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ORIGINAL ARTICLE
Comparison of seismicity declustering methodsusing a probabilistic measure of clustering
Abdelhak Talbi & Kazuyoshi Nanjo & Kenji Satake &
Jiancang Zhuang & Mohamed Hamdache
Received: 2 April 2012 /Accepted: 23 April 2013 /Published online: 24 May 2013# Springer Science+Business Media Dordrecht 2013
Abstract We present a new measure of earthquakeclustering and explore its use for comparing theperformance of three different declustering methods.The advantage of this new clustering measure overexisting techniques is that it can be used for non-Poissonian background seismicity and, in particular, tocompare the results of declustering algorithms wheredifferent background models are used. We use our ap-proach to study inter-event times between successiveearthquakes using earthquake catalog data from Japanand southern California. A measure of the extent ofclustering is introduced by comparing the inter-eventtime distributions of the background seismicity to thatof the whole observed seismicity. Theoretical aspects ofthe clustering measure are then discussed with respect tothe Poissonian and Weibull models for the backgroundinter-event time distribution. In the case of a Poissonian
background, the obtained clustering measure shows adecrease followed by an increase, defining a V-shapedtrend, which can be explained by the presence of short-and long-range correlation in the inter-event time series.Three previously proposed declustering methods (i.e.,the methods of Gardner and Knopoff, Reasenberg, andZhuang et al.) are used to obtain an approximation of theresidual “background” inter-event time distribution inorder to apply our clustering measure to real seismicity.The clustering measure is then estimated for differentvalues of magnitude cutoffs and time periods, takinginto account the completeness of each catalog. Plots ofthe clustering measure are presented as clustering atten-uation curves (CACs), showing how the correlation de-creases when inter-event times increase. The CACsdemonstrate strong clustering at short inter-event timeranges and weak clustering at long time ranges. Whenthe algorithm of Gardner and Knopoff is used, the CACsshow strong correlation with a weak background at theshort inter-event time ranges. The fit of the CACs usingthe Poissonian background model is successful at shortand intermediate inter-event time ranges, but deviatesat long ranges. The observed deviation shows thatthe residual catalog obtained after declustering remainsnon-Poissonian at long time ranges. The apparent back-ground fraction can be estimated directly from the CACfit. The CACs using the algorithms of Reasenberg andZhuang et al. show a relatively similar behavior, with atime correlation decreasing more rapidly than the CACsof Gardner and Knopoff for shorter time ranges. Thisstudy offers a novel approach for the study of differenttypes of clustering produced as a result of varioushypotheses used to account for different backgrounds.
J Seismol (2013) 17:1041–1061DOI 10.1007/s10950-013-9371-6
A. Talbi (*) :K. Nanjo :K. SatakeEarthquake Research Institute, University of Tokyo,1-1-1 Yayoi, Bunkyo-ku, Tokyo 113-0032, Japane-mail: [email protected]
A. Talbie-mail: [email protected]
A. Talbi :M. HamdacheDépartement Etude et Surveillance Sismique,Centre de Recherche en Astronomie Astrophysiqueet Géophysique (CRAAG), BP 63 Route de l’observatoire,Bouzaréah, 16340 Algiers, Algeria
J. ZhuangInstitute of Statistical Mathematics, 10-3 Midori-Cho,Tachikawa, Tokyo 190-8562, Japan
Keywords Seismicity . Correlations . Clustering .
Declustering algorithms . Probability distributions
1 Introduction
Seismicity is characterized by a complex pattern ofevents in space and time (e.g., Utsu 2002). Inparticular, earthquakes exhibit space–time clustering,which is notably enhanced during seismically activeperiods following major events or during a series ofrelated events referred to as “earthquake swarms.”However, it is still problematic to differentiate betweenclustered and background seismicity components due tothe overlap (superposition) of the two components intime and space (e.g., Touati et al. 2009, 2011).
Currently, there is no standard definition of the dif-ferent series of earthquakes, and a working definition isneeded to support the use of the terms “aftershock,”“mainshock,” and “foreshock.” Given this, manyseismological studies have used a series of declusteringalgorithms based on subjective constraints to captureobserved clustering (e.g., Utsu 1969; Gardner andKnopoff 1974; Reasenberg 1985; Frohlich and Davis1990; Davis and Frohlich 1991; Knopoff 2000; see vanStiphout et al. 2012 for a complete review). Theseapproaches are based on constraints derived fromcharacteristic space–time patterns of seismicity follow-ing major events (Omori 1894; Utsu 1969). Seismicitydata following major events are treated with subjectiverules to characterize the clustered seismicity, includingspatial and temporal constraints, the number and size oftriggered events, and constraints linked to the physicalfault model under consideration.
Most of the studies that assume a Poisson distribu-tion of the earthquake catalog, particularly in probabi-listic seismic hazard assessment, use declusteringalgorithms to remove clustering from the original cata-log. The residual catalog is then assumed to reflectPoissonian earthquake occurrences. However, artifactsfrom the use of such residual catalogs have not beenrigorously assessed (Beauval et al. 2006). Furthermore,the process of filtering the original catalog to obtain aPoissonian residual catalog is not unique, but a type of“Poissonianization” rather than “declustering” sensustricto (Luen and Stark 2012).
One successful approach developed in the study ofseismicity clustering is the use of stochastic declusteringtechniques, whereby events are triggered according to a
probability law (Kagan and Knopoff 1976; Zhuang et al.2002, 2004). The algorithm of Zhuang et al. (2004) usesthe standardized intensity rates of triggered andbackground events from the epidemic-type aftershocksequence (ETAS) intensity rate as probabilities for theconstruction of the cluster and background processes,respectively. More recently, non-parametric approacheshave been considered in order to obtain objectivedeclustering algorithms. For example, Hainzl et al.(2006) developed a non-parametric reconstruction ofbackground rates based on the first- and second-ordermean proprieties of inter-event times. This methodcan be used to decluster earthquake catalogs by com-paring the whole and background probability densityfunctions (van Stiphout et al. 2013). Similarly, Marsanand Lengline (2008) used rate weighting to declusterearthquake catalogs using assumptions of a linear trig-gering process and a magnitude-dependent mean fieldresponse to the occurrence of an earthquake. Morerecently, Bottiglieri et al. (2009) used the coefficient ofvariation (ratio of the standard deviation to the mean) ofinter-event times to distinguish periods of pure back-ground activity from coeval cluster periods.
The present study was motivated by the common useof the mean properties of inter-event times in all theseaforementioned approaches. The use of inter-event times,instead of occurrence rates, is motivated by the stabilityof the statistics based on the mean inter-event time, t(defined in Section 3), as compared with those based onthe average occurrence rate (Naylor et al. 2009).
In general, all of the previously described declusteringalgorithms either explicitly or implicitly use a measure ofclustering (or dependence) between successive earth-quakes. However, a comparison between these differentapproaches is not readily possible because different hy-potheses and models are used in each case. This limita-tion motivated us to introduce a clustering measure thatrates each declustering algorithm according to its refer-ence background model, so that the performances of thedifferent algorithms can be evaluated. While a generalclass of clustering measures, such as the Ripley K-function (Ripley 1976) and its variants, already exists,their application is limited because estimation of the K-function is complicated by stationarity and edge effects(e.g., Cressie 1991). Hence, application of this type ofclustering measure to earthquake data is presently lim-ited (e.g., Veen and Schoenberg 2006).
To further contribute to the debate about the meritsof different declustering methods, this study focuses
1042 J Seismol (2013) 17:1041–1061
on scoring the methods using as a reference thehypothesis adopted by each method rather than evalu-ating each method according to the subjective definitionof what is an “aftershock,” “mainshock,” or “foreshock.”As such, we used time clustering at different ranges as themeasure of the relative distance between the empiricalwhole and background inter-event time distributions(Talbi and Yamazaki 2010). Relatively high values ofour measure reflect the presence of clustering in thecorresponding inter-event time range, whereas lowvalues correspond to the dominance of backgroundevents. Our proposed new measure for recognizingclustering offers a number of advantages over existingtechniques, including the ability to theoretically measurethe clustering relative to a hypothetical backgroundmodel under general assumptions and applicability tonon-Poissonian background seismicity. In application,each algorithm is rated by the measure according to thedeviation between its output and the hypothetical back-ground model it assumes. As a result, the proposedmeasure can be used to compare the performance ofdifferent declustering algorithms, even if they usedifferent background assumptions.
