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Applications of Statistics and Probability in Civil Engineering – Kanda, Takada & Furuta (eds)© 2007 Taylor & Francis Group, London, ISBN 978-0-415-45211-3

Modeling spatial correlation of ground motion Intensity Measures forregional seismic hazard and portfolio loss estimation

J. Park & P. BazzurroAIR Worldwide Corporation, San Francisco, USA

J.W. BakerStanford University, Stanford, USA

ABSTRACT: The probabilistic assessment of ground motion intensity measures (IMs; e.g., peak ground accel-eration) at an individual site is a standard practice. Less attention has been devoted to estimating the statisticaldependence of IMs from a single event at multiple sites due to common source and wave traveling paths and tosimilar distance to fault asperities. This paper explores the site-to-site correlation of IMs and demonstrate its usein seismic hazard and loss estimation. The use of a spatial correlation function will be shown in two applications:1) evaluation of the effect of modeling spatially correlated ground motion fields on loss estimates for portfoliosof structures with different spacing patterns; and 2) simulation of U.S. Geological Survey (USGS) ground motionShakeMaps consistent both with the recorded IM values at each station and with the IM correlation structure.This last case could be extremely valuable in USGS-sponsored efforts such as PAGER and ShakeCast.

1 INTRODUCTION

Many private and public stakeholders are stronglyaffected by the impact of earthquakes on a regionalbasis rather than on a single property at a specific site.The stakeholders include government and relief orga-nizations that need to prepare for future events andmanage emergency response, and private organiza-tions that have spatially distributed assets. Whether formitigating future seismic risk or managing responseafter an earthquake, assessment of regional earth-quake impact requires a probabilistic description ofthe ground motion field that an event could or hasgenerated. With knowledge, albeit probabilistic, of thelevel of ground shaking at a regional level one couldmore accurately estimate, for example, (1) the mon-etary losses caused to specific structures or portfolioof structures owned by a corporation or insured by aninsurance company, (2) the number of people killed,injured, or displaced in a given region; (3) whether theaccess to certain critical buildings, such as hospitals,might be restricted due to yellow or red tagging; and(4) the probability that distributed lifeline networks forpower, water, and transportation may be interrupted.

The probabilistic assessment of ground motionintensity measures, IMs (e.g., peak ground acceler-ation or spectral quantities) at an individual site basedon the event magnitude, the source-to-site distance,and the local soil conditions is a standard practice

that started in the late 1960s. Much less attention hasbeen devoted, however, to estimating the statisticaldependence of ground motion IMs from a single eventat multiple sites. This paper provides a contributionto filling this research gap. Furthermore, we intendhere to explore the effects of site-to-site correlationof ground motion IMs in more depth, and demon-strate their use in seismic loss estimation of spatiallyextended portfolios of structures.

2 ESTIMATION OF SPATIAL CORRELATION

In general, three effects contribute to the correlationof ground motion IMs at two sites: (a) they have beengenerated by the same earthquake (e.g., a high stress-drop earthquake may generate ground motions in theregion that are, on average, higher at all sites than themedian values from events of the same magnitude);(b) the seismic waves travel over a similar path fromsource to closely spaced; sites and (c) if the dimensionsof the fault rupture are large compared to the distancefrom the sites to the source, adjacent locations may belocated close to or far from the asperities on the faultrupture.

Modern ground motion prediction equationsimplicitly recognize the first cause of dependency viaa specific inter-event error term, ηi, as follows:

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where Yi,j is the ground motion IM of interest (e.g.,Sa(T1)), ln Yi,j is the median value of the log of Ypredicted by the attenuation equation at site j giventhe magnitude and distance of earthquake i and thelocal site conditions, ηi is the aforementioned inter-event standard normal error term, εi,j is the site-to-siteintra-event standard normal error term, and τ and σare the corresponding standard deviations of the twoerror terms, or “residuals.”

An alternative formulation for Equation 1, whichwas common in older prediction equations, is given by

where ε̃i,j is a random variable representing both theinter-event and intra-event variation at site j fromearthquake i. By comparing Equations 1 and 2, it is canbe seen that σ̃ must equal

√σ2 + τ2 for the variances

of the two equations to be equal, and that

In the context of assessing site-to-site correlation ofground motion IMs, it is convenient to use the modelin Equation 2 for at least three reasons: (a) there isnow only one residual term for each observation (ln Yi,jand σ̃ are provided by ground motion prediction equa-tions, and Yi,j is observed, so ε̃i,j can be computeddirectly); (b) the residual term is easy to compute (thevalues of ηi, i = 1, . . . , N , for all the N earthquakes andthe frequency-dependent values of τ are usually notincluded by the developers of ground motion predic-tion equations in their publications); and (c) Equation 2is also the form commonly used in probabilistic seis-mic hazard analysis (PSHA) computer programs, sospatial correlation models in this format can be moreeasily incorporated into existing software.

