A double branching model for earthquake occurrence

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1 Click Here for Full Article JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113,, doi: /2007jb005472, 2008 A double branching model for earthquake occurrence Warner Marzocchi 1 and Anna Maria Lombardi 1 Received 30 October 2007; revised 12 April 2008; accepted 21 May 2008; published 23 August [1] The purpose of this work is to put forward a double branching model to describe the spatiotemporal earthquake occurrence. The model, applied to two worldwide catalogs in different time-magnitude windows, shows a good fit to the data, and its earthquake forecasting performances are superior to what was obtained by the ETAS (first-step branching model) and by the Poisson model. The results obtained also provide interesting insights about the physics of the earthquake generation process and the time evolution of seismicity. In particular, the so-called background seismicity, i.e., the catalog after removing 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-term seismic interaction. Notably, this branching highlights a long-term temporal evolution of the seismicity that is never taken into account in seismic hazard assessment or in the definition of reference seismicity models for a large earthquake occurrence. Another interesting issue is related to the parameters of the short-term clustering that appear constant in a different magnitude window, supporting some sort of universality for the generating process. Citation: Marzocchi, W., and A. M. Lombardi (2008), A double branching model for earthquake occurrence, J. Geophys. Res., 113,, doi: /2007jb Introduction [2] Modeling the spatiotemporal distribution of moderatelarge earthquakes is the basis for time-dependent seismic hazard assessment and for earthquake forecasting. Despite the pivotal scientific and practical relevance of this issue, so far different and sometimes contradictory models for both spatial and temporal occurrence have been developed. [3] As regards the spatial distribution, earthquake occurrence is usually modeled by using areas (regular grid or seismotectonic zonation), or single seismogenic structures. While the latter should be obviously the optimal choice because it would drastically reduce the spatial coverage of the distribution, there are still several doubts about the completeness 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 this perspective, the use of spatial grids and seismotectonic zonations [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 for the epistemic uncertainty associated to the lack of fault catalogs completeness. Moreover, the use of areas instead of faults makes easier the set up of firm rules for earthquake forecasting that can be tested rigorously [Schorlemmer et al., 2007]. For all of these reasons, in this work we adopt a grid to account for the spatial distribution of earthquake occurrence. 1 Istituto Nazionale di Geofisica e Vulcanologia, Rome, Italy. Copyright 2008 by the American Geophysical Union /08/2007JB005472$09.00 [4] The temporal distribution of moderate-strong events is maybe more uncertain. While a short-term aftershock clustering in small areas is commonly accepted, other kinds of time evolution, on different scales, are still matter of discussion. When aftershocks are removed, it is usually assumed that the events of the detrended catalogs (background events, hereinafter) follow a stationary Poisson distribution [Wyss and Toya, 2000]. Notably, this paradigm is still implicitly accepted in many practical applications, such as in the formulation of probabilistic seismic hazard assessment methodologies on the basis of Cornell s method and 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 background events, ranging from quasi-periodic distribution [Nishenko and Buland, 1987; McCann et al., 1979], to short-term and long-term modulation. Remarkably, while the former is usually hypothesized and never passed rigorous statistical tests with real data [Kagan and Jackson, 1991a, 1995; Rong et al., 2003], significant empirical evidence of time modulation at different time scales were found, ranging from hours/days as in volcanic swarms [Hainzl and Ogata, 2005; Lombardi et al., 2006], to decades and longer [Kagan and Jackson, 1991b, 1994; Rhoades and Evison, 2004; Lombardi and Marzocchi, 2007]. These temporal features of the background events have probably different physical causes compared to the aftershocks, that are usually attributed to the elastic response of the lithosphere. The short-scale modulation found in volcanic swarms are very likely induced by fluid 1of12

2 injections [Hainzl and Ogata, 2005; Lombardi et al., 2006], and therefore it appears to be a distinctive feature of some volcanic swarms. Instead, the variations of seismogenetic capability at longer time scales, lasting more than one decade, are characteristic of tectonic earthquakes, and therefore driven by tectonic rate variations, or viscoelastic interaction between earthquakes [Pollitz, 1992; Piersanti et al., 1995]. [6] In this work, we build a model that is able to capture a complex space-time behavior. In particular, we use a double branching process to account for different scales of the seismic time-space evolution. Basically, the model assumes that each earthquake can generate, or is correlated to, other earthquakes, through different physical mechanisms. In order to explore the behavior for different time-magnitude windows, we consider two worldwide catalogs: the Pacheco and Sykes catalog [1992] (PS92 hereinafter) that reports shallow M 7.0+ earthquakes in the last century, and the Preliminary Determination of Epicenters (hereinafter NEIC) catalog, collected by the National Earthquake Information Service (NEIC/USGS) ( epic.html) that reports a