This study builds on our former efforts to studyinter-event time distributions (Talbi 2009; Talbi andYamazaki 2009, 2010). In particular, the mixedWeibull model of the inter-event time distribution(Talbi and Yamazaki 2010) is used in estimating theproposed clustering measure.
The remainder of this paper is organized as follows.Section 2 introduces the declustering approaches usedto filter the catalogs from aftershocks, while Section 3describes the mathematical formulations used to de-fine the clustering measure, in addition to discussingthe shape and behavior of the proposed measure.The sampling methodology and description of theearthquake catalogs are presented in Sections 4 and5, respectively, and Section 6 introduces the esti-mation of the proposed clustering measure usingreal catalog data and analysis of clustering attenua-tion curves. Finally, Section 7 reports the mainresults of this study.
It is important to distinguish between the theoreticalaspects and subsequent applications of our work inthis paper. The main part of this work introduces thetheoretical basis of our clustering measure and dis-cusses its shape and behavior using two hypotheticalmodels for background seismicity (Section 3 and theAppendix). The objective of this main part of the
paper was to measure directly the physical clusteringprocess. Hence, this part of the study is detailed and isnot concerned with artifacts arising from the use ofdeclustering algorithms as such algorithms are notemployed. Applications of our clustering measure areconsidered in Section 6, where the use of differentdeclustering algorithms is pivotal in comparing theirperformances.
2 Declustering
In this study, the terms “clustered,” “residual,” and“whole” seismicity correspond to “dependent,”“background,” and “all” events in the catalog, respective-ly. The term “residual” is preferred to “background”when used with reference to the outputs of a declusteringalgorithm. In fact, the subjective algorithms usedfor declustering only provide a catalog that approxi-mately (not rigorously) reflects background seismicity;in particular, testing as to whether the declusteredcatalog is Poissonian should be carefully considered(Luen and Stark 2012). The term “background seismic-ity” is the ideal, unknown, independent structure that weseek to estimate in our clustering study. A preliminaryassessment of the background seismicity structure isobtained by processing with the algorithms of Gardnerand Knopoff (1974), Reasenberg (1985), and Zhuang etal. (2002). These algorithms are based on space–time windows, physical fault models, and ETASmodels, respectively. This multifaceted approach isexpected to minimize artifacts and focus on realand dominant characteristics of earthquake clustering.Additionally, the results may be used to test the perfor-mance of these declustering algorithms. Although over-views of seismicity declustering methods are describedelsewhere (e.g., van Stiphout et al. 2012), this sectionprovides a brief description of each of the declusteringalgorithms used later in this study.
Gardner andKnopoff (1974) introduced a declusteringalgorithm that uses the proximity of earthquakes inspace and time as an indicator of clustering. For agiven catalog of data, events are ordered in decreasingmagnitude. Starting from the first event, space–timewindows are measured around each event in the catalog.The size L and duration T of each window vary with themagnitude M of the potential mainshock. The largestevent in each window is identified as a mainshock, whilethe others (foreshocks or aftershocks) are identified and
J Seismol (2013) 17:1041–1061 1043
removed. The window parameters are estimated usingthe following regressions:
logðTÞ ¼ a1M þ b1 ð1Þ
logðLÞ ¼ a2M þ b2 ð2Þ
For a specific earthquake catalog, the parametersa1, b1, a2, and b2 can be estimated by interpolation ofpast aftershock zone extent and aftershock durationdata. However, most studies use standard values ofthese parameters estimated from, for example, the mag-nitude–length–time data (M, L, and T, respectively)given in Table 1 of Gardner and Knopoff (1974).
The algorithm of Reasenberg (1985) assumes an in-teraction zone centered on each earthquake. Earthquakesoccurring within the interaction zone of a priorearthquake are considered aftershocks. The zone isdynamically modeled with spatial (Rfact) and temporal(τmax) parameters. The length scale Rfact is proportionalto the source dimension, and the temporal scale τmax isdetermined using a heterogeneous Poisson process foraftershocks with rate l(t). Given t>0, the probability ofobserving n earthquakes in the time interval [t, t+C[ isgiven by
P @ t; t þ t½ ½ð Þ ¼ nð Þ ¼ e�lðtÞt lðtÞt½ �nn!
ð3aÞ
with l(t) following the Omori law
lðtÞ ¼ k t þ cð Þ�p ð3bÞℵ is the process that counts the number of after-
shocks occurring in the time interval [t, t+τ[, whereask, c, and p are positive constants representing theOmori law parameters. The waiting time interval, τw,required to observe the next event with probability, P,in a given sequence of aftershocks is
tw ¼ �t log 1� Pð Þ102 Mmax�mc�1ð Þ 3=
ð4Þ
Mmax and mc in this equation are the largest mag-nitude in the sequence and completeness magnitude,respectively. The waiting time is constrained as fallingbetween the minimum and the maximum look-aheadtimes (i.e., τmin≤τw≤τmax).
Figures 1a and 2a show the original seismicity databefore declustering for southern California and Japan,
respectively. The earthquake databases used aredescribed in Section 5. Figures 1b and 2b show plotsof the residual seismicity obtained using the algorithmof Gardner and Knopoff (1974) for southern Californiaand Japan, respectively. Figure 1c presents seismicitydata for southern California filtered using the algorithmof Reasenberg (1985). In Fig. 1c, most of the clusteringstructure is not evident in the filtered data, giving rise tolower spatial density, which is concentrated close toactive faults. In contrast, there is less concentrated andmore diffuse southern Californian seismicity in Fig. 1b.For Japan, the residual seismicity is mostly diffuse whenfiltered using Gardner and Knopoff’s (1974) algorithm(Fig. 2b). The main difficulty in the application ofReasenberg’s (1985) algorithm is adjustment of thealgorithm parameters according to the seismicity of theregion. Typically, a selection of parameters is made afterattempting different runs with varying parameters, andthe various residual catalogs that are obtained from theseruns are examined. Even after undertaking such aprocedure, clustering cannot be adequately removed andbackground seismicity is usually overestimated (Hainzl etal. 2006; Tibi et al. 2011). For this reason, we restrict ourtesting of Reasenberg’s (1985) algorithm to the southernCalifornia data where the original parameters have beenset, and for which a set of suitably chosen parameters hasalready been identified (Hutton et al. 2010).
The stochastic declustering procedure of Zhuang etal. (2004) is based on space–time ETAS models(Ogata 1988) that use the following form of the con-ditional intensity:
l t; x; y;M jHtð Þ ¼ l t; x; yjHtð ÞJ Mð Þ ð5aÞl t; x; y Htjð Þ ¼ μ x; yð Þ þ
Xi:ti<t
k Mið Þg t � tið Þf x� xi;y� yi Mij� �ð5bÞ
l is the conditional intensity on the history ofobservation ℋt until time t, whereas μ, κ(M), g, f,and J are the background intensity, the expected num-ber of triggered events from a magnitude M event, theprobability distribution function of the occurrencetimes of the triggered events, the spatial distributionof the triggered events, and the magnitude probabilitydistribution function, respectively. An importantingredient in Eq. 5b is the modified Omori law, g(Utsu 1969; Yamanaka and Shimazaki 1990).
gðtÞ ¼ p� 1
c1þ t
c
� ��p
ð5cÞ
1044 J Seismol (2013) 17:1041–1061
In the above equation, p and c are the Omori lawparameters defined in Eq. 3b.
