J Seismol (2008) 12:65–78DOI 10.1007/s10950-007-9070-2

ORIGINAL ARTICLE

Probabilistic aftershock hazard assessment II: applicationof strong ground motion simulations

František Gallovic · Johana Brokešová

Received: 28 February 2007 / Accepted: 31 October 2007 / Published online: 7 December 2007© Springer Science + Business Media B.V. 2007

Abstract Probabilistic aftershock hazard assess-ment (PAHA) has been introduced by Wiemer(Geophys Res Lett 27:3405–3408, 2000). Themethod, in its original form, utilizes attenua-tion relations in evaluating peak ground velocity(PGV) exceedence probability. We substitute theattenuation relations together with their uncer-tainties by strong ground motion simulations fora set of scenarios. The main advantage of suchan approach is that the simulations account forspecific details of the aftershock source effects(faulting style, slip distribution, position of thenucleation point, etc.). Mean PGVs and their stan-dard deviations are retrieved from the simula-tion results obtained by the new hybrid k-squaredsource model, and they are used for the PAHAanalysis at a station under study. The model cho-sen for the testing purposes is inspired by theIzmit A25 aftershock (Mw = 5.8) that occurred26 days after the mainshock. The PAHA maps arecompared with (1) those obtained by the use ofattenuation relations and (2) the peak values often selected strong-motion recordings written bythe aftershock at epicentral distances <50 km. Weconclude that, although the overall hazard decay

F. Gallovic (B) · J. BrokešováFaculty of Mathematics and Physics,Department of Geophysics, Charles University,V Holešovickách 2, Prague 8, 180 00, Czech Republice-mail: gallovic@karel.troja.mff.cuni.cz

with increasing fault distance is similar, the PAHAmaps obtained by the use of simulations exhibitremanent radiation pattern effect and prolonga-tion in the strike direction due to the directivityeffect pronounced for some of the scenarios. Asregard the comparison with real data, we con-clude that the PAHA maps agree with observedpeak values due to appropriate attenuation modeladopted in the analysis.

Keywords Time-dependent aftershockprobabilistic hazard · Aftershock statistics ·Omori’s law · Strong ground motion simulations

1 Introduction

Large aftershocks may pose a substantial hazardto populated areas. They can cause even moredamage than the mainshock as, e.g., during theM = 6.1 aftershock of the 2002 M7.4 Hindu Kush,Afghanistan, mainshock. Another example of asignificant aftershock is the M5.8 aftershock A25of the 1999 M7.4 Izmit, Turkey, earthquake, whichcaused the death of 7 people and left 420 injured.Therefore, the prediction of aftershock impact isof great interest.

Due to the stochastic character of the after-shock occurrence, a probabilistic approach to af-tershock hazard assessment seems to be the mostappropriate. Based on the classical probabilistic

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seismic hazard approach (Cornell 1968), Wiemer(2000) has suggested the so-called probabilisticaftershock hazard assessment (PAHA). As such,it can statistically take into account hazards fromall likely activated faults producing aftershocksof diverse magnitudes. It can account even foraftershocks due to slip on blind faults. The mainadvantage of the PAHA is that it allows obtainingintegrated, easy to understand, hazard map for allpossible scenarios, including various magnitudesand likely activated fault zones, which may besuitable for responsible authorities.

Traditionally, probabilistic hazard analysis isperformed using empirical attenuation relation-ships. The empirical relations are obtained byregression of data, which consist of those mea-sured during earthquakes of various mechanisms,directions of rupture propagation, final slip dis-tributions, etc. The application of the attenua-tion relations is justified in cases when we haveonly seismicity data at our disposal and no spe-cific information about likely activated faults. Onthe contrary, for cases of known activated faults,waveform modeling could, in principle, lead toimprovements in seismic hazard analysis. In caseswhen one can estimate, e.g., the mechanism ofthe aftershock on an activated fault, the attenua-tion relations represent too rough approximation.Strong ground motion modeling techniques con-sidering a finite-extent source are able to accountfor specific aftershock properties, i.e., to capturethe key features of ground motions from largeearthquakes such as their amplitude dependenceon the azimuth to the observation point (sourcedirectivity). Note that the areas where seismicfaults are mapped are relatively common, espe-cially in Japan, California, or Turkey.

