A double branching model for earthquake occurrence

Warner Marzocchi1 and Anna Maria Lombardi1

Received 30 October 2007; revised 12 April 2008; accepted 21 May 2008; published 23 August 2008.

[1] The purpose of this work is to put forward a double branching model to describe thespatiotemporal earthquake occurrence. The model, applied to two worldwide catalogs indifferent time-magnitude windows, shows a good fit to the data, and its earthquakeforecasting performances are superior to what was obtained by the ETAS (first-stepbranching model) and by the Poisson model. The results obtained also provide interestinginsights about the physics of the earthquake generation process and the time evolution ofseismicity. In particular, the so-called background seismicity, i.e., the catalog afterremoving short-time clustered events, is described by a further (second-step model)branching characterized by a longer time-space clustering that may be due to long-termseismic interaction. Notably, this branching highlights a long-term temporal evolution ofthe seismicity that is never taken into account in seismic hazard assessment or in thedefinition of reference seismicity models for a large earthquake occurrence. Anotherinteresting issue is related to the parameters of the short-term clustering that appearconstant in a different magnitude window, supporting some sort of universality for thegenerating process.

Citation: Marzocchi, W., and A. M. Lombardi (2008), A double branching model for earthquake occurrence, J. Geophys. Res., 113,

B08317, doi:10.1029/2007JB005472.

1. Introduction

[2] Modeling the spatiotemporal distribution of moderate-large earthquakes is the basis for time-dependent seismichazard assessment and for earthquake forecasting. Despitethe pivotal scientific and practical relevance of this issue, sofar different and sometimes contradictory models for bothspatial and temporal occurrence have been developed.[3] As regards the spatial distribution, earthquake occur-

rence is usually modeled by using areas (regular grid orseismotectonic zonation), or single seismogenic structures.While the latter should be obviously the optimal choicebecause it would drastically reduce the spatial coverage ofthe distribution, there are still several doubts about thecompleteness of fault systems [e.g., Marzocchi, 2007],overall for moderate but still destructive earthquakes (i.e.,with magnitude ranging from 6.0 to 7.0). Under thisperspective, the use of spatial grids and seismotectoniczonations [Kagan and Jackson, 1994, 2000; Faenza et al.,2003; Cinti et al., 2004; Gerstenberger et al., 2005;Holliday et al., 2005] can be seen as a way to account forthe epistemic uncertainty associated to the lack of faultcatalogs completeness. Moreover, the use of areas instead offaults makes easier the set up of firm rules for earthquakeforecasting that can be tested rigorously [Schorlemmer etal., 2007]. For all of these reasons, in this work we adopt agrid to account for the spatial distribution of earthquakeoccurrence.

[4] The temporal distribution of moderate-strong eventsis maybe more uncertain. While a short-term aftershockclustering in small areas is commonly accepted, other kindsof time evolution, on different scales, are still matter ofdiscussion. When aftershocks are removed, it is usuallyassumed that the events of the ‘‘detrended’’ catalogs (back-ground events, hereinafter) follow a stationary Poissondistribution [Wyss and Toya, 2000]. Notably, this paradigmis still implicitly accepted in many practical applications,such as in the formulation of probabilistic seismic hazardassessment methodologies on the basis of Cornell’s methodand in evaluating earthquake prediction/forecasting models[e.g., Cornell, 1968; Kagan and Jackson, 1994; Frankel,1995; Varotsos et al., 1996; Gross and Rundle, 1998;Kossobokov et al., 1999; Marzocchi et al., 2003a].[5] On the other hand, other researchers found and

hypothesized more complex distributions for backgroundevents, ranging from ‘‘quasi-periodic’’ distribution[Nishenko and Buland, 1987; McCann et al., 1979], toshort-term and long-term modulation. Remarkably, whilethe former is usually hypothesized and never passed rigor-ous statistical tests with real data [Kagan and Jackson,1991a, 1995; Rong et al., 2003], significant empiricalevidence of time modulation at different time scales werefound, ranging from hours/days as in volcanic swarms[Hainzl and Ogata, 2005; Lombardi et al., 2006], todecades and longer [Kagan and Jackson, 1991b, 1994;Rhoades and Evison, 2004; Lombardi and Marzocchi,2007]. These temporal features of the background eventshave probably different physical causes compared to the‘‘aftershocks’’, that are usually attributed to the elasticresponse of the lithosphere. The short-scale modulationfound in volcanic swarms are very likely induced by fluid

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1Istituto Nazionale di Geofisica e Vulcanologia, Rome, Italy.

