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Stochastic Declustering of Space-Time EarthquakeOccurrencesJiancang Zhuang, Yosihiko Ogata and David Vere-JonesJiancang Zhuang is a graduate student, Department of Statistical Sciences,Graduate University for Advanced Studies, 4-6-7 Minami-Azabu, Minato-ku, 106-8569Tokyo, Japan. Yosihiko Ogata is Professor, Institute of Statistical Mathematics, 4-6-7Minami Azabu, Minato-ku, 106-8569 Tokyo, Japan. David Vere-Jones is EmeritusProfessor, School of Mathematics and Computing Sciences, Victoria University ofWellington, New Zealand. The study was initialized while the first author was avisitor to School of Mathematical and Computing Sciences, Victoria University ofWellington, supported by International Cooperation Fund of the China SeismologicalBureau and an Earthquake Forecasting subcontract with Institute of Geological andNuclear Sciences in New Zealand. It was continued with the support of grant-in-aid 11680334 for the Scientific Research from the Ministry of Education, Science,Sport and Culture, Japan. This study was also supported by a research fund ofYoshiyasu Tamura. The authors thank Ma Li for her continuous encouragement andsupport. The authors also gratefully acknowledge the assistance of Ray Brownrigg,David Harte, Makoto Taiji, and Koichi Katsura, as well as the helpful comments andencouragement from the referees and the editor.

Available online: 31 Dec 2011

To cite this article: Jiancang Zhuang, Yosihiko Ogata and David Vere-Jones (2002): Stochastic Declustering of Space-Time Earthquake Occurrences, Journal of the American Statistical Association, 97:458, 369-380

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Stochastic Declustering of Space-TimeEarthquake Occurrences

Jiancang Zhuang, Yosihiko Ogata, and David Vere-Jones

This article is concerned with objective estimation of the spatial intensity function of the background earthquake occurrences from anearthquake catalog that includes numerous clustered events in space and time, and also with an algorithm for producing declusteredcatalogs from the original catalog. A space-time branching process model (the ETAS model) is used for describing how each eventgenerates offspring events. It is shown that the background intensity function can be evaluated if the total spatial seismicity intensity andthe branching structure can be estimated. In fact, the whole space-time process is split into two subprocesses, the background events andthe clustered events. The proposed algorithm combines a parametric maximum likelihood estimate for the clustering structures using thespace-time ETAS model and a nonparametric estimate of the background seismicity that we call a variable weighted kernel estimate. Todemonstrate the present methods, we estimate the background seismic activities in the central region of New Zealand and in the centraland western regions of Japan, then use these estimates to produce catalogs of background events.

KEY WORDS: Background seismicity; Branching processes; Earthquake declustering; Space-time ETAS model; Thinning method;Variable weighted kernel estimate.

1. INTRODUCTION

The motivation of this article arises from problems in theprediction of large earthquakes with clusters of aftershocks.The earthquake clusters complicate the statistical analysisof the background seismic activity that might be related tochanges in the tectonic � eld. The cluster features differ fromplace to place and typically lie between two extreme types ofspatial clusters of earthquakes: those that eventually decreasein time, such as aftershock sequences and swarms, and thosethat persist in time at the same location. Ultimately, the persis-tent background activity prevails over the aftershock activity.To forecast the location of the large earthquakes, it is neces-sary to analyze the background seismicity, for which removalof temporal cluster members is considered to be of centralimportance.

A number of methods have been proposed for decluster-ing a catalog to obtain the background seismicity. Most ofthese methods remove the earthquakes in a space-time win-dow around a large event called the mainshock; the differ-ences among these methods relate to the choice of the windowsizes and some other detailed procedures (see, e.g., Utsu 1969;Gardner and Knopoff 1974; Kellis-Borok and Kossobokov1986). In general, the bigger the magnitude of the mainshock,the larger the window. An alternative to the window-baseddeclustering methods is the link method based on a space-timedistance between the events (e.g., Resenberg 1985; Frohlichand Davis 1990; Davis and Frohlich 1991).

Jiancang Zhuang is a graduate student, Department of Statistical Sciences,Graduate University for Advanced Studies, 4-6-7 Minami-Azabu, Minato-ku,106-8569 Tokyo, Japan (E-mail: zhuangjc@ism.ac.jp). Yosihiko Ogata is Pro-fessor, Institute of Statistical Mathematics, 4-6-7 Minami Azabu, Minato-ku,106-8569 Tokyo, Japan (E-mail: ogata@ism.ac.jp). David Vere-Jones is Emer-itus Professor, School of Mathematics and Computing Sciences, Victoria Uni-versity of Wellington, New Zealand. The study was initialized while the � rstauthor was a visitor to School of Mathematical and Computing Sciences, Vic-toria University of Wellington, supported by International Cooperation Fundof the China Seismological Bureau and an Earthquake Forecasting subcon-tract with Institute of Geological and Nuclear Sciences in New Zealand. Itwas continued with the support of grant-in-aid 11680334 for the Scienti� cResearch from the Ministry of Education, Science, Sport and Culture, Japan.This study was also supported by a research fund of Yoshiyasu Tamura. Theauthors thank Ma Li for her continuous encouragement and support. Theauthors also gratefully acknowledge the assistance of Ray Brownrigg, DavidHarte, Makoto Taiji, and Koichi Katsura, as well as the helpful comments andencouragement from the referees and the editor.

