Apparent Clustering and Apparent Background Earthquakes Biased by Undetected Seismicity

0.24 MB·English

by  D. Sornette

JOURNAL OF GEOPHYSICALRESEARCH,VOL. ???, XXXX, DOI:10.1029/,
Apparent Clustering and Apparent Background Earthquakes
Biased by Undetected Seismicity
Didier Sornette
DepartmentofEarthandSpaceSciences,andInstituteofGeophysicsandPlanetaryPhysics,Universityof
5
California,LosAngeles,USAandLaboratoiredePhysiquedelaMati`ereCondens´ee,CNRSUMR6622,
0 Universit´edeNice-SophiaAntipolis,France
0
2 Maximilian J. Werner

n DepartmentofEarthandSpaceSciences,andInstituteofGeophysicsandPlanetaryPhysics,Universityof
a California,LosAngeles,USA
J
0 Abstract.
1 In models of triggered seismicity and in their inversion with empirical data, the de-
tection threshold md is commonly equated to the magnitude m0 of the smallest trigger-

] ing earthquake. This unjustified assumption neglects the possibility of shocks below the
h
detection threshold triggering observable events. We introduce a formalism that distin-
p
- guishes between the detection threshold md and the minimum triggering earthquake m0 ≤
o md. By considering the branching structure of one complete cascade of triggered events,
e we derive the apparent branching ratio na (which is the apparent fraction of aftershocks
g in a given catalog) and the apparent background source Sa that are observed when only
s. the structure above the detection threshold md is known due to the presence of smaller
c undetected events that are capable of triggering larger events. If earthquake triggering
i
s is controlled in large part by the smallest magnitudes as several recent analyses have shown,
y this implies that previous estimates of the clustering parameters may significantly un-
h derestimate the true values: for instance, an observed fraction of 55% of aftershocks is
p
renormalized into a true value of 75% of triggered events.
[

1
v 1. Introduction otherearthquakesthroughavarietyofphysicalmechanisms
9 but the effective laws do not allow the identification of a
4 There are numerous evidences that a seismic event can particular mechanism.
0 have a significant effect on the pattern of subsequent seis- The questions suggested by this approach include: 1)
1 micity, most obvious in aftershocks of large events. More what is the fraction of triggered versus uncorrelated earth-
0 recently has emerged an important extension of the con- quakes(which is linked to the problem of clustering)? How
5 cept of earthquake interactions in the concept of triggered can one use this modeling approach to forecast future seis-
0 seismicity, in which the usual distinction, that foreshocks micity? What are the limits of predictability and how are
/ are precursors of larger mainshocks which in turn trig- they sensitive to catalog completeness and type of tectonic
s ger smaller aftershocks, becomes blurred: a parsimonious deformation? In general, to attack any such question, one
c and efficient description of seismicity does not seem to re- needs in one way or another to estimate some key parame-
si quire the division between foreshocks, mainshocks and af- ters of the models of triggered seismicity.
y tershocks, which appear indistinguishable from the point Theroyalpath isin principletousethemaximum likeli-
h of view of many of their physical and statistical proper- hood method to estimate the parameters of the considered
p ties[Helmstetter andSornette,2003a]. Animportantlogical modelfrom acatalogofseismicity (time,location andmag-
nitude) (see for instance Ogata [1988] and Kagan [1991]).
: consequenceisthat cascades of triggered seismicity (“after-
v The calculation of the likelihood function requires evaluat-
shocks,” “aftershocks” of “aftershocks,” ...) may play an
i ingthetheoreticalrateofseismicityattimetinducedbyall
X important role in the overall seismicity budget [Helmstetter
and Sornette, 2003b; Felzer et al.,2002]. past events at times ti < t. The maximization of the like-
r lihood with respect to the parameters of the model, given
a There is thus a growing interest in phenomenological the data, then provides an estimate of the parameters. All
models of triggered seismicity, which use the Omori law
previousstudieshaveconsideredthatsmallearthquakes,be-
as the best coarse-grained proxy for modeling the com-
lowthedetectionthreshold,arenegligible. Thus,therateof
plexandmulti-facetedinteractionsbetweenearthquakes,to-
seismicity is calculated as if triggered only by earthquakes
gether with the other most solid stylized facts of seismic- abovethedetection threshold. However, thismethodis not
ity(clustering in space, theGutenberg-Richter(GR) earth- correct because it does not take into account events below
quakesizedistributionandanaftershockproductivitylaw). the detection threshold, which may have an important role
This class of ETAS (Epidemic-Type Aftershock Sequences) in the triggering of seismicity. Indeed, small earthquakes
models introducedbyOgata[1988] and Kagan and Knopoff have a significant contribution in earthquake triggering be-
[1981] offers a parsimonious approach replacing the classi- causetheyaremuchmorenumerousthanlargerearthquakes
fication of foreshocks, mainshocks and aftershocks by the (Helmstetter [2003]; Felzer et al. [2002]; Helmstetter et al.