In this paper, we first briefly describe main for-mulas employed in PAHA. Then, we describe thehybrid k-squared approach proposed by Gallovicand Brokešová (2007a), combining integral andcomposite source modeling. This methodologyis used in strong-motion prediction for the A25Izmit aftershock (Mw = 5.8, September 13, 1999)example. Peak ground velocities (PGVs) are re-trieved from simulations (assuming a number ofscenarios) and, together with their scatter, com-pared with the attenuation relations. The mean

PGVs and their standard deviations are then usedfor the PAHA analysis. PAHA maps are alsocompared with those obtained by the use of atten-uation relations.

2 Probabilistic aftershock hazard analysis

Assume a region S, in which aftershocks of agiven sequence can take place. The aftershockoccurrence in a sequence is commonly describedby a nonstationary Poisson process. The quantityof main interest in the PAHA is the probability ofexceedence of certain strong motion characteris-tics. For example, the probability of PGV value v

exceedence at receiver x within the time windowT1 − T2 reads

Pv(x, T1, T2) = 1 − exp

⎛⎝−

T2∫

T1

�(v′ ≥ v, x, t)dt

⎞⎠ ,

(1)

where � is the rate of v′ ≥ v occurrence given bythe following integral over area S and a range ofexpected magnitudes from mmin to mmax,

�(v′ ≥ v, x, t)

=mmax∫

mmin

∫∫S

dr(m′ ≥ m, A, t)dm

× λ(v′ ≥ v, A, x, m)dmdS(A). (2)

In Eq. 2, r(m′ ≥ m, A, t) is the occurrence rate ofaftershocks of magnitude m′ ≥ m at position Aand at time t. Quantity λ(v′ ≥ v, A, x, m) is theprobability of exceedence of given PGV value v

at station x due to an event of magnitude m thatoccurred at position A. The double integration∫∫

S dS(A) is performed over area S with generallyspatially dependent rate r, covering all possibleaftershock loci under study.

The rate r in a given area can be approx-imated by the generalized Omori’s law, com-bining classical Omori’s law (Ogata 1988) andGutenberg–Richter relation modified for after-shocks (Shcherbakov et al. 2004). The rate is

J Seismol (2008) 12:65–78 67

described by a set of generally spatially depen-dent parameters m∗, b, p, β ′, and c(m∗). Theirmeaning is following: extrapolated maximum af-tershock magnitude governing aftershock produc-tivity (m∗), parameter controlling number–sizedistribution of events (b), time decay slope (p),and characteristic time for m∗ aftershock (c(m∗))and parameter controlling the ratio of characteris-tic times between higher and lower m’s (β ′). Theseparameters can be obtained empirically. For amore detailed discussion, see the paper Gallovicand Brokešová (2007b) (further referred to as thecompanion paper). In that paper, typical observedranges are given, and a numerical study of theinfluence of these parameters on the probabilistichazard is provided. Note that Eq. 2 in the presentpaper is analogous to Eq. 6 in the companionpaper, with the only difference that we integrateover an area with generally spatially dependentr and λ.

As the predicted PGVs v′ are usually assumedto be log-normally distributed, we can put

λ(v′ ≥ v, A, x, m)

= 1√2πσ(A, x, m)

∞∫

ln v

× exp

[−

(ln v′−ln v(A, x, m)√

2σ(A, x, m)

)2]

d(ln v′).

(3)

The distribution is described by ln v(A, x, m) andσ(A, x, m) that correspond to the mean and thestandard deviation of logarithms of PGVs. Val-ues of v and σ can be obtained either from theattenuation relations (see the companion paper)or from the statistical analysis of strong groundmotion simulations for a set of scenarios (varyinguncertain parameters of the adopted model). Thelatter approach is illustrated in the present paper.

In the numerical example that follows, thederivative dr/dm and integral over t are evaluatedanalytically. The integrations over m and S areperformed numerically substituting the integral bya sum. The bounds mmin and mmax in Eq. 2 arechosen rather formally in such a way that the

resulting hazard probabilities are not much sen-sitive to their slight changes.

3 Application to Izmit aftershock no. A25

In this paper, we show an application in which,after the mainshock, a specific fault is identified asbeing activated, and our task is to assess the haz-ard due to this fault by means of PAHA (Eq. 1).For this purpose, we utilize a strong motion sim-ulation technique to obtain the PGV statisticaldistribution λ(v′ ≥ v, A, x, m) (Eq. 3) instead ofthe too general attenuation relation.