All of the aforementioned conventional methods containarbitrary parameters for de� ning the aftershock window sizesor the link distances in both space and time. Each choice ofparameter gives a different declustered catalog and thus a dif-ferent estimate of the background seismicity. As a result, theconventional declustering algorithms have dif� culty makingan optimal choice of parameters. For a dataset from a speci� c(rather narrow) seismic region and a given threshold magni-tude, these choices are usually colored by the user’s subjectiveimpression of the declustered output. Also, given a dataset,the concept of what constitutes an aftershock has not beenuniquely de� ned. These are among the reasons why so manydifferent declustering algorithms have been proposed.

To avoid such dif� culties, it seems to us inevitable that thedeclustering method should be based on a stochastic model toobjectively quantify the observations, so that any given eventhas a probability to be either a background event or an off-spring (cluster) event generated by others. Therefore, the maintask of a declustering method is to estimate this probabil-ity for each event. Several authors have proposed space-timemodels for the way in which earthquakes generate clusteredevents. To � t their models to Italian historical data, Musmeciand Vere-Jones (1992) assumed that the background seismic-ity was proportional to a kernel estimate obtained from thespatial locations of all earthquakes that include clusters. Ogata(1998) suggested a space-time version of the epidemic-typeaftershock sequence (ETAS) model. He used a conventionaldeclustering algorithm for preliminary estimation of the back-ground rate by � tting bicubic B-spline functions before � ttingthe space-time ETAS model to all the events. However, as weshow in this article, the use of such a preliminary declusteringmethod can be avoided.

In the next section we give a formulation of the space-timepoint process for earthquake occurrences and then investigatesthe relationship between the total spatial rate, the backgroundrate, and the branching structure of this model. This rela-tionship suggests an approach to objective methods for esti-

© 2002 American Statistical AssociationJournal of the American Statistical Association

June 2002, Vol. 97, No. 458, Applications and Case Studies

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mating the background intensity. We discuss several techni-cal methods for carrying out such estimation, including thethinning (random deletion) method and a variable bandwidthkernel function method associated with maximum likelihoodestimation of the space-time ETAS model. We then obtain thebackground intensity using an iterative algorithm and gener-ate the declustered catalogs. Finally, we apply this procedureto datasets from the central region of New Zealand and thewestern and central Honshu area of Japan to demonstrate ourmethods.

2. SPACE–TIME MODELS FOREARTHQUAKE OCCURRENCE

Several space-time point-process models have been pro-posed for describing clustering phenomena in seismic activity(Kagan 1991; Musmeci and Vere-Jones 1992; Rathbun 1993;Ogata 1998). These models have common features that can beoutlined as follows:

(a) The background events are regarded as the “immi-grants” in the branching process of earthquake occurrence;their occurrence rate is assumed to be a function of spatiallocation and magnitude, but not of time.

(b) Each “ancestor” event produces offspring indepen-dently. The expected number of direct offspring from an indi-vidual ancestor is assumed to depend on its magnitude M andis denoted by Š4M5.

(c) The probability distribution of the time until the appear-ance of an offspring event is a function of the time lag fromits direct ancestor and is independent of magnitude (cf. Utsu1962); thus its probability density function is assumed to havethe form g4t—’5 D g4t ƒ ’5, where ’ is the occurrence time ofthe ancestor. Moreover, the function is independent of whathappens between ’ and t.

(d) The probability distributions of the location 4x1 y5 andmagnitude M of an offspring event are dependent on themagnitude M ü and the location 4�1 ‡5 of its direct ances-tor. These probability density functions are denoted by f4x ƒ�1 y ƒ ‡—M ü 5 and j4M —M ü 5, where �, ‡ represents the loca-tion and M ü represents the magnitude of the ancestor.

In general, this class of marked branching point processesfor earthquake occurrences can be represented completely bythe conditional intensity function (e.g., Daley and Vere-Jones1988, chap. 13) de� ned by

Pr©an event in 6t1 t C dt5 � 6x1 x C dx5

� 6y1 y C dy5 � 6M1M C dM5 — ¨t

ª

D ‹4t1 x1 y1M — ¨t5 dt dx dy dM

Co4dt dx dy dM51 (1)

provided that it exists, where ¨t denotes the space-time mag-nitude occurrence history of the earthquakes up to time t.Note that here ¨t is the history of the available earthquakerecords including not only those during the study period andin the study area, but also those before the study period (e.g.,Ogata 1992) and outside of the study area (e.g., this article).In particular, we include information about large earthquakes

from outside the study area if they contribute substantially tothe seismicity of the study region during the study period.

Based on the assumptions a–d, the conditional intensityfunction for the space-time model can be written as

‹4t1 x1 y1M — ¨t5 D Œ4x1 y1M5 CX

8k2 tk<t9

Š4Mk5g4t ƒ tk5

� f 4x ƒ xk1 y ƒ yk—Mk5j4M —Mk50 (2)

In this equation, Œ4x1 y1M5 is the background intensity func-tion and is assumed to be independent of time. The functionsg4t5, f 4x1 y—Mk5, and j4M —Mk5 are the normalized responsefunctions (i.e., pdf’s) of the occurrence time, location, andmagnitude of an offspring from an ancestor of magnitudeMk . From the fact that the kth event excites a nonstationaryPoisson process with intensity function Š4Mk5g4t ƒ tk5f 4x ƒxk1 y ƒ yk

—Mk5j4M —Mk5, it is easy to see that Š4Mk5 is theexpected number of offspring from an ancestor of size Mk.Note also that the probability function g4t5 is independent ofthe magnitude of the ancestor, as mentioned in assumption c.