complete catalog for M 5.5+ since [7] We devote a large part of the paper to carefully assess the reliability of the model. This essential step of the analysis aims also to evaluate the forecasting capability of the model compared to other competitive models, to verify its capacity to describe past and future seismicity, and to test the physical hypotheses/assumptions that stand behind the model set up. 2. Data Sets [8] The catalog PS92 contains epicentral coordinates, origin time, surface magnitude M s, and seismic moment M 0, of 698 events occurred in the period , with M s 7.0 and depth d 70 km. The values of M s can be considered homogeneous in time, because the authors apply some corrections to original estimates in order to compensate the lack of uniformity in recording (see Pacheco and Sykes [1992] for details). In our study we use these corrected M s values but, to avoid the problem of saturation of surface magnitude scale, we prefer to consider the moment magnitude M w for events with M s > 8.0. These values are obtained by seismic moment using the relation of Hanks and Kanamori [1979]. Since all but two of events with M s > 8.0 have independently (i.e., from literature) determined seismic moments, saturation of surface-wave magnitudes should not affect M w estimation. For events with M s 8.0 the M w values are very close to corrected M s values 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 (M max ) among different magnitude values reported. This choice permits to reduce problems coming from saturation of magnitude and is in agreement with NEIC valuation method used in compiling catalog (eighty columns format). We select events occurred from 1 January 1974 to 31 December 2006, with depth 70 km and magnitude M max 6.0 (3590 events). For most events (about 50%) M max is the surface magnitude (M s ), whereas moment (M w ) and body (M b ) magnitude are considered for 30% and 13% of events, respectively. The magnitude of remaining events belongs to minor (local M l, duration M d, energy M e ) or to unknown scales. Clearly some of these magnitude classes have not been uniformly recorded in time. The events for which the magnitude scale is unknown mostly occurred in the first decade. Moreover M w recording practically begin at about 1977, with development of organized networks as Centroid Moment Tensor (CMT) system ( harvard.edu/projects/cmt/). 3. Model Setup 3.1. General Philosophy of the Model [10] Earthquake occurrence process is very likely governed by different and (more or less) independent physical processes, such as tectonic loading, stress variations induced by other earthquakes through elastic and viscoelastic interactions, external perturbations, etc. The spatiotemporal domains involved by these processes can vary several orders of magnitude. For instance, whereas the elastic interaction has most of the effect in the spatiotemporal range of aftershock sequences [Stein et al., 1992, 1994; King and Cocco, 2000], the viscoelastic stress transfer can influence for decades seismicity of a region recovering hundreds of kilometers [Pollitz, 1992; Piersanti et al., 1995]. So, even if different physical processes are all responsible of the seismicity of an area, their relative importance 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 complex model that accounts simultaneously for all the different elementary processes; (2) we can follow a stepwise procedure, fitting step by step simple models that account for all single elementary processes separately. Despite the second strategy seems often somehow ad hoc, it has been demonstrated that it is much better to describe structured problems, such as additive or interactive modeling [Bühlmann, 2003]. [12] One of the most diffuse strategy of this kind is the Boosting technique [Freud and Schapire, 1996]. It provides an aggregation scheme in which a sequentially fitting models are applied on data, heavily weighting at each step those observations poorly predicted by the previous model. Data and the final boosting estimator is then constructed via a linear combination of such multiple estimates (see Freud and Schapire [1996] for details). Here, we follow a similar strategy to describe the aggregation of processes involving different spatiotemporal windows Multiple Branching Process [13] Following the general philosophy of Boosting Methods, we adopt a new procedure combining representative models of the same family which may reveal different aspects of the data. The main difference respect to Boosting Methods is that at each step of our methodology we do not re-weight observations, but we cut off from data set events well explained by current model, keeping only records poorly fitted. These last events form a new smaller data set to model in a following step. 2of12

3 [14] We use, as class of models, a well-known family of point processes, called self-exciting models. Introduced in the early 1970s by Hawkes [1971], these became the first example of stochastic models of general utility for the description of seismic catalogues. The rate l(t) of this class of models is given by lðþ¼f t ðþþ t X gt ð t i Þ ð1þ t i<t where f(t) and g(t) are usually given in parametric form and t i are the occurrence times of observations. Such a model has a branching process interpretation. Any single event can be thought of as the parent of a family of later events, its offspring, which ultimately die out. Therefore f (t) isthe rate of events not generated by any previous event whereas g(t) describes the occurrence of offsprings. Early applications to seismic data of self-exciting models are in the works of Hawkes and Adamopoulos [1973] and Kagan and Knopoff [1987], but they have been greatly improved and extended by Ogata [1988, 1998] with formulation of the ETAS (Epidemic Type-Aftershock Sequences) model, where