An example of observed ε̃i,j residuals for peakground acceleration (PGA) from the 1999 Chi-Chi,Taiwan, earthquake is shown in Figure 1; these resid-uals, whose value is indicated by the color scale, arethe combined residuals from Equations 2 and 3 andrepresent both the inter- and intra-event error terms.

The intra-event site-to-site ground motion correla-tion, namely the correlation between the two randomvariables εi,j and εi,k at two different sites j and k , dueto commonality of wave paths and of distance fromasperities has not yet been extensively investigated.Spatial dependence can be observed in Figure 1, bynoting that residuals at nearby locations have similarvalues. This intra-event site-to-site correlation, whichis of course not addressed in attenuation relationshipsfor single sites, is crucial for the spatially distributedapplications mentioned above.

The limited research on this topic to date indi-cates that the correlation of peak ground acceleration

Figure 1. Observed attenuation residuals from the 1999Chi-Chi, Taiwan, earthquake.

or velocity values decreases with increasing spacingbetween two sites. This correlation can be estimatedby computing empirical correlation coefficients for εi,jvalues at a site separation distance h (plus or minussome tolerance). Because the ηi value is fixed for eachith earthquake, it is effectively a constant when empiri-cal correlation coefficients are estimated from a singleearthquake. Thus, correlation coefficients obtainedfrom εi,j values or ε̃i,j values will be identical, but thesecorrelation coefficients only represent the correlationin the εi,j values. To obtain correlation coefficients forthe ε̃i,j values, one must add the effect of the ηi randomvariable, which is perfectly correlated at all distancesbut cannot be detected from the previous empirical cor-relation coefficients. The total correlation in ε̃i,j valuesis thus

where ρ(h) = ρεi,j,1,εi,j,2 (h) is the empirical correlationcoefficient calculated for intra-event εi,j values sep-arated by a distance h, and ρ̃(h) = ρε̃i,j,1,ε̃i,j,2 (h) is thecorrelation coefficient for the total ε̃i,j values definedin Equation 3. Note that for very close sites (i.e., h → 0)the correlation ρ̃(h) of IMs, of course, tends towardone, whereas for very distant sites (i.e., h → ∞) it issimply given by the ratio of the inter-term-variance tothe total variance, as expected. The variance of the dif-ference of the same IM quantity at two sites, k and l,separated by a distance h is simply

Several models currently exist for these correla-tion functions (e.g., Boore et al., 2003; Kawakami

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and Mogi; 2003; Wang and Takada, 2005; Jeon andO’Rourke, 2005). The models differ in their results,potentially due to differences in the IM parameter (e.g.,peak ground acceleration versus peak ground veloc-ity) or in the earthquake events being studied, andto differences in estimation methodologies. Using acomprehensive study of many IM parameters withinthe framework described here, the authors expect toidentify and resolve these differences shortly.

In the simulation steps that follow below, the corre-lation coefficient ρ̃(h) in Equation 4, which incorpo-rates correlation due to common inter-event residualsand spatial correlation of intra-event residuals, willbe used.

3 SIMULATION OF GROUND MOTION IMSFOR LOSS ESTIMATION

It is a standard practice to model a ground motion IM,Yi,j (e.g., the spectral acceleration Sa(T1), at periodT1), generated by earthquake i at a site j as a log-normally distributed random variable (RV). In thenotation of Equation 2 this implies that its natu-ral logarithm is a Gaussian RV, i.e., that ln Yi,j ∼N (ln Yi,j; σ̃2). We now make the reasonable assump-tion that the ln Yi,j for all j = 1, . . . , M sites affectedby earthquake i are not only marginally normally dis-tributed, but also jointly normally distributed RV’s.Under this assumption, the generation of a M -site ran-dom field of ground motion IM values that are spatiallycorrelated involves the generation of a multivariateGaussian vector of correlated, standard normal randomvariables, X = [X1, X2, . . . , XM], with a (symmetric,positive definite) covariance matrix �, or in shorthandX ∼ NM(0, �). Here, Xj =Qεi,j is the total residual termin the model for the computation of ln Yi,j described byEquations 2 and 3. Note that, because ε̃i,j has a standarddeviation of one due to normalization, the element of� in the kth row and lth column is computed by sim-ply evaluating Equation 4, using the distance betweensite k and site l to evaluate the correlation function.Earthquake i can be either a realization of a futureevent or of a past event for which empirical observa-tions of ground motion IMs at sites k = 1, 2, . . . , K areavailable.