The numerical study is performed for anexample of the well-known Izmit earthquake(August 17, 1999, Mw=7.4, Bouchon et al. 2000).It was followed by a number of large aftershocks,the two strongest were the Duzce earthquake(Mw = 7.2) and aftershock A25 (Mw = 5.8,September 13, 1999, 11:55:30 AM, see Orguluand Aktar 2001). We choose the latter one for itsrelatively smaller magnitude, which allows us tokeep the numerical effort in the scenario studyunder reasonable limits.

The A25 event occurred close to the mainshockfault and shows right-lateral strike-slip faulting onan east–west striking plane (Orgulu and Aktar2001). It occurred 26 days after the mainshockand caused a significant post-seismic stress releasearound the nucleation zone.

The A25 Izmit aftershock was recorded by anumber of strong motion accelerographs. We re-strict ourselves to 15 stations at epicentral dis-tances < 50 km. Their recordings have beendownloaded from the European Strong-MotionDatabase (http://www.isesd.cv.ic.ac.uk). Some ofthe stations are located very close to each other,and they recorded very similar waveforms. Thus,these “duplicate” recordings (five stations) wereremoved from our data set. A map with station lo-cations with respect to the epicenter can be foundin Fig. 1. In this figure, the stations are indicatedby triangles, the size of which is proportional tothe logarithm of the observed horizontal velocity(see Table 1).

For this event, we apply strong ground mo-tion simulation for a large number of finite-extent

68 J Seismol (2008) 12:65–78

km

km

Fig. 1 Map showing selected stations (triangles) that recorded the A25 aftershock (star). In the inset map, the size of eachtriangle is scaled according to the observed PGV (see Table 1)

source models (scenarios) for a set of magnitudes.In this way, we obtain a set of PGVs that is pre-sumably log-normally distributed, being describedby mean v and standard deviation σ necessary toevaluate λ (Eq. 3). Knowing λ, we can evaluate �

(Eq. 2) and, consequently, Pv (Eq. 1), the quantityof our main interest in this study. The predictedhazard maps are compared with (1) those ob-tained by the use of attenuation relations and (2)

Table 1 Observed peak ground velocity (European Strong-Motion Database, http://www.isesd.cv.ic.ac.uk)

Station code PGV (m/s)

TPT 0.600IZT 0.127BHC 0.282TUW 0.093SKR 0.020TYN 0.046KAR 0.091C106 0.008DOR 0.010IZN 0.044

the peak values of the ten selected strong-motionrecordings (Fig. 1).

3.1 Strong motion synthesis

The ground motions are simulated using a hybridk-squared source modeling technique (Gallovicand Brokešová 2007a). For this technique, therupture process is decomposed into slippingon the individual overlapping subsources ofvarious sizes, distributed randomly on the faultplane. The hybrid approach combines (1) theintegral approach at low frequencies, based onthe representation theorem and the k-squared slipdistribution composed by the subsources and (2)the composite approach at high frequencies, basedon the summation of ground motion contributionsfrom the subsources. Let us emphasize that thesame set of subsources is used for both thefrequency ranges.

Scaling properties of the subsources are thesame as used by Zeng et al. (1994). Their number–

J Seismol (2008) 12:65–78 69

size distribution obeys a power law with frac-tal dimension D = 2, and their mean slips areproportional to their dimensions (so-called con-stant stress-drop scaling). The subsource’s scalingimplies (Andrews 1980) that the subsources com-pose a k-squared slip distribution.

Concerning the numerical implementation, wefirst build a subsource database, which includesthe subsources’ positions on the fault, their di-mensions, mean slips (and, consequently, seismicmoments), and corner frequencies. Subsource di-mensions are taken as integer fractions of thefault’s length L and width W; that is, the sub-source length is l = L/n, and its width is w =W/n. Let us call the integer n the subsource level.The number of all the subsources at levels ≤ n(i.e., of size L/n × W/n and larger) is consideredto be n2. More specifically, the number of sub-sources N(n) at level n is N(n) = n2 − (n − 1)2 =2n − 1. At each level, the subsources are assumedto be identical in dimensions, mean slip and cor-ner frequency, and their position is random (and,therefore, subjected to variations).

The mean slip for subsources at level n isgiven by �u(n) = cu/n (obtained from the con-stant stress-drop assumption). We get the constantof proportionality cu, assumed to be independentof n, by matching the seismic moment of the wholeearthquake to the sum of the moments of all the

subsources considered in the calculation, i.e., upto certain nmax.

The corner frequency fc of the subsourcesat level n is considered to be proportional ton, fc = cf n. The inverse of the constant of pro-portionality cf is comparable to the duration ofthe whole earthquake. As fc controls the high-frequency spectral level of the synthetics, cf canbe adjusted by comparing the synthetic PGAs orPGVs with the local attenuation relation and/orwith observed time histories.