the triggered events (aftershocks) represent the short-term clustering due to the elastic response of the upper layers of the Earth. [15] Until now, the self exciting models have been used in seismology to describe the clustering of earthquake occurrence, because of short-term interaction (aftershock sequences). However, this class of models are based on an intuitive motivation that could be extended to different spatiotemporal scales. Here, we explore the potentialities of this type of modeling by applying it at different steps. Specifically, we adopt the following methodology: we apply the well known ETAS model that allows to remove the shortterm triggered events from the catalog. Then, we re-apply the same (or a very similar) branching model to the remaining seismicity, usually called background, searching for interaction at larger spatiotemporal scales. We focus our attention on larger time scale rather than smaller, because, as mentioned before, the former appears to be more relevant for tectonic earthquakes [see, e.g., Kagan and Jackson, 1991b; Lombardi and Marzocchi, 2007]. [16] Notably, the statistical modeling adopted in the second step allows the quantitative check of the stationary Poisson hypothesis of the background events, that stands behind classical ETAS modeling [Ogata, 1998] and procedures for seismic hazard assessment [Working Group on California Earthquake Probabilities (WG02), 2002 and references therein]. An implicit assumption behind the branching modeling is that possible departures from a constant seismic rate of background activity can be described through interaction among events on longer timespace scales respect the short-term clustering. As we explain later, we carefully test this hypothesis through some goodness of fit tests and some analysis about the statistical significance of the estimated parameters and of the forecasting capability of the model First-Step Branching [17] The first step of our methodology consists of applying the well-known ETAS model proposed by Ogata [1988, 1998] to describe the short-term clustering. Hereinafter, we use the terms first-step branching and short-term clustering as synonyms. ETAS model describes the total seismic rate as the sum of two contributions: the rate of spontaneous events, that refers to activity which is not triggered by precursory events and is forced by external physical processes (background events), and the rate of events internally triggered by stress variations of previous earthquakes. The time evolution of seismic rate triggered by each event is assumed in agreement with the modified Omori-law [Utsu, 1961]. The spatial decay on area surrounding the epicenter of source event is modeled by an isotropic power law function. Therefore the total intensity function of model is given by l 1 ðt; x; y=h t Þ ¼ m 1 ðx; yþþ X t i<t " # K 1 ðt t i þ cþ p a1 Mi Mmin e ð Þ C d1;q 1 ri 2 þ d1 2 q1 where H t ={t i, M i,(x i, y i ), t i < t} is the observation history up time t, M min is the minimum magnitude of catalog, C d1,q 1 is normalization constant of triggering spatial function, and r i is the distance between locations (x, y) and (x i, y i )[Ogata, 1998]. We set m(x, y) =n 1 u 1 (x, y), where n 1 is a positivevalued parameter and u 1 (x, y) is the probability density function (PDF) of locations of spontaneous events. [18] To estimate parameters of the stationary ETAS model (n 1, K 1, c, p, a 1, d 1, q 1 ) we use the iteration algorithm developed by Zhuang et al. [2002]. This is based on the Maximum Likelihood Method [Daley and Vere-Jones, 2003] and on kernel estimation of total seismic rate. Specifically this last is obtained by smoothing the observed seismicity with a Gaussian kernel function K d having a spatially variable bandwidth d (see Zhuang et al. [2002] for details). Together with parameters estimation, this procedure also provides an estimation of PDF u 1 (x, y) for location of spontaneous events. Since some physical investigations show that static stress changes decrease with epicentral distance as r 3 [Hill et al., 1993], we impose q 1 = 1.5. This choice is also justified by the recognized trade-off between parameters q 1 and d 1, that is different pairs of q 1 and d 1 values provide almost the same likelihood of the model [Kagan and Jackson, 2000]. [19] To remove the triggered events from original data set we use a strategy based on the stochastic declustering algorithm proposed by Zhuang et al. [2002]. The technique can be described by following steps: [20] 1. Computation of the probability p i (I) that the ith event is a spontaneous event, given by p ðþ I i ¼ m 1 ðx i ; y i Þ l 1 ðt i ; x i ; y i =H ti Þ and of probability p (I) ij that the ith event is triggered by a previous jth event, given by p ðþ I ij ¼ K 1 e a1ðmj MminÞ C p d1;q 1 t i t j þ c rij 2 þ d2 1 l 1 ðt i ; x i ; y i =H ti Þ q ð2þ ð3þ ð4þ 3of12

4 where r ij is the distance between locations (x i, y i ) and (x j, y j ). [21] 2. Computation of the expected number N of background events, by using the probabilities found in equation (3). [22] 3. Selection of the N earthquakes of the seismic catalog with the highest probability p i (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 earthquake that the algorithm identifies as aftershock because it has been anticipated by a foreshock. The declustered catalog obtained by this procedure represents the seismicity filtered by the first-step branching. Hereinafter we call it background data 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 of Zhuang et al. [2002] produce a set of stochastic realizations of declustered catalog. The main rationales behind the use of a single declustered catalog are two: first, a single catalog allows 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 be easily handled (see last point of the procedure