The principle behind this algorithm is the classifi-cation of each event j as a triggered or backgroundevent using the following standardized intensity rates,ρj and φj, respectively, obtained from Eq. 5b:
ρj ¼Xi<j
ρi j ð6aÞ
ρij ¼z tj;xj;yjð Þ
l tj;xj;yjjHt j
� � ; j > i
0 otherwise
8<: ð6bÞ
z i tj; xj; yj� � ¼ k Mið Þg tj � ti
� �f xj � xi; yj � yijMi
� �ð7Þ
8j ¼ 1� ρj ð8Þ
This stochastic declustering algorithm simulatesbackground and triggered events from the whole catalogby scanning all events and treating them according tothe probabilities calculated using Eqs. 6–8. Hence, theresulting catalog is not unique because it uses the like-lihood of such probabilities. Figure 2c, d shows theseismicity in Japan before and after applying the
Fig. 1 Epicenter distribution of whole (a) and residual seismic-ity (b, c) in southern California for magnitude M≥2.5 eventswithin the period 1932–2010. The residual catalog is obtainedusing the algorithm of Gardner and Knopoff (1974), where atotal of 16,848 events are identified as non-clustered out of60,092 events in the whole catalog and most of the clustering
structure is eliminated giving rise to lower spatial density, whichis concentrated close to active fault segments (b) andReasenberg’s (1985) algorithm with τmin=5, τmax=150, P=0.90, and R fact=6, where a total of 26,417 events are identifiedas non-clustered and a similar picture of background seismicityis depicted (c)
J Seismol (2013) 17:1041–1061 1045
declustering algorithm of Zhuang et al. (2004).Background seismicity is concentrated along the JapanTrench, whereas it is relatively diffuse at distancesfarther from the Japan Trench. In contrast with thealgorithm of Gardner and Knopoff (1974) (i.e., theseismicity inside the dashed rectangle in Fig. 2a, b),the overall structure of the spatial distribution ofseismicity is preserved after the application of thedeclustering method of Zhuang et al. (2004).
3 Mathematical background
Assume seismicity to occur as a point process in spaceand time (e.g., Cox and Isham 1980; Daley andVere-Jones 1988). Within a finite region in space, theprocess is inferred to follow an inter-event time prob-ability density function f, which is a perturbed form ofthe probability density function fB of the backgroundseismicity component. It is then possible to write thefollowing equations:
f tð Þ ¼ fB tð Þ þ f tð Þ � fB tð Þ½ � ð9aÞ
f tð Þ � fB tð Þ þ " tð Þ ð9bÞ
The variable τ≥0 is the inter-event time and ε(τ)≈f(τ)− fB(τ) is the corresponding deviation from thebackground distribution. The proprieties of ε arediscussed further in part A of the Appendix.
It is important to note that ε is zero in the case of apure background process. For inter-event times τ≥0 with f(τ)>0, the deviation ε(τ) defines a standardizedprobabilistic measure C(τ) of the distance between thewhole and background inter-event time distributionsas follows:
C tð Þ � " tð Þf tð Þ � f tð Þ � fB tð Þ
f tð Þ ð10aÞ
In this case, both fB and ε appear as filtered versionsof the whole seismicity probability density function f,in which C and 1−C represent the role of time-varyingmultiplicative filters.
" tð Þ � C tð Þf tð Þ ð10bÞ
fB tð Þ � 1� C tð Þð Þ f tð Þ ð10cÞ
Accordingly, f can be divided into two componentsobtained by filtering
f tð Þ � 1� C tð Þð Þ f tð Þ þ C tð Þ f tð Þ ð11Þ
Furthermore, dividing Eq. 10a by f(τ) allows one toreduce the scaling effect of the distributions. For ex-
ample, if ef and ef B are the inter-event time probabilitydensity functions scaled using the inter-event timemean t of the whole series, then
t ¼Z þ1
0t f tð Þ d t ð12aÞ
f tð Þ ¼ 1
tef t t=ð Þ ð12bÞ
fB tð Þ ¼ 1
tef B t t=ð Þ ð12cÞ
The corresponding induced measure eC is definedfor scaled inter-event times as
eC t t=ð Þ �ef t t=ð Þ � ef B t t=ð Þef t t=ð Þ
ð13Þ
It then follows that the two measures are equivalentin the sense that
eC t t=ð Þ ¼ C tð Þ ð14Þ
In Eqs. 12b and 13, the mixed inter-event timedistribution, which has been defined for heteroge-neous seismicity (e.g., Bak et al. 2002; Talbi andYamazaki 2010), is used as the scaled inter-event timeprobability density function ef . This function has ap-proximately the same shape for different time periods,magnitudes, and space scales (Talbi and Yamazaki2010). It should be noted that the local inter-eventtime distribution fitted using a gamma distribution(e.g., Fig. 2 of Corral 2003) is different and holds onlyfor stationary periods. It is important to stress this dif-ference because both are confusingly called the “scalinglaw for earthquake inter-event time distribution” bydifferent authors. In our case, the probability densityfunction ef can be analytically approximated usingpreviously published fitting models (Saichev andSornette 2007; Talbi and Yamazaki 2010). As a result of
1046 J Seismol (2013) 17:1041–1061
this, C depends mainly on the mean scaled backgrounddistributionef B and can be calculated using a hypothetical backgroundmodel. In the following analysis, the approx-
imation of C used in application is given by Eq. 33 in the
Fig. 2 Seismicity map for Japan for magnitude M≥3.5 eventsregistered during the period 1923–2010. a Original data includ-ing 87,037 events. b Declustered catalog using the algorithm ofGardner and Knopoff (1974) including 22,317 events. Dashedrectangle shows the area used by Zhuang et al. (2004) to testtheir stochastic declustering algorithm. In our study, the JMA
catalog data used by Zhuang et al. (2004) are extended to 2009and used in schemes J4a-c (see the following c and d). c, dJapan seismicity with magnitudes M≥4.2 registered within theperiod 1923–2009 using the original data including 31,644events (c) and the declustered catalog of Zhuang et al. (2004)including 12,753 events (d)
J Seismol (2013) 17:1041–1061 1047
case of absolutely continuous probability measures P andP(⋅|B) and also in term of frequencies in Eq. 25. Thesedefinitions are equivalent, and it is according to theproperties of the measures P and P(⋅|B) whether to usethe exact expression (Eqs. 30b and 25) or the approxi-mation (Eqs. 33 and 25). It is important to note thatEqs. 25, 30b, 33, and 38 are general and holds for auser-defined background model such as the gamma orthe inverse Gaussian distributions (Talbi and Yamazaki2010; Matthews et al. 2002).
To provide a working definition of C, it isnecessary to define the probability P(B)=P(τ∈B)that an inter-event time τ belongs to the background.For a sufficiently large number of events, this probabil-ity is comparable to the proportion ν of backgroundevents in the whole catalog, which is commonly calledthe background fraction; that is
P ðBÞ � v ð15Þ
For regions where aftershock activity is not domi-nant, a simple biased nonparametric estimate of ν isthe ratio of the first- to second-order moments of theinter-event times, i.e., inter-event time mean over var-iance (Molchan 2005; Hainzl et al. 2006).
It is possible to distinguish two cases, where thefirst is the general case when the cluster (non-back-ground) inter-event time distribution is unknown andthe second is a specific case where it is driven from aspecific probability density function fN. In both cases,the measure C is calculated explicitly.
3.1 Case 1: Cluster inter-event time distributionis unknown
In this case, C can be defined in the neighborhood ofan inter-event time τ0 as
C t0ð Þ ¼ 1� P B t0 � Δt � t < t0 þ Δtjð Þn
ð16Þ
The variables Δτ, ν, and P B t0�jð Δt � t < t0 þΔt:Þ are a positive time increment, the backgroundfraction, and the posteriori probability of backgroundinter-event times, respectively. The derivation ofEq. 16 is detailed in part B of the Appendix.