Let us describe the assumed time evolution ofthe rupture. At large scales, the subsources act sothat the faulting is equivalent to the classical inte-gral k−2 model. At low scales, the subsources be-have chaotically in such a way that their radiatedwavefield appears effectively to be isotropic. Tosimulate this, for strong motion synthesis, we usetwo methods, the integral and the composite, eachfor a different frequency range. Their applicationis controlled by two bounding frequencies f1 andf2, f1 < f2.

Concerning the low-frequency range (up to f2),the computation is performed according to therepresentation theorem. We discretize the faultdensely enough to evaluate the integral correctlyup to frequency f2. The slip at a point is givenby the sum of the slips of all the subsources fromthe database that contain the point (assuming a

Fig. 2 Left an example of slip distribution constructedfrom a subsource database. Note that level n = 1 is ne-glected. It would correspond to a slip patch over thewhole fault, which is, however, not observed in slip

inversions of medium-to-large sized earthquakes. Rightthree cross-sections of the spatial amplitude Fourier spec-trum of the slip distribution (left). The solid arrow indicatesthe k-squared decay (after Gallovic and Brokešová 2007a)

70 J Seismol (2008) 12:65–78

Fig. 3 A schematic picture of the hybrid combination ofthe integral and composite synthetics in the frequency do-main. Weighted summation of the synthetics is applied inthe crossover frequency zone (between bounding frequen-cies f1 and f2) to simulate smooth transition between thedeterministic and chaotic style of faulting. The weightingfunctions are sin2 and cos2 (from Gallovic and Brokešová2007a)

k-squared slip distribution at each individual sub-source). An example of the slip distribution con-structed in this way is shown in Fig. 2. The rupturetime is given by the distance of the point fromthe nucleation point assuming constant rupturevelocity vr. The slip velocity function is assumedto be Brune’s pulse with constant rise time τ .

In the high-frequency range (above f1), thesubsources from the database are treated as in-dividual point sources with Brune’s source timefunction. Their seismic moments and corner fre-quencies are obtained directly from the database.

Table 2 Source models obtained for various moment mag-nitudes Mw

Mw M0 (Nm) MS (cm) L (km) W (km)

5.0 3.57 · 1016 11 4.6 2.35.5 2.00 · 1017 20 8.4 4.26.0 1.12 · 1018 35 15.0 7.56.5 6.31 · 1018 62 26.4 13.2

Fault length and fault width are obtained from the faultsurface area given by Somerville et al. (1999) to have thelength twice higher than the width, which is an approxima-tion of observed ratio for strike-slip earthquakes.M0 Seismic moment (Hanks and Kanamori 1979), MSmean slip (using relation by Somerville et al. 1999), L faultlength, W fault width

Table 3 Simplified structural model after Orgulu andAktar (2001)

Depth Shear wave Density(km) velocity (km/s) (kg/m3)

0 1.87 2.22 3.32 2.430 3.70 2.7

The rupture time is given by the time the ruptureneeds to reach the subsource’s center (assumingthe same constant velocity vr as for the integral ap-proach). Due to the random subsource positions,the wavefield contributions sum incoherently.

The crossover combination of the computedsynthetics between f1 and f2 is illustrated inFig. 3. In the Fourier domain, we apply weightedaveraging of the real and imaginary parts ofthe spectrum.

To synthesize the final strong ground motionsat a given receiver, one has to calculate Green’sfunctions to involve effects due to the wave prop-agation phenomena. Generally, any method canbe employed, even different for each of the twofrequency ranges. In this paper, however, we uti-lize just one method for both ranges. As we are in-terested in high-frequency peak values, we choosenumerically effective ray method (Cervený 2001;Brokešová 2006), providing a high-frequency as-ymptotic approximation of a wavefield. It is veryflexible, applicable even for more complex (2D,3D) models. An important disadvantage of theray method is that it does not yield a completewavefield (e.g., surface waves and near-field part

Fig. 4 A sketch of 28 scenarios considered in this paper.The large rectangles correspond to the whole faults, thegray rectangles represent the asperities, and the stars arethe hypocenters

J Seismol (2008) 12:65–78 71

a

c

b

d

Fig. 5 Comparison of PGV simulations for magnitude 5.5without (a, c) and with (b, d) focal mechanism variationsprescribed at the individual point sources used in the com-posite part of the simulations. a, b All the simulated PGVs(gray points) plotted with respect to the fault distance. Themean PGVs determined for each station are denoted as

triangles. The solid line represents the attenuation curveand the dashed lines its ±2 standard deviations. c, d Sim-ulated mean PGV map (the values are the ones denotedas triangles in the top plot). The light gray line denotes theconsidered fault, and the triangles represent the stations

72 J Seismol (2008) 12:65–78

of the wavefield are completely missing in raysynthetics). In our computations, we take into ac-count the direct S-wave only, as in the region andtime window of interest, it is assumed to representthe dominant phase of the wavefield.