In the future event case when no empirical observa-tions of ground motions are available, the generationof X can be done simply by recognizing that X = AUwhere U = [U1, U2, . . . , UM] is a vector of indepen-dent standard normal variates and the matrix A is theCholesky decomposition (i.e, matrix square root) of�, that is the unique lower triangular matrix A suchthat AAT = � (e.g., Melchers, 1999).

These ε̃i,j values (see Figure 2 for an example basedon the correlation function developed by Boore etal., 2003) can then be added to the mean ground

-2.0-1.5-1.0-0.50.00.51.01.52.0

Los Angeles

Figure 2. One realization of an ε̃i,j random field whose spa-tial correlation is modeled after Boore et al. (2003). Therectangle is the projection of the 1994 Northridge earthquakerupture in southern California.

0.20.61.01.41.82.22.6

Los Angeles

Figure 3. Map of spatially correlated 5% damped Sa(0.3 s)compatible with the rupture (white rectangle) of the 1994Northridge earthquake. This map was generated using themap of ε̃i,j shown in Figure 2.

motion term, ln Yi,j , in Equation 2 to obtain realiza-tions of spatially correlated ground motion IMs. Thisis precisely what has been done to produce the mapof 5%-damped Sa(0.3s) shown in Figure 3, whichis based on the ε̃i,j field in Figure 2 and is com-patible with the rupture and magnitude of the 1994,Northridge earthquake. For the generation of this mapwe used the digital soil map of Southern California(http://www.consrv.ca.gov/), and the ground motionprediction equation of Abrahamson and Silva (1997).Note that in generating Figure 3, we assumed that thecorrelation function for PGA developed by Boore etal. (2003) is also applicable to the spatial correlation ofSa(0.3s). The map in Figure 3, like the ShakeMap pro-duced by USGS (http://earthquake.usgs.gov/eqcenter/shakemap/sc / shake /Northridge#0.3_sec_Period) isconsistent with the same rupture scenario but, unlikethe “real” ShakeMap, it does not preserve by design

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any of the recordings from the real event and is pro-duced for the geometric mean, not the maximum value,of Sa(0.3s) of the two horizontal components.

Section 5 presents an alternative method for pro-ducing maps of correlated ground motion IMs forpast events that are consistent with ground motionmeasurements at the recording stations.

4 PORTFOLIO LOSS ESTIMATION RESULTS

Using the simulation methodology in Section 3, wehave studied the seismic risk of two portfolios ofstructures in the San Francisco Bay Area in NorthernCalifornia

• Portfolio 1: 1,090 properties (Figure 3) in the samecounty.

• Portfolio 2: 133 properties (inset of Figure 3) in thesame postal code area.

Both portfolios are made of residential 1- to 3-storywoodframe buildings with an assumed replacementvalue of $0.5 M each. We will refer to Portfolio 1 asthe large portfolio and to Portfolio 2 as the small port-folio. Note that here the adjectives “large” and “small”denote the spatial size of the footprint of the portfoliosand not the number of properties contained in them.Similar qualitative conclusions as those presentedbelow would have been found even if the spatially con-fined Portfolio 2 contained more structures than in thewidely spread Portfolio 1.

As with conventional PSHA, the two portfolioswere subjected to a large catalog of future earthquakeevents to estimate the likelihood of losses due to struc-tural and non-structural damage.To study the effects ofspatial ground motion correlation modeling on port-folio loss estimates, we generated a random field ofSa(0.3 s) in the affected region for each event in thecatalog according to the six approaches listed below.In this loss estimation exercise, the random fields weregenerated for Sa(0.3 s) because this ground motionIM is a good predictor of the response of stiff wood-frame buildings. Note that in all six cases, all the RV’swere simulated according to a Gaussian distributiontruncated to ±3σ̃.