The posteriori conditional probability in the formerequation describes approximately the proportion oflocal background inter-event times from all inter-
event times close to τ0, within the range Iτ0 definedby t0 �Δt � t < t0 þΔt. This probability is sub-sequently referred to here as the local backgroundfrequency (LBF). LBF is estimated from the numberof background inter-event times τ within the range Iτ0divided by the total number of inter-event times in thesame range. Given that ν is the mean backgroundfrequency on the whole inter-event time scale, themeasure C(τ0) compares LBF and ν locally aroundτ0. For a given ν, 1−C(τ0) is proportional to LBF, sothat low C(τ0) values correspond to high LBF and viceversa. C(τ0) is zero when the local background fre-quency is equal to the mean ν, whereas it is positivewhen LBF is below the mean and negative when itexceeds the mean. Conversely, C(τ0) tends towardunity for very low background frequencies at the dis-tribution tails. C(τ0) cannot be defined if no inter-eventtime is registered in the vicinity of τ0 (i.e., whenP t0 �Δt � t � t0 þΔtð Þ ¼ 0Þ.
In the following discussion, C has been calculatedanalytically and plotted for a set of inter-event timemodelsto assess its behavior. For this purpose, the use of the exactexpression of C obtained from Eq. 30a is preferred overthe use of the approximation given by Eq. 33 (part B ofthe Appendix). As previously mentioned, the analyticalexpressions of the cumulative distribution function for allthe inter-event times and its corresponding probabilitydistribution function, which are required in Eqs. 30a and33 to calculate P t0 �Δt � t < t0 þΔtð Þand f t0ð Þ,respectively, can be found in, for example, Saichev andSornette (2007) and Talbi and Yamazaki (2010). In thisstudy, Poissonian and non-Poissonian models are testedas background.
Figure 3a shows a plot of two analytical fits of theinter-event time probability distribution function f againstthree exponential distributions with different means μB,symbolizing the background distribution fB. For f, weconsider the mixed Weibull distribution MixWeib(c, p,k, θ) with c=0.2, p=1.5, k=0.5, and θ=0.7 (Eqs. 14, 28,54, and 55 of Talbi and Yamazaki 2010).
f ðxÞ ¼ μ1
θ Γ 1þ 1k
� � Q1ðxÞe� x θ=ð Þk þ 1
μDQ2ðxÞ 1
xþ cð Þ1þa
" #ð17aÞ
Q1ðxÞ ¼ 1� ca
1� að ÞμDxþ cð Þ1�a þ c
1� að ÞμD
� �kθ�kxk�1
þ 2ca
μD
1
xþ cð Þa ð17bÞ
1048 J Seismol (2013) 17:1041–1061
Q2ðxÞ ¼ a ca 1� 1
θ Γ 1þ 1k
� � Jk;θðxÞ !
ð17cÞ
Jk;θðxÞ ¼Z x
0e� t θ=ð Þk d t ð17dÞ
μD ¼ aca
1� atmax þ cð Þ1�a þ c1þa
tmax þ cð Þa � c1�a
a 1� að Þð17eÞ
The variables μ and τmax are equal to 1 and 103, respec-
tively, while α is linked to p as α=1−1/p, 0<α<1.Alternatively, we use a rational interpolation to
mimic the behavior of inter-event times at short andlong ranges, h tð Þ ¼ 1þθnt�1�θ
1þt2þc , with n=0.9, θ=0.03,and χ=0.7 (Eq. 76 of Saichev and Sornette 2007).Figure 3b shows the plots of the probability distribu-tion function f against a set of Weibull distributionswith different parameters. In most cases, departurebetween the whole and the background probabilitydistribution functions, f and fB, respectively, isobserved at the distribution tails, while the two proba-bility distribution functions are relatively similar aroundthe mean t. The deviation between f and fB observed inFig. 3a, b is studied using the measureC in the followingtwo examples.
3.1.1 Example 1: Poisson background
In this example,
C t0ð Þ ¼ 1� e�t0 μB= eΔt μB= � e�Δt μB=� �
F t0 þ Δtð Þ � F t0 � Δtð Þ ð18Þ
The derivation of Eq. 18 is detailed in part C of theAppendix. Although the analytical form of F can becalculated in this case (Talbi and Yamazaki 2010), itssubstitution into Eq. 18 is not carried out here for thesake of simplicity.
The calculated C measures are plotted in Fig. 4a fordifferent μB values. For the mixed Weibull model, Cdecreases gradually to reach its minimum for inter-event times close to the background mean μB. Incontrast, for the interpolating function h, C decreasesirregularly to a minimum value that deviates from thebackground mean. At short inter-event times, C hasvalues around unity, which correspond to very lowproportions of background inter-event times comparedwith the overall series. At long inter-event times, Cincreases rapidly to unity. This behavior is clearlylinked to the steep decrease in the exponential proba-bility distribution function observed in Fig. 3a and isan unsurprising result given that the models for fmimic real data with a non-exponential Weibull orpower law-like long tail (Talbi and Yamazaki 2010).
Fig. 3 Plots of the probability distribution function f of earth-quake inter-event times using a mixed Weibull distributionMixWeib(c, p, k, θ) with c=0.2, p=1.5, k=0.5, and θ=0.7 (Eq.15 of Talbi and Yamazaki 2010), shown as a black solid curve, and
the rational interpolation h tð Þ ¼ 1þθnt�1�θ
1þt2þc with n=0.9, θ=0.03,
and χ=0.7 (Eq. 76 of Saichev and Sornette 2007), shown in
magenta. The distribution f is coupled with plots of the exponentialdistribution with different means μB=2, 1, and 0.5, symbolizingPoissonian background occurrences (shown in red) (a), and plotsof the Weibull distribution with different scale and shape param-eters k=2 and 0.5, θ=2, 1, and 0.5, symbolizing Weibull back-ground occurrences (shown in blue and red) (b)
J Seismol (2013) 17:1041–1061 1049
Figure 3a clearly supports the presence of long-rangeclustering in real data. The two maxima in C visible forthe interpolating function h with μB=1 (solid magentacurve) are caused by the failure of h to account for thebehavior of inter-event times near the mean (Talbi 2009).
3.1.2 Example 2: Weibull background
In this example,
C t0ð Þ ¼ 1� e� t0�Δtð Þ θ=½ �k � e� t0þΔtð Þ θ=½ �k
F t0 þ Δtð Þ � F t0 � Δtð Þ ð19Þ
The derivation of Eq. 19 is detailed in part D of theAppendix. Although the analytical form ofF is provided
in Talbi and Yamazaki (2010), it is not reproduced herefor the sake of simplicity.
The calculated C measures are plotted in Fig. 4b, cfor different values of the parameters k and θ. In the casek=2, which corresponds to the Rayleigh distribution, Cis plotted in Fig. 4b. Its behavior is quite similar toFig. 4a, but with a sharper and deeper V-shaped behav-ior. The minimum C value for all curves is quite close tothe scale parameter θ. Given that θ and μB are compa-rable (θ=0.5, 1, and 2 correspond to μB=0.44, 0.89, and1.77, respectively), the minima are also close to themean background inter-event time μB. The casefor k=0.5 is shown in Fig. 4c. For τ≤1, C is quitestable, with values close to zero, particularly for themixed Weibull distribution (black curves). This type of
Fig. 4 Theoretical plots of the clustering measure C calculatedusing the mixed Weibull distribution (in black) and the rationalinterpolation h (in magenta). a Dotted, solid, and dashed curvescorrespond to an exponential background distribution with
mean μB=0.5, 1, and 2, respectively. b, c The curves correspondto a Weibull background distribution with scale parameter θ andshape parameter k=2 (b) and k=0.5 (c). Dotted, solid, anddashed curves correspond to θ=0.5, 1, and 2, respectively
1050 J Seismol (2013) 17:1041–1061
behavior reflects the dominance of the background pro-cess at short ranges, with a relatively constant LBF closeto the mean νwithin each inter-event time bin. At longerinter-event times (i.e., τ≥1), it is possible to distinguishtwo cases corresponding to θ<1 and θ≥1. In the firstcase, where θ=0.5, which corresponds to μB=1, C isquite stable, varying around zero for a wide range ofinter-event times (dotted curves). This stability reflectsthe contiguity of the probability distribution functionsfB and f and the fact that fB (the stretched exponen-tial distribution) is successful in capturing the long-term behavior of the whole inter-event time series. Inthe second case, where θ≥1 (i.e., θ=1 and 2, whichcorresponds to μB=2 and 4, respectively), C de-creases to a minimum in the case of the linearinterpolation h. The minimum C value is muchhigher than the mean μB value in this case. Thedifferences between the background and the wholedistributions at short and long time ranges indicatetime clustering, whereas the minimum C valueattained at around the mean background inter-eventtime μB identifies the maximum observed LBF, andhence the dominance of inter-event times from back-ground seismicity.