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injections [Hainzl and Ogata, 2005; Lombardi et al., 2006],and therefore it appears to be a distinctive feature of somevolcanic swarms. Instead, the variations of seismogeneticcapability at longer time scales, lasting more than onedecade, are characteristic of tectonic earthquakes, andtherefore driven by tectonic rate variations, or viscoelasticinteraction between earthquakes [Pollitz, 1992; Piersanti etal., 1995].[6] In this work, we build a model that is able to capture a

complex space-time behavior. In particular, we use a doublebranching process to account for different scales of theseismic time-space evolution. Basically, the model assumesthat each earthquake can generate, or is correlated to, otherearthquakes, through different physical mechanisms. Inorder to explore the behavior for different time-magnitudewindows, we consider two worldwide catalogs: the Pachecoand Sykes’ catalog [1992] (PS92 hereinafter) that reportsshallow M 7.0+ earthquakes in the last century, and thePreliminary Determination of Epicenters (hereinafter NEIC)catalog, collected by the National Earthquake InformationService (NEIC/USGS) (www.neic.cr.usgs.gov/neis/epic/epic.html) that reports a complete catalog for M 5.5+ since1974.[7] We devote a large part of the paper to carefully assess

the reliability of the model. This essential step of theanalysis aims also to evaluate the forecasting capability ofthe model compared to other competitive models, to verifyits capacity to describe past and future seismicity, and to testthe physical hypotheses/assumptions that stand behind themodel set up.

2. Data Sets

[8] The catalog PS92 contains epicentral coordinates,origin time, surface magnitude Ms, and seismic momentM0, of 698 events occurred in the period 1900–1990, withMs � 7.0 and depth d � 70 km. The values of Ms can beconsidered homogeneous in time, because the authors applysome corrections to original estimates in order to compen-sate the lack of uniformity in recording (see Pacheco andSykes [1992] for details). In our study we use thesecorrected Ms values but, to avoid the problem of saturationof surface magnitude scale, we prefer to consider themoment magnitude Mw for events with Ms > 8.0. Thesevalues are obtained by seismic moment using the relation ofHanks and Kanamori [1979]. Since all but two of eventswith Ms > 8.0 have independently (i.e., from literature)determined seismic moments, saturation of surface-wavemagnitudes should not affect Mw estimation. For eventswith Ms � 8.0 the Mw values are very close to corrected Msvalues provided by catalog.[9] The catalog NEIC is the most complete worldwide

instrumental data set of the last thirty years [Kagan, 2003].The magnitude scale considered is the maximum (Mmax)among different magnitude values reported. This choicepermits to reduce problems coming from saturation ofmagnitude and is in agreement with NEIC valuation methodused in compiling catalog (eighty columns format). Weselect events occurred from 1 January 1974 to 31 December2006, with depth � 70 km and magnitude Mmax � 6.0(3590 events). For most events (about 50%) Mmax is thesurface magnitude (Ms), whereas moment (Mw) and body

(Mb) magnitude are considered for 30% and 13% of events,respectively. The magnitude of remaining events belongs tominor (local Ml, duration Md, energy Me) or to unknownscales. Clearly some of these magnitude classes have notbeen uniformly recorded in time. The events for which themagnitude scale is unknown mostly occurred in the firstdecade. Moreover Mw recording practically begin at about1977, with development of organized networks as CentroidMoment Tensor (CMT) system (http://www.seismology.harvard.edu/projects/CMT/).

3. Model Setup

3.1. General Philosophy of the Model

[10] Earthquake occurrence process is very likely gov-erned by different and (more or less) independent physicalprocesses, such as tectonic loading, stress variations in-duced by other earthquakes through elastic and viscoelasticinteractions, external perturbations, etc. The spatiotemporaldomains involved by these processes can vary severalorders of magnitude. For instance, whereas the elasticinteraction has most of the effect in the spatiotemporalrange of aftershock sequences [Stein et al., 1992, 1994;King and Cocco, 2000], the viscoelastic stress transfer caninfluence for decades seismicity of a region recoveringhundreds of kilometers [Pollitz, 1992; Piersanti et al.,1995]. So, even if different physical processes are allresponsible of the seismicity of an area, their relative impor-tance can be very different according to the magnitude–spatio–temporal scale considered.[11] From a technical point of view, we can study such

a system in two ways: (1) we can fit a single complexmodel that accounts simultaneously for all the differentelementary processes; (2) we can follow a stepwiseprocedure, fitting step by step simple models that accountfor all single elementary processes separately. Despite thesecond strategy seems often somehow ‘‘ad hoc’’, it hasbeen demonstrated that it is much better to describestructured problems, such as additive or interactive mod-eling [Bühlmann, 2003].[12] One of the most diffuse strategy of this kind is the

Boosting technique [Freud and Schapire, 1996]. It providesan aggregation scheme in which a sequentially fittingmodels are applied on data, heavily weighting at each stepthose observations poorly predicted by the previous model.Data and the final boosting estimator is then constructed viaa linear combination of such multiple estimates (see Freudand Schapire [1996] for details). Here, we follow a similarstrategy to describe the aggregation of processes involvingdifferent spatiotemporal windows.

