PSHA for Distributed Seismicity-Lapajne

2502

Bulletin of the Seismological Society of America, Vol. 93, No. 6, pp. 2502–2515, December 2003

Probabilistic Seismic Hazard Assessment Methodology

for Distributed Seismicity

by Janez Lapajne, Barbara Sket Motnikar, and Polona Zupancic

Abstract This study deals with probabilistic seismic hazard analysis (PSHA)based on earthquakes that cannot be assigned to specific geologic structures (distrib-uted seismicity). To calculate seismic hazard from distributed seismicity, the PSHAmethodology is extended in numerous ways. A simple quantitative seismotectonicmodel enables statistical incorporation of certain seismotectonic knowledge and theapplication of attenuation relationships that are based on fault distance. The influenceof a large historic earthquake on seismic hazard is taken into account with an energy-based seismic activity rate, obtained by spatial distribution of the released seismicenergy.

It will be shown that for at least a rough seismic hazard assessment, the type ofgeometrical seismicity modeling is not a deciding factor. To illustrate this, the clas-sical seismic source zone approach and the improved spatially smoothed seismicityapproach are studied in detail to determine their common and distinctive components.Except the geometry of modeling the seismic activity, there is no essential differencebetween the two approaches.

Seismic activity in Slovenia can be treated as distributed seismicity. For this rea-son, the presented methodology is applied to the calculation of seismic hazard mapsof Slovenia as a case study.

Introduction

This study focuses on modeling seismicity that cannotbe assigned to specific geologic structures due to inaccurateand insufficient seismologic and geologic data. It is gener-ally termed “distributed seismicity” (Stirling, 2000). Suchearthquakes are sometimes referred to as “floating” or “ran-dom” earthquakes (dePolo and Slemmons, 1990). Seismo-tectonic knowledge of a given area often does not allow theexact determination of faults that cause earthquakes of mag-nitude below 5. However, many historical earthquakes withmagnitudes in the range between 5 and 6.5 are also floating,because their records in the earthquake catalog are very in-accurate.

As an example, in the probabilistic seismic hazard anal-ysis (PSHA) of California, earthquakes with a magnitude of4–6.5 have been modeled separately from the well-knownfault sources. In this case, distributed seismicity is named“background seismicity” (Cao et al., 1996), as opposed tothe seismicity of the well-known faults.

McGuire (1993) classified the main procedures of PSHAinto deductive and historic categories. Historic methods useonly information from earthquake catalogues, for example,the nonparametric historic method by Veneziano et al.(1984). On the other hand, deductive methods focus also onthe determination and estimation of the cause and origin of

earthquakes. The most popular deductive method follows theCornell (1968) approach and is based on the determinationof seismogenic source zones with homogeneous seismic ac-tivity rate. The main problem in this approach can be largesubjectivity in the delineation of seismic source zones, par-ticularly where the seismotectonic knowledge of the ob-served area is poor. Some interesting approaches that use theadvantages of both methods are the spatially smoothed seis-micity procedure developed by Frankel (1995) and Frankelet al. (2000), the parametric-historic procedure proposed byKijko and Graham (1998, 1999), and the zoning free kernelapproach proposed by Woo (1996).

All of these methods may be applied to the modeling ofdistributed seismicity. Although the characteristics of dis-tributed seismicity increase the unreliability of deductive ap-proaches, the classical seismic source zones method is stillwidely used. On the other hand, the simplicity and naturaldisposition of the spatially smoothed seismicity approach areconvenient for avoiding the subjective delineation of seismicsource zones. The particular suitability of this approach formodeling distributed seismicity leads us to study the spatialsmoothing in detail and to make improvements in numerousways (e.g., Lapajne, 2000; Sket Motnikar et al., 2000). Animportant goal of this research is also to bring both methods

Probabilistic Seismic Hazard Assessment Methodology for Distributed Seismicity 2503

to a common base and to prove that the geometrical (spatial)modeling of seismicity is in fact the only differentiatingfactor.

Numerous seismic hazard assessment studies are basedonly on the earthquake catalog. However, for detailed andparticularly for site-specific estimates, seismotectonic knowl-edge should also be taken into account. The amount and thequality of the input data influence the assumptions madeabout parameters and on the types of modeling of the seismicactivity, magnitude distribution, attenuation relationship,and seismotectonics. Later in this article, many alternativesin the input data will be proposed with the aim to generalizethe methodology as much as possible.

Common Characteristics in ModelingDistributed Seismicity

Earthquake Time Series

In PSHA, it is common practice to assume that the Pois-son process describes the earthquake occurrence rate. In ar-eas where causal faults are unknown, the Poissonian modelis particularly suitable. But it is justified even in many sit-uations where the fault memory actually exists (Cornell andWinterstein, 1988). To enable the application of the Poissonprocess, all preshocks and aftershocks should be removedfrom the earthquake catalog. Then, at least one completesubcatalog should be determined. In the complete subca-talog, all earthquakes above the corresponding lower boundmagnitude m0 are assumed to be recorded. This lowest valueof the magnitude in the observed complete subcatalog de-termines the level or threshold of completeness.

Magnitude Distribution

The standard log-linear magnitude–frequency relation-ship was established by Gutenberg and Richter (1944),

�b(m�m )0n(m) � n(m ) • 10 ,0 (1a)

and later generalized to the doubly truncated exponentialrecurrence relationship (Cornell and Vanmarcke, 1969),

�b(m�m ) �b(m �m )0 u 010 � 10n(m) � n(m ) ,0 �b(m �m )u 01 � 10

(1b)

where n(m) is the cumulative annual number of earthquakeswith magnitude equal to or greater than m, the seismic ac-tivity rate n(m0) is the total annual number of earthquakeswith magnitude equal to or greater than the lower boundmagnitude m0, mu is the upper bound magnitude, and b isthe decay rate. Usually, the seismic activity rate is normal-ized to an annual rate. Assuming that the magnitude isbounded, the doubly truncated exponential distribution isrecommended, as it naturally follows from the maximumentropy principle (Berrill and Davis, 1980).