Finally, it is important to note the typical V-shapedbehavior observed in Fig. 4a, b. For very large μB
values that are much greater than the mean t, Creaches a minimum outside the observable range; con-sequently, C appears to decrease along the whole inter-event time range and the V-shaped behavior cannot beobserved.
3.2 Case 2: Cluster inter-event time distributionis known
In this case, the probability distribution function f canbe separated into two components as follows:
f tð Þ ¼ 1� nð Þ μμN
fN tð Þ þ nμμ
B
fB tð Þ ð20Þ
The variables ν, μB, μN, and μ are the back-ground fraction (Eq. 15), the mean inter-event timeof the background, and the clustered and wholeseismicity components, respectively. fB and fN arethe conditional probability distribution functions ofthe background and clustered seismicity compo-nents, respectively. The derivation of Eq. 20 isdetailed in part E of the Appendix. The proportion
ν can be estimated from the normalization conditionimposed by weights in Eq. 20 as follows:
1� nð Þ μμN
þ nμμB
¼ 1 ð21Þ
Subsequently, for μ>0 and μB−μN>0,
n ¼ μB
μμ� μN
μB � μN
� �ð22Þ
Combining Eqs. 10a, 20, and 22, an expression forthe approximation of C can be derived as
C tð Þ � 1� 1μB�μμB�μN
� �fD tð ÞfB tð Þ þ μ�μN
μB�μN
� � ð23Þ
This expression can be simplified to
C tð Þ � 1� 1
1� bð Þ fN tð ÞfB tð Þ þ b
;
with b ¼ μ� μN
μB � μN
ð24Þ
Equation 24 shows that for a given β, C tracks thesame variations (increase and decrease with τ) as theratio of cluster to background probability distributionfunctions.
4 Inter-event time sampling and inference
The approach adopted in this study uses earth-quake random sampling (ERS) to produce inter-event time series. The algorithm performs an iter-ative sampling by computing seeds from a networkof local disks with radius R in each run (Talbi andYamazaki 2010). R is increased with larger mag-nitudes when sampling is judged to be poor.Subsequently, the obtained time series are mixedand scaled using the arithmetic mean inter-eventtime t obtained from the whole series. In additionto this approach being able to reveal an approxi-mately unique shape of the inter-event time distri-bution (e.g., Bak et al. 2002; Christensen et al.2002; Corral 2007), such a scaling procedure provides astandard dimensionless reference where the terminology
J Seismol (2013) 17:1041–1061 1051
“short term” versus “long term” can be used relative tothe mean value. “Short term” is used in reference tointer-event times with t and “long term” is used inreference to inter-event times with t >> t. The unique-ness of the scaled inter-event time distribution f fordifferent regions and completeness magnitudes was pre-viously used in the theoretical calculation of C inSection 3. In this section, the same scaled distributionis used for consistency.We also use the mean inter-eventtime scaling because it is more stable than the meanevent rate scaling. It should be noted that the occurrenceof an extreme event adds a large number to the sum ofpast events rates and a small amount to the sum of pastinter-event times (Naylor et al. 2009).
The degree or effect of clustering is calculated usingthe measure C as defined in Eq. 10a. As the independentseismicity structure is unknown, fB cannot be estimatedreliably. However, the measure C allows us to constrain,at least in part, how time correlation between events islinked. Even in the case where the declustering methodfails to remove all correlations, C should be able toidentify at least large and small deviations between thewhole and background distributions along differentinter-event timescales.
In Section 6, C is computed using the inter-eventtime frequencies fr and frB, corresponding to the em-pirical whole distribution obtained from the originaldata and the residual empirical distribution obtainedusing a chosen declustering approach, respectively.The frequencies fr and frB are calculated given a par-ticular inter-event time bin I; then, for each inter-eventtime τ∈I, C(τ) is approximated as
C tð Þ � 1� frBfr
; fr > 0 ð25Þ
Since broad timescales are involved (from secondsand minutes to years), an appropriate bin size along theinter-event time axes can be obtained by considering thelogarithmic scale (e.g., Bak et al. 2002) or an exponen-tially increasing series ci, where i is the label of consec-utive bins and c>1 (e.g., Corral 2004, 2005). In thisstudy, we set c=1.5; other choices give similar results.
5 Earthquake data
Earthquake catalog data from southern California andJapan were used in this analysis. For southern California,
the catalog for the period 1932–2010 was compiled fromthe Southern California Seismic Network (http://www.data.scec.org/ftp/catalogs/SCSN/). The region be-tween latitudes 32°–37° N and longitudes 114°–122° Wwas considered (Fig. 1a). Seismicity in southernCalifornia is dominated by mainshock–aftershock se-quences. Dense clusters are formed by aftershocks con-centrated on mainshock rupture zones, in addition toindividual swarms. Some of the largest earthquake se-quences recorded were the 1952 Mw=7.5 Kern County,the 1992Mw=7.3 Landers, and the 1999Mw=7.1 HectorMine earthquakes, which all ruptured slow-slippingfaults east of the San Andreas Fault. These events in-duced long-lasting aftershocks that continue today,which are distributed over tens of kilometers within thecorresponding mainshock epicentral zones (Hutton et al.2010). The catalog also includes smaller but still damag-ing earthquakes such as the 1933Mw=6.4 Long Beach,the 1971 Mw=6.7 San Fernando, and the 1994 Mw=6.7Northridge earthquakes. Hutton et al. (2010) showed thatthe magnitude of completeness of the SCSN catalogexceeds roughly 3.2 and 1.8, starting from 1932 and1981, respectively, whereas it reaches 1.0 within the coreof the network coverage area.
In our case studies, the magnitudes of complete-ness, the corresponding time periods, together with thenumber of events and depths are shown in Table 1 forboth southern California and Japan. The letters SC andJ in the first column refer to southern California andJapan, respectively. In the case of southern California,the magnitude of completeness and their correspond-ing time periods (Table 1) are taken from Talbi andYamazaki (2009). These magnitudes of completenessfor southern California are higher than those given byHutton et al. (2010).
For Japan, we used the Japan MeteorologicalAgency (JMA) catalog covering the period 1923–2010 and within the region between latitudes 24°–50° N and longitudes 122°–152° E. This datasource is a modern seismic network operated andmaintained by the JMA. Most seismicity is typi-cally concentrated along the Japan Trench in thesubduction zone between the Pacific and Eurasianplates (Fig. 2a). A recent study by Nanjo et al.(2010) showed that relatively low completenessmagnitudes, mc≥1.9, characterize the mainland re-gion, which is consistent with the high density ofseismic stations in this region; in contrast, com-pleteness magnitudes are high in offshore areas
1052 J Seismol (2013) 17:1041–1061
where few seismic stations are present. As is thecase for southern California, the magnitudes ofcompleteness listed in Table 1 for Japan, excludingscheme J4, were derived in our former studyencompassing time periods up to 2005 (Talbi andYamazaki 2009). Here, the time periods in Table 1of Talbi and Yamazaki (2009) were extended to2010. In addition to the study of Talbi andYamazaki (2009), the schemes J1a–b, J2a–b, J3a–b have magnitudes of completeness and time pe-riods comparable with the magnitudes of complete-ness derived by Nanjo et al. (2010). Scheme J4corresponds to the JMA catalog data used in thestudy of Zhuang et al. (2004), but with the timeperiod extended in our study to 2009. Scheme J4covers the region with latitudes 30°–46° N andlongitudes 128°–148° E. This scheme is only usedin this study to test the stochastic declusteringalgorithm.
