total correlation coefficients (Equation (4)) and the correlationmatrix R.

For the generation of the k-site random field of ground-motion values that are spatially correlated,it is necessary to generate a Gaussian vector of correlated standard normal variables (total residualterm) X= [X1, X2, . . ., Xk] with a symmetric correlation matrix R, or X∼ Nk(0,�). The correlationmatrix R is defined as follows:

�=

⎛⎜⎜⎜⎜⎜⎜⎜⎝

1 �12 . . . �1k

�21 1... �2k

......

......

�k1 �k2 . . . 1

⎞⎟⎟⎟⎟⎟⎟⎟⎠

, (6)

where �ij is the empirical correlation coefficient calculated for the sites separated by a distance�. The procedure of the generation of k-site random field of ground-motion error values thatare spatially correlated equals the generation of random variables X= [X1, X2, . . ., Xk] with acorrelation matrix R from the k-dimensional Gaussian copula. Descriptions of the procedure maybe found in many sources [12, 44]. The generation consists of the following steps. First, a vectorof independent standard normal variates U= [U1,U2, . . .,Uk] with standard deviation, or U∼Nk(0,�2

T ), is generated. Then, a correlation matrixR is constructed and a Cholesky decomposition isapplied to represent the correlation matrix R as the matrix product of matrix B and its transpositionBT, i.e. R=B BT. The required vector X is obtained as X=BU. These X j values are added to themedian ground-motion term ln Yij to obtain a realization of spatially correlated ground motions.

In this study, we generated ground-motion parameters (peak ground acceleration) from a singlescenario earthquake across a wide area (e.g. a city). Figure 2(a) shows the area of 22km×18km,which is divided into cells of 1km×1km. Ground-motion parameters were estimated in the centreof the cells (marked by triangles) from an M 7.0 earthquake located near the area. The groundmotion-prediction equation for the calculation was used in the following form, which was recentlyproposed for Taiwan [29]:

lnPGA=−3.07+0.83MW−1.33ln[R+0.15exp(0.54MW)]+0.0023R±�T , (7)

where PGA is measured in units of g and R is the hypocentral distance in km. The total standarddeviation of ground-motion prediction for the earthquake was taken as �T =0.4 (total variance�2

T =0.16). This relatively small value of �T is based on the results obtained by Sokolov et al.[29] and it represents a hypothetical case when the total standard deviation was reduced by usingthe region- and site-specific correction factors. The median-expected PGA values, which were



Figure 2. (a) Studied area divided into cells of 1km×1km and the location of the scenario earthquake.(b) Distribution of peak ground acceleration, with median values estimated using Equation (7).

calculated using Equation (7), are shown in Figure 2(b). The Monte Carlo technique was appliedfor the generation of correlated PGA values (10 000 generations) for every cell using the proceduredescribed above. Examples of realizations of strong motion distribution, which were estimatedusing particular parameters of correlation, are shown in Figure 3.

3. DAMAGE AND LOSS ESTIMATION

The loss estimation was performed using the generation of the correlated ground-motion field.Peak ground acceleration was used to evaluate ground shaking-induced structural damage. A setof hypothetical buildings mimicking a real building stock was constructed based on three typesof buildings that are typical for Taiwan [45]: (1) concrete of one to three stories; (2) steel-bracedframes of four to seven stories and (3) frames with shear walls more than eight stories. The wholearea of 22km×18km was divided into cells of 1km×1km, and different numbers of buildingswere assigned, more or less randomly, to every cell. Figure 4(a) shows the distribution of buildingsalong the considered territory. The total number of buildings in the area was 5478.

