statistical modelling of seismicity patterns before and after the...

New Zealand Journal of Geology & Geophysics, 2000, Vol. 43: 447-4600028-8306/00/4303-0447 $7.00/0 © The Royal Society of New Zealand 2000

447

Statistical modelling of seismicity patterns before and after the 1990 Oct 5Cape Palliser earthquake, New Zealand

JIANCANG ZHUANG

Centre for Analysis and PredictionChina Seismological BureauP.O. Box 166Beijing 10036, China

Abstract The earthquake sequence for this study occurredoff Cape Palliser at the southeastern tip of the North Islandof New Zealand. Two main events occurred, on 1990 Oct 5and 1990 Oct 6, both with local magnitude M/,5.3. They wereaccompanied by a large number of aftershocks. The sequenceis remarkable in that it broke the region of low seismicity inthe area between the Hikurangi Trough and the main faultsin the Wellington region. This paper studies the seismicityduring the period 1978-96. The ETAS model is applied tothe data. The whole period can be divided into four stages:early background period, relatively quiescent period,mainshock and the aftershock sequence, and active periodof post-aftershocks. In order to detect the quiescence,residual analysis from the ETAS model was applied tominimise the effect due to previous aftershock clusters. Toexplain all features in this sequence, a seismicity phasehypothesis is proposed.

The paper discusses aspects of the sequence includingbackground, spatial-temporal analysis, ETAS model, relativequiescence, residual analysis, and its application to the data.

Keywords Cape Palliser; earthquake; ETAS model;quiescence; residual point process

INTRODUCTION

The Cape Palliser earthquake sequence was chosen foranalysis because, in the Wellington area, it stands apart fromthe main active fault (the Wairarapa Fault) and occurred inthe aseismic background. One main purpose of this study isto analyse the aftershocks for this sequence, particularly inrelation to their fit with the Omori law. In seismologicalstudies, the Omori law, proposed by Omori in 1894 (Utsu etal. 1995), is one of the few basic empirical laws. This lawdescribes the decay of aftershock activity with time. It andits modified forms have been used widely as a fundamentaltool for studying aftershocks (Utsu et al. 1995).

A second main task is to study the seismicity patterns inthe region before and after the sequence. For this purpose,in addition to graphical display, we make extensive use ofOgata's ETAS model (Ogata 1983, 1988, 1989, 1992). Inits various forms, the ETAS (epidemic type aftershock

G98004Received 17 December 1998; accepted 27 January 2000

sequence) model seems to be the best available repre-sentation of the main features of seismicity. It is a stochasticversion of the modified Omori law. Based on the theory ofbranching processes or self-exciting point processes, it hasonly six independent parameters, and can be used to describeboth background activity and multistage aftershocks. Beforethe results from fitting the model to the data are discussed,a brief presentation of the ETAS model and its technicalpoints is given.

The tectonic structure in the region studied and the dataused are described in the section following. Then the mainfeatures of the sequences are described, and the ETAS modeland its use in detecting seismic quiescence are explained.The main results, discussion, and concluding remarks follow.

SEISMICITY AND DATA

Tectonic structureSeismicity in New Zealand is dominated by the plateboundary. This boundary runs down from the Tonga-Kermadec trench to the northeast of New Zealand, continuesdown the east coast of the North Island as the HikurangiTrough, passes under the branch faulting systems of the CookStrait-Marlborough Sounds region, and continues down thewest coast of the South Island as the Alpine Fault. To thesouthwest of New Zealand it takes the form of the PuysegurTrench. Seismic activity is at its greatest to the northeast ofthe North Island. The Hikurangi Trough forms part of awestward-dipping subduction zone, the seismicity followingthe boundaries of the subducting plate down to depths of300-400 km. The seismicity also spreads out in a shallowbelt (0-80 km) below the northeast-trending mountainranges of the North Island and the Taupo Volcanic Zonebehind them. The subduction zone continues below CookStrait, but appears to come to an end, or at least the deeperearthquakes peter out, somewhere south of the MarlboroughSounds. The northern part of the South Island, includingthe West Coast, has been among the most seismically activeregions of New Zealand during this century; by contrast,the Alpine Fault itself, the major (transcurrent) faultaccommodating plate motion in the South Island, has beenseismically quiet. Seismic activity picks up rapidly in theFiordland region and to the southwest of the South Island,and marks the beginning of a further, but eastward-dipping,subduction zone, with earthquakes recorded to depths of<200 km.