1. Independent ground motions at each site (i.e., ε̃i,jand ε̃i,k are independent RV’s if j �= k , andρ(h) = 0).

2. Spatially correlated ground motion field withreduced correlation ρ̃(h) = ρ̃ = τ2 / (σ2 + τ2). Thisexpression is derived from Equation 4 after zero-ing the ρ(h) term. The spatial correlation modeledin this case is only due to the common value of theinter-event term ηi (see Equation 1) at all sites. Thevalues of σ2 = 0.66σ̃2 and τ2 = 0.34σ̃2 for Sa(0.3 s)were taken from Lee et al. (2000).

Figure 4. Location of the large and small (inset) portfoliosconsidered for the loss estimation analysis.

Figure 5. Four ground motion correlation models for ρ̃(h) asa function of inter-site distance, h, used in the loss estimationanalysis.

3. Spatially correlated ground motion field, wherethe intra-event correlation term ρ(h) = 1 −[1 − e−√

0.6 h]2

in Equation 4 is obtained accordingto the function by Boore et al. (2003).

4. Spatially correlated ground motion field witha preliminary alternative correlation functionρ(h) = e−(h / 6) developed by Baker (2006) basedon an analysis of the residuals for the Chi-Chiearthquake only.

5. As in Case 3 but with the separation distance atwhich the correlation coefficient has decreased toρ = e−1 (Vanmarcke 1983) suggested by Boore etal. doubled (i.e., 0.6 is replaced by 0.3 in the equa-tion above). This case is considered as an upperbound of ground motion correlation.

6. ε̃i,j and ε̃i,k are assumed to be perfectly positivelycorrelated RV’s.

The trend of ρ̃(h) with site-to-site distance, h, forCases 2–6 is shown in Figure 5. Cases 3–5 are fully

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Figure 6. Mean rate of exceedance curves for losses to thelarge portfolio (top) and small portfolio (bottom) computedusing the six proposed models for simulating random fieldsof ground motion IMs for each earthquake.

consistent with the methodology in Section 3, Case 2 isconsidered because, for its simplicity, it has been usedto model ground motion correlation in many appli-cations. Case 1 has a place in here because in theoverwhelming majority of cases ground motion spa-tial correlation is simply disregarded. Finally, Case 6 isan unrealistic, extreme case included for comparisonpurposes only.

The losses expressed as a fraction of the replace-ment cost, called damage ratio (DR) experienced by abuilding at given site were simulated from a so-called“damage function” (DF), which is not shown here.A DF is a probabilistic relationship that presents theDR as a function of the severity of the ground motionIM, here Sa(0.3 s), observed at the site. In all six casesthe portfolio losses for each event were obtained bysimply adding the losses predicted at each site, if any.The losses caused by all the events in the catalog andthe annual rate of occurrence of each earthquake werethen assembled to produce a curve showing the annualmean rate of exceedance (MRE) of portfolio losses of

Figure 7. Ratio of MRE curves from Cases 1–6 to MREcurve for Case 3, where Boore et al. (2003) correlation func-tion is used. The left panel refers to the large portfolio andthe right panel to the small portfolio.

different amounts. The MRE curves for the six casesabove are shown in Figure 6 for both the large and thesmall portfolios displayed in Figure 4.

The loss exceedance curves in Figure 6 show thatthe trend of the MRE curve is systematically dis-torted if the spatial correlation of ground motion IMsis ignored (Case 1). The rare losses are consistentlyunderestimated and the frequent losses are overesti-mated compared to those produced by Cases 3 and4, which we consider to be the benchmarks for thisexercise. This assertion becomes even more evidentwhen considering the ratio of the MRE curves fromCases 1-6 to the MRE curve of the reference Case 3as a function of MRE (Figure 7). For the large andsmall portfolios, the underestimations occur for lossescorresponding to MREs less than or equal to 7 × 10−3

and 1 × 10−2, or mean return periods (MRPs) of about150 or 100 years and longer, respectively.

As expected, neglecting the intra event site-to-sitecorrelation of IMs, namely setting ρ(h) = 0 in Equa-tion 4 (Case 2), produces loss estimates that are lessbiased than those from Case 1. The underestima-tion/overestimation of losses is still significant andbegins at about the same MRE (and MRP) values asin Case 1. As expected, the bias introduced by neglect-ing or underestimating the spatial correlation of IMsis more severe for small portfolios than it is for largeones. Neglecting ground motion correlation makes theoccurrence of extremely high (or low) ground motionvalues everywhere in the footprint of the portfolioessentially an impossible occurrence. When a portfo-lio footprint (here about 4 km for the small portfolio) iswithin the separation distance of the correlation model(4 km for Case 3 and 6 km for Case 4), then a sce-nario with significantly higher (lower) than expectedground motions at all sites in the portfolio is a much

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more likely event than it is for portfolios with largefootprints. At an MRP of 1,000 years, for example,the large portfolio losses are underestimated by 19%in Case 1 and by 7% in Case 2. For the small portfo-lio the bias in percentage terms increases to 22% forCase 1 and 13% for Case 2.