In this study, we use the methodology, which is based on fragility functions (e.g. HAZUS [47]),for loss estimation. A fragility function defines the exceedance of a damage state for a given levelof ground shaking. Five damage states are considered: None, Slight, Moderate, Extensive andComplete. The probability of being or exceeding a damage state DSi is modelled with a cumulativelognormal distribution,

P[DSi |IM]=�

[1

�DSi

(IM

SMEDIAN

)], (8)

where IM is the given level of ground shaking, �DSiis the standard deviation of the natural

logarithm of spectral amplitudes of damage state DSi , SMEDIAN is the median value of the ground-motion parameter at which the building reaches the threshold of the damage state DSi and � isthe standard normal cumulative distribution function. Every damage state is characterized by arepair and replacement cost ratio RRCi , expressed as a percentage of the building replacementcost RC. If we know the replacement cost RC of every type of building we are considering, thenthe expected direct loss EDL is the product of the damage factor and the replacement cost:

EDL=RC∑

allDSi

RRCi × P[DSi |IM], (9)

where P[DSi |IM] is the probability of being in damage state DSi for a given ground-motion levelIM.



Figure 3. Examples of strong motion distribution (PGA, units of g) along the consid-ered territory (see Figure 2(a)) generated using Equation (7) and various characteristicsof correlation. Total variance �2

T =0.16, in units of natural logarithm; ratio ��/�ε =0.3(between-earthquake variation/within-earthquake variation): (a) mean values from atten-uation equation 7; (b) no correlation, only within-eqrthquake component; (c) correlation

distance 25 km; and (d) correlation distance 5 km.

The methodology used in this study for the computation of the direct loss involves several uncer-tainties. The total variability of each structural damage state �SDS is modelled by the combinationof the following three contributors to damage variability [47]: uncertainty in the damage-statethreshold of the structural system �M(SDS); variability in the capacity (response) properties ofthe model �C and variability in the response due to the variability of ground motion (seismicdemand) �D

�SDS =√

(CONV[�C ,�D, SMEDIAN,DS])2 +(�M(SDS))2. (10)

Thus, fragility functions include both aleatory uncertainty (e.g. uncertainty of the ground-motionlevel given the characteristics of the earthquake) and epistemic uncertainty (e.g. uncertainty relatedto the structural and geometric characteristics of the structure as well as to the models developing



Figure 4. Studied area: (a) distribution of buildings (total numbers) within the cells of 1km×1km and(b) distribution of bridges (vulnerable elements of a lifeline network). The numbers denote the type of

the bridge in Liao and Loh [46] (see also Table II).

them). The components of variability related to the demand and the capacity are discussed in[38, 48–50].

In our study, we used the PGA-based fragility curves developed by Liao et al. [45]. Theauthors reviewed the types of building structures in Taiwan and proposed the classification tobe implemented in the Taiwan Earthquake Loss Estimation System (TELES). Analytical fragilitycurves were developed by nonlinear static analysis using computer models of the structures. Onlythe uncertainty of ground motion demand was used by Liao et al. to represent the total variability�SDS of the damage state. In our analysis, the vulnerability curves should contain only variabilityrelated to characteristics of structures to avoid the double counting of ground-motion variability �D[38]. The ground-motion variability is modelled by the use of Monte Carlo simulation. Porter et al.[49] showed that the influence of uncertainties in ground shaking (intensity of shaking and detailsof ground motion at a given level) on the overall uncertainty in seismic performance (repair cost) isapproximately equal to the influence of uncertainty in the capacity of building assemblies to resistthe damage. Therefore, because the analysis of the relation between ground-motion variability andthe variability related to the construction properties is outside the scope of our study, here weassumed that the values of standard deviations � given by Liao et al. [45] may be used to representonly uncertainty in the seismic capacity. The parameters of fragility curves, which were taken from[45] (Table IV, high-code variant), are listed in Table II. The values of replacement cost (RC) forevery type of building were assigned arbitrarily (see Table II); the total replacement cost for theportfolio containing 5478 buildings was 5077 Mln$.