To simplify the subsequent discussion, we also make thefollowing stronger assumption:

(e) The magnitude distribution of a background eventis independent of its location, so that Œ4x1 y1M5 DŒ4x1 y5jŒ4M5; the magnitude distribution of a direct off-spring event is independent of the size of its ancestor, sothat j4M —M ü 5 D j4M5; and the magnitude distributions for thebackground events and their offspring are identical, that is,jŒ4M5 D j4M5.

Under these conditions, the conditional intensity function forthe model can be decomposed as

‹4t1 x1 y1 M — ¨t5 D j4M5‹4t1x1 y — ¨t51 (3)

where

‹4t1 x1 y — ¨t5 D Œ4x1 y5 CX

8k2 tk<t9

Š4Mk5g4t ƒ tk5

� f4x ƒ xk1 y ƒ yk—Mk50 (4)

Later, in applications, we make use of the speci� c functionforms

Š4M5 D Ae�4MƒM05

g4t5 D(

4p ƒ 15cpƒ14t C c5ƒp for t > 0

0 otherwise1

and

f 4x1 y—M5 D 1

2� de�4MƒM05exp ƒ 1

2

x2 C y2

de�4MƒM05

as an example. These constitute one form of the space-timeETAS model (Ogata 1998; see also Rathbun 1993 for a similarform).

Given an estimated intensity function u4x1y5, we set

Œ4x1 y5 D �u4x1y5

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Zhuang, Ogata, and Vere-Jones: Stochastic Declustering 371

for the background rate in (4), where � is a positive-valuedparameter, and maximize the log-likelihood function

logL4È5 DNX

kD1

log‹È4tk1 xk1 yk— ¨tk

5

ƒZ T

0

ZZ

S

‹È4t1 x1 y — ¨t5 dx dy dt (5)

to obtain the maximum likelihood estimates (MLEs), OÈ D4 O�1 bA1 O�1 Oc1 Op1 Od5, where the subscript k runs over all of theevents occurring in the study region S and the study timeinterval 601T 7. Computational details have been given byOgata (1998).

Now, assuming stationarity and ergodicity of the space-timepoint process, the total spatial intensity function (� rst-ordermoment density) is equal to

m14x1 y5 D limT !ˆ

1T

Z T

0‹4t1 x1 y — ¨t5 dt1 (6)

where T is the length of the observation period. Replacing thelimit in (6) by a � nite approximation and substituting (4) in(6), we obtain

m14x1 y51T

Z T

0Œ4x1 y5 C

X

8k2 tk<T 9

Š4Mk5g4t ƒ tk5

� f 4x ƒ xk1 y ƒ yk—Mk5 dt

D Œ4x1 y5 C 1T

X

8k2 tk<T 9

Š4Mk5f 4x ƒ xk1 y ƒ yk—Mk5

�Z T

tk

g4t ƒ tk5 dt

D Œ4x1 y5 C 1T

X

8k2 tk<T 9

Š4Mk5

� f 4x ƒ xk1 y ƒ yk—Mk50 (7)

Therefore, an estimator of the background intensity can bederived from

Œ4x1 y5 m14x1 y5 ƒ 1T

X

8k2 tk<T 9

Š4Mk5f 4x ƒ xk1 y ƒ yk—Mk51

(8)

where both the total rate m14x1 y5 and the transfer functionŠ4M5f 4x1 y—M5 can be estimated.

3. THINNING PROCEDURE

To obtain a practical version of (8), we consider the fol-lowing thinning operation (or random deletion). Consider aseries of events 84tj1 xj1 yj 52 j D 1121 : : : 1N 9 associated witha series of probabilities 8�j 2 j D 1121 : : : 1N 9. Suppose thateach event j of this process is deleted with probability �j .Then the remaining events represent a new point process,called the thinned process. The simplest form of the thin-ning operation is to delete each point with a � xed probability(e.g., Daley and Vere-Jones 1972). The thinning method has

also been used for the simulation of point processes (see e.g.,Lewis and Shedler 1979; Ogata 1981, 1998; Musmeci andVere-Jones 1992).

Here we interpret �j as the probability of the jth earthquakebeing an offspring in the process, de� ned as follows. In thesecond term in (4), we set

�i1jD Pr 8the jth event is an offspring of ith event—¨tj

9

D Š4Mi5g4tjƒ ti5f 4xj

ƒ xi1 yjƒ yi

—Mi5

‹4tj1 xj1 yj— ¨tj

5(9)

and

�jD Pr 8the jth event to be an offspring—¨tj

9

Djƒ1X

iD1

�i1j1 (10)

where �j µ 1 because it represents the ratio of the sum termin (4) to the whole of (4). Thus the probability that the jthevent belongs to the background is

�jD Pr 8the jth event is an immigrant — ¨tj

9

D 1ƒ �jD

Œ4xj1 yj—¨tj

5

‹4tj1 xj1 yj— ¨tj

50 (11)

If we delete the jth event in the process with probability �j

for all j D 1121 : : : 1N , then the thinned process should real-ize a nonhomogeneous Poisson process of the spatial intensityŒ4x1 y5 (see Ogata 1981 for the justi� cation). We call this pro-cess the background subprocess and call the complementarysubprocess the cluster subprocess or the offspring process.