describe above). The use of a single catalog may lead to biased results only if our procedure is not able to capture the main statistical features of the declustered catalog. In order to verify this assumption, in the following analyses we check the sensitivity of the results by using also a set of 1000 stochastic declustered catalogs generated by the original technique of Zhuang et al. [2002] Second-Step Branching [26] The second step of our procedure consists in reapplying a branching process to the background data set obtained in the previous step to describe the long-term clustering. Hereinafter, the terms second-step branching and long-term clustering are considered synonyms. Through this step, we can check if the seismicity of a region, filtered by the short-term clustering, shows statistically significant subtler features that may suggests a longer time clustering or trend. In particular, the model assumes that the time evolution of the background can be described by a second-step branching process, working at larger space-time scales compared to the short-term cluster removed 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 is the elapsed time from the occurrence of the earthquake that generates stress variations, and t is the characteristic time of the interaction. This parameterization is a simplification of the model proposed by Piersanti et al. [1995, 1997] to describe temporal evolution of postseismic stress variations. In such a model, the relaxation is composed by a sum of exponential decays, that mimic different relaxation modes. Our parameterization implies that, for seismic interaction on a time scale covered by a seismic catalog, one relaxation mode is predominant. This prevalent relaxation time t depends on the viscosity of the mantle, for which very different values, ranging from Pas [Pollitz et al., 1998] to Pas[Piersanti, 1999], have been proposed. [28] To describe the spatial decay of stress variation, with distance r from epicenter of perturbing event, we choose an inverse power law PDF. The dependence of hazard function with the magnitude M of exciting event is assumed of exponential type, i.e., proportional to e a 2M. Therefore the time-dependent conditional rate of earthquake occurrence for the second step of our procedure is given by l 2 ðt; x; y=h t Þ ¼ m 2 ðx; yþþ X K 2 e t i<t a2 ð Mi Mmin Þ e ð t t iþ t C d2;q 2 q2 ri 2 þ d2 2 ð5þ This branching model ascribes the occurrence of background events to the superposition of two effects: the tectonic loading, assumed stationary and with a timeindependent spatially variable rate m 2 (x, y) =v 2 u 2 (x, y), and a long-term coupling that we ascribe to postseismic stress transfer produced by each previous earthquake. The parameters of the model are estimated by the same procedure used in step 1 [Zhuang et al., 2002]. Its application needs of estimation of total mean rate ^m(x, y). As in the first step, this is again obtained using Gaussian kernel functions K d with a spatially variable bandwidth d and is given by ^m ðx; yþ ¼ 1 T X j K dj x x j ; y y j where T is the length of the observation period (see Zhuang et al. [2002] for details). [29] As for the first-step branching (see equation (3)), we can define the probability i ¼ m 2ðx; yþ l 2 ðt; x; y=h t Þ p ðiiþ that the ith event is not triggered by any previous shock. This probability can be used to distinguish events due to tectonic loading (^s(x, y)), and to the second-step triggered events (^c(x, y)). These rates are respectively given by ^m 2 ðx; yþ ¼ 1 T ^cðx; yþ ¼ 1 T ð6þ ð7þ X p ðiiþ j K dj x x j ; y y j j X 1 p ðiiþ ð8þ K dj x x j ; y y j j [30] This interpretation requires some important additional remarks. Sornette and Werner [2005] have shown that the implicit (and certainly unjustified) assumption of ETAS models, as well as the branching models used here, that the threshold magnitude is also the minimum magnitude for triggering event leads to a significant bias in estimating the background rate. In brief, this is due to the fact that there could be events in the background that are actually triggered by events with a magnitude below the completeness magnitude threshold. For this reason, in the following we do not interpret the values obtained in an absolute sense (that are certainly biased), but we only focus on the relative varia- j 4of12

5 Table 1. Parameters of the ETAS Model for PS92 and NEIC Learning Data Sets a PS92 (M min = 7.0) NEIC (M min = 6.0) Parameter (629 Events) (3064 Events) n ± 0.3 yr 1 74 ± 2 yr 1 K 1 (4 ± 1) 10 3 yr p 1 (1.0 ± 0.1) 10 2 yr p 1 p 1.1 ± ± 0.01 c (2 ± 1)10 4 yr (4.0 ± 1.0)10 5 yr a ± ± 0.1 d 1 22 ± 4 km 12 ± 0.5 km q a Maximum likelihood parameters of the ETAS model (first-step branching; see equation (2)) for PS92 and NEIC learning datasets. tions of the background and triggered rate in terms of catalog used, branching stage, and spatial distribution. 4. Checking the Goodness of Fit and the Performance of the Model [31] A careful testing of any model is a basic step to decide on whether or not the model provides an adequate description of data. Here, to achieve this goal, we use a procedure consisting of two steps. At first, we set up the model on a subset of the catalog (the learning data set); then, we evaluate the goodness of fit and the forecasting capability of the model on an independent subset of the catalog (the testing data set), which has not considered at any step of modeling. Since the feasibility of short-term clustering has been already tested in the past, also for large earthquakes [Lombardi and Marzocchi, 2007], here we focus our attention on testing the feasibility of the secondstep branching, or, in other words, if the background of the ETAS model is a stationary Poisson process, or if it contains second-step branching structures. At this purpose, the