As has been noted in [37], the development of damage-loss functions for various types of struc-tures is a demanding task. Such conversions have been performed in some of the above-mentionedstudies using the regional damage ratio models based either on (a) Modified Mercalli intensitiesthat were calculated directly [14] or estimated from the PGA values [35] or (b) pseudo-spectralacceleration [12, 37]. In our study, the conversion of damage to monetary loss was performedusing Equation (9). The cost ratios of loss to rebuilding for each damage state were adopted fromHAZUS, with 0.02 for slight damage, 0.1 for moderate damage, 0.5 for extensive damage and1.0 for complete damage. Crowley et al. [48] discussed the sensitivity of the losses to variationsof damage-loss conversion by applying another set of cost ratios: 0.15 for slight damage, 0.3 formoderate damage and 1.0 for both complete and extensive damage. We applied both sets in ouranalyses, and the variants of the damage-loss conversion are called DLC1 and DLC2, respectively.

For a given ground-motion amplitude generated for a cell, a single value of expected directloss EDLi, j was estimated for every building. The loss values, which had been estimated for allbuildings within a particular cell, were summarized to obtain a cell-specific loss. Then, a single



Table II. Parameters of fragility curves for constructions, as considered in this study.

Median PGA (g) for damage statesType of construction, Replacementtheir number N Slight Moderate Extensive Complete cost, Mln$

Buildings, Liao et al. [45]Concrete moment, 1–3 stories, N =2064 0.33 0.53 0.65 0.71 0.5Steel braced frame, 4–7 stories, N =2783 0.30 0.53 0.71 0.83 1.0Frame with shear wall, more 0.33 0.53 0.73 0.85 2.0than 8 stories, N =631Deviation, � 0.50 0.43 0.40 0.40

Bridges, Liao and Loh [46]Type 4, multiple-span, simply 0.20 0.30 0.38 0.50supported super-structure,pier wall, N =3Type 6, multiple-span, continuous 0.23 0.38 0.53 0.70super-structure, multiplecolumn belt, N =2Deviation, � 0.50 0.43 0.40 0.40

total loss amount for a given generation of the ground-motion distribution was obtained as the sumof losses from all the cells,

LOSS=NC∑i=1

NB∑j=1

EDLi, j , (11)

where NB is the number of buildings within the cell j and NC is the total number of cells. Thegenerated set of total loss values (10 000 generations) is used for the estimation of the PDF andCumulative Probability Function (CPF) and the analysis of the parameters of loss distribution.In this study, we did not take into account the structure-to-structure loss correlation of damage[34, 50] and the uncertainty of replacement costs [49].

Some applications require assessing the probability that a specific event will occur during acertain condition at least once or that several such events will occur simultaneously. For example,there may be an interest in knowing whether the vulnerable elements of a lifeline network (e.g.bridges) are likely to be simultaneously damaged during an earthquake or at least whether oneelement will be damaged [14]. To analyse such cases, we considered five bridges located within thearea (Figure 4(b)). The corresponding characteristics of the bridges were taken from Liao and Loh[46]. The damage for bridges was estimated using HAZUS recommendations (see also [51, 52])as follows:

RCRT = ∑all DSi

RCRi × P[DSi |IM], (12)

where RCRT is the total repair cost ratio, RCRi is the repair cost ratio or damage factor (i.e. thefraction of the replacement cost for the i th damage mode) and P[DSi |IM] is the probability ofbeing in damage state DSi for a given ground-motion level IM. Each damage state expresses arange of repair cost ratios, and the so-called best mean repair cost ratio or central damage factoris used in the calculations. Stergiou and Kiremidjian [52] provided a methodology for modellingthe uncertainty in repair cost ratios; however, in this work, we use only the best mean values.

4. RESULTS AND DISCUSSION

We considered the following parameters of loss distribution: the mean value of loss LOSSMEAN =∑Ni=1 LOSSi/N , where LOSSi is the total loss value for the i simulation and N is the total number

of simulations; the standard deviation of the loss distribution �LD; the coefficient of variation



CV=�LD/LOSSMEAN; the median value for which the CPF equals 0.5 and the particular valuesof loss with a certain probability of not being exceeded, e.g. 90% (LP90) or 99% (LP99).