3.2. Multiple Branching Process

[13] Following the general philosophy of Boosting Meth-ods, we adopt a new procedure combining representativemodels of the same family which may reveal differentaspects of the data. The main difference respect to BoostingMethods is that at each step of our methodology we do notre-weight observations, but we cut off from data set eventswell explained by current model, keeping only recordspoorly fitted. These last events form a new smaller dataset to model in a following step.

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[14] We use, as class of models, a well-known family ofpoint processes, called ‘‘self-exciting’’ models. Introducedin the early 1970s by Hawkes [1971], these became the firstexample of stochastic models of general utility for thedescription of seismic catalogues. The rate l(t) of this classof models is given by

l tð Þ ¼ f tð Þ þXti

where rij is the distance between locations (xi, yi) and (xj, yj).[21] 2. Computation of the expected number �N of

background events, by using the probabilities found inequation (3).[22] 3. Selection of the �N earthquakes of the seismic

catalog with the highest probability pi(I) to belong to the

background.[23] 4. A background event is replaced by one of its

aftershocks if the latter has a larger magnitude.[24] The latter step avoids the removal of a large earth-

quake that the algorithm identifies as aftershock because ithas been anticipated by a foreshock. The declustered catalogobtained by this procedure represents the seismicity filteredby the first-step branching. Hereinafter we call it backgrounddata set, and it will be analyzed in the following step.[25] It is worth noting that our procedure produces a

single declustered catalog, while the original technique ofZhuang et al. [2002] produce a set of stochastic realizationsof declustered catalog. The main rationales behind the useof a single declustered catalog are two: first, a single catalogallows the parameters of the model to be set univocally;second, the use of the original method of Zhuang et al.[2002] does not allow the problem of foreshocks to beeasily handled (see last point of the procedure describeabove). The use of a single catalog may lead to biasedresults only if our procedure is not able to capture the mainstatistical features of the declustered catalog. In order toverify this assumption, in the following analyses we checkthe sensitivity of the results by using also a set of 1000stochastic declustered catalogs generated by the originaltechnique of Zhuang et al. [2002].

3.4. Second-Step Branching

[26] The second step of our procedure consists in re-applying a branching process to the background data setobtained in the previous step to describe the long-termclustering. Hereinafter, the terms ‘‘second-step branching’’and ‘‘long-term clustering’’ are considered synonyms.Through this step, we can check if the seismicity of aregion, filtered by the short-term clustering, shows statisti-cally significant subtler features that may suggests a longertime clustering or trend. In particular, the model assumesthat the time evolution of the background can be describedby a second-step branching process, working at largerspace-time scales compared to the short-term cluster re-moved after the first step.[27] To establish some constraints on time evolution of

the damping factor, that controls the long-term evolution,we use an inverse exponential distribution e�t/t, where t isthe elapsed time from the occurrence of the earthquake thatgenerates stress variations, and t is the characteristic time ofthe interaction. This parameterization is a simplification ofthe model proposed by Piersanti et al. [1995, 1997] todescribe temporal evolution of postseismic stress variations.In such a model, the relaxation is composed by a sum ofexponential decays, that mimic different relaxation modes.Our parameterization implies that, for seismic interaction ona time scale covered by a seismic catalog, one relaxationmode is predominant. This prevalent relaxation time tdepends on the viscosity of the mantle, for which verydifferent values, ranging from 5 1017 Pas [Pollitz et al.,1998] to 5 1020 Pas [Piersanti, 1999], have been proposed.

[28] To describe the spatial decay of stress variation, withdistance r from epicenter of perturbing event, we choose aninverse power law PDF. The dependence of hazard functionwith the magnitude M of exciting event is assumed ofexponential type, i.e., proportional to ea2M. Therefore thetime-dependent conditional rate of earthquake occurrencefor the second step of our procedure is given by

l2 t; x; y=Htð Þ ¼ m2 x; yð Þ þXti

tions of the background and triggered rate in terms ofcatalog used, branching stage, and spatial distribution.

4. Checking the Goodness of Fit and thePerformance of the Model

[31] A careful testing of any model is a basic step todecide on whether or not the model provides an adequatedescription of data. Here, to achieve this goal, we use aprocedure consisting of two steps. At first, we set up themodel on a subset of the catalog (the learning data set);then, we evaluate the goodness of fit and the forecastingcapability of the model on an independent subset of thecatalog (the testing data set), which has not considered atany step of modeling. Since the feasibility of short-termclustering has been already tested in the past, also for largeearthquakes [Lombardi and Marzocchi, 2007], here wefocus our attention on testing the feasibility of the second-step branching, or, in other words, if the background of theETAS model is a stationary Poisson process, or if it containssecond-step branching structures. At this purpose, the anal-yses are carried out on the background testing data sets, i.e.,the catalog after to have removed the short-term clusteringdescribed by the first-step branching.[32] As regards the goodness of fit test, we consider the

time evolution of the integral of the conditional intensity

L tð Þ ¼Z tTstart

dt0ZRdxdyl2 t0; x; y=Htð Þ ð9Þ

where Tstart is the starting time of observation history. Bythe time transformation ~t = L(t), the occurrence times ti aretransformed into new values ~ti. If the model describes wellthe temporal evolution of seismicity, the transformed data ~tiare expected to behave like a stationary Poisson processwith the unit rate [Papangelou, 1972; Ogata, 1988]. Anydeviation from expected Poisson behavior indicates asignificant factor underlying the data which is not capturedby the model.[33] We test the Poisson hypothesis for transformed times