The Cape Palliser region is located on the southeasterntip of the North Island near the front of the Australian plate.The plate convergence here is accommodated by thenorthwest subduction of the Pacific plate and deformationof the overlying Australian plate. To the east lies HikurangiTrough, which trends northeast and defines the leading edgeof the subduction zone. To the west is the major strike-slipWairarapa Fault, which extends northeast and approximately

448 New Zealand Journal of Geology and Geophysics, 2000, Vol. 43

parallel to the trough. Most of the earthquakes within 15 kmdeep in the Wellington region lie close to and between theWairarapa and Wellington Faults. The area between theWellington Fault and Kapiti Island is relatively aseismic.

Luo (1992) studied the subduction structure of this regionusing geophysical inversion of seismic waves from nineaftershock events. His results show that the convergence partis c. 14 km deep, and that a layer of low velocity lies on theupper boundary of the subducting plate.

DataThe data used for analysis are from the Wellington cataloguerecorded by the Institute of Geological & Nuclear Sciences.For 1978-86 the events are based on readings from filmrecords, with a distance cut-off of 12 s S-P time (i.e., a halfsphere of radius c. 108 km). For 1978-82 all beatableevents are included. For 1983-86 a rough magnitudecriterion was applied, so events of magnitude ML < 2.3 werenever analysed. For all the rest of the period, 2.0 can beused as the magnitude threshold to study the seismicity. For1987-96 the events are from the CUSP analysis (seeMaunder 1999). All locatable events with epicentres withinthe box defined by latitudes 40.5-42.2° and longitudes173.6-176.0° are included, irrespective of depth. Themagnitudes of the film events (1978-86) have been adjustedto conform with the CUSP magnitudes.

A cylinder centred at 41.686°S, 175.508°E with a radiusof 36 km and a depth of 40 km was chosen as the study area.The main reason for choosing such a region was to includefull information on the sequence. If the radius or the heightof the cylinder is changed by 2 or 4 km, the total number ofevents does not change much (<60 in nearly 1500 events).Another reason concerns the location error for depth.Because the sequence is not far away from the network (thenearest station is <20 km away), the location error shouldbe <5 km.

1437 events {Mi > 1.0) were recorded in the study areafrom 1 Jan 1978 to 31 May 1996. To ensure completenessof the catalogue, studies are mainly based on the 996 eventswith ML > 2.0. While the ETAS model is being fitted to thedata from the period 1983-86 to check the existence ofquiescence, a higher magnitude threshold of 2.3 is used.

MAIN FEATURES OF THE CAPE PALLISEREARTHQUAKE SEQUENCE

The two events in the Cape Palliser earthquake sequenceboth had magnitude 5.3 and occurred on 1990 Oct 5d 23h

48m and 1990 Oct 6d 2h 4 lm. After the first of the above twoshocks, more than 1000 events were recorded by the localmonitoring network.

OutlineFrom the magnitude-time plot (Fig. 1), a first impressioncan be got that, apart from the sequence itself, the patternsfor the seismicity in the study area can be divided into threestages: average, quiet, active. From the depth histogram(Fig. ID), most of the events are at a depth of 10—30 km,especially 10-24 km. To ensure completeness of thecatalogue, magnitudes corresponding to the straight part ofthe curve in the magnitude-time plots in Fig. 1 (i.e., withMi > 2.0) were used for the basic study.

Epicentre distribution in time and spaceFigure 2 shows the epicentre distribution. The whole timeinterval was divided into four periods, shown in Fig. 3.

1978.1.1-1982.10.30 (Pj): 103 events with ML > 2.0.This period can be regarded as background seismicitypatterns.

1982.10.31-1990.10.2 (P2): 87 events with ML > 2.0.This may be the preparation period of the mainshocks.

1990.10.3-1990.12.1 (P3): 389 events with ML > 2.0.This is the period of the immediate aftershock sequence.

1900.12.2-1996.5.31 (P4): 417 events with ML > 2.0.Post-aftershock period. Even at the end of this period, therate is still much higher than in the first period.

In the first period, the seismicity patterns can bedescribed as "little clusters against a low activitybackground". The activity is mainly concentrated to thenorthwest of the epicentre of the two ML 5.3 events. In thesecond period, the occurrence rate of events with ML > 2.3decreases, some events occur to the southeast of the futurelarge events, and possibly a gap forms, even if not clearly.The direct aftershock sequence occurs in the third periodand forms two groups in space. The two mainshocks are inthe eastern group. For the last period, the pattern is evenmore clustered than in the first period. The main activity isnot at the site of the main sequence but forms a northwest-southeast-trending belt, somewhat to the west of the mainevents. The general activity level is much higher than ineither of the two first periods.