The loss exceedance curves for Cases 3, 4, and 5,namely those that include more realistic models of theintra-event spatial correlation of IMs, are quite inter-estingly fairly similar for both portfolios. The lossesare within ±10% for all MRE values even for Case 5where the spatial correlation is, by design, overes-timated. Additional MRE curves (not shown here)computed for a portfolio of about 25 structures con-fined within a 1 km-diameter circle confirm thesefindings (i.e., ±20% in that case for all MRE valuesof practical interest).

The similarity of MRE curves produced by differ-ent models for spatial correlation of IMs implies that,at least for this application and these two portfolios,the details of the correlation function do not have asignificant impact on the losses of portfolios of struc-tures with footprints ranging from one to more thanone hundred kilometers. For widely spread portfolios,a significant departure from this cluster of consistentMRE curves can only be obtained either when thesite-to-site spatial correlation of IMs is disregarded(Case 1), as discussed above, or when it is unrealisti-cally overestimated (Figure 7 left), such as in Case 6where the ground motion IMs at different sites are con-sidered perfectly positively correlated RV’s regardlessof inter-site distance. For portfolios with small foot-prints, on the other hand, assuming perfect positivecorrelation of IMs appears to produce a less significantbias in the loss estimates

5 CHARACTERIZATION OF UNCERTAINTYIN SHAKEMAPS

The tools described above can also be modified to sim-ulate spatially correlated ground motions that are alsoconsistent with observed ground motion IMs at indi-vidual sites with recording instruments. This approachis useful for rapid post-earthquake ground motion esti-mation tools such as ShakeMap. By more completelycharacterizing the ground motion IMs that may haveoccurred at locations with no recordings, PAGER andother similar tools developed by USGS will be ableto more accurately assess the uncertainty in post-earthquake rapid estimates of fatalities and losses.

In this case, the proposed alternative approachuses conditional simulation of ε̃i,j values at individuallocations, given observations from recording stationsand previously simulated locations. These ε̃i,j valuescan then be added back to mean attenuation predic-tions in Equation 2 to obtain realizations of groundmotion IMs.

The joint realizations of ε̃i,j values in a region ofinterest can be generated using a conditional simu-lation approach. To do this, the approach describedin Section 3 is modified as follows. The vector X isstill used to describe the joint Gaussian distributionof ε̃i,j values at the M sites of interest (the sites whereproperties are located as well as the sites where record-ings were obtained). We now partition this vector intotwo sub-vectors X’ (containing the sites of interest forloss analysis) and Xobs (consisting of the sites withobserved ground motion intensities).

By re-ordering the elements of X and partitioningthe relevant matrices, the distribution of the vector canbe denoted

where, as before, the elements of � are evaluated usingEquation 4 using the separation distance between thetwo associated sites. Nothing in this equation requiresknowledge of observed ground motion intensities –the ordering is merely for notation and convenience atthe next step.

The observed values of ε̃i,j are incorporated intothe model by noting that the values of Xobs, are nowknown. With this information, the conditional distri-bution of X’, given knowledge that the vector of ε̃i,jvalues at Xobs, are equal to x is given by

where [A|B] is used to denote the distribution of Agiven B. Whereas the original (marginal) mean andcovariance of X’ were 0 and �22, respectively, thosehave been updated based on the observed values. Itis interesting to note that with this multivariate nor-mal model, the conditional covariance of X’, �22 −�21�

−111 �12, is always less than �22, and that it is

independent of the observed values x.Once the mean and standard deviation of this con-

ditional distribution have been computed using Equa-tion 7, the Cholesky decomposition and simulationapproach described above can proceed as before. Sim-ulated values at the sites with recordings will alwaysequal the recorded value (because the conditionalmean is equal to the observed value and the condi-tional standard deviation is equal to zero). Conditionalsimulations at sites nearby the recordings will alsohave reduced standard deviations and adjusted meansin accordance with the specified spatial correlation.