~ti of the background testing data set, by using two nonpara-metric statistical tests: the Runs test and the one-sampleKolmogorov-Smirnov test (KS1) [Gibbons and Chakraborti,2003]. Whereas the Runs test verifies the reliability of theindependence of the earthquake occurrence, the KS1 checksthe hypothesis that the interevent times are exponentiallydistributed.

[34] As regards the evaluation of the forecasting capabil-ity of the model, we compute the information gain per event(IGpe) [Daley and Vere-Jones, 2003] on the backgroundtesting data set. IGpe measures the performance of a modelH1 relative to a reference model H0 and is given by thedifference D ln L of log likelihoods of two processesdivided by the number of events N

IGpe D lnLN

¼ lnL1 � ln L0ð ÞN

ð10Þ

The value of IGpe should be relatively negative if thereference model is the best and relatively positive if H1 isthe best performing. Hence a greater IGpe (relative to thenull model) describes how much more predictable the fittedmodel is than the null model. This measurement, basicallyequivalent to the R-test from Kagan and Jackson [1995], isuseful to compare the predictability of various competingmodels.[35] In our case, H1 is the second-step branching, and H0

is the Poisson model, both for background testing data set.In practice, this comparison consists of verifying if thebackground events, selected after the first step of the model(i.e., after removal of the short-term clustering), have aPoisson distribution or are still clustered. To this purpose,we compare IGpe values obtained by the real backgroundtesting data set (from now on IGpe?), and by two differentsets of synthetic catalogs. The first set consists of 1000Poisson synthetic catalogs (with the same background rateand the same length of the testing data set). The second setis composed by 1000 synthetic catalogs generated by ourmodel. The comparison of IGpe of the real and syntheticcatalogs allows the two hypotheses H0 and H1 to be tested.Specifically, if the hypothesis to be tested (H0 or H1) is true,IGpe? can be seen as a random realization of the IGpeobtained by the model under testing; in this case, hypothesisH0 is rejected if IGpe

? is above the 95th percentile of the1000 values obtained for the Poisson model (one tail testwith a significance level of 0.05); the hypothesis H1 isrejected if the same value is below the 5th percentile of the1000 IGpe values obtained by the second-step branchingmodel (one tail test with a significance level of 0.05).Finally, in order to verify the sensitivity of the results to thechoice of the declustered catalog, we calculate also the IGpefor 1000 declustered catalogs by using the stochasticprocedure suggested by Zhuang et al. [2002]. In this way,it is straightforward to verify if possible rejection of thehypotheses mentioned above can be explained by ourdeclustering procedure.

5. Results of the Analysis for PS92

[36] For PS92, we set the learning data set as the part ofthe catalog in the time interval 01/01/1900–12/31/1979(629 events), and the testing data set as the part in the timeinterval 01/01/1980–12/31/1989 (69 events).[37] The first step of our procedure involves estimation of

parameters (n1, K1, c, p, a1, d1, q1) of ETAS model(equation (2)) on learning data sets. The values obtainedby the procedure of Zhuang et al. [2002], together withrelative errors and maximum likelihood value, are reportedin Table 1. As a first remark, it is worth noting that such

Table 1. Parameters of the ETAS Model for PS92 and NEIC

Learning Data Setsa

ParameterPS92 (Mmin = 7.0) NEIC (Mmin = 6.0)

1900–1980 (629 Events) 1974–2002 (3064 Events)

n1 6.8 ± 0.3 yr�1 74 ± 2 yr�1

K1 (4 ± 1) 10�3 yr p�1 (1.0 ± 0.1) 10�2 yr p�1

p 1.1 ± 0.1 1.05 ± 0.01c (2 ± 1)10�4 yr (4.0 ± 1.0)10�5 yra1 1.2 ± 0.2 1.1 ± 0.1d1 22 ± 4 km 12 ± 0.5 kmq1 1.5 1.5

aMaximum likelihood parameters of the ETAS model (first-stepbranching; see equation (2)) for PS92 and NEIC learning datasets.