Cross-sectionsDepth cross-sections for different periods are shown in Fig. 4and 5. The magnitude threshold for drawing cross-sectionsis ML> 2.0. The section plane is vertical and is perpendicularto the converging line of the Australian plate and the Pacificplate. Its intersection with the Earth's surface has an azimuthof 135°. According to Luo (1992), the interface between theAustralian plate and Pacific plate is c. 13 km deep. The twomain clusters and the clusters that occurred in the earlyperiod (Fig. 4) are all beneath this depth.

In the first period, the earthquakes occur mainly to thewest of the future earthquake. The earthquakes mainly occuron one side of the Australian plate, and two little clustersoccur near the interface.

The second period is different from the previous period.The earthquakes cross the section line and appear on theother side. A gap is formed at the area for future aftershocks.

The earthquakes in the direct aftershock sequence formtwo clusters. One is in the upper boundary of the subductingplate, the other on the lower boundary. Not many earthquakesoccur in the inner plate near the mid-line. This can beexplained by geodynamics: bending a thin plate causesextension in the outer arc of the plate, while the undersideof the plate is in compression. The extension zone is separatedfrom the compressive zone by a neutral surface. The bendingis caused by the subduction of the Pacific plate under theAustralian plate. The events in this period occur in explosiveclusters, and can be divided into three classes according totheir locations (Fig. 4): (1) background shallow events withdepth < 13 km; (2) events in the cluster on the upper boundary;and (3) events in the cluster on the lower boundary.

In the final period, the events are located somewhat tothe north and west of the site of the main sequence andremain highly clustered.

Zhuang—Statistical modelling of seismicity

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From the figures, the thickness of the Pacific plate atthis region is c. 14 km. This agrees with the results obtainedby Luo (1992) and Reyners et al. (1997).

Doughnut pattern—spatial-temporal distributionTo discuss the changing seismicity patterns with time, aspatial-temporal plot was drawn (Fig. 6). The horizontal axismeans time of events, the vertical axis represents thehypocentral distance to the first mainshock. For the distanceswithin 20 km, a linear scale is used, while a cubic scale isused for distances >20 km. The cut-off magnitude isMi = 2.0. One must be careful when looking at this figure,for the catalogue is not complete during the period 1983—86, as mentioned above.

Before 1990, events are concentrated near the boundaryof a circle of radius 15-20 km, centred on the futureepicentres. A few events come into this circle during 1982—84, but this may be caused by location error.

Figure 6 shows:(1) Most of the events in the region of study occur after the

mainshocks.

(2) Most of the events are at distances between 15 and 23 kmfrom the mainshocks.

(3) The events are more clustered in the period 1978-82 thanfrom 1986 to the occurrence time of the mainshocks."More clustered" means not only in time but in space aswell. In the first period, the clusters extend as far as

34 km from the centre of the circle, while in the secondperiod, the range is <30 km.

(4) The events are most highly clustered in the third andfourth periods, and mainly occur within 23 km of thecircle centre. This again suggests that the size of the studyregion is appropriate.

(5) The doughnut pattern, thought to be one of the well-known precursory patterns, is reasonably clear,particularly in the period 1984-90. Between 1984 andthe main events in 1990, no events occur within 10 kmof the epicentre of the main events.

MODELS AND METHODS OF ANALYSIS

Intensity functionsBoth models considered in this analysis belong to the classof point process models with time-varying intensity. In theanalysis of the direct aftershock sequences, we shall use anon-stationary Poisson process with rate proportional to theusual Omori law form. In the analysis of the periodspreceding and following the main aftershock sequence, weshall use versions of Ogata's ETAS model to study the ratesof occurrence and levels of clustering in the backgroundactivity.

The intensity associated with the simple Omori law willbe taken in the form (Utsu 1961)

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where g(t) is an Omori-type decay function in time, where t\ and ti are the occurrence times of the two mainnormalised so that jg(t)dt = 1, namely events and

and 9, c, p are parameters to be determined. Because the andCape Palliser sequence is a typical sequence with double , Nmain events, a double Omori law will also be applied, i.e., i,\ __ Pi~^\ \ + J_