The corresponding probability density function of themagnitude for the log-linear case is

�b(m�m )0p(m) � bln10 • 10 (2a)

and

�b(m�m )0bln10 • 10p(m) � �b(m �m )u 01 � 10

(2b)

for the doubly truncated exponential recurrence relationship.Parameters n(m0), m0, mu, and b of this relationship may bedetermined for the entire observed area (global parameters)or for its smaller parts (local, variable parameters), for ex-ample, for a particular source zone or a set of grid cells.However, a too small or a very low hazard area may not beappropriate for statistical evaluation. Of course, the esti-mates depend on the chosen estimation method and on thetime span of the catalog.

As observed earlier, the lower bound magnitude m0 gen-erally equals the lowest magnitude in the observed completesubcatalog and is therefore taken as a global parameter forthe whole area. For the estimation of decay rate b, it wasshown (Page, 1968; Weichert, 1980) that the maximum like-lihood estimate (MLE) has the best properties. It may be alocal or a global estimate.

A common way to determine the seismic activity raten(m0) is to count the events in the complete earthquake sub-catalog. This method usually fails for very large earthquakesdue to historical records that are generally too short. Some-times, the rate of large earthquakes may be determined fromthe geological record, for example, from fault slip rates(Youngs and Coppersmith, 1985). When such geological in-formation is not given, even large earthquakes can be treatedas part of distributed seismicity. Later, we propose an alter-native way to emphasize the influence of larger earthquakeson the resulting seismic hazard, which is based on the re-leased seismic energy (Lapajne et al., 1997).

For the well-known fault sources, the upper bound mag-nitude mu can be based on the maximum observed size ofruptures using empirical relationships between the rupturedimension and the magnitude. For distributed seismicity, thecausal faults are unknown, which is why other approachesshould be used. Statistical evaluation of the upper boundmagnitude mu was most thoroughly treated by Kijko (e.g.,Kijko and Sellevoll, 1989; Kijko and Graham, 1998). How-ever, for an approximate estimation of the upper boundmagnitude mu, the maximum observed magnitude in the(sub)catalog might be increased by a chosen proportion ofthe magnitude unit, for example, 0.5. A simple criterion forthe determination of the magnitude increment was proposedby Lapajne and Sket Motnikar (1996). In their paper, a well-defined complete earthquake subcatalog was chosen to es-timate the decay rate b with various values of the upperbound magnitude mu. The value of mu, at which the corre-

2504 J. Lapajne, B. Sket Motnikar, and P. Zupancic

sponding standard error of decay rate is minimal, was takenas an estimate of the upper bound magnitude for the periodof the chosen subcatalog. The difference between this esti-mate and the largest observed magnitude in the given com-plete subcatalog was taken to be a magnitude increment alsofor other subcatalogs in the given area.

Grid of Cells

In the PSHA procedures, it is convenient to use a hori-zontal flat surface grid of rectangular cells. All input andoutput values are given in the centers of grid cells. All epi-centers and fault centers inside a particular grid cell are as-signed to its center. Due to the inaccuracy of input data andnumerical rounding the grid can be rather coarse, for ex-ample, rectangles of length sides 0.1� or squares of lengthside 10 km. Finer grids give smoother maps, but the com-putation takes more time.

A grid of cells is suitable for representing different kindsof seismic sources. The center of a cell may either representa point source or a point on a fault (usually a center on asurface fault trace), while an areal source zone may be con-sidered as a set of corresponding grid cells.

Distribution of Faults: QuantitativeSeismotectonic Model

One of the main problems in PSHA is how to handle theavailable seismotectonic knowledge. Also the classificationof a given approach into a deductive or historical groupstrongly depends on this. When modeling distributed seis-micity, seismotectonics is usually incorporated by using sub-jective engineering judgment, or only implicitly through theearthquake catalog. Nevertheless, if in a given area at leasta general knowledge about the main types and directions ofseismogenic structures exists, some quantitative statisticalmodeling is possible.

We suggest a simple seismotectonic model that statis-tically connects earthquakes and faults and enables the useof attenuation relationships for epicenters and faults. In ap-plying a grid of cells, each center of a cell has the followingattributes: type of dominating structures (i.e., strike slip, nor-mal or reverse fault, thrust), their directions (azimuths), andthe corresponding weights. Grid cells with the same attrib-utes may be grouped into regions of homogenous seismo-tectonic characteristics, such as in Poljak et al. (2000b).

Different Geometry of Seismicity Model

The geometry of distributed seismicity may be modeledwith the delineation of areal source zones (e.g., EPRI, 1988;Bernreuter et al., 1989). Areal source zones are defined withpolygons and have a uniform distribution of epicenters.Areal source zones may or may not contain earthquakesfrom the neighboring area.

An alternative to the classical source zone approach isGaussian smoothing of seismicity. It allows fuzzy modelingof activity rate, where grid cells are the only geometricalelements. Simple one-stage circular Gaussian smoothing

was introduced and further improved by Frankel (1995),Frankel et al. (2000), Cao et al. (1996), and Lapajne et al.(1997). Two-stage, circular and elliptical smoothing was im-plemented by Lapajne (2000) and Sket Motnikar et al.(2000) and applied also by Hamdache et al. (2001). Thecorresponding procedure and smoothing equations are givenin detail later.

Computer programs for seismic hazard assessment, us-ing the classical source zone approach (e.g., FRISK88 [RiskEngineering, 1988] and Seisrisk III [Bender and Perkins,1987]), as well as their later versions, are widely used. How-ever, there has been a lack of software for computing seismichazard using the spatially smoothed approach. OHAZ (Ori-ented HAZard) is a program for computing seismic hazardfrom distributed seismicity. The version OHAZ 2.0 was de-veloped in Java (Zabukovec, 2000) and includes a user-friendly interface, while a more generalized version, OHAZ4.0, is written in C�� Builder to allow faster computation(it has not yet been published). OHAZ emphasizes the spa-tially smoothing approach, but it can also manage the seis-micity ascribed to regional or local areal seismogenic zoneswith or without neighborhood. It can model and maintainmaps of spatially smoothed seismicity, taking into accounta simple seismotectonic model.