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parameters are consistent with values computed for singleaftershocks sequences in tectonic zones [e.g., Ogata, 1999],corroborating the hypothesis of universality of physical lawsdriving short-term triggering [Lombardi and Marzocchi,2007].[38] Before going to the second-step branching, we re-

move from original learning data sets the short-term trig-gered shocks, through the procedure described in theprevious section. From Figure 1a, we note that most ofevents have a probability pi

(I) to belong to background closeto 0 and 1, and only about 10% of all events have aprobability ranging between 0.1 and 0.9. In this case, theexpected number of background events according to theprocedure of Zhuang et al. [2002] is �N = 547. The back-ground learning data set is composed by the �N events havingthe highest value of pi

(I).[39] The application of second-step branching model (see

equation (5)) on the PS92 background learning data setprovides parameters reported in second column of Table 2.The first interesting remark regards the value of relaxationtime t, equal to = 36?7 years: it reveals a significant long-term clustering behavior of seismicity, different from theone identified in the first stage of the model. Comparing thelog likelihoods (lnL) of Poisson and the double branchingmodel of the background learning data set (see Table 2), wefind that the latter is significantly better in light of AkaikeInformation Criterion (AIC) [Akaike, 1974]. This last isdefined by AIC = �2 lnL + 2np, where np is the number ofparameters. The lower value of the AIC identifies the modelthat better represents the data. We have AIC = 19441.2 forthe Poissonian Model and AIC = 18886.4 for the second-step branching model, revealing a best performance of thislast model.[40] Because of the long-lived nature of triggering activ-

ity, the seismicity in the learning time interval may beaffected by earthquakes which occurred before this period.To take into account this effect, we fit again the secondbranching model by considering a time interval precursoryto the period used to estimate the parameters. Seismicityoccurred in this last target period and triggered by earth-quakes belonging to precursory interval is taken into com-putation. Considering the value of relaxation time tpreviously estimated, we consider 30 years for the precur-

sory learning period (1900–1930; 195 events) and 50 yearsfor the target learning data set (1930–1980; 352 events). Inthis run we set t 30 years. Results are shown in lastcolumn of Table 2. The log likelihood per event points outthe better performance of this last model. Hereinafter, weconsider these values as the most reliable parameters of ourmodel for PS92 catalog. We stress that the choice of settingt a priori does not entail any bias in fitting the model. If wedrop this assumption, the maximum likelihood procedureprovides a compatible value of t, but with a much largeruncertainty because of a too short learning data set. More-over, the difference of maximum log likelihood values,about 5.0, obtained by setting (lnL = �5933.1) and opti-mizing (lnL = �5928.3) t, corroborates the negligibleinfluence of this assumption.[41] In Figure 2 we show the histogram of probability pi

(II)

that the ith event is caused by tectonic loading for targetlearning data set (1930–1980; see equation (7)). Whereasthe analogous histogram of step 1 (Figure 1a) is stronglybimodal, revealing a well-defined identification of sponta-neous and short-term triggered events, in this case we have

Figure 1. Histogram of probability pi(I) of belonging to ‘‘spontaneous’’ seismicity for events collected

into (a) PS92 and (b) NEIC learning catalogs. The values of pi(I) are computed by the ETAS model by

using the procedure proposed by Zhuang et al. [2002] (see text for details).

Table 2. Estimated Parameters of the Second-Step Branching

Model for the Background Learning PS92 Data Set (1900–1980)a

Parameter

Poisson Model Second Branching Model

1900–1980(547 Events)

1900–1980(547 Events)

1930–1980(352 Events)

n2 6.8 ± 0.3 yr�1 2.4 ± 0.2 yr�1 2.5 ± 0.4 year�1

K2 0.029 ± 0.005 0.026 ± 0.003t 35 ± 7 yr 30 yra2 0.2 ± 0.2 0.3 ± 0.2d2 140 ± 30 km 140 ± 40 kmq2 1.9 ± 0.3 2.3 ± 0.5Log likelihoodper event

�17.8 �17.2 �16.8

AIC 19,441.2 18,886.4aMaximum likelihood parameters of Poisson and second-step branching-

model (see equation (5)) for background learning PS92dataset. To estimateparametersof branching model we do two runs. In the first (second column)we set up the modelby using whole background learning dataset of 80 years(1900–1980) . In the second (lastcolumn) we consider a precursory periodof 30 years (1900-1930) to the interval timeof 50 years (1930–1980) usedto apply the estimating procedure (see text for details).These last parameters(marked in boldface) are used to following testing of the model.

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a more uncertain recognition of long-term triggered effect.About 70% of all events have a probability ranging between0.1 and 0.9. This result can be due to two reasons: aninefficiency of the model to distinguish for all events themain cause of their occurrence or to an actual jointcomparable action of tectonic loading and long-term stresstransfer.