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The ETAS (epidemic type aftershock sequence) modelwas developed from the modified Omori law, based on theassumptions set out below: (1) The background rate of eventswithin the study region is a constant n; that is, thebackground events occur according to a stationary Poissonprocess with a constant rate fi. (2) All events, includingaftershocks themselves, produce their own offspring, andeach of their offspring produces secondary offspringindependently. The total number of direct offspring from anancestor of size M follows a Poisson distribution with meanA{M). Later we shall give A{M) the explicit form

where MQ is the threshold magnitude and K and a areparameters to be determined. (3) The offspring from anancestor of size M occur in time according to a Poissonprocess with rate A(Af)g(t-t0) where g(t) is the Onion-typedecay function in Equation (2) and to denotes the occurrencetime of the ancestor. (4) The magnitude distribution isindependent of the occurrence rate. Later we shall use theexplicit form of the Gutenberg-Richter relation as theprobability density function of magnitudes with

where j3 is linked to b by p = 230b.In summary, the ETAS model has a conditional intensity

function of the form

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with parameters /I, K, a, c and p. An important parameterfor understanding the behaviour of the process is theexpected number of direct offspring per ancestor, averagedover the magnitude distribution

p= \A(M)f(M)dM.

When A(M) and J{M) are given the explicit forms referredto above,

This p plays the role of the criticality parameter for thebranching process interpretation of the ETAS model: if p <1, the process is stable (subcritical); if p > 1, it is unstableand will grow indefinitely in time, p can also be interpretedas the proportion of all events which are aftershocks: if m isthe overall mean rate of events, it can easily be seen that therate of background events is

(l-p)m = (l,so that the rate of offspring events (i.e., aftershocks) is m-[i=p/n.The above form for the conditional intensity functionof the ETAS model is slightly different in parameterisationfrom the original form suggested by Ogata (1989, 1992).The original form is

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Among the parameters of the ETAS model, a and/? areparticularly useful for characterising the temporal patternsof seismicity. The p value indicates the decay rate ofaftershocks and the a value measures the efficiency of themagnitude of an earthquake in generating its offspring(Ogata 1989, 1992). From the discussion above, we can seethat the derived parameter p, proportional to K, also playsan important role in determining cluster structure of theearthquakes.

Likelihoods and AIC proceduresDetermination of the coefficients is carried out by the methodof maximum likelihood. The full likelihood is the likelihoodfor the set of pairs (?,-, M,) where tt is the occurrence timeand Mi is the magnitude of the ;-th event. Because of theassumed independence between time and magnitude, thislikelihood reduces to the product of two components—alikelihood for time and a likelihood for magnitude, that is,after taking logarithms,

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Fig. 6 Spatial-temporal distri-bution of earthquakes. Thehorizontal axis represents time.The vertical axis represents theepicentral distance to the centre ofthe studied region. The differentsized circles indicate differentmagnitudes from 2.0 to 5.3.

where [0, 7] denotes the whole observation interval. Thefirst part,

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log£2= ]Tlog/(M;),i:Q<t,<T

becomes the usual likelihood for estimating /}.More than one model is applied to the aftershock

sequence of the Cape Palliser earthquake: the simple Omorilaw, the double Omori law, and the ETAS model. Moreover,two alternative forms for the double Omori law areconsidered: the model with equal decay rates (p\ = p2 inEquation (3)) and the model with different decay rates (p\ *

Pi)-To find which model fits the data best, the AIC (Akaike

Information Criterion) model selection procedure is used(Akaike 1977). With this procedure, models with differentforms and parameters are compared by computing for eachthe statisticAIC =-2logL +2k, (7)

where L is the likelihood and k is the total number of fittedparameters, for the particular model being assessed. Themodel with smallest AIC is considered to be the best fit tothe data. Numerical examples of the use of this procedureare given in the IASPEI software (Lee 1995; Ma & Vere-Jones 1997) or SSLIB (Harte 1998).

The change-point problemThe change-point problem considers whether the occurrencepatterns of events changes before and after a time TQ. Anextension of the AIC procedure can be used to determine

whether the temporal patterns of seismicity changed beforeand after a time point TQ in a given time interval [0, 7]. Ifthe combination of different models applied to both periods[0, TQ] and [TQ, T\ is regarded as one model, this matter canbe reduced to the problem of model selection between thiscombined model and a single model fitted to the whole period[0, T\. Here, we introduce Ogata's work on the applicationof AIC procedures to the change-point problem. FollowingOgata (1992), we calculate the following AIC values

v47C0=-21ogZ,[0,7] + 2 x 5

AICX = -2 log L[0, TO ] + 2 x 5

AIC2 = - 2 log L[T0, T] + 2 x 5

where log L[0, 7], log L[0, TQ], and log L[T0, T] are themaximum log-likelihood for each single ETAS modelapplied to the periods [0, 7], [0,7b], and [TQ, 7], respectively,and AICQ, AIC\ &ndAIC2 are the corresponding AICs.