Characteristics of loss distribution for the extreme cases, where all the variability is betweenearthquakes (�ε =0,�� =1.0) and where all the variability is within earthquakes (�� =0,�� =0.0),were analysed in several papers (see the foregoing short review and [48]). We can also mentionthat, if all of the ground-motion variability is treated as within-earthquake (�� =0.0), the medianvalues of loss, which are affected by particularly great losses, are larger than those for the case ofonly between-earthquake variability. The DLC2 variant of damage-loss conversion, which assumesthe higher cost ratios, causes larger mean and median values of total loss, as well as a largerstandard deviation, than the DLC1 variant (see also [48]). However, the coefficient of variation issmaller in the case of the higher cost ratios (DLC2), which means a smaller risk of relatively largevalues of loss, compared with the mean values.

Here we jointly consider both the components of variability (within-earthquake and between-earthquake) and the site-to-site (spatial) correlation. Here, we accepted a total variance of �2

T =0.16and various ratios of RBW =�2

�/�2ε (between-earthquake variation/within-earthquake variation),

namely, 0.1, 0.2, 0.3, 0.45, 0.75 and 1.0, which correspond to the following values of between-earthquake correlation �� =�2

�/�2T : 0.09, 0.17, 0.23, 0.31, 0.43 and 0.5. The correlation distances

CD vary from 5 to 30 km. We also consider a case of prefect correlation (i.e. �ε(�)=1) for allseparation distances �, and a case of spatially uncorrelated ground motion (i.e. �ε(�)=0) forall separation distances � except for �=0km. Obviously, the perfect spatial correlation shouldbe considered an unrealistic and extreme case, and it was included here only for comparisonpurposes.

The results of the calculations are shown in Figure 5. While the mean values of possibleloss do not show any dependence on the ratios of RBW and the correlation distances, the otherparameters of loss distribution vary over wide ranges when considering different characteristicsof the inter-event correlation and spatial correlation. Table III presents a quantitative descriptionof variations in the characteristics of loss distribution relative to the base values. The base valuescorrespond to the smallest between-earthquake correlation (�� =0.09) and spatially uncorrelatedground motion. When considering the observed ranges of the characteristics of ground-motioncorrelation, the variations in the loss characteristics due to a combined influence of the between-earthquake correlation and the spatial correlation may be as high as 40% for median values, 330%for standard deviations and 300% or more for particular values of loss with certain probabilitiesof not being exceeded (e.g. P<99%).

However, by separately comparing the variations of loss characteristics due to between-earthquake correlation �� alone and due to spatial correlation �ε(�) alone, it can be observed thatvariations in the �ε(�) functions (correlation distances) lead to larger changes in the loss charac-teristics than variations in the �� values. Thus, for example, when the correlation distance changesfrom 0.0 to 30 km, the median loss decreases by 40%, while changes of the between-earthquakecorrelation from 0.09 to 0.5 reduce the median loss by only 28%. The correspondent variationsof standard deviation are about 300 and 240%, and the variations of the loss values for the 0.99probability of not being exceeded are about 270 and 210%.

The DLC2 variant of damage-loss conversion (Figure 5(b)), which assumes the higher costratios, caused larger mean and median values of the total loss, as well as a larger standard deviation,than the DLC1 variant (Figure 5(a)). Again, the accepted full-scale variations (i.e. changes withinconsidered limits) in site-to-site correlation lead to larger changes in the loss characteristics thanthe accepted full-scale variations in the between-earthquake correlation. However, the range of thechanges is smaller than that for the DLC1 variant.