Distribution of Past Epicenters

The estimation of the hazard is most often derived fromthe distribution of past earthquake epicenters, given as lati-tude and longitude coordinates in one or more complete sub-catalogs. Usually, this information is shown in the epicentermap where epicenter location errors are not considered. Fordistributed seismicity and for historical earthquakes, the lo-cation uncertainty may be quite considerable. In catalogs,many earthquakes with epicenters from a larger area mayeven have the same coordinates, which represent large set-tlements that historical earthquakes have been ascribed to.

Both source zoning and smoothing approaches of seis-mic activity rate modeling take this into account. In the seis-mic source zone approach, earthquakes may be subjectivelyclassified into a particular source zone and are uniformlysmoothed throughout the source zone area. In the one-stagecircular smoothing approach, location error is encounteredin the subjectively defined radius of smoothing (Frankel,1995; Lapajne et al., 1997), while the estimated maximumlocation error directly defines the first-stage radius in thetwo-stage smoothing procedure (Lapajne, 2000).

Seismicity Models: Models of Seismic Activity Rate

It is commonly assumed that future earthquakes willoccur in the vicinity of past earthquakes. The input for haz-ard calculations is the model of seismic activity rate, derivedfrom the distribution of past epicenters and from seismotec-tonic knowledge. Earthquakes occur in (known or unknown)seismogenic zones; thus some seismotectonic information isalready indirectly contained in the catalog. However, wemay often benefit from additional seismotectonic informa-


tion, a map or a model that is not included in the earthquakecatalogs. We would also like to clarify the steps in the pro-cedure and avoid subjective decisions as much as possible,or at least make them easily tracked.

In seismic source zone modeling, seismotectonic infor-mation is considered subjectively in the determination ofsource geometry, depending therefore on the author’sknowledge. The one-stage circular smoothing procedure isbased only on the earthquake catalog. Thus, it is a historicmethod. Seismic source zone modeling and the one-stagecircular smoothing procedure do not explicitly define thedistribution of past epicenters that would include locationerror. In the two-stage smoothing approach, such a distri-bution of past epicenters is obtained with circular smoothingof the seismic activity rate. Further elliptical smoothing,based on a seismotectonic model, produces a seismicitymodel (i.e., model of seismic activity rate), used as the maininput for the calculation of seismic hazard.

Multimodel Approach

Due to the uncertainty in all kinds of data, more thanone seismicity model is usually proposed. Different modelsmay reflect different earthquake subcatalogs, different au-thors (teams or single experts), or alternatives in parameters,organized in the logic tree.

Usually there is more than one complete subcatalog,which enables different models of distributed seismicity.Rare large earthquake areas often do not coincide with theareas of more frequent smaller earthquakes. Subcatalogs andthe corresponding models have a different time span, differ-ent lower bound magnitude, and corresponding seismic ac-tivity rate, while the upper bound magnitude and the decayrate may or may not remain the same. To be comparable,models should be normalized to the same total annual seis-mic activity rate (above the same magnitude and in the samearea). Due to the most accurate and complete data, the sub-catalog of the latest time span is usually the most appropriatebase for the reference seismic activity rate above the chosenthreshold magnitude in the given influence area.

Even when only one subcatalog is given, we may de-velop various seismicity models to consider the uncertaintyin parameters and in source zone geometry. Subjective de-termination of alternative parameter values should encom-pass an uncertainty range in seismological, geological, andtectonic data. Such parameter alternatives, as well as globalalternatives of different authors, are usually modeled in alogic tree. Typical parameters, for which uncertainty is con-sidered in this way, are source zone coordinates, seismicactivity rate, b-value, upper bound magnitude, and attenua-tion model.

As earthquakes from distributed seismicity cannot beassigned to particular faults, a multimodel approach is evenmore advisable. Different subcatalogs as well as the logictree procedure may be applied to the classical source zoneapproach and to all variants of smoothing. However, multi-modeling of source zones is usually based on the logic tree

approach and is performed by different authors, whilesmoothing approaches are usually based on different sub-catalogs.

For a given return period, civil engineers usually wantto have only one site estimate of a given ground-motionparameter. The final result of a multimodel approach shouldbe one estimate of seismic hazard, which is either obtainedfrom a compromised seismotectonic model and seismic ac-tivity rate or, more often, calculated as a weighted averageof different estimates. Different approaches give differentfinal results:

1. From a compromised model only one seismic hazard es-timate is calculated. An example of a seismic hazard mapbased on the compromised seismotectonic model and thecorresponding seismic source zone model is the latestpublished seismic hazard map of Italy (Slejko et al.,1998).

2. For each model of seismic activity rate or for each selec-tion of parameter values, the corresponding seismic haz-ard estimate is calculated, and their weighted estimate isthe final result. Many seismicity-smoothing applicationsare good examples of this (Lapajne et al., 1997; Frankelet al., 2000; Sket Motnikar et al., 2000). A prototype ofsuch weighting in the seismic source zone approach isthe study of seismic hazard for 69 sites of nuclear powerplants in the central and eastern United States (EPRI,1988; Bernreuter et al., 1989).

Both approaches can be applied to a single site estimateand to a map (grid of sites).