6 Results and analysis
The measure C is computed and plotted for each sam-pling scheme listed in Table 1, and then the usefulnessof the results in the characterization of clustering and
background processes, and in identifying differencesbetween declustering algorithms, are discussed. For thisdiscussion, different declustering algorithms are appliedto reduce possible artifacts and then C is computed ineach case, combined with its corresponding samplingscheme as given in Table 1. Only major and persistenttrends are used for the characterization of the clusteringand background seismicity structures. Although the in-terpretation of the results is sensitive to artifacts, it stillclearly distinguishes between the clustering behaviorsproduced by the different declustering techniques.
Given that C typically decreases with increasinginter-event times, it is convenient to introduce the term“clustering attenuation curves” (CACs) in reference tothe behavior ofC. Figure 5a, b shows the CACs obtainedfor Japan and Southern California seismicity, respective-ly, using the algorithm of Gardner and Knopoff (1974).The last column of Table 1 identifies the schemes andparameters used in each figure. In Fig. 5a, the CACs forshallow depth events (d<100 km) are plotted with redmarkers, and the theoretical fit of C is shown using themixedWeibull distribution as the whole distribution f andan exponential background distribution fB with meanμB=4. This fit suggests a background fraction of about25 %, which is comparable with the ratio of 22,317independent events out of 87,037 total events identified
Table 1 Parameters of the sampling schemes
Scheme Period mc Depth N R (km) Figure Dec
SC1 1990–2010 2.5 All 27,693 50 Fig. 5b, d GK, R
SC2 1947–2010 3.5 All 6,308 50 Fig. 5b, d GK, R
SC3a 1932–2010 4.5 All 781 50 Fig. 5b GK
SC3b 1932–2010 4.5 All 781 100 Fig. 5d R
J1a 1990–2010 3.5 All 47,686 50 Fig. 5a GK
J1b 1990–2010 3.5 <100 km 39,718 50 Fig. 5a GK
J2a 1975–2010 4.5 All 12,741 50 Fig. 5a GK
J2b 1975–2010 4.5 <100 km 10,985 50 Fig. 5a GK
J3a 1923–2010 5.5 All 3,864 100 Fig. 5a GK
J3b 1923–2010 5.5 <100 km 3,180 100 Fig. 5a GK
J4 1926–2009 4.2 <100 km 31,644 50 Fig. 5e Z
Period, mc, N, R, and Dec denote the start and end year of events considered in the scheme, the magnitude of completeness, the numberof all events with magnitudes M≥mc, the ERS sampling radius, and the declustering method used, respectively. These parameters aregrouped and labeled according to the sampling schemes defined in the first column. SC and J correspond to southern California andJapan, respectively. GK, R, and Z correspond to the declustering algorithms proposed by Gardner and Knopoff (1974), Reasenberg(1985), and Zhuang et al. (2004), respectively. The last two columns (Figure and Dec) list the figures in this paper that correspond to theuse of each of the sampling schemes and declustering methods. For example, for SC1, Fig. 5b corresponds to the GK algorithm,whereas Fig. 5d corresponds to the R algorithm
J Seismol (2013) 17:1041–1061 1053
by the declustering algorithm. An increase in C is notobserved in the real data because of the shortage of dataat long range, t > 10tð Þ. The data in Fig. 5b show high
fluctuations at long range and cannot be adequatelyfitted as in Fig. 5a. However, it is possible to considerthe best fit to the data with the models presented in
1054 J Seismol (2013) 17:1041–1061
Section 3. Data in Fig. 5b are fitted in a similarfashion to Fig. 5a, but with μB=2 and 3 as dashedand dotted curves, respectively. This model suggestsa background fraction ranging between 33 and 50 %,although only about 25 % of the data are identified asbackground by the declustering algorithm (i.e.,16,848 events out of 60,092 total events). This in-consistency between the real data and the fitted mod-el for t > t can be explained by the deviation of theresidual events identified by the declustering algo-rithm from the assumed Poissonian behavior. Let usshow that the behavior of C for t > t can bereproduced using a mixed exponential–Weibull back-ground distribution, i.e., fB~ pr Exp μð Þ þ 1� prð ÞWeib k; θð Þ; 0 � pr � 1. The solid line in Fig. 5bcorresponds to a mixed exponential–Weibull back-ground distribution with pr=0.45. It fits quite wellthe data for t > t. The whole data set is the best fittedby an exponential background with μB=3 (dottedcurve) for t � t and the mixed exponential–Weibullbackground distribution (solid curve) for t > t. Aconsistent way to test point process fitting is to usediagnostic methods applicable to point processeswith known conditional intensity (Stoyan andGrabarnik 1991; Lawson 1993; Baddeley et al.2005, 2008, 2011; Zhuang 2006). However, in ourcase, these methods are not applicable because theconditional intensity is not defined. Alternatively, to
test the fitting models in Fig. 5b, we use the meansquare residual function defined as
RMS tð Þ ¼ 1
n� m
Xi=xi�t C xið Þ � bC xið Þ
� �2ð26Þ
C and bC are the measure predicted by the models inFig. 5b and the measure calculated from Eq. 25 for theschemes in Fig. 5b, respectively. The variables n andm are the sample size and the number of free param-eters in the model, respectively. In our case, we sup-pose f fixed and count the number of free parametersin the model for fB. Therefore, m=1 when fB is expo-nential and m=4 when fB follows the mixed exponen-tial–Weibull model. From the RMS plot in Fig. 5c, weconfirm that the best fit is obtained using an exponen-tial background distribution with μB=3 (red markers)for t � t, while for t > t, the use of the mixed expo-nential–Weibull background distribution (blackmarkers) provides the best fit to our data. In Fig. 5b,the slight increase of C visible at long range forscheme SC2 supports the validity of the theoreticalfit. It should be noted that, in general, C trends to zeroaround the mean inter-event time t in the case ofshallow depth events (red markers in Fig. 5a), andotherwise at shorter times (black markers in Fig. 5a,b). Exceedingly small values of C indicate that LBF isreaching the mean ν (LBF≈ν). For t � 0:1t, C(τ) hasvalues around 1, indicating a very low backgroundfrequency. The clustering component is expected tobe dominant because of the very low backgroundfrequency registered at a short inter-event time range.For t � t in Fig. 5a and t � 0:1t in Fig. 5b, C(τ)decreases rapidly to values less than −1, whichcorresponds to a background inter-event time fre-quency more than twice the whole inter-event timefrequency. In this case, the clustering is expected tobe low and to decrease with increasing LBF. Bothpositive and negative C values indicate the pres-ence of clustered pairs of events in the correspond-ing time ranges.