4. VARIABLE KERNEL ESTIMATESOF SEISMICITY

Estimating the mean rate m14x1 y5 of the total seismic activ-ity in (8) can be carried out by several methods, includingusing kernel estimates (e.g., Vere-Jones 1992) or spline func-tions (e.g., Ogata and Katsura 1988). Here we adopt the ker-nel estimate method. The simple kernel estimate with a � xedbandwidth has a serious disadvantage, however. For a spatiallyclustered point dataset, a small bandwidth gives a noisy esti-mate for the sparsely populated area, whereas a large band-width mixes up the boundaries between the densely populatedand sparsely populated areas. Therefore, instead of the kernelestimate

Om14x1 y5 D 1T

NX

jD1

kd4x ƒ xj1 y ƒ yj51 (12)

where kd4x1 y5 denotes the Gaussian kernel function

kd4x1 y5 D 12� d

exp ƒ x2 C y2

2d2

with a � xed bandwidth d, we adopt

Om14x1 y5 D 1T

NX

jD1

kdj4x ƒ xj 1 y ƒ yj 51 (13)

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where dj represents the varying bandwidth calculated for eachevent j in the following way. Given a suitable integer np

between 10 and 100, � nd the smallest disk centered at thelocation of the jth event that includes at least np other earth-quakes and with a radius larger than some small value (e.g., adistance within 002 degree, which is of the order of the loca-tion error), and let its radius be dj (e.g., Silverman 1986, chap.5). Similar ideas have been given by Choi and Hall (1998)and Musmeci and Vere-Jones (1986).

Using the same kernel function kdjas in (13), the occur-

rence rates of the cluster and background subprocesses canthen be estimated by

Oƒ4x1 y5 D 1T

Xj

�jkdj4x ƒ xj1 y ƒ yj51 (14)

where �j is derived from (10) and

OŒ4x1y5D Om14x1y5ƒ Oƒ4x1y5D 1T

Xj

41ƒ�j5kdj4xƒxj1yƒyj50

(15)

We call the estimates in (14) and (15) the variable weightedkernel estimates.

There remains the problem of optimal selection of np forthe variable bandwidth for the statistics in (13), (14), and (15).Remembering that the clustering intensity term on the rightside of (8),

IT 4x1 y5 D 1T

X

j

Š4Mj5f 4x ƒ xj1 y ƒ yj—Mj51 (16)

is also an image of the clustering intensity in space, but not sosmoothed, we can in principle select a suitable np for Oƒ4x1 y5

by minimizing the discrepancy between the low-frequency(smoothed) component of IT 4x1 y5 in (16) and Oƒ4x1 y5 in (14).In practice, the estimates are rather insensitive to the choiceof np , so that a rough estimate is generally suf� cient. Atthe same time, the mean rate Om14x1 y5 is also compared toR T

0O‹4t1 x1 y—¨t5 dt=T in view of (7), where 601T 7 is the obser-

vation time interval.

5. ITERATION ALGORITHM

In Sections 2–4, we outlined methodologic solutions to thefollowing issues: (a) how to obtain the MLE of the space-timeETAS model for the branching structure when the backgroundrate is given, (b) how to estimate the total spatial occurrence

Table 1. Convergence Steps of the Evaluation Procedure in Applying the Space-Time ETAS Modelby Algorithm 1 to the Earthquake Data From the Central New Zealand Region

log L � A c � p d

1 ƒ7987049 043594 .43378 .024737 .73554 1.1573 039395e ƒ 022 ƒ7716051 1025850 .31482 .016264 .93397 1.1671 019980e ƒ 023 ƒ7710075 095684 .33944 .017698 .88535 1.1638 023097e ƒ 024 ƒ7696038 1000945 .33370 .017370 .89776 1.1644 022326e ƒ 025 ƒ7695060 099755 .33492 .017441 .89518 1.1643 022487e ƒ 026 ƒ7694069 1000011 .33465 .017426 .89574 1.1643 022452e ƒ 027 ƒ7694088 099955 .33471 .017429 .89562 1.1643 022459e ƒ 028 ƒ7694084 099966 .33470 .017428 .89565 1.1643 022458e ƒ 02

rate, and (c) how to estimate the background seismicity in thecase where the branching structure is provided. But in real-ity we know only the earthquake data—namely, the records ofoccurrence times, locations, and magnitudes of the observedevents. To estimate the background rate from these data only,we carry out the following iterative algorithm that simultane-ously estimates the background rate and the branching struc-ture:

Algorithm 1

1. Given a preliminary parameter np , say 20, calculatethe bandwidth dj for each event 4tj1 xj1 yj1Mj5, j D11 21 001N .

2. Set l D 0 and u4054x1 y5 D 1.3. Using the maximum likelihood procedure described by,

for example, Ogata (1998), � t the conditional intensityfunction

‹4t1 x1 y—¨t5 D �u4l54x1 y5 CX

k2 tk<t

Š4Mk5g4t ƒ tk5

� f 4x ƒ xk1 y ƒ yk—Mk5 (17)

to the earthquake data, where k, g, and f are de� nedafter (4).