Bearing in mind the importance of site-to-site correlation, we also analysed the influence ofthe shape of the correlation function. Figure 1(a) shows the examples of correlation models thatare characterized by almost equal correlation distances (about 27 km) but in which the shapes ofthe functions are different. Table IV summarizes the characteristics of loss distribution calculatedusing these models of spatial correlation. Goda and Hong’s model, which is characterized by arelatively rapid decrease of correlation with separation distance, resulted in larger median loss



Figure 5. Parameters of loss distribution estimated for various correlation distances CD andratios RBW =��/�ε (between-earthquake variation/within-earthquake variation). VariantsDLC1 and DLC2 denote the damage-loss conversion models adopted from HAZUS and

proposed by Crowley et al. [48], respectively.

values, smaller standard deviations and smaller values of loss for certain probabilities of not beingexceeded than Wang and Takada’s model. The difference may reach 23% for the median values,14% for the standard deviation and 15% for the 0.99 probability of not being exceeded. Theinfluence of the shape of the spatial correlation function on characteristics of loss decreases withthe increase of between-earthquake correlation. The relation between the results of applying thesetwo models is not surprising. The high rate of decrease in the site-to-site correlation with theseparation distance would increase the influence of within-earthquake variability, which in turnincreases the possibility of large losses at particular locations.

Now let us analyse the case when several critical elements (bridges) are located within theconsidered territory (Figure 4(b)). The classification of typical bridges in Taiwan is given by Liaoand Loh [46], and we used two types of bridges, namely, Types 4C and 6C (Table I in [46]). Note



Table III. Changes in the characteristics of loss distribution relative to the base values, which correspondto the smallest between-earthquake correlation (�� =0.09) and spatially uncorrelated (correlation distance

0 km) ground motion (variants DLC1/DLC2).

Correlation distance (km)RBW =�2�/�2

ε

(�� =�2�/�2

T ) 0.0 5.0 15.0 30.0 Perfect

Median loss0.10 (0.09) 1.00/1.00 0.83/0.91 0.72/0.85 0.61/0.78 0.47/0.680.20 (0.17) 0.97/0.98 0.81/0.89 0.71/0.83 0.59/0.77 0.46/0.680.30 (0.23) 0.92/0.94 0.77/0.86 0.67/0.81 0.56/0.75 0.44/0.670.45 (0.31) 0.85/0.91 0.74/0.85 0.66/0.80 0.56/0.75 0.43/0.670.75 (0.43) 0.77/0.86 0.67/0.82 0.59/0.77 0.53/0.73 0.42/0.661.00 (0.50) 0.72/0.84 0.63/0.78 0.58/0.75 0.51/0.71 0.41/0.66

Standard deviation0.10 (0.09) 1.00/1.00 1.96/1.63 2.47/2.00 2.98/2.38 3.87/3.000.20 (0.17) 1.35/1.13 2.18/1.75 2.62/2.09 3.06/2.44 3.76/2.940.30 (0.23) 1.59/1.34 2.34/1.89 2.79/2.21 3.24/2.83 3.82/2.970.45 (0.31) 1.86/1.56 2.42/1.97 2.78/2.23 3.14/2.50 3.71/2.880.75 (0.43) 2.24/1.88 2.76/2.19 3.09/2.42 3.41/2.66 3.82/3.001.00 (0.50) 2.36/1.96 2.76/2.24 3.03/2.43 3.29/2.61 3.81/2.94

Loss for probability P<90%0.10 (0.09) 1.00/1.00 1.28/1.26 1.42/1.38 1.56/1.50 1.68/1.660.20 (0.17) 1.16/1.13 1.37/1.33 1.50/1.41 1.59/1.54 1.69/1.670.30 (0.23) 1.22/1.20 1.41/1.36 1.51/1.43 1.61/1.56 1.70/1.670.45 (0.31) 1.30/1.26 1.47/1.41 1.54/1.49 1.64/1.57 1.71/1.680.75 (0.43) 1.38/1.33 1.48/1.44 1.57/1.52 1.66/1.60 1.72/1.681.00 (0.50) 1.44/1.39 1.50/1.49 1.58/1.54 1.68/1.62 1.74/1.68