Distribution of Released Seismic Energy: Energy-Based Seismic Activity Rate

“Thus, I conclude that, while proximity to previoussmall earthquakes is generally an indicator of areas wherelarger earthquakes are likely to occur, small earthquake seis-micity alone is not sufficient to identify areas of concernregarding the occurrence of larger earthquakes” (Kafka,1999). From this quote it follows that larger historical earth-quakes, not registered in the complete subcatalogs, need tobe considered. The problem is how to handle old strong his-torical events. Even for well-documented data, the time spanof the complete catalog cannot be extended over centuries.Sometimes, the activity rate for larger earthquakes can bedetermined from geological information, for example, fromfault slip rates. Such detailed knowledge is often not avail-able. To take into account the hazard from large reportedhistorical earthquakes, which occurred centuries ago, we in-troduced an approach for the estimation of the seismic ac-tivity rate based on the released seismic energy. The area oflarge released seismic energy coincides with the epicentralarea of a big earthquake. It is assumed that all big earth-quakes are registered in the catalog time span. In this sense,the catalog is complete. The exact level of completeness(lower bound magnitude) does not need to be determined,


as the released seismic energy from small and moderateearthquakes can be neglected.

The distribution of the released seismic energy will beused to develop additional models of seismic activity rate(distribution of epicenters). For this purpose, the releasedseismic energy of a given area has been transformed to acorresponding normalized activity rate. The released seismicenergy E has a simple empirical relation to the earthquakemagnitude (Gutenberg and Richter, 1956a, 1956b; Richter,1958; Bath, 1973; Willmore, 1979):

logE � A • m � B. (3)

The regression coefficient A is sometimes estimated to be1.5 (Richter, 1958; Willmore, 1979; Reiter, 1990). Coeffi-cient B is not important for further evaluation. With the as-sumption that earthquakes fit ideally the log-linear magni-tude–frequency law (equation 1a), an energy-based seismicactivity rate n(m0) is obtained (Lapajne et al., 1997):

A(m �m )R 0(A � b) • 10n(m ) � ,0 (A�b)(m �m )u 0b • (10 � 1)

(4a)

or, generalized for the doubly truncated exponential law(equation 1b) case:

A(m �m ) �b(m �m )R 0 u 0(A � b) • 10 (1 � 10 )n(m ) � ,0 (A�b)(m �m )u 0b • (10 � 1)

(4b)

where mR denotes the magnitude of an earthquake that wouldhave all the released seismic energy in the given time span:

N1 Amkm � log 10 . (5)R �� A k�1

Note that equations (4a) and (4b) are valid either for thecomputation of the seismic activity rate for a small area (acell of a grid or a source zone) or for the entire region. Anarbitrary complete subcatalog or the whole earthquake cat-alog may be considered. The calculated energy-based seis-mic activity rate is sensible only for very large historic earth-quakes, so the completeness of the whole catalog for smallerearthquakes is not required.

Distributions of past epicenters and seismicity modelsthat are based on the described transformation from the re-leased seismic energy can be used together with ordinaryseismicity models in a multimodel approach.

Attenuation Model, Epicentral, and Fault Distance

To estimate seismic hazard for a given site, besides theseismicity model also an attenuation model of a chosenground-motion parameter u (e.g., peak ground acceleration[PGA], peak ground velocity [PGV], pseudo spectral accel-eration [PSA], or pseudo spectral velocity [PSV]) should beknown. The general form of an attenuation equation is

ln u � f(m, d) � e, (6)

where d is the nearest epicentral or fault distance to thesite and e is a normally distributed error with standard de-viation r.

When the territory of interest is divided into grid cells,centers of grid cells may be taken as seismic point sources.As mentioned earlier, epicenters in a cell are assigned to itscenter. Epicentral distance is defined from the site to thecenter of the corresponding grid cell. On the other hand, eachcell center can be assumed to be the center of a hypotheticalfault. Its type, direction, and length are determined from thedescribed quantitative seismotectonic model (distribution offaults). Fault distance defines the distance from the site tothe nearest point of the hypothetical fault (see Fig. 1). Equa-tion (6) may therefore be used also with coefficients that aredetermined for fault sources.

Circular and Elliptical Smoothing of SeismicActivity Rate

The Original Circular Smoothing Procedure

The method of spatially smoothed seismicity (Frankel,1995; Frankel et al., 2000) is very suitable for regions wheredetailed knowledge of the related seismotectonics is notavailable (distributed seismicity) and where the delineationof seismic source zones involves too much subjectivity.Geometrical delineation of seismic source zones is not amathematical procedure. It is based on subjective engineer-ing judgment. On the other hand, the mathematical characterof smoothing automatically reduces subjectivity. CircularGaussian smoothing of seismic activity rate ni(m0) is appliedwith

2�(D /c)ijn (m )e� j 0jn (m ) � , (7)i 0

2�(D /c)ije�j

where c is a correlation distance and Dij is the distance be-tween the ith and jth cells. In each grid cell i, the seismicactivity rate ni(m0) is counted from the earthquake catalogor, as suggested earlier, calculated from the released seismicenergy (equation 4a or 4b). The radius of smoothing equals3c. In the original procedure the correlation distance, andconsequently the radius of smoothing, was defined more orless arbitrarily following certain qualitative criteria. Thesmoothing comprised the error of the epicenter location andthe assumption that future earthquakes will take place in thevicinity of past epicenters. A smoothed map of the activityrate represents the seismicity model.

Two-Stage Smoothing Procedure

To precisely determine various sources of smoothingand to estimate the radius in a qualitative way, the smoothing


oo

grid cell

site site

w2

w3

w1

fault ruptures

for a given magnitude

epicentral distance weighted fault distances

grid cell

Figure 1. Epicentral and fault distance from the site.

procedure is separated in the following two stages (Lapajne,2000). In the first stage, the seismic activity rate is smoothedcircularly as in the original approach (equation 7), but theradius of smoothing is the function of epicenter location er-ror. Simply, the radius may equal the maximum estimatedlocation error in the considered subcatalog. The circularlysmoothed map of the activity rate represents the distributionof past epicenters in which the location error is considered.