concept of earthquake triggering: earthquakes may trigger [2004]). This can simply be seen from the competition
between the productivity law ∼ 10αM giving the number
of events triggered by a mainshock of magnitude M and
the relative abundance ∼ 10−bM of such mainshocks given
Copyright2008bytheAmericanGeophysical Union. by the Gutenberg-Richter law: the contribution of earth-
0148-0227/08/$9.00 quakes of magnitude M to the overall seismic rate is thus
1
X - 2 SORNETTEANDWERNER: APPARENT CLUSTERINGANDBACKGROUNDEVENTS
∼ 10−(b−α)M, which is dominated by small M’s for α < b θ >0, the normalized pdf of the Omori law can be written
[Helmstetter, 2003] or equally contributed by each magni- as
tude class for α = b (Felzer et al. [2002]; Helmstetter et
al. [2004]). Therefore, oneneeds to takeinto account small θcθ
Ψ(t)= . (2)
events that are not observed in order to calibrate correctly (t+c)1+θ
modelsofseismicityandobtainreliableanswerstoourques-
tions stated above. This is an essential bottleneck for the Third, the number of direct aftershocks of an event of
development of earthquakeforecasts based on such models. magnitude m is assumed tofollow the productivitylaw:
The purposeof thisnoteis topresent ageneral theoreti-
caltreatmentoftheimpactofunobservedseismicity within ρ(m)=k10α(m−m0),m0 ≤m≤mmax. (3)
the framework of models of triggered seismicity. We show
byanalyzing thebranchingstructureof a complete cascade Notethat theproductivitylaw (3) is zero below thecut-
(cluster)triggeredbyanindependentbackgroundeventthat off m , i.e. earthquakes smaller than m do not trigger
0 0
the unobserved seismicity has the effect of decreasing the other earthquakes, as is typically assumed in studies using
real branching ratio n and of increasing the number of in- the ETAS model. The existence of the small magnitude
dependentbackgroundeventsS intoapparentquantitiesna cut-off m0 is necessary to ensure the convergence of these
and Sa. This bias may be very significant. We therefore typesofmodelsoftriggeredseismicity(instatisticalphysics
claimthatpreviousworkshouldbereanalyzedfromthenew of phase transitions and in particle physics, this is called
perspective of our approach. This leads also to important
an “ultra-violet” cut-off which is often necessary to make
consequencesfor themethodspresentlyused toforecast fu- thetheory convergent). Ina closely related paper, Sornette
tureseismicity based only on incomplete catalogs.
and Werner [2004] showed that the existence of the cut-off
m hasobservableconsequenceswhichconstrainitsphysical
0
2. The ETAS model and the smallest value.
triggering earthquake The key parameter of the ETAS model is defined as the
number n of direct aftershocks per earthquake, averaged
2.1. Definition of the ETAS model overallmagnitudes. Here,wemustdistinguishbetweenthe
two cases α=b and α6=b:
To make this discussion precise, let us consider the
epidemic-typeaftershock sequence(ETAS)model, in which mmax
anyearthquakemaytriggerotherearthquakes,whichinturn
n ≡ P(m)ρ(m)dm
maytriggermore,andsoon. Introducedinslightlydifferent
Z
forms by Kagan and Knopoff [1981] and Ogata [1988], the
m0
model describes statistically the spatio-temporal clustering
kb 1−10−(b−α)(mmax−m0)
of seismicity. = ( ), (4)
The triggering process may be caused by various mecha- b−α 1−10−b(mmax−m0)
nismsthateithercompeteorcombine,suchaspore-pressure
for thegeneral case α6=b. The special case α=b gives
changes due to pore-fluid flows coupled with stress varia-
tions, slow redistribution of stress by aseismic creep, rate-
and-statedependentfrictionwithinfaults,couplingbetween n= kbln(10)(mmax−m0) (5)
the viscoelastic lower crust and the brittle upper crust, 1−10−b(mmax−m0)
stress-assisted micro-crack corrosion, and more. The ETAS
Threeregimescanbedistinguishedbasedonthevalueof
formulationamountstoatwo-scaledescription: theseabove
n. Thecasen<1correspondstothesubcritical,stationary
physical processes controlling earthquake interactions enter
regime,whereaftershocksequencesdieoutwithprobability
in the determination of effective triggering laws in a first
one. The case n > 1 describes unbounded, exponentially
step and the overall seismicity is then seen to result from
growing seismicity [Helmstetter and Sornette, 2002]. In ad-
the cascade of triggering of events triggering other events
dition,thecaseb<αleadstoexplosiveseismicitywithfinite
triggeringothereventsandsoon[Helmstetter andSornette,
timesingularities[SornetteandHelmstetter,2002]. Thecrit-
2002].
ical case n=1 separates the two regimes n<1 and n>1.
The ETAS model consists of three assumed laws about
Helmstetter andSornette[2003b]showedthatthebranching
the nature of seismicity viewed as a marked point-process.
ratio n is also equal to the fraction of triggered events in a
We restrict this study to the temporal domain only, sum-
seismic catalog.
ming over the whole spatial domain of interest. First, the
The fact that we use the same value for the productiv-
magnitude of any earthquake, regardless of time, location,
ity cut-off and the Gutenberg-Richter (GR) cut-off is not
ormagnitudeof themothershock,isdrawn randomly from
the exponential Gutenberg-Richter (GR) law. Its normal- a restriction as long as the real cut-off for the Gutenberg-
ized probability density function (pdf) is expressed as Richter law is smaller than or equal to the cut-off for the
productivity law. In that case, truncating the GR law at
bln(10)10−bm the productivity cut-off just means that all smaller earth-
P(m)= 10−bm0 −10−bmmax,m0≤m≤mmax, (1) quakes, which do not trigger any events, do not participate
in the cascade of triggered events. This should not be con-
where theexponent b is typically close to one, and thecut- fused with the standard incorrect procedure in many pre-
offs m0 and mmax serve to normalize the pdf. The upper vious studies of triggered seismicity of simply replacing the
cut-off mmax is introduced to avoid unphysical, infinitely GR and productivity cut-off m0 with the detection thresh-
large earthquakes. Its value was estimated to be in the old md in equations (1) and (3) (see, for example, Ogata
range8−9.5[Kagan,1999]. Astheimpactofafinitemmax [1988]; Kagan [1991]; Ogata [1998]; Console et al. [2003];
isquiteweakinthecalculationsbelow,replacingtheabrupt Zhuang et al. [2004]). The assumption that md =m0 may
cut-off mmax by a smooth taper would introduce negligible lead to a bias in theestimated parameters.