In order to test the consistency of the former obser-vations, C has been calculated using the algorithms ofReasenberg (1985) and Zhuang et al. (2004) for south-ern California and Japan, respectively. Figure 5d is agraph of the CACs for southern California calculatedusing Reasenberg’s (1985) declustering algorithmwith the parameters τmin=5, τmax=150, P=0.90, and
Fig. 5 Plots of the clustering measure C calculated using thealgorithm of Gardner and Knopoff (1974) for Japan (a) andsouthern California (b). Red markers in (a) show C calculatedfor shallow (d<100 km) events. Dashed and dotted curves in (b)show theoretical fits calculated using the mixed Weibull distri-bution as the whole inter-event time distribution and an expo-nential background distribution with μB=2 and 3, respectively.Solid curves in (a) and (b) show the fits calculated using themixed Weibull distribution as the whole inter-event time distri-bution and an exponential background distribution with μB=4and a mixed exponential−Weibull distribution, respectively. cPlot of the residual mean squares, RMS, corresponding to the fitin (b). Blue, red, and black markers correspond to the residualsof the dashed, dotted, and solid curves in (b). d Plots of theclustering measure C calculated using the algorithm ofReasenberg (1985) for southern California using the parametersτmin=5, τmax=150, P=0.9, and Rfact=6. Solid line shows loga-rithmic fits to the data between 10�2t and t, whereas the 95 %limiting curves are shown by dashed lines. e Plots of theclustering measure C calculated using the algorithm of Zhuanget al. (2004) for Japan seismicity with magnitudesM≥4.2 withinthe period 1923–2009. The solid curve shows the mean C curveobtained from 100 runs, whereas the dashed curves show the95 % confidence limits of the mean curve
R
J Seismol (2013) 17:1041–1061 1055
Rfact=6. These values are slightly below the middlerange of acceptable values used by Hutton et al.(2010) for the SCSN catalog. This choice of parame-ters accounts for low-magnitude events since thedeclustering algorithm is applied to M≥2.45 eventsinstead of M≥2.95 events, as used by Hutton et al.(2010). The same periods and magnitude of complete-ness used in Fig. 5b are considered with a samplingradius R=100 km instead of R=50 km for M≥4.5events. A linear fit with 95 % confidence limits showsthat C decreases logarithmically for 10�2t � t � t.After the mean, C appears to be relatively constant, withvalues between approximately −0.8 and −0.6. This be-havior is partly a function of the use of Reasenberg’s(1985) algorithm for aftershocks (assuming a non-homogeneous Poissonian with an Omori law rate, Eqs.3a and 3b), which results in a power law inter-event timedistribution (Utsu et al. 1995; Shcherbakov et al. 2005,2006; Yakovlev et al. 2005). Figure 5d shows the meanCAC obtained using 100 stochastically declustered cat-alogs with corresponding 95 % confidence limit curves.This figure is highlighted for two reasons. Firstly, theCAC demonstrates the clustering in the ETAS model,which is one of the most successful approaches previ-ously used to model seismicity. Secondly, the stochasticdeclustering method does not use any subjective param-eters, but objectively estimates them from the observedseismicity. In Fig. 5e, the plot of C shows three differenttypes of behavior. A rapid logarithmic decrease in C thatis similar to that observed in Fig. 5d is evident fort � 0:1t. C then shows a slower decrease for0:1 t � t � t. At long ranges, a stationary-like regimeappears to exist, but it is masked by marked fluctuationsin C. As is the case in Fig. 5d, the behavior of C for inter-event times t � 0:1t is linked to the Omori law hypoth-esis for cluster events in the ETAS. This illustrates theregular disruption of the clustering structure and thegradual settling of a relative increase in backgroundinter-event time frequency. LBF reaches the mean ν ataround t � 0:1t as C trends to zero.
Finally, the CACs plotted in Fig. 5a, b can bemodeled quite successfully using the mixed Weibullmodel with exponential background distribution, withthe exception of the discrepancy observed in Fig. 5b atlong ranges, which is fitted using a mixed exponential–Weibull background distribution. The need of suchmixed background distribution can be explained by thefailure of the declustering algorithm to closely model
Poissonian residual inter-event time distribution. Thecorrelation identified using our clustering measurevaries with the declustering algorithm used. While thealgorithm of Gardner and Knopoff (1974) shows astrong correlation with a weak background componentat short ranges (Fig. 5a, b), time correlation decreasesmore rapidly at short term and saturates at long rangeswith an almost constant LBF when the algorithms ofReasenberg (1985) and Zhuang et al. (2004) are used(Fig. 5d, e). This discrepancy can be explained by thebasic assumptions used by each of the algorithms.Reasenberg (1985) and Zhuang et al. (2004) assumean Omori law non-homogenous Poisson process foraftershocks (Eqs. 3b and 5c), whereas Gardner andKnopoff (1974) do not use such assumption. The algo-rithm of Gardner and Knopoff is removing all eventsthat occur close in time around each event (i.e., diggingtime holes), resulting in the overestimation of clustering(aftershocks), especially at short time ranges.
Finally, it is interesting to note the typical behavior ofC in both Fig. 5d, e. At short range, Omori law holds foraftershocks worldwide and, thus, constitutes a bench-mark hypothesis. The Omori law aftershock model ismore than a simple hypothesis; it is an empirical lawfitting broad aftershock sequences worldwide. If webelieve the Omori law, C should typically decrease reg-ularly (almost linearly) on the semi-logarithmic scalegoverned by the aftershocks that constitute most of theclustered events. At long range, C should be more stableor vary slowly. Figure 5a, b relative to the algorithm ofGardner and Knopoff are more fitting our model withPoissonian background, but are missing important infor-mation about the clustered events that is the Omori law.In consequence, the algorithm of Gardner andKnopoff isdeficiently overestimating the clustering at short rangeand underestimating the clustering at long range. Thefact that the different behaviors of our measure are linkedto the assumptions made for the aftershock models ineach algorithm is not a limitation since our measure candeal with any available hypothetical aftershock model.
7 Concluding remarks
The first objective of this study was to introduce aclustering measure for seismicity by assuming a givenbackground seismicity model. The clustering measureis simply defined as the standardized difference
1056 J Seismol (2013) 17:1041–1061
between the whole and background inter-event timefrequencies. The proposed clustering measure doesnot make any preconceived assumptions about thenature of the clustered seismicity, but instead describesit using a simple clustering attenuation curve. In par-ticular, the Omori law assumed by the methods ofReasenberg (1985) and Zhuang et al. (2004) inducesa linear decrease in the CACs (Fig. 5d, e). The pro-posed clustering measure is then used in the analysisof earthquake clustering at different inter-event time-scales and for the interpolation of the results to describeclustered and background seismicity. Our results showthat the clustering measure successfully quantifies localperturbations of seismicity produced by time clustering.In particular, it distinguishes between a strong clusteringat short inter-event time ranges versus a weak and lessapparent clustering at long time ranges that is dependenton the declustering algorithm used.
The second objective was to use our clusteringmeasure to compare the results obtained with threedifferent declustering methods. The algorithms ofReasenberg (1985) and Zhuang et al. (2004), whichassume an Omori-type aftershock decay, show thattime clustering decreases more rapidly as a powerlaw decay at short inter-event times and saturates atlong inter-event times. These different behaviors arelinked to the assumptions made for the aftershockmodels in each algorithm. The algorithm of Gardnerand Knopoff (1974), which assumes a finite space–time range of aftershocks, shows a strong time clus-tering with weak background seismicity at short inter-event times. If we believe the Omori-type decrease forthe number of events after the mainshock, the algo-rithm of Gardner and Knopoff (1974) only removesevents close in space and time, resulting in theoverestimation of clustering at short range and under-estimation of clustering at long range.
In addition to the characterization of earthquaketime clustering over different inter-event time ranges,our proposed clustering measure is particularly usefulin the detection of long-term clustering. Future workusing this approach could be extended to the study ofspace clustering and the combination of space and timeclustering analysis to construct a stochastic declusteringstrategy.
Acknowledgments This work was supported by a fellowshipfrom the Japanese Society for the Promotion of Science. The authors
are grateful to the Japanese Meteorological Agency and the South-ern California Seismic Network for providing the catalog data usedin this study. The authors thank Rodolfo Console from INGV, ananonymous reviewer and the editor Torsten Dahm for theircomments, which improved an earlier version of the manuscript.
Appendix
A. Proprieties of the deviation ε
Earthquake clustering is typically observed after theoccurrence of intermediate- to high-magnitude events.A series of events called “aftershocks” that occur closelyin space and time are triggering phenomena. Our generalhypothesis assumes that the overall pattern of seismicitycan be described by a stable seismicity componentoverprinted by local perturbations that reflect clusteringin space and time. The stable component is linked tosome background process reflecting regional constraintsthat are mainly tectonically driven seismicity, whereasthe local perturbations produce clustering through thetime-dependent relaxation of the Earth’s crust.
The deviation ε has the noise propriety that sums tozero.
E C tð Þð Þ ¼Z þ1
0CðsÞf ðsÞ d s ¼ 0 ð27Þ
These perturbations contribute to the observation ofclustering at different time ranges, with the clusteringeffect or extent calculated using the measure C asdefined in Eq. 10a.