3.2 Source scenarios

To obtain the function λ(v′ ≥ v, A, x, m), seeEqs. 2 and 3; we need to evaluate the mean PGVv and its standard deviation σ for a set of mag-nitudes at any point x in the region of interest.In this respect, for each magnitude, we synthesizethe PGVs in a regular grid of 26 × 18 receivers,covering an area of 150 × 102 km2, assuming a setof scenarios. At each station, we calculate v andstandard deviation σ of all the simulated PGVs.The receivers are distributed around the rupturedfault center, fixed in the center of the area.

We assume that the only information we have isthat the aftershock can be expected on a verticalright-lateral strike-slip fault with the hypocenterat a depth of 10 km (typical depth of strike-slip earthquakes). The magnitudes taken into ac-count are 5.0, 5.5, 6.0 and 6.5 (all of them arevalues around the magnitude of the expectedaftershock). For a given magnitude, the faultdimensions and mean slips are obtained from thescaling relations by Somerville et al. (1999) (seeTable 2). For our range of magnitudes, Somervilleet al. (1999) found empirically one or two asper-ities on average. They cover about one-fourth ofthe fault area, and the slip on the asperities isapproximately two times larger than the meanslip of the whole earthquake. In this paper, weassume, for simplicity, one asperity only. To ob-tain the slip distribution in our model with suchan asperity, we fix two of the three subsourcesat the second level at the given asperity posi-tion. The rupture propagates from a nucleationpoint radially at constant velocity vr = 3.0 km/s,which is assumed the same for all magnitudes.The bounding frequencies were chosen f1 = 0.5and f2 = 2.0 Hz. This frequency interval coversthe limiting frequency often used in present slipinversions from local data (about 1 Hz), where theintegral approach is employed. The rise time, nec-essary for the low-frequency calculation only, isconstant and the same for all magnitudes, τ =1 s.

�Fig. 6 Mean PGV map for magnitude 6.0 (a) and itsstandard deviation map (c), assuming varying radiationpattern at high frequencies. d All the simulated PGVs(gray points) plotted with respect to the fault distance andtheir means (triangles) plotted in the same way. The solidline represents the attenuation curve and the dashed linesits ±2σ . b The mean PGV map obtained by the use ofattenuation relations. Note that its standard deviation mapwould be of constant value, equal to approximately 0.55.The considered fault is represented by the light gray linein (a–c)

The crustal model used for the ray calculationsis described in Table 3. Parameter cf (controllingthe corner frequency of subsources) is chosen tohave synthetic PGVs around the mean attenua-tion curve (see below).

For a given magnitude, 28 different scenarioswith varying position of the nucleation point andthe asperity are considered (see Fig. 4). For eachscenario, three different realizations of the sub-source positions (and subsource slip distributions)were produced. The strong-motion synthesis wasperformed for all the total 84 scenarios. Using thisset of results, the mean value v and its standarddeviation σ were determined at each receiver.

3.3 Scenario simulation results

Figure 5 shows, for magnitude m = 5.5, the meanPGV map (Fig. 5c) and a graph where all thesimulated PGVs (points) and their mean values(triangles) are plotted with respect to the faultdistance (from the closest point of the surface faultprojection) and are compared with the attenua-tion curve (Si and Midorikawa 1999) and its ±2standard deviations (Fig. 5a). One can see that thesimulation yields relatively large scatter. We havefound that the mean standard deviation is approx-imately 0.7. This synthetic scatter is larger thanthat expected from the empirical relation (0.55).This can be explained as an overestimated effectof the radiation pattern in the high-frequency partof the computation, exhibited as four lobes on themean PGV map in Fig. 5c.