corrections to our results. Without loss of generality, we consider one independent
Second,themodelassumesthatdirectaftershocksaredis- branch (cluster or cascade of aftershocks set off by a back-
tributed in time according to the modified “direct” Omori groundevent)oftheETASmodel. Letthusanindependent
law(seeUtsuetal. [1995]andreferencestherein). Assuming background event of magnitude M occur at some origin
1
SORNETTEANDWERNER: APPARENTCLUSTERINGANDBACKGROUNDEVENTS X - 3
of time. The mainshock will trigger direct aftershocks ac- dueto tectonic loading, may each independentlytrigger di-
cording to the productivity law (3). Each of the direct af- rectaftershocks,eachofwhichmayinturntriggersecondary
tershocks will trigger their own aftershocks, which in turn shocks, which in turn may trigger more. Because every
produce their own, and so on. Averaged over all magni- triggered event,excludingofcoursethenon-triggered back-
tudes, an aftershock produces n direct offsprings according ground events, has exactly one mainshock (mother), but
to (4). Thus, in infinite time, we can write the average of the mother may have many direct aftershocks (children),
thetotalnumberNtotal ofdirectandindirectaftershocksof the model can be thought of as a simple branching model
the initial mainshock as an infinite sum over terms of (3) without loops. The background eventsare assumed to bea
multiplied by nto thepower of thegeneration [Helmstetter
stationary Poisson process with a constant rate. The rate
and Sornette, 2003b], which can be expressed for n<1 as:
oftheaftershocksofabackgroundeventisanon-stationary
Poisson process that is updated every time another after-
N =ρ(M )+ρ(M )n+ρ(M )n2+...
total 1 1 1 shock occurs until the cascade dies out. The intensity is
k10α(M1−m0) thusconditionedonthespecifichistoryofearthquakes. The
= (6)
1−n expectation of the conditional intensity is an average over
an ensemble of histories. The predicted number of after-
However,sincewecanonlydetecteventsabovethedetection shocks of an independent background event of magnitude
thresholdmd,thetotalnumberofobservedaftershocksNobs M1 as in expression (8) is thus averaged over the ensemble
of the sequence is simply Ntotal multiplied by the fraction of possible realizations of the aftershock sequence, and it is
of eventsabove thedetection threshold, given by alsoaveragedoverallpossiblemagnitudesoftheaftershocks.
The branchingratio n istherefore an average not only over
10b(mmax−md)−1 magnitudes but also over an ensemble of realizations of the
f = (7)
obs 10b(mmax−m0)−1 non-stationary Poisson process. The model thusconsists of
statistically independent Poisson clusters of events, which
according to the GR distribution. The observed number of are, however, dependent within onecluster.
eventsin the sequenceis therefore
The second view of the ETAS model does not allow a
unique identification of the mother or trigger of an earth-
N =N f
obs total obs quake. Rather,eachaftershockwastriggeredcollectivelyby
k10α(M1−m0) 10b(mmax−md)−1 allpreviousearthquakes,eachofwhichcontributesaweight
= . (8)
1−n (cid:18)10b(mmax−m0)−1(cid:19) determined by the magnitude-dependent productivity law
ρ(m) that decays in time according to the Omori law ψ(t)
Equation (8) predicts the average observed number of di- andinspaceaccordingtoaspatialfunctionR(r),oftencho-
rect and indirect aftershocks of a mainshock of magnitude sen to be an exponential or a power law centered on the
M > m . Sornette and Werner [2004] showed that m event. Theinstantaneousconditionalintensityrateatsome
1 d 0
maybeestimatedusingfitsofN givenby(8)toobserved time t at location r is given by
obs
aftershock sequences and B˚ath’s law. The essential param-
eterneededtoconstrainm0 isthebranchingration. Aswe λ(t,r)=µ+ ρ(mi)ψ(t−ti)R(r−ri) (9)
demonstrate below, typical estimates of n in the literature
obtained from a catalog neglect undetected seismicity and iX|ti<t
therefore cannot be used directly toconstrain m .
0 where the sum runs over all previous events i with mag-
Naturally, there is no justification for assuming that m
should equal m , as is done routinely in inversions of catad- nitude mi at time ti at location ri. Thus the triggering
0 contributionofapreviouseventtoalatereventattimetis
logs for the parameters of the ETAS model (see, for exam-
givenbyitsownweight(itsspecificentryinthesum)divided
ple, Ogata [1988]; Kagan [1991]; Ogata [1998]; Console et
by the total seismicity rate, including the background rate.