4. Calculate �j from (9) and (10) for each j D 1, 21 : : : 1N .5. Calculate OŒ4x1 y5 from (15) and record as u4lC154x1 y5.6. If max4x1y5

—u4lC154x1 y5 ƒ u4l54x1 y5— > ˜, where ˜ is asmall positive number, then set l D lC1 and go to step 3.Otherwise, take O�u4lC154x1 y5 as the background rate andstop.

The np is tuned so as to � nd the value that can minimizethe difference between the low-frequency part of IT 4x1 y5 andOƒ4x1 y5. In this study we � nd that the adjustment of np is notso sensitive for the � nal estimates of Om14x1 y5 and Oƒ4x1 y5,comparing to

R T

0O‹4t1x1 y5 dt and IT 4x1 y5. The estimates only

change slightly when np changes in the range of 15–100. Inthis article we adopt np

D 20 as a standard value for use withthe New Zealand and Japanese data.

This algorithm converges quickly. Table 1 shows anexample of the convergence steps of this algorithm when thespace-time ETAS model in (4) was applied to the earthquakedata from the central New Zealand region, which is discussedin the next section.

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Zhuang, Ogata, and Vere-Jones: Stochastic Declustering 373

6. APPLICATIONS

6.1 Central New Zealand Region

The data used in this calculation are from the New Zealandlocal catalog recorded by the Institute of Geology and NuclearSciences. For this study, we select the earthquake data for theperiod January 1970–August 1999 from the rectangular area38�–43�S and 171�–179�E (the central New Zealand region),which extends from the boundary of the Bay of Plenty toArthur’s Pass (Fig. 1). We take the magnitude threshold ML

D400 and consider shallow events down to the depth of 40 km.

The � nal MLE values of the space-time ETAS modelfor this subregion are bA D 033470 event/(deg2� day), O� D0895651 Oc D 0017428 day, Op D 101643, and Od D 00022458/deg2.From here on, we use deg2 to denote the unit of the geograph-ical area, where the unit degree is that of latitude (111.11 km).The converging steps of the calculation are shown in Table 1.

Figure 2(a) shows the temporal change of the function

p4t5 DRR

SO‹4t1 x1 y—¨t5 dx dyRR

SOŒ4x1 y5 dx dy

1 (18)

that is, the integrated conditional intensityRR

SO‹4t1 x1

y—¨t5 dx dy over the whole study region S relative to theoverall background rate

RRS

OŒ4x1 y5 dx dy. The pattern is sim-ilar to that from the simple ETAS model (Ogata 1988).Figure 2(b) shows the total intensity Om14x1 y5 of the centralNew Zealand region. Figure 2(c) shows the spatial clusteringintensity Oƒ4x1 y5, calculated by (14). To enhance the contrastbetween Figures 2(b) and (c), we introduce the concept ofrelative clustering effect, which is shown by the contours inFigure 2(d). We do this to measure the cluster effect relativeto the overall rate and de� ned by the ratio of the spatial clus-tering intensity to the total intensity, that is,

C4x1 y5 DOƒ4x1 y5

Om4x1 y5D

Pj �jkdj

4x ƒ xj1 y ƒ yj5Pj kdj

4x ƒ xj1 y ƒ yj50 (19)

Here C4x1 y5 can be regarded as a smoothed value of �j .

(a)

43

o4

2o

41

o4

0o

39

o3

8oS

170o

172o

174o

176o

178oE ’70 ’75 ’80 ’85 ’90 ’95

(b)

43

o4

2o

41

o4

0o

39

o3

8oS

Figure 1. Epicentral Locations in the Central New Zealand Region (a) and Space-Time Plots for the Latitudes Against the Times of All of theEvents (b).

Figures 2(b)–(d) show that most places in the study areahave a total intensity of less than 50 events/deg2 through-out the whole period of 28.75 years (T D 101500 days). Theclustering intensity is less than 20 events/deg2 throughout theperiod, and the relative clustering effect is less than .3. Somespots get high values for these three quantities, however. Theyare located mainly in the plate boundary between the Paci� cand Australian plates or in the inland volcanic zones.

By subtracting the clustering intensity from the overallintensity as indicated in (15), we can get the estimate of thebackground intensity function shown in Figure 2(e). It can beseen that the background rate is much smoother than the totalrate in Figure 2(b). Most places have values in the range of20–50 events/deg2, except for the plate boundary, where thehigh values form up a strip with values of 50–100 events/deg2

and some peaks up to 200 events/deg2. Also, some high valuesare seen in the volcanic zones and offshore from Wanganui.

6.2 Central and Western Japan

The second example used data from the central and westernHonshu area of Japan from the hypocenter catalog compiledby the Japan Meteorological Agency (Fig. 3). We selecteddata for the period 1926–1995 in the rectangular area 34�–39�N and 131�–140�E with magnitudes MJ ¶ 400 and depthsµ100 km. Outside the southern boundary were two greatearthquakes, the 1944 Tonankai earthquake of MJ 7.9 with epi-center (136.62�E, 33.80�N) and the 1946 Nankai earthquake ofMJ 8.0 with epicenter (135.62�E, 33.03�N). These events andtheir large aftershocks apparently affected the seismic activ-ity in the aforementioned selected inland area. However, thedetection rate of earthquakes offshore is quite low comparedto that inland, and thus it is hard to widen the region of theanalysis without raising the threshold magnitude substantially.Instead, we keep the same area and same magnitude thresholdas before for the analysis, but allow for boundary effects aris-ing from the occurrence of the earthquakes not more than 1degree outside the study area, to include the in� uence of thetwo great events with magnitude ¶708. Thus the likelihoodcalculation is implemented for the enlarged history ¨t con-taining data from the extended region.