[42] In Figure 3 we show the learning and testing datasets (a), and the regions in which the triggering effect has apredominant role for seismicity occurred in the targetlearning data set (b); specifically, we show the map ofproportion of seismic rate due to long-term triggering effect.Keeping in mind that these estimations are probably biased[Sornette and Werner, 2005], we can have an idea of thespatial variations of the triggered events. In particular, theplot shows that the seismicity due to long-term stresstransfer is mostly apparent in the Pacific Ring, in thenorthern part of Sumatra Island and in Turkey.[43] A first check of the reliability of the model is given

by application of goodness of fit tests on transformed times~ti (see equation (8)) for PS92 testing data set. For both KS1and Runs test we cannot reject the Poisson model at asignificance level of 0.1. As regards the evaluation of theforecasting capability of the model, the results are reportedin Figure 4. The graph shows IGpe? (the IGpe of the realbackground testing data set; vertical dashed line at 1.2), and1000 IGpe obtained from stochastic declustered realcatalogs through the original procedure of Zhuang et al.[2002] (dark gray bars), synthetic stationary Poissonbackgrounds (light gray bars), and synthetic backgroundsgenerated by our model (light gray bars). Both IGpe? andthe 1000 IGpe obtained by the declustered catalogs throughthe procedure of Zhuang et al. [2002] have values largerthan the ones relative to the synthetic Poisson background.This test permits to reject the Poisson model at 0.01significance level. On the other hand, the same valuespartially overlap the IGpe distribution obtained by ourmodel. In particular, we have that 89% of synthetic values

Figure 2. Histogram of the probability pi(II) of being

caused by tectonic loading for events belonging to the PS92background learning target catalog (events that occurredbetween 1930 and 1980).

Figure 3. (a) Maps of the seismicity reported in the PS92 catalog. The circles represent the seismicity ofthe learning (in black) and testing (in red) data sets; the dimension of circles is proportional to themagnitude. (b) Map of the ratio between (bottom) long-term triggered seismic rate ĉ(x, y) and totalbackground rate m̂(x, y) for the PS92 background learning target (1930–1980) catalog.

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are smaller than IGpe?, indicating that we cannot reject ourtime-dependent model, at least for a significance level of0.1.

6. Results of the Analysis for NEIC

[44] For NEIC catalog, we set the learning data set as thesubset of events occurred from 01/01/1974 to 12/31/2002(3064 events); therefore the testing subset is the part ofcatalog spanning the time interval 01/01/2003–12/31/2006(526 events).[45] The last column of Table 1 reports the parameters of

the ETAS model (first-step branching) n1, K1, c, p, a1, d1,q1 (see equation (2)) estimated on learning data set. Also inthis case, we stress that such parameters are similar to whatfound for single aftershock sequences.[46] As before, we proceed in our analysis removing from

original learning data set the short-term triggered shocks,through the procedure described before. Also in this case,we note a marked bimodal distribution with the two modesclose to 0 and 1; only about 10% of all events have aprobability pi

(I) ranging between 0.1 and 0.9 (see Figure 1b).In this case, the expected number of background eventsaccording to the procedure of Zhuang et al. [2002] is �N =2099. The background learning data set is composed by the�N events having the highest value of pi

(I).[47] The value of parameters (v2, K2, t, a2, d2, q2) for

NEIC catalog, obtained by considering 19 years for theprecursory period (1974–1992; 1270 events) and 10 yearsfor the learning target data set (1993–2002; 829 events) arereported in Table 3. Considering results obtained by PS92catalog and the temporal coverage of NEIC target data set,we set t 30 years. In order to check this choice, we havenoted that the use of other reasonable values of t (10, 20,and 40 years) do not lead to a better fit of data in terms ofmaximum likelihood. Moreover, we anticipate that thischoice will be justified by the better forecasting perfor-mance on the background testing data set of our modelcompared to Poisson model.[48] From the histogram of pi

(II) for NEIC (Figure 5)catalog, we find that about 20% of events are likely nottriggered by any previous earthquake (pi

(II) > 0.9), and onlyabout a 5% of events that are very likely due to a long-term

interaction (pi(II) < 0.1). In other words for more than 70% of

events we cannot clearly decide if they have a ‘‘tectonicallydriven’’ or ‘‘long-term triggered’’ feature (see Figure 5). Asfor PS92, the long-term stress transfer is predominant inPacific Ring (see Figure 6).[49] By applying the Runs and KS1 tests on transformed

times ~ti (see equation (8)) of NEIC background testing dataset, we do not reject the Poisson model (by both test) at 0.05significance level, corroborating the reliability of our mod-eling. The test about the forecasting capability of thebranching model on NEIC background testing data setshows similar results respect to PS92 catalog (see Figure 7).Here, IGpe? is equal to 0.6. By adopting the samestrategy described for PS92 catalog, we see that this value,as well as the IGpe values obtained by 1000 stochasticdeclustered catalogs by means of the procedure of Zhuanget al. [2002], are larger than all values derived fromsynthetic Poisson catalogs. This stands for a rejection of thePoisson model hypothesis at a 0.01 significance level. Atthe same time, the same values overlap well the IGpedistribution obtained by the second-step branching model.In other terms, we cannot reject this model at a significancelevel of 0.1.[50] Results of all performed tests mean that the back-

ground events do not have a time-independent (Poisson)distribution, and that our model is able to capture basic