al. [2003]; Zhuang et al. [2004]). First, detection thresh-
A non-zero background rate then contributes evenly to all
olds change overtime as instruments and network coverage
eventsandcorrespondstoanomnipresentloadingcontribu-
become better, while thephysical mechanisms in theEarth
presumablyremain thesame. Nosignificant deviationfrom tion. Inthisway,earthquakesareseentobetheresultofall
the Gutenberg-Richter distribution or the productivity law previous activity including the background rate. This cor-
has been recorded as the detection threshold m decreased responds to a branching model in which every earthquake
d
overtime. Second,studiesofearthquakeoccurrenceatsmall links to all subsequent earthquakes weighted according to
magnitude levels below the regional network cut-offs show thecontributiontotriggering. Abranchingratiocanthenbe
thatearthquakesfollowthesameGutenberg-Richterlaw(for interpretedasacontributionofapastearthquaketoafuture
a recent study of mining-induced seismicity, see, for exam- earthquake, averaged over an ensemble of realizations and
ple, Sellers et al. [2003]), while acoustic emission experi- allmagnitudes. Thissecondviewbecomestheonlypossible
ments have shown the relevance of the Omori law at small onefornonlinearmodelswhosetriggering functionsdepend
scales (see, for instance Nechad et al. [2004] and references nonlinearly on previous events (see e.g. the recently intro-
therein). Withintheassumptionofself-similarity,i.e. acon- ducedmulti-fractal earthquaketriggering model by Ouillon
tinuationoftheGRandproductivitylawsdowntoacut-off, and Sornette [2004] and references therein).
evidence thus points towards a magnitude of the smallest Thesetwoviewsareequivalentbecausethelinearformu-
triggering earthquakeand a Gutenberg-Richtercut-off that lation of the seismic rate of the ETAS model together with
lie below the detection threshold and are thus not directly the exponential Poisson process ensures that the statisti-
observable.
cal properties of the resulting earthquake catalogs are the
same. Thelinearsumovertheindividualcontributionsand
2.2. Two interpretations of the ETAS model
thePoissonprocessformulationarethekeyingredientsthat
The ETAS model may be viewed in two mathematically allow the model to beviewed as a simple branchingmodel.
equivalent but interpretionally different ways. In this sec- This duality of thinking about the ETAS model is re-
tion,wedevelopbothviewstounderlinethatourresultsap- flected in the existence of two simulation codes in thecom-
ply in both cases and to stress theequivalence of these two munity, each inspired by one of the two views. A program
views. The first describes the model as a simple branching written by K. Felzer and Y. Gu (personal communication)
model without loops: The independent background events, calculates the background events as a stationary Poisson
X - 4 SORNETTEANDWERNER: APPARENT CLUSTERINGANDBACKGROUNDEVENTS
process and then simulates each cascade independently of magnitude down to 3 (Helmstetter [2003]; Helmstetter and
the other branches as a non-stationary process. The sec- Sornette [2003a]) suggeststhatm issmaller thanthecom-
0
ondcodebyOgata[1998], ontheotherhand,calculatesthe pletenessmagnitudem andisthusnotdirectlyobservable.
d
overallseismicity ateachpointintimebysummingoverall Thus, m is the size of the smallest triggering earthquake,
0
previous activity. The latter code is significantly slower be- which most likely differs significantly in size from the cur-
cause the independence between cascades is not used, and rent detection threshold m . By considering the branching
d
theentirecatalogismodeledasthesumofastationaryand structure of the model, we derive the apparent branching
a non-stationary process. Despite the different approach,
ratio and the apparent background source that are found
both resulting earthquake catalogs share the same statisti-
if only the observed (detected) part of the ETAS model is
cal properties and are thus equally acceptable. In the in-
analyzed.
terpretation of the model as a simple branching model, the
Asstatedabove,theETASmodelconsistsofstatistically
parameter n defined in (4) would correspond to a branch-
independentPoisson-distributedclusters. Withineachclus-
ing ratio, while the view that aftershocks are triggered col-
ter,eachshockhasexactlyonetrigger,apartfromtheinitial
lectively by all previous earthquakes and the background
backgroundevent(mainshock)thatsetsoffthecascade. We
ratewouldinterpretnasanaveragecontributionofasingle
restrictthisstudytothecasen<1formathematicalconve-
earthquake on future earthquakes. The important point is
nience and because this range of branching ratios gives rise
that thestatistical properties are thesame.
While the simulation or forward problem is straight- to statistically stationary seismic sequences.
forward when adopting the view of the ETAS model as a Since aftershock clusters are independent of each other,
branching model with one assigned trigger for any after- averages of one cluster are equal to ensemble averages, as
shock, the inverse problem of reconstructing the branching nothing but the inherent stochasticity of the model deter-
structure from a given catalog can at best be probabilistic. mines theproperties of theclusters. Oneclusterconsists of
Because aftershocks of one mother cannot be distinguished one independent background event (source) and its direct
fromthoseofanothermotherexceptbyspatio-temporaldis- and indirect aftershocks. However, if not all events of the
tance, we have no way of choosing which previous earth- sequence are detected, then there will appear to be less di-
quake triggered a particular event, or whether it is a back- rect (and indirect) aftershocks, i.e. thebranching ratio will
ground event. Rather, we must resort to calculating the appeardifferent. Furthermore,someobservedeventswillbe
probability of an event at time t to be triggered by any triggeredbymother-earthquakesbelowthedetectionthresh-
previouseventaccordingtothecontributionthattheprevi- old,resultinginapparentlyindependentbackgroundevents.
ouseventhasat timetcompared totheoverall intensityat We will refer to the observed branching structure above
time t. This probability is of course equal to the weight or the detection threshold as the apparent structure. As a
triggering contribution that a previous event has on a sub-
useful visualization of the effect of the detection threshold
sequent event when adopting the collective-triggering view.
on the branching structure, one can think of the branching
However,theinterpretationremainsdifferentsincetheprob-
structure as a mountain range that is submerged to some
ability specifiesa uniquemotherin afraction of manyreal-
fraction of its height in water (see Figure 1). The height of
izations.