In the second stage it is assumed that earthquakes occuron faults or in fault zones of past earthquakes. Althoughthere is for distributed seismicity no direct relationship be-tween the known faults and earthquakes, the empirical re-lationship may be applied to obtain the hypothetical faultlength. The type and the direction of faults can be deter-mined statistically from the geological data. In the quanti-tative seismotectonic model (Poljak et al., 2000b), faulttypes and fault direction are determined in selected regions.Seismic activity rate is further smoothed according to thedetermined directions of seismogenic faults. For this pur-pose, fault-rupture-oriented elliptical Gaussian smoothing isproposed:

r rT T�1/2d V RVdil ilen (m ) � , (8)il 0 r rT T�1/2d V RVdij ij�j e

where nil(m0) is the number of earthquakes in cell i that isduring the elliptical smoothing shifted to cell l. Indexes l andj denote all cells in the smoothing area around cell i. T meansthe transposition and the vector ij defines the distance fromrdcell i to cell j. The correlation matrix

1 02rR � (9)1� �0 2s

defines the length of the first (r) and the second (s) principalhalf-axis of the ellipse of smoothing. Their meaning is anal-ogous with the correlation distance c; the elliptical smooth-

ing goes to 3r and to 3s, respectively. The matrix V deter-mines the direction of the first principal axis of the ellipseand is defined with azimuth �:

cos� �sin�V � . (10)� �sin� cos�

The first principal axis lies in the direction of the seismo-genic fault. It is reasonable to define its length proportionallyto the fault rupture length L (or even simpler: r � L). Faultlength may be calculated from the estimated upper boundmagnitude mu:

log L � a � b m , (11)1 1 u

where a1 and b1 are regression coefficients, which dependon the fault type and may be determined from empiricalrelationships (e.g., Wells and Coppersmith, 1994). The axiss represents the width of a seismogenic zone and is assumedto be proportional to L: s � kL, k � 1.

The circularly and elliptically smoothed activity ratemap represents the seismicity model.

Equivalent One-Stage Smoothing Procedure

When the smoothing procedure is based only on theearthquake catalog, an equivalent one-stage circular Gaus-sian smoothing (Lapajne, 2000) is an alternative to the origi-nal circular smoothing procedure (Frankel, 1995). When nostatistical seismotectonic model is available, this procedureapproximately substitutes the two-stage smoothing ap-proach. It enables one to objectively determine the radius ofone-stage circular smoothing. It is assumed that the equiv-alent-smoothing surface approximately equals the surface ofthe two-stage smoothing area, which is approximately equalto the surface of the ellipse with principal axis re � c � rand se � c � s (see also Fig. 2). Consequently, the equiv-alent correlation distance ce is determined with

2c � r s (12)e e e


Circular smoothing Elliptical smoothing

Equivalent circular smoothingTwo-stage smoothing

C

C

C

Ce

�

ττ

τ

�

�

Figure 2. Smoothing areas and radii.

and therefore depends on the maximum estimated locationerror (given in the catalog) and on the fault rupture lengthL. Its evaluation from equation (11) requires the value of theupper bound magnitude mu and the fault type. When the typeis not known, either the most common structure or the av-erage of different types may be assumed. The worst casemay also be taken into account. Once the correlation distanceis determined, the procedure is the same as the original one-stage circular smoothing procedure.

Seismic Hazard Calculation

The expected annual rate of exceedance of ground-motion level u0 at a site is usually calculated using (e.g.,Reiter, 1990)

k(u � u ) � n (m )0 � i mini (13)

mu

P[u � u | m, r]p (m)p (r) dr dm,0 i i� �m rmin

where ni(mmin) is the possibly smoothed annual seismic ac-tivity rate above the starting magnitude mmin � m0 in a seis-mic source i, P[u � u0 | m, r] is the conditional probabilitythat an earthquake of magnitude m at a distance r from thesite produces a ground-motion level u which is greater thanlevel u0, pi(m) is the probability density function of magni-tude (2a) or (2b) within source i, and pi(r) is the probabilitydensity function of the distance between the point of sourcei and the site, for which the hazard is being estimated. Theconditional probability P[u � u0| m, r] is calculated from

lnu � lnu(m, r)1 0P[u � u | m, r] � U* , (14)0 � �2 r 2�

where U* represents the complementary error function andln u is an attenuation model, for example, given by equation(6). For small values, the annual rate of exceedance is prac-tically equal to the annual probability of exceedance. Aftercalculating values k for several levels of ground motion u0,the ground-motion value for a given probability of exceed-ance can be obtained with the interpolation of k values.

For point sources, equation (13) has a simplified form:

k(u � u ) � n (m )0 � i mini (15)mu

P[u � u | m, r ]p (m) dm.0 i i�mmin

Equation (15) may be used for a grid of point sources (sourcei represents the center of the grid cell), thus it is applicablefor an arbitrary geometrical presentation of epicenter distri-bution. When grid cells are small enough, equation (15)gives practically the same results for areal sources as equa-tion (13), but the computation time with equation (13) isshorter. On the other hand, significant advantages of equa-tion (15) are its simplicity and its applicability, irrespectiveof the type of geometry of the seismic activity rate model.

Equation (15) may also be used for fault sources, whenattenuation model (6) with fault distance is given. In thiscase, the center of the ith grid cell is the center of a hypo-thetical fault, and distance ri is determined from the site tothe closest point on that fault.


determiningsubcatalogues

zoning

start

geometricalseismicitymodeling

earthquakecatalogue

delineation ofsource zones

epicenterlocation error

smoothing

statisticalseismotectonic

model

circularsmoothing

radius

NO

ellipticalsmoothing

distributions ofpast epicenters

models ofexpected

seismicity

equivalentcircular

smoothing

hazardcalculation

seismic hazardestimates / maps

weights

seismotectonicknowledge

attenuationmodel

dividing regioninto grid of cells

classifyingepicenters

into grid cells

weightedseismic hazardestimate / map

statisticalseismotectonic

model

YES

Figure 3. A flow chart of the proposed methodology for the PSHA of distributedseismicity.