analyses are carried out on the background testing data sets, i.e., the catalog after to have removed the short-term clustering described 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 Z t Z LðÞ¼ t dt 0 dxdyl 2 ðt 0 ; x; y=h t Þ ð9þ T start R where T start is the starting time of observation history. By the time transformation ~t = L(t), the occurrence times t i are transformed into new values ~t i. If the model describes well the temporal evolution of seismicity, the transformed data ~t i are expected to behave like a stationary Poisson process with the unit rate [Papangelou, 1972; Ogata, 1988]. Any deviation from expected Poisson behavior indicates a significant factor underlying the data which is not captured by the model. [33] We test the Poisson hypothesis for transformed times ~t i of the background testing data set, by using two nonparametric statistical tests: the Runs test and the one-sample Kolmogorov-Smirnov test (KS1) [Gibbons and Chakraborti, 2003]. Whereas the Runs test verifies the reliability of the independence of the earthquake occurrence, the KS1 checks the hypothesis that the interevent times are exponentially distributed. [34] As regards the evaluation of the forecasting capability of the model, we compute the information gain per event (IGpe) [Daley and Vere-Jones, 2003] on the background testing data set. IGpe measures the performance of a model H 1 relative to a reference model H 0 and is given by the difference D ln L of log likelihoods of two processes divided by the number of events N IGpe D ln L N ¼ ð ln L 1 ln L 0 Þ N ð10þ The value of IGpe should be relatively negative if the reference model is the best and relatively positive if H 1 is the best performing. Hence a greater IGpe (relative to the null model) describes how much more predictable the fitted model is than the null model. This measurement, basically equivalent to the R-test from Kagan and Jackson [1995], is useful to compare the predictability of various competing models. [35] In our case, H 1 is the second-step branching, and H 0 is the Poisson model, both for background testing data set. In practice, this comparison consists of verifying if the background events, selected after the first step of the model (i.e., after removal of the short-term clustering), have a Poisson distribution or are still clustered. To this purpose, we compare IGpe values obtained by the real background testing data set (from now on IGpe? ), and by two different sets of synthetic catalogs. The first set consists of 1000 Poisson synthetic catalogs (with the same background rate and the same length of the testing data set). The second set is composed by 1000 synthetic catalogs generated by our model. The comparison of IGpe of the real and synthetic catalogs allows the two hypotheses H 0 and H 1 to be tested. Specifically, if the hypothesis to be tested (H 0 or H 1 ) is true, IGpe? can be seen as a random realization of the IGpe obtained by the model under testing; in this case, hypothesis H 0 is rejected if IGpe? is above the 95th percentile of the 1000 values obtained for the Poisson model (one tail test with a significance level of 0.05); the hypothesis H 1 is rejected if the same value is below the 5th percentile of the 1000 IGpe values obtained by the second-step branching model (one tail test with a significance level of 0.05). Finally, in order to verify the sensitivity of the results to the choice of the declustered catalog, we calculate also the IGpe for 1000 declustered catalogs by using the stochastic procedure suggested by Zhuang et al. [2002]. In this way, it is straightforward to verify if possible rejection of the hypotheses mentioned above can be explained by our declustering procedure. 5. Results of the Analysis for PS92 [36] For PS92, we set the learning data set as the part of the catalog in the time interval 01/01/ /31/1979 (629 events), and the testing data set as the part in the time interval 01/01/ /31/1989 (69 events). [37] The first step of our procedure involves estimation of parameters (n 1, K 1, c, p, a 1, d 1, q 1 ) of ETAS model (equation (2)) on learning data sets. The values obtained by the procedure of Zhuang et al. [2002], together with relative errors and maximum likelihood value, are reported in Table 1. As a first remark, it is worth noting that such 5of12

6 Figure 1. Histogram of probability p (I) i of belonging to spontaneous seismicity for events collected into (a) PS92 and (b) NEIC learning catalogs. The values of p (I) i are computed by the ETAS model by using the procedure proposed by Zhuang et al. [2002] (see text for details). parameters are consistent with values computed for single aftershocks sequences in tectonic zones [e.g., Ogata, 1999], corroborating the hypothesis of universality of physical laws driving short-term triggering [Lombardi and Marzocchi, 2007]. [38] Before going to the second-step branching, we remove from original learning data sets the short-term triggered shocks, through the procedure described in the previous section. From Figure 1a, we note that most of events have a probability p i (I) to belong to background close to 0 and 1, and only about 10% of all events have a probability ranging between 0.1 and 0.9. In this case, the expected number of background events according to the procedure of Zhuang et al. [2002] is N = 547. The background learning data set is composed by the N events having the highest value of p i (I). [39] The application of second-step branching model (see equation (5)) on the PS92 background learning data set provides parameters reported in second column of Table 2. The first interesting remark regards the value of relaxation time t, equal to = 36?7 years: it reveals a significant longterm clustering behavior of seismicity, different from the one identified in the first stage