each mountain corresponds to the magnitude of an earth-
Having determined from catalogs a branching structure
quake. To the left (i.e. backward in time) of each peak
weightedaccordingtotheprobabilityoftriggering,onemay
is one peak (the trigger) and to the right (i.e. forward in
ofcoursechoosetoalwayspickassourceofaneventthemost
time) of each peak may be several (the direct aftershocks),
probable contributor, be that a previous event or the back-
ground rate. Anotheroption is to choose randomly accord- connected from left to right via a ridge linking trigger to
ingtotheprobability distributionand thusreconstructone offspring. Sincetheheight ofthepeaksvaries(according to
possible branching structure among the ensemble of many the GR law), some of the peaks will be below water (below
other possible ones. The latter approach has been used by the detection threshold). A whole part of the branch may
Zhuang et al. [2004] and labeled stochastic reconstruction. be submerged in water (i.e. unobserved) until eventually a
The key point is that equating the detection threshold peak rises above the surface and appears to have no ridge
withthesmallesttriggeringearthquakewillmostlikelylead connecting it to a previous peak, because the ridge of the
to a bias in the recovered parameters of a maximum likeli- last observable mountain simply descends intothe water.
hood analysis as performed by Zhuang et al. [2004] and in This view leads to the conclusion that the average num-
many other studies. Therefore, the weights or probabilities ber of direct aftershocks that are observed will be less than
of previous events triggering subsequent events were calcu- the real branching ratio, since some of the triggered events
lated from biased parameters. of an observed shock will fall below m and hence not be
d
In the following, we show that the branching ratio and included inthecount. Onlythefraction f from equation
obs
the background source events are significantly biased when (7)abovem ofthetotaldirectaftershocksρ(m)willbeob-
d
they are estimated from the apparent branching structure
served. Moreover, the pdf P(m|m ≥ m ) of mother events
d
observed above the detection threshold m instead of the
d conditioned on being larger than m is zero for m < m
complete tree structure down to m . We adopt the view of d d
the simple branching model to ma0ke the derivations more and equal to P(m)/fobs for mmax >m≥md. Wecan thus
definethe apparent branchingratio as
illuminatingbutallresultscanbereinterpretedascontribu-
tions in thecollective-triggering view. mmax
na ≡ P(m|m≥md)ρ(m)fobsdm
3. The Apparent Branching Structure of Z
the ETAS Model m0
mmax
3.1. The apparent branching ratio na = P(m)ρ(m)dm (10)
Z
Seismiccatalogs areusuallyconsideredcompleteabovea
md
threshold m , which varies as a function of technology and
location. Fordinstance,md ≈2formodernSouthernCalifor- = kb 10−(b−α)(md−m0)−10−(b−α)(mmax−m0)
nia catalogs (and for earthquakesnot too close in time toa b−α(cid:18) 1−10−b(mmax−m0) (cid:19)
large mainshock [Kagan, 2003]). The analysis of the statis-
ticsoftheOmoriandinverseOmorilawsforearthquakesof for thecase α6=b. The special case α=b gives
SORNETTEANDWERNER: APPARENTCLUSTERINGANDBACKGROUNDEVENTS X - 5
where mmax = 8.5, md = 3, θT = 0.08, AT = 0.116
days−θT, b = α = 1, c = 0.014 and M = 6.04. Using
1
na = kb1l−n(1100−)(bm(mmmaaxx−−mm0)d). (11) tShoerndeetctleusatnedrinWgeprnerefror[m20e0d4]boybRtaeianseednberg and Jones[1989],
Usingequation(4)andeliminatingk,wehavenainterms n θcθ10−a
of n: m0 =mmax− 1−n bln(10)(1−10−b(mmax−md)) (16)
10(b−α)(mmax−md)−1
na =n(cid:18)10(b−α)(mmax−m0)−1(cid:19), (12) where mmax = 8.5, md = 3, θ = 0.08, a = −1.67, c = 0.05
and b=1. Finally, using B˚ath’s law, Sornette and Werner
[2004] found
when α6=b, and
n (1−10−b(mmax−md))
na =n mmax−md , (13) m0=mmax−(1−n) bln(10) 10b(M1−ma)
(cid:16)mmax−m0(cid:17) (17)
when α=b.
Accordingtoexpression (12),na ≤n,wheretheequality where M1 − ma = 1.2 according to the the law, b = 1,
holds for m equal to m . In principle, equation (12) also mmax =8.5, and md =3.
d 0
holdsfor n>1,but werestrict thisstudytothestationary Substituting these four estimates of m0 from equations
regime n < 1. Figure 2 shows na as a function of n for a (14), (15), (16), and (17) intoequation (12) forna provides
range of values of m0 for the case α = b. The values of fourestimatesofnaversusnallintermsofknownconstants.
m0 are m0 = md = 3 (solid), m0 = 0 (dashed), m0 = −5 These four estimates of n as a function of na can be used
(dotted)andm =−10(dash-dotted). Weassumedm =3 to find the correct fraction of aftershocks from the measur-
0 d
and mmax =8. Specifying m0 then fixes the linear slope of able apparent fraction of aftershocks. Figure 4 shows these
the dependence between n and na. Figure 2 demonstrates fourestimateswiththeaboveconstants. Asnotedabove,n
that the apparent (measurable) fraction of aftershocks may varieslinearly with na withaslopedeterminedbym0. The
significantly underestimate the true fraction of aftershocks four estimates of nas afunction of na allow for all possible
evenfor m0 not verysmall. Forexample, m0=−5roughly values of m0. The solid line n = na, corresponding to the
translates a real branching ratio of n = 0.9 into an appar- slope 1whenm =m ,separatestheleft sideof thegraph,
0 d
ent branching ratio na = 0.3. Decreasing α below b places wherem0<md,fromtherightside,wherem0 ≥md,which
more importance on the triggering from small earthquakes
can be ruled out based on observed aftershocks from mag-
and therefore strongly amplifies this effect.