If a change point can be set, based on information fromoutside the data, AICQ is compared with the following

(8)

AICn = +AIC2 (9)

Otherwise, the change point is set from the data, that is, theproposed point is obtained by running TQ through the timeinterval and finding a position that gives the minimum sumof AIC\ aaAAIC2- In this case, we have to be more careful,for the same information from the data will be used twice ifAICQ is compared with AfC^ in Equation (9). Instead, Ogata(1992) showed that^47Co should be compared with

AICU = AICX + AIC2 + 2q(n), (10)where the quantity q(n) is the contribution of TQ as anadjustment parameter and is dependent on the total numberof events, n, in the interval [0, 7] as shown in Fig. 7. AIC\2< AICQ indicates a significant distinction between theseismicity patterns in the two time spans.

Relative quiescence and the residual point processAn important problem is to check if there is a quiescentperiod before a large earthquake. Quiescence is a commonly

Zhuang—Statistical modelling of seismicity 455

Fig. 7 Penalty factor of the AICfor the change-point parameter.The factor q(N) versus sample sizeN is obtained by a Monte Carlosimulation experiment. See Ogata(1992).

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Number of points N

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accepted precursory phenomenon before major earthquakes.Mogi (1968) showed that before several large earthquakes(1944 Tonankai earthquake, 1946 Nankaido earthquake) thefocal region became calm. Ma et al. (1990) discussed thequiescent seismicity pattern before major earthquakes inChina. But naive quiescence may be caused by two reasons:one is stopping of a previous aftershock sequence, and theother is real quiescence. To reduce the effect of previousclusters, the usual method is to decluster the catalogue. Thekey problem for the above method is that there is no generalrule to decluster the catalogue from regions with variousseismicity properties.

AIC procedures can be use to detect whether occurrencepatterns change with time. But even if a change is detected,more is needed to determine whether such changes representquiescence or not. Here, the method of relative quiescence,introduced by Ogata (1992), is used. This method uses hisso-called "residual analysis" for point processes. Thismethod depends on the fact that a point process in time withvariable rate function X(t) can be transformed into astationary Poisson process by making use of the timetransformation

: = A(t)=j X(u)du (11)

(Daley & Vere-Jones 1988). The sequence {T,}={A(?,)} iscalled the transformed time sequence. It is known that thestandard stationary Poisson process results if the conditionalintensity corresponds to the true model. Since we do notknow the true model, nor its parameter values, we use themaximum likelihood estimate for the parameter values ofthe assumed conditional intensity model in (11). Then, thetransformed data of the occurrence times using (11) is calledthe "residual point process (RPP)".

If the ETAS model provides a good fit to the seismicity,then RPP is well approximated by the standard stationaryPoisson process. For a good fit to the standard process, thecurve of the cumulative numbers should be close to a straightline with a unit slope in the plot of the cumulative frequencyversus the transformed time. If, on the contrary, there is asignificant deviation from the unit rate Poisson process in

any characteristic property of the residual point process, thissuggests some discrepancy between the model and the data,such as inhomogeneity of the data or the existence ofseismicity change, which is not captured by the estimatedETAS model, no matter how high or low the originalseismicity.

The above discussion suggests a method to detect relativequiescence. First of all, by inspection of the cumulativenumber curve of either the ordinary data or the correspondingRPP for the whole time interval [0, S], we hypothesise achange-point TQ where the onset time of the relativequiescence is defined. This point is clear in some cases, asdescribed in Wyss & Habermann (1988). We then calculatethe maximum log-likelihoods to confirm whether AlC\i <AICQ or not, where AIC 12 is calculated with Equation (9) orEquation (10) according to how the hypothesised change-point is set. If the above inequality holds, we calculate theresidual point process with the parameters for the timeinterval [0, TQ] but apply it to the whole time interval [0, T\;then we plot the cumulative curve versus transformed time.If there exists quiescence before the mainshock, the ratedrops in the quiescent period, and the cumulative curve fallsbelow its expected straight-line form. As this quiescence isin the residual point process, and may not be visible in theoriginal seismicity, it is called relative quiescence. To detectrelative quiescence, the catalogue need not be declustered,for the features of clustering are already modelled in theETAS model.

APPLICATION OF THE OMORI LAW AND ETASMODELS TO THE CAPE PALLISEREARTHQUAKE SEQUENCE

Studies on the direct aftershocksTo study the properties of the direct aftershock sequence, acomparison was made between fitting the Omori law andthe ETAS model to events with ML > 2.0. Here, the simpleOmori law (see Equation (1)) is first considered. As thereare two mainshocks in this sequence, two more complicatedforms of the Omori law—the double Omori law in Equation


in <o

LUo

8-

Single Omori LawDouble Omori Law1Double Omori Law2ETAS Model

10 20 30 40 50

Fig. 8 Comparison between thefitted results for the direct after-shock period (P3) with severalstatistical models. "DoubleOmori Lawl" means the modelof the double Omori law with thesame decay rate for aftershocksfrom the two mainshocks, and"Double Omori Law2" meanswith different decay rates.