of the model. Comparing the log likelihoods (lnl) of Poisson and the double branching model of the background learning data set (see Table 2), we find that the latter is significantly better in light of Akaike Information Criterion (AIC) [Akaike, 1974]. This last is defined by AIC = 2 lnl +2n p, where n p is the number of parameters. The lower value of the AIC identifies the model that better represents the data. We have AIC = for the Poissonian Model and AIC = for the secondstep branching model, revealing a best performance of this last model. [40] Because of the long-lived nature of triggering activity, the seismicity in the learning time interval may be affected by earthquakes which occurred before this period. To take into account this effect, we fit again the second branching model by considering a time interval precursory to the period used to estimate the parameters. Seismicity occurred in this last target period and triggered by earthquakes belonging to precursory interval is taken into computation. Considering the value of relaxation time t previously estimated, we consider 30 years for the precursory learning period ( ; 195 events) and 50 years for the target learning data set ( ; 352 events). In this run we set t 30 years. Results are shown in last column of Table 2. The log likelihood per event points out the better performance of this last model. Hereinafter, we consider these values as the most reliable parameters of our model for PS92 catalog. We stress that the choice of setting t a priori does not entail any bias in fitting the model. If we drop this assumption, the maximum likelihood procedure provides a compatible value of t, but with a much larger uncertainty because of a too short learning data set. Moreover, the difference of maximum log likelihood values, about 5.0, obtained by setting (lnl = ) and optimizing (lnl = ) t, corroborates the negligible influence of this assumption. [41] In Figure 2 we show the histogram of probability p i (II) that the ith event is caused by tectonic loading for target learning data set ( ; see equation (7)). Whereas the analogous histogram of step 1 (Figure 1a) is strongly bimodal, revealing a well-defined identification of spontaneous and short-term triggered events, in this case we have Table 2. Estimated Parameters of the Second-Step Branching Model for the Background Learning PS92 Data Set ( ) a Parameter Poisson Model (547 Events) Second Branching Model (547 Events) (352 Events) n ± 0.3 yr ± 0.2 yr ± 0.4 year 1 K ± ± t 35 ± 7 yr 30 yr a ± ± 0.2 d ± 30 km 140 ± 40 km q ± ± 0.5 Log likelihood per event AIC 19, ,886.4 a Maximum likelihood parameters of Poisson and second-step branchingmodel (see equation (5)) for background learning PS92dataset. To estimate parametersof branching model we do two runs. In the first (second column) we set up the modelby using whole background learning dataset of 80 years ( ). In the second (lastcolumn) we consider a precursory period of 30 years ( ) to the interval timeof 50 years ( ) used to apply the estimating procedure (see text for details).these last parameters (marked in boldface) are used to following testing of the model. 6of12

7 (II) Figure 2. Histogram of the probability p i of being caused by tectonic loading for events belonging to the PS92 background learning target catalog (events that occurred between 1930 and 1980). a more uncertain recognition of long-term triggered effect. About 70% of all events have a probability ranging between 0.1 and 0.9. This result can be due to two reasons: an inefficiency of the model to distinguish for all events the main cause of their occurrence or to an actual joint comparable action of tectonic loading and long-term stress transfer. [42] In Figure 3 we show the learning and testing data sets (a), and the regions in which the triggering effect has a predominant role for seismicity occurred in the target learning data set (b); specifically, we show the map of proportion 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 the spatial variations of the triggered events. In particular, the plot shows that the seismicity due to long-term stress transfer is mostly apparent in the Pacific Ring, in the northern 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 ~t i (see equation (8)) for PS92 testing data set. For both KS1 and Runs test we cannot reject the Poisson model at a significance level of 0.1. As regards the evaluation of the forecasting capability of the model, the results are reported in Figure 4. The graph shows IGpe? (the IGpe of the real background testing data set; vertical dashed line at 1.2), and 1000 IGpe obtained from stochastic declustered real catalogs through the original procedure of Zhuang et al. [2002] (dark gray bars), synthetic stationary Poisson backgrounds (light gray bars), and synthetic backgrounds generated by our model (light gray bars). Both IGpe? and the 1000 IGpe obtained by the declustered catalogs through the procedure of Zhuang et al. [2002] have values larger than the ones relative to the synthetic Poisson background. This test permits to reject the Poisson model at 0.01 significance level. On the other hand, the same values partially overlap the IGpe distribution obtained by our model. In particular, we have that 89% of synthetic values Figure 3. (a) Maps of the seismicity reported in the PS92 catalog. The circles represent the seismicity of the learning (in black) and testing (in red) data sets; the dimension of circles is proportional to the magnitude. (b) Map of the ratio between (bottom) long-term triggered seismic rate ^c(x, y) and total background rate ^m(x, y) for the PS92 background learning target ( ) catalog. 7of12