nitudes 2. Thus, only the region to the left of the diagonal
In Figure 3, we plot the ratio na/n as a function of the should beconsidered.
unknown m0. Again, we assume md =3, mmax =8, b=1, Figure 4 can be used to find the real fraction of after-
butnowweletα=0.5(dash-dotted),α=0.8(dashed)and
shocks from the measured apparent fraction by assuming
α = b = 1.0 (solid). As expected, when m = m , the ra-
0 d one of the four estimates of m as a function of n. For ex-
tio is one because there is no unobserved seismicity. As m 0
0 ample, Helmstetter et al. [2004] find that 55 percent of all
goes to minus infinity, na approaches zero since almost all
seismicity occurs below the threshold. We see clearly that earthquakesare aftershocks above md =3. Using their val-
unobserved seismicity results in a drastic underestimate of uestoestimatem0asafunctionofn,wecandeterminethat
thefraction of aftershocks. therealfraction ofaftershocksiscloserto75percent. Thus
Given an estimate of the magnitude of the smallest trig- thesizeofthiseffectissignificant. Furthermore,havingde-
gering earthquakem0 (see Sornette and Werner [2004] and termined a point on the line estimating n from na for all
referencestherein),onecancalculatethetruebranchingra- values of m0 fixes theslope of n(na) and therefore m0. Us-
tiofromtheapparentbranchingratio. Infact,Sornette and ingtheirvalues,wefindm =1.2. Similarestimatescanbe
0
Werner[2004]obtainedfourestimatesofm0asafunctionof madeusingtheapparentfractionofaftershockvaluesfound
n by comparing the ETAS model prediction of the number by Felzer et al. [2002] and Reasenberg and Jones [1989].
of observed aftershocks (8) from fits to observed aftershock Assuming that current maximum likelihood estimation
sequences and from the empirical B˚ath’s law. Their equa- methods of the ETAS model parameters, which assume
tions (10), (13), (16) and (18) are the estimates of m0 as a m0 = md, determine a branching ratio that corresponds
function of n and a number of known constants specific to
to the present apparent branching ratio, we can similarly
thefitsto observed aftershock sequences. Wecan usethese
correct these values to find the true fraction of aftershocks
relationsofm asafunctionofntoeliminatem fromequa-
0 0 using Figure 4. For example, Zhuang et al. [2004] find a
tion(12)toobtaindirectestimatesofnasafunctionofthe
“criticality parameter” of about 45 percent, which we take
fimnedaisnugrsabtoletnhae.caFsoerαsi=mpbl.icTithye, wesetimreasttreicrtestuhletinugsefroofmthtehier as a proxy for na. Figure 4 shows that the true branching
ratio then lies between 0.45 and 0.80, depending on which
fitsperformed byHelmstetter et al. [2004] yielded
estimate(among thefourmodels(14), (15),(16), and(17))
n θcθ 1−10−b(mmax−md) of m0 as a function of n is chosen. These calculations sug-
m0=mmax−(1−n)K bln(10) (14) gest that previous estimates of the fraction of aftershocks
fit obtained by various declustering methods significantly un-
derestimated its value.
with the values mmax = 8.5, md = 3, θ = 0.1, c = 0.001,
b=1andK =0.008. ThestudyinitiatedbyFelzer et al.
fit
3.2. Determination of apparent background events
[2002] provided anotherestimate
Sa of uncorrelated seismicity
n (1−10−b(mmax−md))
m0 = mmax− 1−n bln(10) cadIentohradterarteondoetrivtreigtgheerendumbybearmofosthhoerckasbwovitehtinheotnherecsahs--
×θTcθT 10b(M1−md), (15) old and thus appear as independent background events, we
AT needtodistinguishbetweenthecasewheretheinitial(main)
X - 6 SORNETTEANDWERNER: APPARENT CLUSTERINGANDBACKGROUNDEVENTS
shock of magnitude M is observable (i.e. M ≥ m ) and n for an example aftershock cascade set off bya magnitude
1 1 d
thecase where it is undetected (i.e. M1 <md). m = 5 initial shock. We assumed the values mmax = 8.5,
If M1 ≥ md, then the initial mainshock produces md = 3 and α = b = 1.0. The figure shows that for one
ρ(M )f observed direct aftershocks. On average, these cascade, i.e. one independent background event, hundreds
1 obs
will in turn collectively produce ρ(M1)fobsna observed sec- ofearthquakesappearasapparentbackgroundeventswhen
ond generation aftershocks. Wespecifically do not consider m0 <md.
events above m triggered from below m , which we deal InFigure6,weinvestigatetherelativeimportanceofthe
d d
apparent background events with respect to the observed
with below in the definition of the apparent background
number of aftershocks of one cascade. According to equa-
sources. By continuing this “above-water” cascade for all
tion (22)
generations of aftershocks, we can calculate the number of
triggered eventsthatare indirect lineage abovethethresh-
old back to the mainshock as the infinite sum of terms of Sa/Nobs =n−na, (23)
ρth(Mep1o)wfoebrsomfuthlteipgleinederbaytiotnh.eIafp,ponartehnetobtrhaenrchhainngd,rathtieoinniatitaol i.e. asignificantfractionn−na ofeventsoftheactuallyob-
servedaftershocksarefalselyidentifiedasbackgroundevents
mainshock is below m , then no such direct “above-water”
d (sincealleventsarereallytriggeredfromasinglemainshock
cascade will be seen. Any observed shock will be triggered
in our example). For m = m , the ratio is zero, since no
by an event below the water. Thus, for the two cases, the d 0
events trigger below the detection threshold. However, as
“above-water” sequenceis expressed as:
m decreases and more and more events fall below m , the
0 d
Nabove=(cid:26) ρ(M1−1n)faobs =Nobs11−−nna0 ,,MM1 ≥<mmd (18) vpfraraolcuatecisohneosfiαnnc.greeTanhseeirssaleluffyneptciltlacinneacmrgeooarseeessimtwopitzoherrtdoaneaccrneedaosntinhtgeheαrac.utimSomualapal--l
1 d
tive triggering of small earthquakes.