Time

(3)—are also applied to the data. The difference betweenthese two forms is that in one the decay rates (p values) forthe frequency of aftershocks of the two main events aredifferent (p\ ^ pi), while in the other they are equal (p\ =Pi)-

The numerical results are shown in Table 1 and Fig. 8,where ETAS models and several versions of the Omori lawwere applied to the data. All the models are fitted to thedata over time intervals of 30,50, and 90 days. From Table 1,we can see that the ETAS model gives the minimum valueof AIC and fits the data better than any of the other modelsused. Figure 8 shows that the sequence consists of multipleaftershock sequences, which are apparent even in the patternsfor the first 3 days after the first mainshock.

The better fit of the ETAS model is not because it hasmore parameters, for the form of the double Omori law withdifferent decay rates has one more parameter than it. Onereason why the ETAS model fitted the data best is that allmodels from the Omori law are fixed in form from thebeginning of the mainshock, whereas the ETAS model

changes its conditional intensity as events occur. Thus, theETAS model is more flexible and so gives a better fit to thedata.

The comparison between fitting the data with the simpleOmori law and the ETAS model shows that the Cape Palliserearthquake sequence contains many secondary aftershocks.From the viewpoint that an aftershock is a readjustment ofthe stress field after a large earthquake, every earthquakecan change the stress field in its neighbourhood and socreates the need for further readjustment.

Detection of precursory relative quiescence before themainshocksFrom the studies in the earlier sections, we can see apparentchanges during the period of preparation of the mainshocks,such as the doughnut pattern, or the motion of the most activezone. To confirm this impression, we use the procedures fordetecting relative quiescence described in the previoussection. To be sure that the pattern is not due to the missingsmall events in the catalogue during the period 1983-86,

Table 1data.

Difference in parameters and the AIC values between models fitted to the direct aftershock

No. of events:

Simple Omori law

Double Omori lawwith differentdecay rates

Double Omori lawwith identicaldecay rates

ETAS model

AICMICMICIN

AICMICAAIC/N

AICMICAAIC/N

AICMICAAICIN

30 days377

-1883.8850.0000.000

-1935.698-51.813

-0.137

-1914.678-30.793

-0.0817

-2210.208-326.323

-0.866

50 days389

-1791.5570.0000.000

-1883.336-91.779

-0.236

-1865.826-74.269

-0.191

-2179.849-388.292

-0.998

90 days415

-1719.3100.0000.000

-1810.747-91.437

-0.220

-1795.603-76.293

-0.184

-2173.398^154.088

-1.094


1978 1981 1983 1986 1989

ncy

a>3D"

refr

em

uiat

i'

3

o

o

soCVI

o

1978 1981 1984 1987

Time

1990 20 40 60 80 100

Transformed time

1978 1981 1983 1986 1989

|'c(0

LO

mCO

in

I

1978 1981 1984

Time

1987 1990

in

3.5

2.b

I i

0

ill]2C

, , In ill., I: nil 111

40 60

Transformed

Ii.

„

80

time

ill i l l100

Fig. 9 Detection of quiescence before the mainshocks. The four panels show, in order, cumulative number of earthquakes againsttime, cumulative number of earthquakes against transformed time, magnitude-time plot, and magnitude-transformed time plot. Eventsof Mi > 2.3 are used in this plot.

we use a higher magnitude threshold, 2.3, which is thoughtto be reliable (R. Robinson pers. comm.).

From inspection of the data in Fig. 1, the proposedchange-point was taken as 1982.10.31, corresponding to thedivisions into the time intervals P\ and Pi described earlier.Since this change-point was neither fixed entirely byconsiderations from outside the data, nor chosen by selectingthe optimal AIC, Equation (10) was used as a conservativechoice to assess the AIC value of the joint model.

Using this change-point, the ETAS model was fitted tothe data in P\ and P2 separately, then to the data in P\ andPi combined into a single period. It is found thaXAIC{P\) +AIC(P2) < AIC(P\i) + 2q(N). The parameters from the modelfitted to P\ were then used to calculate the transformed timesequence over the whole time interval of P\ and Pi. Thecumulative curve for the residual point process is displayed

in Fig. 9, which shows a readily distinguishable drop of therate. The dashed curves represent 99% confidence bands. Itcan therefore be concluded that the balance of evidencesuggests the existence of a relative quiescence period beforethe two Mi 5.3 earthquakes.