8 Figure 4. Plot of IGpe for the real PS92 background testing data set (vertical solid line) and for synthetic catalogs obtained by the Poisson model and the branching model. are smaller than IGpe?, indicating that we cannot reject our time-dependent model, at least for a significance level of Results of the Analysis for NEIC [44] For NEIC catalog, we set the learning data set as the subset of events occurred from 01/01/1974 to 12/31/2002 (3064 events); therefore the testing subset is the part of catalog spanning the time interval 01/01/ /31/2006 (526 events). [45] The last column of Table 1 reports the parameters of the ETAS model (first-step branching) n 1, K 1, c, p, a 1, d 1, q 1 (see equation (2)) estimated on learning data set. Also in this case, we stress that such parameters are similar to what found 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 modes close to 0 and 1; only about 10% of all events have a probability p (I) i ranging between 0.1 and 0.9 (see Figure 1b). In this case, the expected number of background events according to the procedure of Zhuang et al. [2002] is N = The background learning data set is composed by the N events having the highest value of p (I) i. [47] The value of parameters (v 2, K 2, t, a 2, d 2, q 2 )for NEIC catalog, obtained by considering 19 years for the precursory period ( ; 1270 events) and 10 years for the learning target data set ( ; 829 events) are reported in Table 3. Considering results obtained by PS92 catalog and the temporal coverage of NEIC target data set, we set t 30 years. In order to check this choice, we have noted 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 of maximum likelihood. Moreover, we anticipate that this choice will be justified by the better forecasting performance on the background testing data set of our model compared to Poisson model. (II) [48] From the histogram of p i for NEIC (Figure 5) catalog, we find that about 20% of events are likely not triggered by any previous earthquake (p (II) i > 0.9), and only about a 5% of events that are very likely due to a long-term interaction (p i (II) < 0.1). In other words for more than 70% of events we cannot clearly decide if they have a tectonically driven or long-term triggered feature (see Figure 5). As for PS92, the long-term stress transfer is predominant in Pacific Ring (see Figure 6). [49] By applying the Runs and KS1 tests on transformed times ~t i (see equation (8)) of NEIC background testing data set, we do not reject the Poisson model (by both test) at 0.05 significance level, corroborating the reliability of our modeling. The test about the forecasting capability of the branching model on NEIC background testing data set shows similar results respect to PS92 catalog (see Figure 7). Here, IGpe? is equal to 0.6. By adopting the same strategy described for PS92 catalog, we see that this value, as well as the IGpe values obtained by 1000 stochastic declustered catalogs by means of the procedure of Zhuang et al. [2002], are larger than all values derived from synthetic Poisson catalogs. This stands for a rejection of the Poisson model hypothesis at a 0.01 significance level. At the same time, the same values overlap well the IGpe distribution obtained by the second-step branching model. In other terms, we cannot reject this model at a significance level of 0.1. [50] Results of all performed tests mean that the background 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 ( ) a Poisson Model Second Branching Model Parameter (2099 Events) (829 Events) n 2 72 ± 2 yr 1 47 ± 3 yr 1 K ± t 30 yr a d 2 45 ± 9 km q ± 0.4 Log likelihood per event a Maximum likelihood parameters of Poisson and second-step branching model for background learning NEIC dataset. To set up the model we consider a precursory period of 19 years ( ) and a target period of 10 years ( ) (see text for details). 8of12

9 Figure 5. Same as Figure 2 but for the background learning target NEIC data set ( ). features of spatiotemporal evolution of seismicity recorded in NEIC catalog. 7. Discussion of the Results [51] The results reported above highlight that the double branching model fits well the data of the NEIC and PS92 catalogs, and that its earthquakes forecasting capabilities on testing data sets (independent from the data used to set the model) are better than single branching model (ETAS). This outperforming of the model has important implications on the physics of the earthquake occurrence process. [52] The most remarkable aspect is the existence of a second-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 seismic background, that stands behind most of the seismic hazard assessment and most of the models for earthquake forecasting [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. Particularly interesting is the characteristic time of such a second-step branching. We find a value of about 30 years, that is higher than the few years/one decade found in recent papers for worldwide and Italian seismic catalogs [Parsons, 2002; Faenza et al., 2003].Wearguethatthisdiscrepancycan be due to the fact that both Parsons [2002] and Faenza et al. [2003] analyze the catalogs with a model that allows a single characteristic time for the clustering; in this case, the ten years might represent the average between the characteristic times of short-and long-term clustering found here. [53] Remarkably, the characteristic time of the secondstep branching is compatible to the post-seismic relaxation [Kenner and Segall, 2000]. For this reason, we argue that post-seismic perturbations could be the most likely physical driving mechanism to explain such a long-term clustering. Notwithstanding, we find a low value for a 2 that is not statistically significant from zero (see Tables 2 and 3), implying the remarkable and unexpected peculiarity that the postseismic triggering capability of an event is independent by its magnitude. This result could raise some doubts on advisability to use branching model to represent the long-term evolution of the background seismic rate. Actually, although a well-set positive value of a 2 would strengthen the reliability of the second-step branching process, the uncertain estimation of a 2 -value is not a sign Figure 6. Same as Figure 3 but for the background learning target NEIC data set ( ). 9of12