In the paper by Satoh (2002), the author showsthat the radiation pattern vanishes at high fre-quencies. To simulate the vanishing radiationpattern, we consider random variations of thefocal mechanism (±90◦ for strike, dip, and rake)prescribed for the individual point sources in the

J Seismol (2008) 12:65–78 73

a b

d

c

74 J Seismol (2008) 12:65–78

composite part of the computation. This weakensthe radiation pattern effect at high frequencies,which results in a lower scatter of the simulatedPGVs as shown in Fig. 5b, with mean syntheticstandard deviation being about 0.35. Note thatsuch a number may seem too small, but we haveto keep in mind that our computation does notinvolve any other phenomena that may boost thescatter of the simulations (e.g., 3D structural ef-fects, site effects, etc.). The PGV lobes on thecorresponding mean PGV map (Fig. 5d) are nowdiminished, especially at short fault distances. The“remanent” lobes are present, as the radiationpattern without variations is taken into account inthe integral part of the modeling. Note that, in thefollowing, we assume these variations of the focalmechanism in the composite approach for all themagnitudes considered.

As an example, in Fig. 6, one can find themean PGV map (Fig. 6a) and its standard devi-ation (Fig. 6c) for magnitude 6.0 computed by thehybrid k-squared approach. Note that the mapin Fig. 6a is analogous to that in Fig. 5d, withthe only exception that this one corresponds toa higher magnitude. Figure 6b shows the PGVmap obtained by the use of the attenuation rela-tion. Both maps in Fig. 6a, b are mutually con-sistent in terms of overall decay of PGVs withincreasing fault distance, as also confirmed in theplot in Fig. 6d, where all the simulated PGVsfollow the mean attenuation curve. At distancesclosest to the fault, the mean PGV values, pro-vided by the simulations, are lower than thosefrom the attenuation relation. Such comparisonis, however, difficult, as the attenuation relationsare poorly constrained at these distances due tothe lack of strong motion data. Despite the over-all similarity of the mean PGV maps, there areremarkable differences. The simulated PGV mapexhibits a “remanent” radiation pattern effect incontrast to the attenuation PGV map. Moreover,the simulated PGV map is slightly more prolatealong the fault in both directions, which is dueto the directivity effect pronounced for some ofthe scenarios.

Comparing the mean PGV map for magni-tude 5.5 (Fig. 5d) and the map for magnitude 6.0(Fig. 6a), one can see, as expected, that the simu-lated PGVs are higher for higher magnitude. The

PGV contours are, in the former case, less prolatealong the considered fault than in the latter, whichis a clear consequence of the fault length. Notethat the PGV maps for magnitudes 5.0 and 6.5(not shown here) exhibit analogous features.

3.4 PAHA maps

To compute the PAHA maps, we utilize Eqs. 1and 2. Using the trapezoidal rule, the integralover m in Eq. 2 is replaced by a sum over thecontributions from the four magnitudes, for whichthe mean PGVs and their uncertainties are de-termined (see above). We assume the parame-ters of the generalized Omori’s law (Eq. 2 in thecompanion paper) to be set to b = 1.1, p = 1.25,β ′ = 1.0, and c(m∗) = 30 s (reference values in thecompanion paper). Parameter m∗ is constrainedby the modified form of Bath’s law (Shcherbakovand Turcotte 2004) stating that the difference be-tween the mainshock magnitude and m∗ is approx-imately 1.2, i.e., m∗ = 6.2 in our case of the A25Izmit aftershock. Note that the parameters are,for simplicity, assumed constant along the fault,although their variations are suggested by somestudies (e.g., Wiemer and Katsumata 1999).

For evaluation of Pv , we consider the timeinterval 0–100 days, i.e., T1 = 0 and T2 = 100 inEq. 1. Note that, according to the parametric studypresented in the companion paper, the excee-dence probabilities are largest immediately afterthe mainshock (within the first day), and they arenot so sensitive to the extent of the time range (seeFigs. 2–6, 9 in the companion paper).

For a strike angle typical for the vicinity ofthe North-Anatolian fault, α = 270o, Fig. 7 showsthe probability that the PGV will exceed 0.1 m/s.The PAHA map on the left hand side of Fig. 7 isobtained by the use of simulated PGVs, whereason the right hand side, it is obtained by the useof the attenuation relation. The main differencebetween the maps is that the one on the lefthand side (obtained by the use of simulations)is again affected by the remanent radiation pat-tern lobes and directivity (prolongation along thefault strike).