Loss for probability P<99%0.10 (0.09) 1.00/1.00 1.74/1.52 2.20/1.82 2.66/2.12 3.62/2.740.20 (0.17) 1.22/1.15 1.85/1.60 2.24/1.87 2.70/2.14 3.62/2.740.30 (0.23) 1.40/1.29 2.06/1.74 2.46/1.99 2.75/2.19 3.61/2.750.45 (0.31) 1.62/1.45 2.13/1.80 2.45/2.02 2.87/2.23 3.62/2.750.75 (0.43) 1.98/1.71 2.40/1.98 2.67/2.15 2.91/2.32 3.61/2.761.00 (0.50) 2.06/1.76 2.44/2.08 2.69/2.22 2.94/2.35 3.62/2.76

that here we did not consider uncertainties arising from the damage factor [52] and correlateddamage states conditioned in ground motion [53].

Figure 6 shows the probabilities of joint damage that are equal to or greater than 20% of thereplacement cost (damage ratio 0.2) and the probabilities that at least one bridge will be damaged atthis level. The expected time of bridge outage for this damage level (extensive damage) is between3 weeks and 3 months (see [51]). As can be seen from the figure, in the considered case of severaldifferent bridges located a few kilometres from each other, the site-to-site correlation plays amore important role in such estimations than the between-earthquake correlation. When estimatingthe probability that at least one bridge will be damaged (Figure 6(a)), the possible variationsin the between-earthquake correlation may be completely neglected if the spatially correlatedground motions are used. The between-earthquake variability reveals a slight influence on theprobability only when considering spatially independent ground motion. In contrast, an increasein the correlation distances, as well as an increase in the between-earthquake correlation, wouldincrease the probability of joint damage (Figures 6(b) and (c)); however, the relative influences ofthe correlation characteristics are not the same.

By separately comparing the variations of damage probabilities due only to between-earthquakecorrelation �� and spatial correlation �ε(�), it can be observed that variations in the �ε(�) functions(correlation distances) lead to larger changes in the probabilities of joint damage than variationsin the �� values. Thus, for example, when the correlation distance changes from 0.0 to 30 km,



Table IV. Characteristics of total loss distribution evaluated using two models of spatial correlation withsimilar correlation distances (see Figure 1(b)).

RBW =�2�/�2

ε(�� =�2�/�2

T )

Reference 0.10 (0.09) 0.20 (0.17) 0.30 (0.23) 0.45 (0.31) 0.75 (0.43) 1.00 (0.50)

Median lossGoda and Hong [25] 118 116 113 108 102 98Wang and Takada [11] 96 94 92 88 84 82Ratio 1.229 1.234 1.228 1.227 1.214 1.195

Standard deviationGoda and Hong [25] 230 240 250 260 265 280Wang and Takada [11] 275 280 290 295 298 300Ratio 0.836 0.857 0.862 0.881 0.889 0.933

Loss for probability P<90%Goda and Hong [25] 470 480 490 500 510 515Wang and Takada [11] 507 510 515 520 525 525Ratio 0.927 0.941 0.951 0.962 0.971 0.981

Loss for probability P<99%Goda and Hong [25] 1100 1170 1210 1250 1320 1380Wang and Takada [11] 1340 1380 1400 1430 1470 1490Ratio 0.821 0.848 0.864 0.874 0.898 0.926

Figure 6. Analysis of damage for critical elements of a network (bridges) that are located within theconsidered territory (Figure 4(b)): Probabilities that (a) at least one bridge will be damaged (damage ratio

of 0.2) and (b),(c) all bridges of different types will be damaged simultaneously.

the probability of joint damage for the bridges of Type 4 would increase by 1.77 times, while thechange of the between-earthquake correlation from 0.09 to 0.5 increases the probability of jointdamage only by 1.51 times (Figure 6(b)). The corresponding changes in the probability of jointdamage of all bridges are about 20 times and 13 times (Figure 6(c)).