10 Figure 7. Same as Figure 4 but for the background testing NEIC data set ( ). 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 data could be not sufficient to test the hypothesis a 2 = 0 in a robust way. In fact, whereas the coseismic stress transfer is a phenomenon spanning all magnitude scales, the postseismic effects are likely mostly caused by the strongest events [Piersanti et al., 1997; Pollitz et al., 1998]. The magnitude range recovered by analyzed catalogs is rather small and the proportion of giant events (M > 8.0), especially for NEIC data set, is negligible. This could be the origin of the large error of a 2 -value. [54] The feasibility of seismic long-term interaction has been largely debated in recent years. Despite the high number of single post-seismic interaction cases reported in literature [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 and Marzocchi, 2005; Piersanti et al., 1995, 1997; Piersanti, 1999; Kenner and Segall, 2000], and numerical models supporting it [Ziv, 2003; Marzocchi et al., 2003b; Kenner and Simons, 2005], this issue can be considered still open, mostly because of the lack of clear robust statistical evidence of this hypothesis. In order to fill this gap, recent efforts have been devoted to forecast the seismic rate changes induced by the postseismic effects of the two giants Andaman-Sumatra earthquakes of [Marzocchi and Selva, 2008] on a large portion of the earth in the next few decades. [55] The spatial distribution of the long-term triggered events makes clear an important contribution of long-term triggering on global seismicity, in most seismogenetic zones (Figures 3b and 6b). Note that these percentages represent lower bounds of the real values [Werner and Sornette, 2005]. Remarkably, the areas with the highest clustering do not overlap the ones with the highest seismic rate; moreover many of these regions have been identified by previous studies on the basis of physical models as single examples of long-term interaction between faults, such as Mexico [Mikumo et al., 2002; Santoyo et al., 2005], California [Pollitz, 1992; Kenner and Segall, 2000; Selva and Marzocchi, 2005], Aleutian and Kurile-Kamchatka trenches [Pollitz et al., 1998], Alaska [Piersanti et al., 1997], Japan [Rydelek and Sacks, 2003], Chile and South Peru [Piersanti, 1999; Casarotti and Piersanti, 2003], and Turkey [Stein et al., 1997; Barka, 1999]. [56] Another interesting issue that comes out from our model is some sort of universality of physical processes behind the elastic interaction. The comparison between the first-step branching (ETAS) parameters of PS92 and NEIC catalogs (see Table 1) does not highlight any significant variation for p and a values. These parameters are the most strictly linked to basic physical features of seismogenetic process [Utsu et al., 1995]. Specifically the a-value measures the efficiency of a shock to generate triggered activity by its magnitude; the parameter p is related to time evolution of triggering effect. Their similarity for PS92 and NEIC catalogs, as well as for values obtained by the analysis of single aftershock sequences [Ogata, 1999], points out that the basic physical features of short-term clustering is almost independent by the time-space-magnitude window considered [Corral, 2004; Lombardi and Marzocchi, 2007]. [57] As regards the other first-step branching parameters, the most remarkable difference is relative to d 1, significantly larger for PS92 catalog. This result, so as the weak differences of K 1 and c, are due to lower threshold magnitude considered for NEIC catalog respect to PS92 data set. The smaller value of d 1 reflects the smaller mean source dimension of NEIC earthquakes and, therefore, the smaller area covered by aftershock clusters embedded in the NEIC data set. The most reliable explanation for larger value of K 1 of NEIC catalog is that the number of events generated by each earthquake, regardless of magnitude of this, becomes larger if we lower the minimum magnitude of triggered events. On the other side, it is well known that the value of c, representing a measure of incompleteness of the catalog in the earliest part of each cluster, decreases at decreasing of threshold magnitude [e.g., Kagan, 2004]. 8. Final Remarks [58] The main goal of this paper has been to outline the features of a double branching process to describe the spatiotemporal distribution of earthquakes. The main advantage of the model is the possibility to account for different spatiotemporal scales of the processes involved. 10 of 12

11 Remarkably, the model is suitable for computing earthquake forecasting in real-time. [59] The model has been applied to two worldwide catalogs with a different time-magnitude coverage (the PS92 and NEIC catalogs). The results obtained put in light two main issues: [60] The background seismicity, obtained by removing the short-term clustering by seismic catalogs, is characterized by a second-step long-term clustering. The characteristic time is compatible with post-seismic relaxation that we propose as the most likely driving mechanism. This feature has a major importance in practice, because it raises several doubts on the feasibility of the stationary Poisson hypothesis that stands behind almost all classical seismic hazard assessment, as well as many reference models for evaluating the performance of earthquake forecasting model or earthquake predictions. 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