Note that, according to the parametric study(see the companion paper), the only parameter

J Seismol (2008) 12:65–78 75

Fig. 7 Map of probability (in percents) that PGV willexceed 0.1 m/s obtained by the use of simulations (left) andattenuation relations (right). The stations considered in this

study are marked by the triangles, the size of which is scaledaccording to the observed PGV (Table 1)

that could substantially change the PAHA mapsis m∗. However, varying m∗ would not changethe “visual” differences between the two kindsof the above-discussed PAHA maps (lobes andprolongation). Let us also emphasize that Table 1indicates that the PGV of 0.1 m/s was exceededat three stations only (namely, TPT, IZT, andBHC). The map in Fig. 7 shows that all of themare situated in a region where the probability ofexceeding this value is larger than 90%.

Figure 8 shows a complementary type of thePAHA maps: PGV that has 90% probability ofbeing exceeded. In this way, the 90% contourin Fig. 7 has to be the same as the contour for0.1 m/s in Fig. 8. As in the previous discussionabout Fig. 7, the main difference between themaps obtained by the use of simulations (left)and by the use of attenuation relations (right) isagain that the map on the left is affected by theremanent radiation pattern and directivity.

Fig. 8 PGV (in m/s) that has a 90% probability of beingexceeded, obtained by the use of simulations (left) andattenuation relations (right). The triangles represent the

stations considered in this study. The size of the trianglesis scaled according to the observed PGV (Table 1)

76 J Seismol (2008) 12:65–78

Fig. 9 Profile of PAHA PGV map in Fig. 8 (right) per-pendicular to the fault (the curve labeled 90–90% in thelegend). Curves 50–50% and 20–20% are analogous, butfor 50 and 20% probability of exceedence prescribing 50and 20% occurrence probability (adjusted by the choice ofm∗) for the m ≤ 5.8 aftershocks, respectively. Attenuationcurve for magnitude 5.8, its ±2σ uncertainty and observeddata are also shown

Let us discuss an interesting property of thePAHA. Note first that, for the studied time inter-val of 0–100 days, the aftershock statistics impliesthat there is 90% probability that an aftershock ofmagnitude larger than or equal to the magnitudeof the A25 aftershock, i.e., m ≥ 5.8, will occur(Eq. 1 in the companion paper). Let us considera profile perpendicular to the fault in the cen-ter of the 90% exceedence probability PGV map(Fig. 8, right). The corresponding PGV distancedecay curve in this profile is shown in Fig. 9(labeled 90–90%). Changing the value of m∗, wecan constrain the occurrence probability to anygiven value, for example, 50 and 20% and thus ob-tain analogous curves for these probabilities (seecurves labeled as 50–50% and 20–20%, respec-tively, in Fig. 9). All the three curves are almostidentical. This means that the maps would give al-most the same PGV values provided that the samepercentage is considered for both the exceedenceprobability and the probability of occurrence of agiven aftershock.

Moreover, in the same Fig. 9, the curves arecompared with attenuation curve for magnitude5.8 and PGV values observed during the after-shock A25 (Table 1). The agreement between

the attenuation curve and the PAHA map cross-sections suggests that hazard at a given probabilitylevel is governed by ground motions from an after-shock with the same probability of occurrence.

4 Discussion and conclusions

In this paper, we have studied and tested numeri-cally the application of the strong ground motionmodeling to the PAHA. Traditionally, empiricalattenuation relations are used for the translationfrom occurrence probabilities to the probabilitiesof exceedence in the probabilistic hazard analy-sis. Such an approach is suitable for applicationsbased on seismicity, such as in the STEP program(available at http://pasadena.wr.usgs.gov/step) forCalifornia (Gerstenberger et al. 2005), whichprovides the probability of strong shaking inCalifornia within the next 24 h.

However, in cases when we know a fault is be-ing activated during the aftershock sequence (i.e.,we can assume aftershock predominant mecha-nism), the attenuation relation represents perhapstoo unspecific description of the strong groundmotions. This follows from the fact that the re-lations are obtained by regression of data, whichconsist of those measured during earthquakes ofvarious faulting mechanisms, directions of rup-ture propagation, etc. On the other hand, strongground motion modeling techniques consideringa source of finite extent are able to account forspecific aftershock properties, i.e., to capture thekey features of ground motions from large earth-quakes such as the directivity.

In this paper, we present a synthetic study,based partially on the Izmit A25 aftershock. Wecompare hazard assessment results obtained uti-lizing the attenuation relations with those basedon the strong ground motion simulations. Notethat the two hazard models are tuned with theaim to have the two approaches fully comparable:Besides having all the Omori’s law parametersthe same, we first adjust the employed hybridk−2 model to have simulated PGVs in agreementwith the attenuation relations. The agreement ofthe calculated PGVs and those observed duringthe A25 aftershock shows that the attenuation

J Seismol (2008) 12:65–78 77

model (attenuation relation) utilized in our paperis appropriate.