The proposed PSHA methodology for distributed seis-micity is described with a flow chart in Figure 3.

Case Study: Seismic Hazard Map of Slovenia

To illustrate the PSHA methodology for distributed seis-micity, a seismic hazard map of Slovenia is calculated usingtwo-stage smoothing of the seismic activity rate. In Slovenia,no large earthquakes of magnitude 7 or above have beenobserved, and no evidence of characteristic earthquakes hasbeen found. According to current geological and seismolog-

ical knowledge, most seismic activity may be considered asdistributed seismicity.

Input Data

The earthquake catalog covers the period of 567–1998and an area between 12.5� and 17.5� E and 44.5� and 47.5�N of approximately 100,000 km2. The borders of this areaare approximately 100 km away from the international bor-ders of Slovenia. The catalog consists of 4687 earthquakes(2426 mainshocks and 2261 foreshocks or aftershocks).Foreshocks and aftershocks are removed from the catalog as


the Poissonian process is assumed. Most events are datedfrom the last two centuries, while older recordings are ofincreasingly inferior quality and limited only to large earth-quakes.

In the observed area, the doubly truncated exponentialrecurrence relationship (equation 1b) fits very well the datafrom the last 120 years above magnitude 3.7 (Lapajne andSket Motnikar, 1996).

Five seismicity models have been proposed (Sket Mot-nikar et al., 2000). Two of them are based on counting earth-quakes from two complete subcatalogs. The catalog may beconsidered complete from 1880 for m0 � 3.7 (M1, referencesubcatalog) and from 1690 for m0 � 5.0 (M2). An upperbound magnitude of 6.5 for both subcatalogs is estimated asrecommended by Lapajne and Sket Motnikar (1996): themaximum observed magnitude in 1690–1998 is increasedby 0.2. The decay rate b of the magnitude–frequency rela-tionship (equation 1b) is estimated from the reference sub-catalog to 0.84 (MLE), and this value is also adopted in the

M2 model. The maximum location error for M1 is estimatedto 22 km and for M2 to 33 km.

Three additional models (M1e, M2e, and M3e) of theenergy-based seismic activity rate are proposed: two for bothcomplete subcatalogs and one for the whole catalog. ModelM1e has the same parameters as model M1 (subcatalog1980–1998, magnitude range 3.7–6.5, b-value 0.84, maxi-mum location error 22 km), and model M2e has the sameparameters as model M2 (subcatalog 1690–1998, magnituderange 5.0–6.5, b-value 0.84, maximum location error 33km). Model M3e covers the whole earthquake catalog periodof 567–1998, with a magnitude range of 3.7–7.0. The mag-nitude of the maximum observed historical event (in 1511)was estimated to 6.8. The maximum location error in thecatalog is estimated to 56 km, and the MLE of decay rate bis 0.85. The seismic activity rate for all three models is cal-culated using equation (4b), and it depends only on largeearthquakes. For all models, b and mu values are chosen tobe constant in the whole observed region.

13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5

44.6

44.8

45.0

45.2

45.4

45.6

45.8

46.0

46.2

46.4

46.6

46.8

47.0

47.2

47.4x 10

-2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

3.7 <= M < 5.05.0 <= M < 6.9

Figure 4. Past epicenters and circularly smoothed seismic activity rate for modelM1: annual number of past earthquakes (M � 4) in a grid cell 5 km � 5 km.


The entire observed region is divided into a grid of cells(5 km � 5 km). The proposed five models of seismic activ-ity rate are smoothed in the described two-stage procedure.The estimated maximum epicenter location error is used asthe radius of circular Gaussian smoothing. The correlationdistance is by definition (Frankel, 1995) one-third of the ra-dius. The circularly smoothed map of activity rate (see ex-ample in Fig. 4) represents the distribution of past epicenterswith considered location error.

A simple seismotectonic model of Slovenia that meetsthe requirements for the elliptical fault-rupture-orientedsmoothing is shown in Figure 5 and was described in Poljaket al. (2000b). The areas of similar seismotectonic charac-teristics were delineated according to the available seismo-tectonic maps of Slovenia and neighboring areas. The re-quired attributes of each delineated area have beendetermined with the following statistical analysis of struc-tures. Structures were divided into 5-km segments and thencounted and grouped according to their azimuth of a chosen

span (e.g., 10�). A few dominant directions were chosen ac-cording to their position within the recent maximum prin-cipal stress of the region. Their weights were determined bythe relative frequency of the selected structure segments. Therecent stress regime in the investigated area is compressionalwith generally north–south–oriented maximum stress axis(Grunthal and Stromeyer 1986; Poljak et al., 2000a). Thiscan produce several structural deformations, that is, thedextral strike-slip movement along northwest–southeast–oriented Dinaric faults, the sinistral strike-slip movementalong northeast–southwest–oriented faults, and thrustingalong east–west–oriented faults of the southern Alps.

Circularly smoothed models of seismic activity rate arefurther elliptically smoothed and represent seismicity mod-els (see example in Fig. 6). All models are also normalizedto the reference model M1. The five normalized seismicitymodels are used as input of the seismic hazard calculation(Sket Motnikar et al., 2000). In Figure 7, the correspondingmaps for all five models are presented.

13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5

44.6

44.8

45.0

45.2

45.4

45.6

45.8

46.0

46.2

46.4

46.6

46.8

47.0

47.2

47.4

0.10

0.110.16

0.30

0.33

0.17

0.32

0.56

0.44

0.56

0.330.11

0.50

0.50

0.500.08

0.42

0.58

0.29

0.13

0.24

0.54

0.22

0.360.47

0.16

0.52

0.50

0.50

A1 A2

B1

B2

B3

C2

E

D B4

C1

Figure 5. Quantitative seismotectonic model of Slovenia. The numbers next to thestructural elements represent weights.