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Figure 2. Results From the Central New Zealand Region. (a) Plot of the conditional intensity function of time integrated over the studied region.The horizontal axis represents time in years, and the vertical axis represents the ratio of the conditional intensity to the background intensity overthe whole region. (b) Contours of the estimated total rate Om1( x, y) [see (13)]. (c) The clustering intensity Oƒ( x,y ) [see (14)]. (d) The estimatedrelative clustering effect bC( x, y) [see (19)]. (e) The estimated background seismicity rate OŒ(x ,y ) [see (15)]. In (b), (c), and (e), the values are inevents/deg2 throughout the whole period (T D 10, 500 days), and contours are drawn for approximately equal intervals in a logarithmic scale. Thethick solid lines represent the coastlines.

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Figure 3. Epicentral Locations and Selected Subregions in the Western and Central Honshu Area of Japan (a) and Space-Time Plots for theLongitudes Against the Times of All of the Events (b).

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The MLEs for this region are bA D 020376148 event/(deg2�day), O� D 101355606, Oc D 0040436107 day, Op D 10250478, andOd D 0001033081/deg2. Figures 4(a)–(e) show estimates of the

time variation of the function p4t5 in (18), total intensity,relative clustering effect, and background intensity. We cansee that the clustering effects arising from typical aftershocksare eliminated in the estimates of the background intensity inFigure 4(d). Those include the 1927 Kita-Tango, 1943 Tot-tori, 1945 Mikawa, 1948 Fukui, 1964 Niigata, 1993 Noto-Hanto-Oki, and 1995 Kobe earthquakes. In contrast, the back-ground rates are almost the same as the total intensities in theWakayama and Ibaragi regions, which show almost pure spa-tial occurrence of nonclustered events.

7. PROBABILITY–BASEDDECLUSTERING ALGORITHM

With the estimated background intensity and the branchingstructure (the space-time ETAS model), we can carry out thefollowing declustering algorithm:

Algorithm 2

1. For each earthquake j D 1121 : : : 1N , calculate probabil-ity �j in (10) from the � nal solution in algorithm 1.

2. Generate N uniform random numbers U11 U21 : : : 1UN in601 17.

3. If Uj < 1ƒ �j , then keep the jth event; otherwise, deleteit from the catalog as an offspring. The remaining eventscan be considered the background events.

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Figure 5. Histograms of Aftershock Probabilities [cf. (10)] for (a) the Central New Zealand Region and (b) the Western and Central HonshuRegion.

The essence of the algorithm is the probability �j associatedwith each event. Figure 5 shows a histogram of these proba-bilities for the two datasets. It can be seen that with very highprobability, most of the events are either independent eventsor aftershocks, but that quite a few events are somewhere inbetween; about 31.7% of the New Zealand events and 18.3%of the Japanese events have probability of being an aftershockranging from .1 to .9.

Using algorithm 2, we obtained examples of catalogs ofbackground events from the two datasets. Typical examplesare shown in Figures 6 and 7. Figures 6(a)–(b) and 7(a)–(b)

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Figure 6. Results From Applying Algorithm 2 to the Central New Zealand Region. (a) Epicenter locations of the background events. (b) Epi-center locations of the offspring events. (c) Space-time plots of latitudes against times of the background events. (d) Space-time plots of latitudesagainst times of the offspring events.

show the epicentral locations of the independent events andthe aftershock events; Figures 6(c)–(d) and 7(c)–(d) includespace-time plots of the independent events and the aftershockevents. Here we used the latitude for the New Zealand dataand the longitude for the Japanese data.

Note that in Figure 7(c) for the declustered events, two clus-ters of linearly aligned events appear in 1945 and 1976 and inthe longitude ranges 136�E–138�E and 137�E–140�E (see thearrows in the � gure). The � rst cluster follows the December1944 Tonankai earthquake of MJ 7.9 and precedes the January1945 Mikawa earthquake of MJ 6.8 that occurred at 137.07�E

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(a)

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Figure 7. Results From Applying Algorithm 2 to the Western and Central Honshu Area. (a) Epicenter locations of the background events.(b) Epicenter locations of the offspring events. (c) and (d) Space-time plots for the longitudes against the times of the background events andthe offspring events. The two arrows mark the occurrence time of the remote triggered events after 1944 and 1975.

and 34.68�N. We have learned that these events include quitea few events remotely triggered by the Tonankai earthquake,whose epicenter (136.18�E, 33.57�N) is outside the studiedarea. In other words, the remotely triggered clusters in timehave not been removed, even though the log-likelihood (5)takes into account the effect of the boundary condition of theTonankai event. The magnitudes of all of the events in thesecond cluster are relatively small, and a possible triggeringevent is the 1975 deep great event of MJ 7.7, which occurredat (130.09�E, 38.79�N) at a depth of 549 km. The main rea-son for the detection of the remote triggering here is probablythe short-range range decay of the spatial response functionde� ned after (4) that conforms to the conventional understand-ing of an aftershock area (Utsu 1969).