Existence of distinct seismic phasesWe now examine different combinations of models and timeperiods to determine which combinations best fit the data.Using the division into time intervals P1-P4, we fit ETASmodels for the whole time interval, the four periodsseparately, and some intermediate combinations. To providea reference for the improvements made by the ETAS models,the Poisson model is fitted at the same time, as shown inTables 2 and 3. The comparison between the AICs is shownin Table 4.

Table 2 Parameters estimated from fitting the ETAS model. Pn means a combination of subperiodsPi and P2 into a single subperiod for fitting the ETAS model, and so on.

Period

PiPiPiPAP\2P34P\nPiuP1234

N

10387

390416190806580893996

0.03700.02400.27310.03530.02610.04910.02710.02170.0268

K

0.27320.16500.37690.46950.45480.31780.28490.41550.3858

c

0.01960.20020.00630.00330.01440.00490.00870.00370.0043

a

1.23840.35051.21111.23750.00001.36521.53951.43091.4517

P

1.28101.47181.30731.11331.33841.19221.23991.12671.1391

P0.40070.19090.89320.97150.45480.81740.82711.11610.9947

b

1.691.120.911.041.370.971.020.991.03


From these tables we can see:(1) The ETAS model performs much better than the Poisson

model. The values of AAICpei event range from 0.139to 7.47, which can be regarded as a big improvementfor fitting the data.

(2) Using one dividing point only, the best subdivision pointfor the subperiods is before the mainshocks, which makesa net improvement in AIC of about 32.6, which issubstantial.

(3) If we divide the whole time interval into four subperiods,the best model from among the various combinationsfor the subdivision is that with four separate subperiods,which results in an improvement of 50 in AIC.

(4) For the subdivision of direct aftershocks and post-aftershocks, the change of AIC values is 7.

(5) The background rate fi is 0.0370 events/day in P\ anddrops to 0.0240 events/day in P2, then increases to0.2731 events/day, and last in the period Pn it becomes0.0353 events/day, back to its original level.

(6) The a values are approximately 1.2 in the periods P\,P$, and P4, which indicate the seismicity patterns ofordinary swarms, whereas a = 0.3505 in P2, whichindicates a pattern of foreshock swarms. It is anotherindication that the period P2 differs from the backgroundperiod P\.

(7) The parameter c reaches its highest value, 0.2002 day,in P^. A higher c suggests that the events in a cluster aremore spread out in time.

(8)Thep values in each period are 1.28, 1.47, 1.30, 1.11;they suggest a change of the decay rate of aftershocksfor the final period after the two big events.

(9) The p values are 0.4007, 0.1909, 0.8932, 0.9715 in foursubperiods, which suggest it is more clustered or closerto criticality in the subperiods P3 and P4. In particular,there is a big difference in the clustering characteristicsbetween the first and last periods.

(10) The ^-values are 1.69, 1.12, 0.91, 1.04, which showthe changes in magnitude structure.

These results and the results outlined before suggestthat even a small region can exist in several differentphases of activation, with different clustering character-istics, as well as different levels of activity, at differenttimes. From Fig. 5, it appears that the changes in modelparameters may be related to the migration of hypocentrelocations to different parts of the plate boundaries indifferent periods.

The above ideas can be outlined as a four-stage seismicphase hypothesis (Fig. 10).

(i) In the first stage, the stress is under the critical leveland is being built up. The seismic activity is low in thisperiod. We call this period the interseismic stage.

(11) Next comes the preseismic stage. This is a quiescentperiod (perhaps because of dilatancy) with fewforeshocks and small clusters. The activity is lower thanin the interseismic period.

(iii) The main earthquakes come after the preseismic periodand only when the stress levels reach or exceed thecritical level. They always are accompanied by largeclusters or aftershock sequences. This is called thecoseismic stage.

(iv) After most of the accumulated stress is released, thestress field comes into an adjustment period. Seismicactivity is still at a high level. Many earthquakes occurin larger clusters, but these clusters are smaller in sizeand event magnitudes than in the coseismic period. Wecan call this period the post-seismic period.

After the adjustment has been finished, anotherinterseismic period begins.

According to the results obtained, it is reasonable to drawthe conclusion that the four subperiods P\, P2, P$, and P4correspond to the four phases in a seismic cycle.