The ground-motion horizontal attenuation model,

2 2lnu � c � c m � c ln d � h � e , (16)�1 2 3

with coefficients from Pugliese and Sabetta (1989) and Sa-betta and Pugliese (1996), is chosen, where c1, c2, c3, and hare regression parameters. It is developed for all the mainground-motion parameters (PGA, PGV, PSA, and PSV) andfor both types of distance, that is, epicentral distance (de-noted by PS-e) and nearest fault distance (PS-f). Many ac-celerograms used in the estimation of coefficients were fromthe neighboring area (Friuli).

The PGA map of Slovenia for rock and firm soil, for areturn period of 475 years and for attenuation model PS-f,is calculated using equation (15). The starting magnitudemmin is chosen to be 4.0, as at the given accuracy all lowermagnitudes may be neglected. All calculations were ob-tained using the OHAZ program (Zabukovec, 2000). PGAvalues in Figure 8 were averaged from five PGA values ob-

tained from five seismicity models. For comparative reasons,equivalent one-stage circular smoothing (Lapajne, 2000) andclassical seismic source zone modeling (Fajfar et al., 2000)were also performed. The results showed that at least for arough seismic hazard assessment, the type of geometricalseismicity modeling is not a deciding factor.

Conclusions

The proposed PSHA methodology for distributed seis-micity is described with a flow chart in Figure 3. All stepsof the improved spatially smoothed seismicity approach arediscussed in detail and compared with the classical seismicsource zone approach. The following advantages are pre-sented:

• The two-stage circular and elliptical smoothing procedureand equivalent one-stage smoothing are particularly suit-able for distributed seismicity modeling.

13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5

44.6

44.8

45.0

45.2

45.4

45.6

45.8

46.0

46.2

46.4

46.6

46.8

47.0

47.2

47.4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x 10-2

3.7 <= M < 5.05.0 <= M < 6.9

Figure 6. Past epicenters and two-stage smoothed seismic activity rate for modelM1: annual number of expected earthquakes (M � 4) in a grid cell 5 km � 5 km.


Figure 7. (a) Five models of normalized circular smoothed seismic activity rate.(b) Five models of normalized two-stage smoothed seismic activity rate. (c) PGA mapsfor five spatial seismic activity models.


Cornell, C. A., and S. R. Winterstein (1988). Temporal and magnitudedependence in earthquake recurrence models, Bull. Seism. Soc. Am.78, 1522–1537.

dePolo, C. M., and D. B. Slemmons (1990). Estimation of earthquake sizefor seismic hazards, in Neotectonics in Earthquake Evaluation, Re-views in Engineering Geology VIII, E. D. Krinitzsky and D. B. Slem-mons (Editors), Geological Society of America, Boulder, Colorado,1–28.

Electric Power Research Institute (EPRI) (1988). Seismic hazard method-ology for the central and eastern United States, EPRI NP-4726-A,revision 1, Final Report, Palo Alto, California.

Fajfar, P., J. Lapajne, I. Perus, B. Sket Motnikar, P. Zupancic, and M.Sostaric (2000). Projektni parametri za potresno odporno projektiranjegradbenih objektov po Eurocode 8, Ljubljana, Univerza v Ljubljani,FGG, IKPIR and MOP, URSG (in Slovene).

Frankel, A. (1995). Mapping seismic hazard in the central and easternUnited States, Seism. Res. Lett. 66, 8–21.

Frankel, A., C. Mueller, T. Barnhard, E. Leyendecker, R. Wesson, S. Harm-sen, F. Klein, D. Perkins, N. Dickamn, S. Hanson, and M. Hopper(2000). USGS national seismic hazard maps, Earthquake Spectra 16,1–20.

Grunthal, G., and D. Stromeyer (1986). Stress pattern in central Europeand adjacent areas, Gerlands Beitr. Geophys. 95, no. 5, 443–452.

Gutenberg, B., and C. F. Richter (1944). Frequency of earthquakes in Cali-fornia, Bull. Seism. Soc. Am. 34, 185–188.

Gutenberg, B., and C. F. Richter (1956a). Earthquake magnitude. intensity,energy, and acceleration, Bull. Seism. Soc. Am. 46, 105–145.

Gutenberg, B., and C. F. Richter (1956b). Magnitude and energy of earth-quakes, Ann. Geofis. 9, 1–15.

Hamdache, M., J. K. Lapajne, and B. Sket Motnikar (2001). Seismic hazardassessment in North Algeria, in Proc. of the Workshop Sur la sismiciteet la gestion du risque sismique dans la region Euro-Mediterra-neenne, Agadir, Morocco, 18–19 June, 33.

Kafka, A. L. (1999). Testing the hypothesis that locations of larger earth-quakes can be forcasted based on the spatial distribution of smallerearthquakes, in Proc. of the 71st Annual Meeting of the Eastern Sec-tion of the Seismological Society of America, 16–20 October, Mem-phis, Tennessee.

• Statistical modeling of seismogenic faults allows the in-corporation of general seismotectonic knowledge and thederivation of the seismicity model.

• The influence of large historical events on seismic hazardis enhanced by introducing the calculation of the activityrate from the released seismic energy.

• The attenuation models based on the closest fault distancemay be used even when faults are not determined. Withstatistical inclusion of seismotectonic data, point sourcesmay be regarded as the centers of fault sources.

• Using a grid of cells, seismic hazard is calculated from thesimplified equation for point sources. An arbitrary geom-etry of epicenter and fault distribution is allowed.

References

Bath, M. (1973). Introduction to Seismology, Birkhauser, Basel, 395 pp.Bender, B., and D. M. Perkins (1987). Seisrisk III: a computer program for

seismic hazard estimation, U.S. Geological Survey Bulletin 1772,Washington, D.C., 48 pp.

Bernreuter, D. L., J. B. Savy, R. W. Mensing, and J. C. Chen (1989).Seismic hazard characterization of 69 nuclear power plant sites eastof the Rocky Mountains, Lawrence Livermore National Laboratory,NUREG/CR-5250.