5. CONCLUSION

In this paper, we compared the effects of variations in the between-earthquake correlation and inthe spatial (site-to-site) correlation of seismic ground motion on the uncertainty of earthquake lossestimations for distributed portfolios and on the probability of damage for several critical elementsof extended structures, e.g. bridges within a lifeline system. It has been recently found that theparameters of ground-motion correlation may vary over wide ranges. To describe the variations,we considered the following boundary values that have been reported in the literature: correlationdistances (site-to-site correlation) varying from 0 to 30 km and between-earthquake correlation



varying from 0.09 to 0.5. A single event, a so-called scenario earthquake, was used as the sourceof seismic influence, and a set of hypothetical buildings representing a real building stock wasconstructed based on typical buildings for the Taiwan area.

The results of the modelling show that the proper consideration of (a) the between-earthquakecorrelation and (b) the parameters of spatial correlation (correlation function �ε(�)) is veryimportant when estimating seismic losses for distributed portfolios. Both types of ground-motioncorrelation affect the probability of joint damage and the characteristics of loss distribution; thepeculiarities of the influence were analysed and are described in literature. In our work, we showedthat the relative influence of each type of correlation is not equal. Thus, at least for the consideredcase of a scenario earthquake, the full-scale variations (i.e. changes within considered limits) in thecorrelation distance (spatial correlation) lead to larger variations in the characteristics of damageand loss than the full-scale variations of between-earthquake correlation. The difference is espe-cially pronounced when considering the median values of loss distribution or the probability thatat least one critical element of a lifeline will be damaged. When the spatial correlation of groundmotion is used for such estimations, the possible variations in the between-earthquake correlationmay be neglected.

The shape of the site-to-site correlation function, i.e. the rate of decrease of the coefficient ofspatial correlation with separation distance also seems to be important for the loss assessmentswhen modelling spatially correlated ground-motion fields. As expected, a high rate of decreaseof the coefficient of spatial correlation with separation distance would increase the influence ofwithin-earthquake variability and, in turn, would increase the possibility of large losses at particularlocations but decrease the variability of total loss.

If information about correlation models is not available, it is necessary either to obtain upperand lower bound estimates by assuming the extreme characteristics of correlations or to makesome assumptions about correlations based on reported models. In this situation, it is necessaryto bear in mind that the spatial correlation of ground motion, in principle, depends on the chosenground-motion model. On the other hand, the correlation structure depends on the local geology aswell on peculiarities of the propagation path (azimuth-dependent attenuation) [29]. Thus, a singlegeneralized model of spatial correlation may not be adequate for large areas.

We realize, of course, that there are several limitations of our study. Our findings relate toa single earthquake and a relatively small area, the dimensions of which are comparable to themaximum value of correlation distance in the spatial correlation model. A comprehensive analysisrequires considering territories of various sizes and various building densities. The influence ofthe geographical resolution of the exposure data (as in [54]) should be analysed. The conclusions,which were made for the case of a single earthquake, should be verified considering the influencefrom multiple earthquakes. The damage analysis was based only on peak ground acceleration, andit included some arbitrary assumptions about the parameters of fragility curves and replacementcosts. Obviously, for practical calculations requiring the estimation of the absolute values of loss,these simplified assumptions have to be reconsidered.

ACKNOWLEDGEMENTS

The authors thank Julian J. Bommer for his thoughtful review comments and suggestions, which signifi-cantly helped to improve the earliest version of the article. The constructive comments from anonymousreviewers are gratefully acknowledged. We also thank Chin-Hsun Yeh for providing necessary data usedin this study, valuable comments and suggestions. This work was sponsored by Deutsche Forschungsge-meinschaft (DFG), Germany, project WE 1394/14-1.

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