Table 3 Comparison between fitting the ETAS model and the Poisson model to the data. (1Q representsthe occurrence rate estimated for a Poisson model; logZ and AIC are the likelihood and AIC value forfitting the ETAS model; logl^, and AICp are the likelihood and the AIC value for fitting the Poissonmodel; fif/J^ is the ratio of the backgroundrate estimated from the Poisson model to the total rate estimatedfrom fitting the ETAS model; AAIC = AIC-AICp represents the AIC gain of the ETAS model from thePoisson model; and AAIC/N means AIC gain per event.

P\PiPiP4P\2

^123^234

Table A

AIC

0.05850.03237.64710.20640.04080.49880.12310.17990.1481

logZ

315.18-375.701097.53-533.99-699.67

553.11377.67152.23

-167.45

AIC

640.35761.40

-2185.061077.981409.37

-1096.22-745.32-294.47

344.9

logLp

-395.47-385.76

403.39-1072.32

-797.91-1366.67-1794.76-2424.66-2898.20

AICp

792.93773.51

-804.772146.641597.822735.353591.524851.315798.40

0.63250.74300.03570.17100.63970.09840.22010.12060.1810

\ Comparison between AICs for different divisions of the period.

AIC-AIC(Pl2i4)

•Pl+2+3+4

294.67-50.23

•P12+3+4

302.29-39.75

^12+34

313.15-32.61

•fl+2+34

305.53-39.37

•P123+4

332.66-12.24

AAIC

-152.58-12.11

-1380.29-1068.66

-188.45-3831.57^336.84-5145.78-5453.50

-Pl+234

345.88-0.98

AAIC/N

-1.481-0.139-3.539-2.569-0.992-4.291-7.477-5.762-5.475

•^1234

344.900


Fig. 10 Illustration of seismicphases hypothesis. interseismic stage

preseismic stagecoseismic stagepostseismic stage

critical stress level

CONCLUSIONS

Before the Cape Palliser sequence, not many earthquakesoccurred between the Hikurangi Trough and the Wairarapafaults. These few earthquakes were mainly concentratedaround the main faults. During the aftershock period,however, the earthquakes were concentrated on bothboundaries of the subducting plate. They formed threesubclusters, located near the upper and lower boundaries ofthe subducting Pacific plate, which is c. 20 km thick, and inthe converging part of the Australian plate. The depth cross-section analysis shows how the activity changed from onepart of the plate boundary to another.

Several statistical models were fitted to the data. Amongthem, the ETAS model fits the direct aftershocks better thanthe Omori law, which suggests that the aftershock sequenceconsists of multistage clusters.

The ETAS model also shows the existence of a quiescentperiod before the two major earthquakes. It shows that thereis a quite clear change of seismicity patterns before and afterthe immediate aftershock sequence, not only in the meanrate, but also in cluster structure. Individually, these changesmay not be caused by random fluctuation due to the smallnumbers of events in some of the investigated regions, foroverall they are beyond the noise level. The change ofseismicity patterns suggests changes to the stress field orthe mechanical properties of the plate following the twomagnitude 5.3 events.

The seismicity phase hypothesis seems to be able tointerpret all the changes in the seismicity patterns before,during, and after the Cape Palliser earthquakes in October1990. Further studies need to be done to test thishypothesis.

Cape Palliser is the turning point of the subduction zone.The two main events were only of magnitudes 5.3, but theyproduced a large number of aftershocks and influence theseismic patterns in the region. These aspects are much likethe behaviour of large mainshocks. It is worth asking whetherthe occurrence of two events with such strange features, inthis particular region, could be a signal that the seismicitywould change in the nearby regions.

ACKNOWLEDGMENTS

These studies were carried out while the author was visiting theInstitute of Geological & Nuclear Sciences and the Institute ofStatistics and Operations Research, Victoria University ofWellington, New Zealand. Ma Li and David Vere-Jones gave a lotof direction during the research work. The author is also gratefulto Russell Robinson and Yoshihike Ogata, who sent the latestcatalogue and gave lots of suggestions. Parts of the software werefrom the statistical section of the IASPEI software, developed byYoshihike Ogata, and from the Statistical Seismological Library,which is designed by David Harte and his colleagues. This researchis funded by FRST, Asia 2000, and the fund for international co-operation from the China Seismological Bureau.

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Ogata, Y. 1989: Statistical models for standard seismicity anddetection of anomalies by residual analysis. Tectonophysics169: 159-174.

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Reyners, M.; Robinson, R.; McGinty, P. 1997: Plate coupling inthe northernmost South Island and southernmost NorthIsland, New Zealand, as illuminated by earthquake focalmechanisms. Journal of Geophysical Research 102:15197-15210.

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