Furthermore, since in the ETAS model, a small earthquake
maytriggerlargeearthquakes,aneventbelowm maypro- 3.3. Consistency check: Nobs as the sum of “above-
d
duce an observed event above m . An inversion method water” cascades triggered by the mainshock and by
d
thatreconstructstheentirebranchingstructureofthemodel the apparent background events
from an earthquake catalog will identify these shocks as
Tocompletethecalculationsandshowconsistencyofthe
background events. But since in reality these events were
results, wenow demonstratethat theobserved cascades set
triggered by earthquakes below the detection threshold, we
off by the apparent background events, when added to the
will refer to them as apparent background events. These
original“above-water”cascade,adduptothetotalobserved
events can of course trigger their own cascades. We thus
numberofaftershocksofthewholesequence. Eachapparent
defineapparentbackgroundsourceSa asthenumberofob- sourceeventwilltriggeritsowncascadeabovethethreshold
served events above m that are apparently not triggered,
i.e. have “mothers” bedlow m . Again, we distinguish be- mdwithbranchingrationa. Thetotalnumberofeventsdue
d totheapparentbackgroundeventsandtheircascadesabove
tweenthecaseswherethemainshockmagnitudeisM1 ≥md the threshold is
andM1 <md. Forthefirst,Sa isgivenbythetotalnumber
oafgeafntuermshboecrkrs bofeldowiretchteafttherresshhooclkdsmthueltyiptlriiegdgebryatbhoeveavtehre- Nsource=Sa+Sana+San2a+...= 1−Sana. (24)
threshold. Forthesecondcase,wemustalsoincludethedi-
rect aftershocks of the initial mainshock that are observed: Substitutingexpression (22) and using (8) gives
Sa =(cid:26) ρ1(M−n1)(1−fobρs1()M−rn1+)(ρ1(−Mf1)ofbso)brs ,,MM11 ≥<mmdd(19) Nsource=(cid:26)Nobs(n1−−nnaa) + ρ(NMo11b−−smn(n10a−−)fnnoaabs) ,,MM11 ≥<mmdd
(25)
Now, the number r of observable direct aftershocks above
m averaged over unobserved mothers between m and m
d 0 d Combining the direct “above-water” cascade (18) with the
is given by thefollowing conditional branchingratio:
apparentsourcecascades (25)givesthetotalamountofap-
md parent eventsobserved after the initial event
r ≡Z P(m|m<md)ρ(m)fobsdm (20) Na = Nsource+Nabove
m0 = Nobs(n1−−nnaa) + ρ(M11−−mn0a)fobs ,M1 ≥md
f (cid:26)N (n−na) + ρ(M1−m0)fobs +0 ,M <m
=(n−na) obs , (21) obs 1−na 1−na 1 d
(cid:18)1−fobs(cid:19) = Nobs, (26)
where we have used P(m|m < md) = P(m)/(1−fobs) for where Nobs is given by (8). The last equality confirms the
m<md and zero otherwise. Substituting (21) into the ex- consistency of our decomposition into apparently-triggered
pressionfortheapparentsource(19)andre-arrangingusing earthquakesand apparent sources.
(8), we obtain Expressions (10) and (22) show that analyzing the tree
structure of triggered seismicity only above the detection
Sa = (cid:26) ρ1(M−n1)fobs(n−ρ1(n−Man1))+fobρs((Mn1−)fnoabs) ,,MM11 <≥mmdd iatzhnerdetshahnaotladbpopletaahrdeasnrtteobrtrehanneocrihmnitnarolgidzreuadctitsoiiomnnauo.lftIaatnniesaoipumpslpayroebrntyatnmstodutor6=cremeaS0la-.
IfthecurrentinversiontechniquesfortheETASparameters
= Nobs(n−na) ,M1 ≥md (22) wereperfectandcorrectlyreconstructedthetreestructureof
(cid:26)Nobs(n−na)+ρ(M1)fobs ,M1 <md all sequences,theinvertedvalueswould beequaltoouran-
alytical results (10) and (22). Accordingly, the value of the
Equation (22) shows that, for each genuine background backgroundsourcewouldbeoverestimated andthebranch-
event, a perfect inversion method would count Sa apparent ing ratio underestimated. In fact, one single true sequence
background events. Figure 5 plots the number of apparent willappearasmanydifferentsequences,eachapparentlyset
background events Sa as a function of the branching ratio off by an apparent background event.