Figures 6(d) and 7(d) include quite a few single events fromwhich one may feel that the background patterns have notbeen well separated from the clusters. However, the single ordouble events that remain in the cluster catalog are the remainsof the numerous double or triple events in the original catalog,most of which represent small clusters from mainshocks withmagnitudes close to the lower magnitude threshold.

The following algorithm provides a stochastic realization offamily trees based on the estimated space-time ETAS model:

Algorithm 3

1. For each pair of earthquakes i1 j D 1121 : : : 1N 4i < j5,calculate probability �i1 j in (9) and �j in (11) from the� nal solution in algorithm 1.

2. Set j D 1.3. Generate a uniform random number Uj in 60117.4. If Uj < �j , then the jth event is considered to be an

immigrant (background event).5. Otherwise, select the smallest I such that Uj < �j

CPIiD1 �i1 j . Then the jth event is considered to be a

descendant of the I th event.6. If j D N , then terminate the algorithm; otherwise, set

j D j C 1 and go to step 3.

This algorithm enables us to identify a set of cluster fami-lies each starting from an immigrant. The earthquake datasetis stochastically separated into clusters and sets consisting ofsingle events; the separation is stochastic because it dependson the series of uniform random numbers used in the forego-ing algorithm. It should be noted that the events selected forthe background here (and, consequently, those generated byalgorithm 2) are always the � rst events occurring in the clus-ter family, but are not always the largest events in the family.However, if preferred, one can take the set of the largest eventin each cluster as an alternative representation of the back-ground seismicity.

8. DISCUSSION

In this section we discuss various additional issues relatingto the methods proposed in the previous sections and the pre-liminary results obtained there.

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8.1 Justi’ cation of the Underlying Model

It is obvious that the success of the algorithms proposedhere depends in the � rst instance on a successful choice of themodel for the underlying clustering process. The ETAS modelused in this article has a long history of use for this purpose,initially in the time domain (see, e.g., Ogata 1988, 1999 andUtsu, Ogata, and Matsu’ura 1995 for reviews of earlier work)and more recently in the space-time domain. In particular, dif-ferent possible forms of the response functions arising withinthe general framework of the space-time branching clustermodel were carried out by Ogata (1998), using the AkaikeInformation Criterion (Akaike 1974) to discriminate betweenmodels.

More generally, when looking to justify a particular modelfor use as the basis of a declustering algorithm, it is worthbearing in mind that in addition to such likelihood-based tests,several conventional graphic statistics are available for testingthe absolute goodness of � t of the model to speci� c spaceand time patterns of the data. These include the variance-timecurve and the log-survivor function for time sequences (e.g.,Ogata 1988; Andersen, Borgan, Gill, and Keiding 1993) andthe K statistic and nearest-neighbor statistics for spatial pointpatterns (e.g., Ripley 1981; Diggle 1983). Graphic compar-isons of results from the original data with those from repeatedsimulations from the model are particularly useful (see Ogata1998 for details of simulation algorithms for space-time data).More sensitive graphic comparisons can be carried out usingsimilar statistics for the transformed times (e.g., Ogata 1988)and possibly for the transformed higher-dimensional data,including locations (e.g., Schoenberg 1999) to test against thestationary Poisson process.

8.2 Dealing With Boundary Effects

As observed in Section 6.2, boundary effects can be impor-tant. Taking these into account, even if in only a rough way,can substantially improve the estimate of the backgroundactivity. In the present analysis, only the large events out-side the region had a signi� cant effect on estimates within thestudy region, because of the relatively short range of the spa-tial clustering terms. Thus it was suf� cient to include in theaugmented history only the coordinates of the larger eventsfrom outside the region. This represented quite a useful com-promise, insofar as it reduced both the computational workand the demands on the underlying catalog, without causingsigni� cant loss of accuracy.

8.3 Interpretation of Temporal Fluctuations inthe Estimated Background Activity

The term Œ4x1 y5 in (4) for the background seismicityimplies that the space-time characteristics of the declusteredprocess are close to those of a Poisson process that is station-ary in time even if it is nonhomogeneous in space. In partic-ular, the proposed algorithm assumes a constant mean rate intime for the background seismicity at each location. Thus, ifthe thinned outcomes for the background events do not appearuniform in time, then this should indicate some departurefrom the base structure assumed for the clustering effects. For

example, this may be caused by temporal activation or quies-cence that has some physical cause outside those causes thatcreate the standard ETAS clusters, or it may be due to somehitherto undetected fault in the catalog data. Revealing suchphenomena and evaluating their signi� cance, or uncertainty,are the eventual aims of the present declustering method. Ineffect, the model used for declustering plays the role of a nullmodel, and departures from it represent effects worthy of fur-ther study and interpretation.

For example, in our study the previously discussed aligneddeclustered events in the space-time plots in Figure 7(c)clearly indicate the presence of remotely triggered events.This in turn suggests that the short-range decay function (2)adopted as the spatial response function of the model mayrequire modi� cation, perhaps replacement by a power lawdecay function, as has been indicated by Ogata (1998).