Table 3. Estimated Parameters of the Second-Step Branching

Model for the Background Learning NEIC Data Set (1974–2003)a

Parameter

Poisson Model Second Branching Model

1974–2003(2099 Events)

1993–2003(829 Events)

n2 72 ± 2 yr�1 47 ± 3 yr�1

K2 0.030 ± 0.002t 30 yra2 �0.0d2 45 ± 9 kmq2 2.3 ± 0.4Log likelihood per event �14.4 �14.0

aMaximum likelihood parameters of Poisson and second-step branchingmodel for background learning NEIC dataset. To set up the model weconsider a precursory period of 19 years (1974–1993) and a target periodof 10 years (1993–2003) (see text for details).

Figure 4. Plot of IGpe for the real PS92 background testing data set (vertical solid line) and forsynthetic catalogs obtained by the Poisson model and the branching model.

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features of spatiotemporal evolution of seismicity recordedin NEIC catalog.

7. Discussion of the Results

[51] The results reported above highlight that the doublebranching model fits well the data of the NEIC and PS92catalogs, and that its earthquakes forecasting capabilities ontesting data sets (independent from the data used to set themodel) are better than single branching model (ETAS). Thisoutperforming of the model has important implications onthe physics of the earthquake occurrence process.

[52] The most remarkable aspect is the existence of asecond-step long-term clustering [cf. Kagan and Jackson,1991b; Rhoades and Evison, 2004; Lombardi and Marzocchi,2007]. This means that the Poisson hypothesis of the seismicbackground, that stands behind most of the seismic hazardassessment and most of the models for earthquakeforecasting [e.g., Cornell, 1968; Kagan and Jackson, 1994;Frankel, 1995; Gross and Rundle, 1998; Wyss and Toya,2000; Marzocchi et al., 2003a], could be wrong. Particularlyinteresting is the characteristic time of such a second-stepbranching. We find a value of about 30 years, that is higherthan the few years/one decade found in recent papers forworldwide and Italian seismic catalogs [Parsons, 2002;Faenza et al., 2003]. We argue that this discrepancy canbe due to the fact that both Parsons [2002] and Faenza et al.[2003] analyze the catalogs with a model that allows a singlecharacteristic time for the clustering; in this case, the tenyears might represent the average between the characteristictimes of short-and long-term clustering found here.[53] Remarkably, the characteristic time of the second-

step branching is compatible to the post-seismic relaxation[Kenner and Segall, 2000]. For this reason, we argue thatpost-seismic perturbations could be the most likely physicaldriving mechanism to explain such a long-term clustering.Notwithstanding, we find a low value for a2 that is notstatistically significant from zero (see Tables 2 and 3),implying the remarkable and unexpected peculiarity thatthe postseismic triggering capability of an event is indepen-dent by its magnitude. This result could raise some doubtson advisability to use branching model to represent thelong-term evolution of the background seismic rate. Actu-ally, although a well-set positive value of a2 wouldstrengthen the reliability of the second-step branchingprocess, the uncertain estimation of a2-value is not a sign

Figure 5. Same as Figure 2 but for the backgroundlearning target NEIC data set (1993–2002).

Figure 6. Same as Figure 3 but for the background learning target NEIC data set (1993–2002).

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of inadequacy of such modeling that, as mentioned before,describes the data better than a single branching (ETAS)model. Moreover, we note that the number of available datacould be not sufficient to test the hypothesis a2 = 0 in arobust way. In fact, whereas the coseismic stress transfer is aphenomenon spanning all magnitude scales, the postseismiceffects are likely mostly caused by the strongest events[Piersanti et al., 1997; Pollitz et al., 1998]. The magnituderange recovered by analyzed catalogs is rather small and theproportion of giant events (M > 8.0), especially for NEICdata set, is negligible. This could be the origin of the largeerror of a2-value.[54] The feasibility of seismic long-term interaction has

been largely debated in recent years. Despite the highnumber of single post-seismic interaction cases reported inliterature [Chéry et al., 2001a, 2001b; Mikumo et al., 2002;Pollitz, 1992; Pollitz et al., 1998, 2003; Rydelek and Sacks,2003; Corral, 2004; Santoyo et al., 2005; Selva andMarzocchi, 2005; Piersanti et al., 1995, 1997; Piersanti,1999; Kenner and Segall, 2000], and numerical modelssupporting it [Ziv, 2003; Marzocchi et al., 2003b; Kennerand Simons, 2005], this issue can be considered still open,mostly because of the lack of clear robust statistical evi-dence of this hypothesis. In order to fill this gap, recentefforts have been devoted to forecast the seismic ratechanges induced by the postseismic effects of the two giantsAndaman-Sumatra earthquakes of 2004–2005 [Marzocchiand Selva, 2008] on a large portion of the earth in the nextfew decades.[55] The spatial distribution of the ‘‘long-term triggered’’