Berrill, J. B., and R. O. Davis (1980). Maximum entropy and the magnitudedistribution, Bull. Seism. Soc. Am. 70, 1823–1831.

Cao, T., M. D. Petersen, and M. S. Reichle (1996). Seismic hazard estimatefrom background seismicity in southern California, Bull. Seism. Soc.Am. 86, 1372–1381.

Cornell, C. A. (1968). Engineering seismic risk analysis, Bull. Seism. Soc.Am. 58, 1583–1606.

Cornell, C. A., and E. H. Vanmarcke (1969). The major influences onseismic risk, in Proc. of the Fourth World Conference on EarthquakeEngineering, January 1969, Santiago, Chile, A-1, 69–93.

Figure 8. Peak ground acceleration map of Slovenia.


Kijko, A., and G. Graham (1998). Parametric–historic procedure for prob-abilistic seismic hazard analysis. Part I: Estimation of maximum re-gional magnitude mmax, Pure Appl. Geophys. 152, 413–442.

Kijko, A., and G. Graham (1999). Parametric–historic procedure for prob-abilistic seismic hazard analysis. Part II: Assessment of seismic haz-ard at specified site, Pure Appl. Geophys. 154, 1–22.

Kijko, A., and M. A. Sellevoll (1989). Estimation of earthquake hazardparameters from incomplete data files. Part I. Utilization of extremeand complete catalogs with different threshold magnitude, Bull.Seism. Soc. Am. 79, 645–654.

Lapajne, J. K. (2000). Some features of the spatially smoothed seismicityapproach, in Proc. of the Workshop Seismicity Modeling in SeismicHazard Mapping, Poljce, Slovenia, 22–24 May, 27–33.

Lapajne, J. K., and B. Sket Motnikar (1996). Estimation of upper-boundmagnitude in earthquake hazard assessment, in Earthquake Hazardand Risk, V. Schenk (Editor), Kluwer, Hingham, Massachusetts, 39–48.

Lapajne, J. K., B. Sket Motnikar, B. Zabukovec, and P. Zupancic (1997).Spatially smoothed seismicity modelling of seismic hazard in Slo-venia, J. Seism. 1, 73–85.

McGuire, R. K. (1993). Computation of seismic hazard, Ann. Geofis. 36,181–200.

Page, R. (1968). Aftershocks and microaftershocks of the great Alaskaearthquake of 1964, Bull. Seism. Soc. Am. 58, 1131–1168.

Poljak, M., M. Zivcic, and P. Zupancic (2000a). The seismotectonic char-acteristics of Slovenia, Pure Appl. Geophys. 157, 37–55.

Poljak, M., P. Zupancic, J. K. Lapajne, and B. Sket Motnikar (2000b).Seismotectonic input for spatially smoothed seismicity approach, inProc. of the Workshop Seismicity Modeling in Seismic Hazard Map-ping, Poljce, Slovenia, 22–24 May, 117–124.

Pugliese, A., and F. Sabetta (1989). Stima di spettri di risposta da registra-zioni di forti terremoti italiani, Ingegneria Sismica 6, no. 2, 3–14.

Reiter, L. (1990). Earthquake Hazard Analysis: Issues and Insights, Co-lumbia U Press, New York, 254 pp.

Richter, C. F. (1958). Elementary Seismology, W. H. Freeman, New York,768 pp.

Risk Engineering (1988). FRISK88 User’s Manual, Version 1.2, Golden,Colorado.

Sabetta, F., and A. Pugliese (1996). Estimation of response spectra and

simulation of nonstationary earthquake ground motions, Bull. Seism.Soc. Am. 86, 337–352.

Sket Motnikar, B., J. K. Lapajne, P. Zupancic, and B. Zabukovec (2000).Application of the spatially smoothed seismicity approach for Slo-venia, in Proc. of the Workshop Seismicity Modeling in Seismic Haz-ard Mapping, Poljce, Slovenia, 22–24 May, 125–133.

Slejko, D., L. Peruzza, and A. Rebez (1998). Seismic hazard map of Italy,Ann. Geofis. 41, 183–214.

Stirling, M. W. (2000). A new probabilistic seismic hazard model for NewZeland, in Proc. of 12 WCEE, Wellington, New Zealand 2362.

Weichert, D. H. (1980). Estimation of the earthquake recurrence parametersfor unequal observation periods for different magnitudes, Bull. Seism.Soc. Am. 70, 1337–1346.

Wells, D. L., and K. J. Coppersmith (1994). New empirical relationshipsamong magnitude, rupture length, rupture width, rupture area, andsurface displacement, Bull. Seism. Soc. Am. 84, 974–1002.

Veneziano, D., C. A. Cornell, and T. O’Hara (1984). Historic Method forSeismic Hazard Analysis, Electric Power Research Institute, ReportNP-3438, Palo Alto, California.

Willmore, P. L. (Ed.) (1979). Manual of Seismological Observatory Prac-tice, World Data Center A for Solid Earth Geophysics, Washington,D.C., 165 pp.

Woo, G. (1996). Kernel estimation method for seismic hazard area sourcemodeling, Bull. Seism. Soc. Am. 86, 353–362.

Youngs, R. R., and K. J. Coppersmith (1985). Implication of fault slip ratesand earthquake recurrence models to probabilistic seismic hazard es-timates, Bull. Seism. Soc. Am. 75, 939–964.

Zabukovec, B. (2000). OHAZ: A computer program for spatially smoothedseismicity approach, in Proc. of the Workshop Seismicity Modelingin Seismic Hazard Mapping, Poljce, Slovenia, 22–24 May, 135–140.

Environmental Agency of the Republic of SloveniaSeismology OfficeDunajska 47/VII1000 Ljubljana, Slovenia

(J.L., B.S.M., P.Z.)

Manuscript received 22 August 2002.