It is also possible that in some smaller regions, apparentdifferences from stationarity in time may be caused by arti-facts of the dataset, in particular the substantial numbers ofsmaller aftershocks that may be missed (particularly in theearly part of the catalog) in the time immediately after a main-shock. The rate of missed events around the threshold mag-nitude decreases as time elapses after the mainshock; further,this type of incompleteness of data has been improving as theability to detect small earthquakes more rapidly has improved.Consequently, the estimates of c and p in the modi� ed Omorifunction in (2) for the aftershock sequences may be biased bythe missing events in early years, giving rise to some under-declustering of the later aftershocks. Some indications of thiseffect can be seen in the space-time plot of the declusteredevents in Figure 7(e) for the earthquakes during the period of1943–1946 and with the longitude range of 135�–138�E.

8.4 Nonuniqueness of the Declustered Catalog

A stochastic declustered catalog is not unique, because itis dependent on the random numbers used in selecting theevents to form the background seismicity. Unlike conventionaldeclustering methods, the stochastic declustering method doesnot make a � xed judgment on whether or not an event is anaftershock. Instead, it gives the probability of how likely eachevent is to be an aftershock. Each catalog produced by thestochastic declustering method should be understood as a sim-ulation, or a resampling from the original catalog. This maybe considered an advantage of the method rather than a disad-vantage, because it allows uncertainty about the declusteringto be quanti� ed. Thus, by repeating random declustering, wecan easily produce many stochastic copies of the declusteredcatalog. From these copies, graphic or other statistics can becalculated to evaluate the uncertainty or signi� cance of a par-ticular feature of the declustered catalog.

8.5 Preservation of Information

One may argue that throwing aftershocks out by the thin-ning (random deletion) method loses potentially useful infor-mation for carrying out seismicity analysis. Although this maybe true of conventional declustering methods or for one par-ticular realization of the stochastically declustered catalog, theessential output of the stochastic declustering algorithm is notthe particular realization of the declustered catalog; rather,

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they are probabilities associated with decomposition of theintensity into background and clustering terms. No data is lostin this process; rather, the process adds a new item, the proba-bility �j , to the parameters of the events in the original catalog.It should not be forgotten that the basic purpose of decluster-ing is not the production of declustered catalogs for their ownsake, but rather the investigation of features of the data thatotherwise may be obscured by the clustering effects. For thispurpose, the best representation of the combined informationin both data and model is attaching these probabilities to theoriginal events, whether studied directly or studied through thestochastically declustered catalogs to which they give rise.

8.6 Avoiding the Selection of Mainshocksas Aftershocks

As we can see from the declustering algorithm, the mag-nitude of an earthquake does not play a role in the randomdeletion. If a mainshock is preceded shortly by one or moreforeshocks, then it has some probability of being interpretedas an offspring of one of the foreshocks, and in this case maynot be selected as a background event. If this is regarded as adisadvantage, then we can separate the clusters by algorithm 3and then adopt the largest event in each cluster instead of theinitial event as the representative for inclusion in the catalogof background events.

8.7 Prospects for Future Study

Possibilities for further study of the stochastic declusteringmethod can be discussed in two areas: theoretic problems andapplications. On the theoretic side, some important aspects ofthe model remain to be considered. For example, we did notconsider the important problem of deciding when the use of anonhomogeneous background is necessary, either in space orin time, nor of how to estimate such time-varying effects inthe background if these indeed are present. On the applicationsside, the stochastic declustering algorithm gives us a usefultool for the study of seismicity patterns. For example, in thisarticle we have found some interesting differences between theNew Zealand catalog and the Japanese catalog, including (a)the Japanese catalog has denser clusters in space than the NewZealand catalog, as can be seen from the parameters estimatedby MLE, and (b) the New Zealand catalog has more eventsthat cannot be clearly classi� ed (31.7% events with probabil-ity .1–.9 of being aftershocks) than the Japanese catalog (only18.3% events with probability .1–.9 of being aftershocks). Thereasons for such differences require further study. For exam-ple, a hierarchical Bayesian space-time ETAS model has beendeveloped recently (Ogata 2001).

Moreover, stochastic declustering can also help answer suchquestions as whether relative quiescence appears before alarge earthquake, whether the occurrence of a large earthquakechanges the seismicity patterns in the neighboring region, andwhat is the probability for an event to produce a larger off-spring, that is, the probability of the occurrence of a foreshock.All of these important problems require careful evaluation ofuncertainties, and although they all require further study, thestochastic declustering algorithm suggests one way in whichsuch problems might be tackled in a more objective fashionthan has been possible in the past.

9. CONCLUSIONS

In this article we have outlined a procedure for objectivelyestimating the spatial intensity of the declustered or back-ground seismicity from an earthquake catalog. The method isbased on an assumed stochastic model (spatial ETAS model)for the clustering patterns. Its core is a two-stage iterative algo-rithm based on the thinning method presented in Section 3and estimation of the background seismicity in (13). The pro-cedure stochastically splits the whole process into backgroundevents and offspring events grouped into clusters or familytrees. Each such splitting can be regarded as on possible real-ization of how the original catalog could have been produced,taking into account that individual events can be classi� ed asbackground events or cluster events only with a certain proba-bility. We can estimate the signi� cance and uncertainties of thefeatures of the catalog not explainable in terms of the assumedmodel for clustering by running the procedure many times.

Using this method, preliminary studies were made of earth-quake datasets from the central region of New Zealand and thefrom the central and western regions of Japan. Although pre-liminary, the results have revealed several features of interestthat could be the subject of future study and interpretation.

[Received November 2000. Revised November 2001.]

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