events makes clear an important contribution of long-termtriggering on global seismicity, in most seismogenetic zones(Figures 3b and 6b). Note that these percentages representlower bounds of the real values [Werner and Sornette,2005]. Remarkably, the areas with the highest clusteringdo not overlap the ones with the highest seismic rate;moreover many of these regions have been identified byprevious studies on the basis of physical models as singleexamples of long-term interaction between faults, such asMexico [Mikumo et al., 2002; Santoyo et al., 2005],California [Pollitz, 1992; Kenner and Segall, 2000; Selvaand Marzocchi, 2005], Aleutian and Kurile-Kamchatkatrenches [Pollitz et al., 1998], Alaska [Piersanti et al.,1997], Japan [Rydelek and Sacks, 2003], Chile and South

Peru [Piersanti, 1999; Casarotti and Piersanti, 2003], andTurkey [Stein et al., 1997; Barka, 1999].[56] Another interesting issue that comes out from our

model is some sort of ‘‘universality’’ of physical processesbehind the elastic interaction. The comparison between thefirst-step branching (ETAS) parameters of PS92 and NEICcatalogs (see Table 1) does not highlight any significantvariation for p and a values. These parameters are the moststrictly linked to basic physical features of seismogeneticprocess [Utsu et al., 1995]. Specifically the a-value meas-ures the efficiency of a shock to generate triggered activityby its magnitude; the parameter p is related to timeevolution of triggering effect. Their similarity for PS92and NEIC catalogs, as well as for values obtained by theanalysis of single aftershock sequences [Ogata, 1999],points out that the basic physical features of short-termclustering is almost independent by the time-space-magnitudewindow considered [Corral, 2004; Lombardi and Marzocchi,2007].[57] As regards the other first-step branching parameters,

the most remarkable difference is relative to d1, significantlylarger for PS92 catalog. This result, so as the weak differ-ences of K1 and c, are due to lower threshold magnitudeconsidered for NEIC catalog respect to PS92 data set. Thesmaller value of d1 reflects the smaller mean source dimen-sion of NEIC earthquakes and, therefore, the smaller areacovered by aftershock clusters embedded in the NEIC dataset. The most reliable explanation for larger value of K1 ofNEIC catalog is that the number of events generated by eachearthquake, regardless of magnitude of this, becomes largerif we lower the minimum magnitude of triggered events. Onthe other side, it is well known that the value of c,representing a measure of incompleteness of the catalog inthe earliest part of each cluster, decreases at decreasing ofthreshold magnitude [e.g., Kagan, 2004].

8. Final Remarks

[58] The main goal of this paper has been to outline thefeatures of a double branching process to describe thespatiotemporal distribution of earthquakes. The main ad-vantage of the model is the possibility to account fordifferent spatiotemporal scales of the processes involved.

Figure 7. Same as Figure 4 but for the background testing NEIC data set (2003–2006).

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Remarkably, the model is suitable for computing earthquakeforecasting in real-time.[59] The model has been applied to two worldwide

catalogs with a different time-magnitude coverage (thePS92 and NEIC catalogs). The results obtained put in lighttwo main issues:[60] The background seismicity, obtained by removing

the short-term clustering by seismic catalogs, is character-ized by a second-step long-term clustering. The character-istic time is compatible with post-seismic relaxation that wepropose as the most likely driving mechanism. This featurehas a major importance in practice, because it raises severaldoubts on the feasibility of the stationary Poisson hypoth-esis that stands behind almost all classical seismic hazardassessment, as well as many reference models for evaluatingthe performance of earthquake forecasting model or earth-quake predictions.[61] The parameters found relative to the first-step

branching (short-term clustering) are very similar to whatobtained for seismic sequences in very different time-space-magnitude windows. This stands for a ‘‘universality’’ of thephysical laws governing the short-term triggering.[62] Our results also imply that future analysis on long

(covering at least few centuries) regional seismic catalogscould find similar long-term modulation of seismicity.Moreover, it is also foreseen that the seismicity of theIndonesian arc in the next decades will be significantlyhigher than what was observed in the few decades beforethe giant Andaman-Sumatra earthquakes of 2004–2005.

[63] Acknowledgment. This paper has been partially funded by theEU project NERIES (JRA2).

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�����������������������A. M. Lombardi and W. Marzocchi, Istituto Nazionale di Geofisica e

Vulcanologia, Via di Vigna Murata 605, I-00143, Rome, Italy. (lombardi@ingv.it; warner.marzocchi@ingv.it)

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