The joint realizations of ε̃i,j values in a region ofinterest can also be generated using a series of succes-sive conditional simulations (Goovaerts 1997). First,the conditional distribution at an arbitrary unsampledlocation, X’1, is determined, conditional upon values

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Figure 8. Conditional simulations of ε̃i,j from the Chi-Chiearthquake, conditional upon the observed values in Figure 1.

of the originally sampled data points. This conditionaldistribution is obtained using Equation 7, but now X’ isonly a scalar rather than a vector. A simulated value ofε̃i,j is generated using this conditional distribution, andthis simulated value is fixed while additional ε̃i,j valuesare simulated conditional upon this observed value.The final simulations are mathematically identical tothe one-step simulation approach described in moredetail here, but the conditional simulation approachsometimes has computational advantages because theconditioning covariance matrices can be reduced toaccount for only locations nearby the individual loca-tion of interest, thus reducing the size of the �11 matrixthat must be inverted.

Three example ε̃i,j simulated random fields that areconsistent with observed ε̃i,jvalues from Figure 1 areshown in Figure 8. At every point in the simulationsof Figure 8 where an observation is available in Figure1, the simulations are equal to the observed values.At locations near to recordings, the variance of ε̃i,jvalues from the simulations is reduced because of thecorrelation with the nearby observed value.

The simulations of Figure 8 were produced usingan implementation of Equation 7 in the open-sourceGSLIB geostatistics software package (Deutsch andJournel 1997). These ε̃i,j simulations can be incorpo-rated into Equation 2 to produce simulations of groundmotion IMs that are spatially dependent and alsoconsistent with observed intensities at the recordingstation locations. The simulations can then be used toquantify uncertainty in ShakeMap-type ground motion

intensity maps, or to perform post-earthquake portfo-lio loss analysis that incorporates measured groundmotion intensities.

6 CONCLUSIONS ANDRECOMMENDATIONS

This paper has shown how empirically-calibrated spa-tially correlated random fields of ground motion IMs(e.g., PGA or spectral accelerations), can be developedand utilized for hazard analysis and loss estimation.

The sources of correlation in ground motion inten-sities were discussed, and treatment of these uncertain-ties in current ground motion prediction models wasdescribed. Within this treatment approach, a model forcorrelation on inter- and intra-event ground motionintensity residuals was described, and an approach formodeling these correlations was presented. Inter-eventresiduals are perfectly correlated within a specificearthquake, while intra-event residuals are only par-tially correlated, with a correlation model that must bedetermined empirically.

Once the correlation model for these residuals wasdetermined, an approach for simulating random fieldsof correlated ground motion IMs was described. Underthe reasonable assumption that the IM residuals have ajoint normal distribution, and by utilizing the Choleskydecomposition of the covariance matrix for this jointnormal random vector, it is possible to efficiently sim-ulate realizations of this spatially correlated IM field.The correlation modeling and ground motion simula-tion results were then effectively used in two importantapplications:

• Generation of spatially correlated random fieldsof given IMs (similar to the USGS-sponsoredShakeMaps) for both future scenario earthquakesand past events.

• Loss estimation for spatially extended portfolios ofstructures.

As an example of the first application, we generatedmaps of Sa(0.3 s) consistent with the Northridge 1994earthquake fault rupture but not with the Northridgeavailable recordings and others consistent with boththe rupture and the recordings of the Chi-Chi, Taiwan,1999 earthquake. In the latter case the simulated IMmaps at sites nearby the recording stations also havereduced standard deviations and adjusted means inaccordance with the specified spatial correlation.A setof such maps could be generated in a post-earthquakeenvironment as alternative realizations of the USGSShakeMaps, which away from the recording stationsrely on median values of IMs.

The second application has shown that only by mod-eling the spatial correlation of IMs can one avoid intro-ducing a bias in portfolio loss estimates. Neglecting or

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underestimating the spatial correlation of IMs tends tooverestimate frequent losses and underestimate rareones. The opposite holds if the spatial correlation ofIMs is severely overestimated with respect to empiri-cal observations. This study has shown, however, thatat least for the cases considered here the portfolio lossestimates seem to be fairly insensitive to the details ofthe mathematical model adopted to describe the spatialcorrelation of IMs. This suggests that the choice of thespecific correlation model for simulation and loss esti-mation studies may not be critical, but it does not implythat such correlations can be completely neglected.Note that these findings may not hold true in all cases.More research is needed to evaluate their generality.

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