The comparison of the two approaches demon-strates that the prediction maps based on finite-extent fault simulations display overall similaritywith those obtained by the use of the attenuationrelations (assumed usually in probabilistic hazardanalysis). However, in certain aspects, the mapsdiffer: The maps based on the synthetic simu-lations exhibit the “remanent” radiation patterneffect and directivity effect that are specific forthe fault mechanism considered. The scenario-specific effects involved in the synthetic maps aresmoothed due to averaging over scenarios, whichmakes the maps apparently symmetric. This is sobecause we employ simple 1D medium and, infact, assign the same probability (or weight) toeach scenario. Note that the aim of this compar-ison is rather to see the possible differences in thePAHA maps obtained by the two approaches andnot to judge which approach is more reliable inpractical use.

In certain situations, some parts of a given faultcan be more likely to rupture due to static anddynamic stress changes and/or due to processesset in motion by those stress changes, such ascrustal fluid flow and plastic deformation (for areview see Steacy et al. 2005). Then, only someof the assumed scenarios (when the nucleationpoint lies in the stress increase part of the fault)should be preferred (by a proper weight). In thiscase, simulations (instead of attenuation relations)should be utilized, as they can provide the statis-tical distribution of strong motion characteristicsaccounting for scenarios with different weights.We expect that, in such a case, a possible direc-tivity effect could be emphasized and the PAHAmaps would no longer be symmetric as in Figs.7 and 8. However, at present, modeling of suchtriggering processes yields qualitative results only.

Finally, let us discuss another possible develop-ment of PAHA regarding not only a given singlefault, but also a family of faults. Mathematically,it is possible to introduce weights for individ-ual faults into the hazard analysis (Eq. 2), whichwould control the aftershock rates. In this way,after a slight modification, this approach couldbe combined with some other, independent,

information, preferring some faults (or part offaults) to be more likely to rupture. Such infor-mation could be related to stress changes as inthe previous case. Again, at present, modeling ofthese processes is not capable of providing quan-titative rate changes (needed by PAHA). Never-theless, in the future, such combinations wouldincrease the value of the hazard forecast.

Due to an insufficient number of strong mo-tion data in our target zone, the validation ofour resulting PAHA maps is difficult. Moreover,as presented in the parametric study providedin the companion paper, the PAHA maps aremostly sensitive to changes of magnitude of thelargest aftershock m∗. In our illustrative examplefor the A25 aftershock, we have set it consider-ing the mean difference between the mainshockmagnitude and m∗ (Bath’s law). However, for aparticular mainshock, the actual difference maybe far from the mean, which means that, for theparticular aftershock sequence under study, thevalue that we have adopted is rather incorrect. Forthese reasons, we can discuss only reliability of ourresults instead.

As we have found, the hazard at a given prob-ability level is governed by ground motions froman aftershock with the same probability of occur-rence. We have illustrated this (see the previoussection) for an example of PAHA PGV maps for90% probability of exceedence with such Omori’slaw parameters that the A25 aftershock has 90%probability of occurrence. We show in Fig. 9 thatthe same PAHA PGV values are obtained pro-vided that the same percentage is considered forboth the exceedence probability and the proba-bility of a given aftershock occurrence (50 and20% in our example). In another words, for agiven generalized Omori law parameter set (fixedm∗-value, etc.), the 50–50% curve (and 20–20%)should give, of course, larger values than the 90–90% curve (more seldom large magnitude eventscombined with lower ground motion exceedanceprobability gives larger PGV values). However,the PGV values will remain the same in the caseof an above-explained special change of them∗-value. As regard the actual values of synthe-sized PGVs, they agree both with the attenuationcurve for the A25 aftershock magnitude (5.8) and

78 J Seismol (2008) 12:65–78

observed data, suggesting reliability of the PAHAmaps due to proper attenuation model that hasbeen utilized in our study.

Acknowledgements Reviews of two anonymous refereeshelped to improve the manuscript. The research was sup-ported by the Grant Agency of Charles University(292/2005/B-GEO/MFF), MSM 0021620800, Grant Agencyof the Czech Republic (205/07/0502) and Marie Curietraining network SPICE in the 6th Framework Program ofthe European Commission (MRTN-CT-2003-504267).

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