A simple case where the deviation ε can be calcu-lated is the case of a simple mixed distribution be-tween background and cluster events. In such a case,Eq. 20 holds and f can be written as
f tð Þ ¼ bfB tð Þ þ 1� bð ÞfN tð Þ ð28aÞ
With α defined as in Eq. 24,
f tð Þ ¼ fB tð Þ þ 1� bð Þ fN tð Þ � fB tð Þ½ � ð28bÞ
Subsequently, ε can be derived fromEqs. 9b and 28b.
" tð Þ � 1� bð Þ fN tð Þ � fB tð Þ½ � ð29ÞEquation 29 shows that ε is proportional to the differ-
ence between the two component distributions fN and fB.
J Seismol (2013) 17:1041–1061 1057
B. Derivation of Eq. 16
In terms of probability, the measure C can be definedin the vicinity of an inter-event time τ0 as follows:
C t0ð Þ ¼ 1
P t0 � Δt � t < t0 þ Δtð Þ P t0 � Δt � t < t0 þ Δtð Þ � P t0 � Δt � t < t0 þ Δt Bjð Þ½ � ð30aÞ
Where Δτ>0 is a time increment for whichP t0 �Δt � t < t0 þΔtð Þ > 0. Simplifying Eq. 30agives
C t0ð Þ ¼ 1� P t0 � Δt � t < t0 þ Δt Bjð ÞP t0 � Δt � t < t0 þ Δtð Þ ð30bÞ
Equation 30b gives the exact expression C(τ0) thatis used in parts C and D of the Appendix.
If the probability P and its conditional form P � Bjð Þare absolutely continuous and the increment Δt issmall enough, then two probability distribution func-tions, f and fB, exist such that
P t0 � Δt � t < t0 þ Δtð Þ � f t0ð ÞΔt ð31Þ
P t0 � Δt � t < t0 þ Δt Bjð Þ � fB t0ð ÞΔt ð32Þ
In this case, the following approximation isobtained:
C t0ð Þ � 1� fB t0ð Þf t0ð Þ ; f t0ð Þ > 0 ð33Þ
Equation 33 is equivalent to Eq. 10a for τ=τ0.In practice, C(τ0) is estimated using the whole andbackground inter-event time frequencies in the vi-cinity of τ0 instead of f(τ0) and fB(τ0). In fact, ourclustering measure simply compares the back-ground and whole frequencies along the inter-event time axes.
Alternatively, a useful expression of C(τ0) can becalculated in terms of the background fraction ν. Theconditional probability in Eq. 30b is equivalent to
P t0 � Δt � t < t0 þ Δt Bjð Þ ¼ P B t0 � Δt � t < t0 þ Δtjð ÞP t0 � Δt � t < t0 þ Δtð ÞPðBÞ ð34Þ
Finally, the expression of the measure C in Eq. 16 isobtained from Eqs. 15, 30b, and 34.
C. Calculation of C in the case of Poissonianbackground seismicity (Eq. 18)
This example is important because it should mimic Ccalculated using real residual catalogs, which are as-sumed to be Poissonian. In this case, the backgroundinter-event time distribution is exponential. fB∼Exp(μB), with a probability distribution function fBand a cumulative distribution function FB, respectively,written as
fB tð Þ ¼ 1
μBe�t=μB ð35aÞ
FB t0ð Þ ¼ 1� e�t0μB ð35bÞ
The variable μB is the mean inter-event time forbackground events.
The probabilities in Eq. 30b are calculatedusing the cumulative distribution functions F andFB of the whole and background inter-event times,respectively.
P t0 � Δt � t < t0 þ Δt Bjð Þ¼ FB t0 þ Δtð Þ � FB t0 � Δtð Þ ð36Þ
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P t0 �Δt � t < t0 þΔtð Þ¼ F t0 þΔtð Þ � F t0 þΔtð Þ ð37ÞSubstituting Eqs. 36 and 37 into Eq. 30b yields
C t0ð Þ ¼ 1� FB t0 þΔtð Þ � FB t0 þΔtð ÞF t0 þΔtð Þ � F t0 þΔtð Þ ð38Þ
Finally, by using Eq. 35b, the expression of themeasure C is that given in Eq. 18.
D. Calculation of C in the case of Weibull backgroundseismicity (Eq. 19)
The Weibull distribution (Weibull 1951) is able tocapture the long-term behavior of inter-eventtimes. For example, it has been used in a numberof recurrence time models for large earthquakes(e.g., Newman et al. 2005; Yakovlev et al. 2006;Turcotte et al. 2007; Zoller and Hainzl 2007). It isused here as a model for the background inter-event time distribution. In this case, the probabilitydistribution function fB and cumulative distributionfunction FB are
fB tð Þ ¼ kθ�kxk�1e� x θ=ð Þk ð39aÞ
FB t0ð Þ ¼ 1� e� t0 θ=ð Þk ð39bÞ
The variables k and θ>0 are the shape and scaleparameters, respectively. The Weibull distribution isnoted hereafter Weib(k,θ), and we write fB∼Weib(k,θ)in reference to its density fB defined in Eq. 39a. Themean μB of the Weibull distribution is a function of theWeibull parameters.
μB ¼ θ Γ 1þ 1
k
� �ð40Þ
Using Eq. 39b, the following conditional probabil-ity is calculated:
P t0 �Δt � t < t0 þΔt Bjð Þ
¼ e� t0�Δtð Þ θ=½ �k � e� t0þΔtð Þ θ=½ �k ð41Þ
Similarly to the former case, C can be calculatedusing the whole inter-event cumulative distributionfunction F as given in Eq. 19.
E. Derivation of Eq. 20
If we assume that seismicity can be separated intobackground and clustered components both describedby corresponding independent processes, the probabil-ity P(τ1>τ) that the time to the next event from a fixedorigin 0 exceeds a given time τ>0 can be calculatedusing the total probability theorem. This is given bythe sum of the probabilities for the two components.
P t1 > tð Þ ¼ P t1 > t Bjð Þ P ðBÞþ P t1 > t Njð Þ P ðNÞ ð42Þ
B and N correspond to the following events:
B: “A background event occurs.”N: “A clustered event occurs.”
P t1 > t Bjð Þ and P t1 > t Njð Þ are the conditionalprobabilities of the forward recurrence time τ1 on theevents B and N, respectively (i.e., the probabilities thatstarting from an arbitrary time, there occur no back-ground events or no cluster events, respectively, in thefollowing time interval of length t).
In this case, P(B) is the former background fractiondefined in Eq. 15.
PðBÞ ¼ n; PðNÞ ¼ 1� n ð43ÞApplying the linear derivation operator to both
sides of Eq. 42 gives
dP t1 > tð Þd t
¼ ndP t1 > t Bjð Þ
d tþ 1� nð Þ
� dP t1 > t Njð Þd t
ð44Þ
The derivatives in Eq. 44 can be obtained using Palm–Khintchine equations (Cox and Isham 1980; Daley andVere-Jones 1988). This also holds for the background andclustered and whole seismicity components.
dP t1 > t Bjð Þd t
¼ � 1� DB tð ÞμB
ð45aÞ
dP t1 > t Njð Þd t
¼ � 1� DN tð ÞμN
ð45bÞ
J Seismol (2013) 17:1041–1061 1059
dP t1 > tð Þd t
¼ � 1� D tð Þμ
ð45cÞ
The substitution of the derivatives defined in Eqs. 45a,45b, and 45c into Eq. 44 yields
1� D tð Þμ
¼ 1� nð Þ 1� DN tð ÞμN
þ n1� DB tð Þ
μBð46Þ
The variables D, μ, DB, μB, DN, and μN are thewhole seismicity distribution, its mean inter-eventtime, the conditional distribution, and the mean inter-event time of the background and cluster series ofevents, respectively. Finally, the following analyticalexpression of the distribution D is obtained:
D tð Þ ¼ 1� μμN
1� nð Þ 1� DN tð Þð Þ þ μμB
n 1� DB tð Þð Þ�
ð47ÞEquation 20 then follows by derivation.
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