SORNETTEANDWERNER: APPARENTCLUSTERINGANDBACKGROUNDEVENTS X - 7
4. Conclusions Helmstetter, A., and D. Sornette, (2003), Importance of
direct and indirect triggered seismicity in the ETAS
We have shown that unbiased estimates of the fraction model of seismicity, Geophys. Res. Lett., 30(11), 1576,
of aftershocks and the number of independent background doi:10.1029/2003GL017670.
events are simultaneously renormalized to apparent values Kagan, Y.Y. (1991), Likelihood analysis of earthquake catalogs,
when thesmallest triggering earthquakem0 is smaller than Geophys. J. Int. 106,135-148.
thedetectionthresholdm . Insummary,mainshocksabove
d Kagan, Y. Y. (1999), Universality of the seismic moment-
thethreshold will appear to haveless aftershocks, resulting frequencyrelation,Pure and Appl. Geophys. 155,537-573.
in a smaller apparent branching ratio. Meanwhile, unob-
Kagan, Y.Y.(2003), Accuracy ofmodernglobalearthquake cat-
served events can trigger events above the threshold giv-
alogs,Phys. Earth Plan. Inter. 135, 173-209.
ing rise to apparently independent background events that
Kagan, Y. Y., and L. Knopoff (1981), Stochastic synthesis of
seem to increase the constant background rate to an ap-
earthquake catalogs,J. Geophys. Res., 86,2853-2862.
parent rate. Assuming that current techniques which are
Nechad, H., A. Helmstetter, R. El Guerjouma and D. Sor-
used to invert for the parameters of the ETAS model (for
nette (2004), Andrade and critical time-to-failure laws in
example, the maximum likelihood method) under the as-
fiber-matrix composites: Experiments and model, in press
sumption md = m0 are unbiased estimators of na and Sa, in Journal of Mechanics and Physics of Solids (JMPS)
then theobtained values for thefraction of aftershocks and
(http://arXiv.org/abs/cond-mat/0404035)
thebackgroundsourceratecorrespond torenormalized val-
Ogata, Y. (1988), Statistical models for earthquake occurrence
ues because of the assumption that the detection threshold
andresidualanalysisforpointprocesses,J.Am.Stat.Assoc.,
m equals the smallest triggering earthquake m . We pre-
d 0 83,9-27.
dictthatnwillbedrasticallyunderestimatedandSstrongly
Ogata,Y.(1998),Space-timepoint-processmodelsforearthquake
overestimated for m much smaller than m .
0 d occurrence,Ann.Inst.Statist.Math.,Vol.50,No.2,379-402.
Acknowledgments. We acknowledge useful discussions Ouillon,G.andD.Sornette(2004),Magnitude-dependentOmori
with A. Helmstetter and J. Zhuang. This work is partially sup- law: Theoryandempiricalstudy,inpressinJ. Geophys. Res.
ported by NSF-EAR02-30429, and by the Southern California (http://arXiv.org/abs/cond-mat/0407208).
Earthquake Center (SCEC). SCEC is funded by NSF Coopera-
Sellers, E. J., M. O. Kataka, and L. M. Linzer (2003), Source
tiveAgreementEAR-0106924andUSGSCooperativeAgreement
parameters of acoustic emission events and scaling with
02HQAG0008. The SCEC contribution number for this paper
mining-induced seismicity, J. Geophys. Res., 108(B9), 2418,
is xxx. MJW gratefully acknowledges financial support from a
doi:10.1029/2001JB000670.
NASAEarthSystemScienceGraduate StudentFellowship.
Sornette,D.,andA.Helmstetter(2002),Occurrenceoffinite-time
singularities in epidemic models of rupture, earthquakes and
starquakes,Phys. Rev. Lett.Vol. 89, No. 15,158,501.
References Sornette, D. and M. J. Werner (2004), Constraints on
the Size of the Smallest Triggering Earthquake from
Athreya, K. B. and P. E. Ney (1972), Branching Processes, the ETAS Model, B˚ath’s Law, and Observed Af-
SpringerVerlag,Berlin,Germany. tershock Sequences, J. Geophys. Res., submitted.
Console, R., M. Murru, and A. M. Lombardi (2002), Refining (http://www.arxiv.org/abs/physics/0411114)
earthquake clustering models, J. Geophys. Res., 108(B10), Utsu, T., Y. Ogata, and R. S. Matsu’ura (1995), The centenary
2468,doi:10.1029/2002JB002123. oftheOmoriformulaforadecaylawofaftershockactivity,J.
Felzer, K. R., T. W. Becker, R. E. Abercrombie, G. Ek- Phys. Earth, 43,1-33.
s7t.1ro¨mHe,ctaonrdMJi.nRe.eaRrtichequ(a2k0e02b)y, Tafrtiegrgsehroinckgsoofftthhee11999992 MMWW Zhuqaunagk,eJc.l,uYst.eOrignagtafe,aatnudreDs.bVyeursei-nJgonsteosc(h2a0s0t4ic),rAecnoanlsytzriuncgtieoanr,thJ-.
7.3 Landers earthquake, J. Geophys. Res., 107(B9), 2190,
Geophys. Res. 109, B05301,doi:10.1029/2003JB002879.
doi:10.1029/2001JB000911.
Helmstetter, A. (2003), Is earthquake triggering driven by
small earthquakes?, Phys. Rev. Lett. 91, 10.1103/Phys- D. Sornette, Department of Earth and Space Sciences, and
RevLett.91.058501. InstituteofGeophysicsandPlanetaryPhysics,UniversityofCal-
Helmstetter,A.,Y.Y.Kagan,andD.D.Jackson(2004), Impor-
ifornia, Los Angeles, California 90095, USA and Laboratoire de
tanceofsmallearthquakes forstresstransfersandearthquake
Physique de la Mati`ere Condens´ee, CNRS UMR6622, Univer-
triggering,J. Geophys. Res.,submitted.
sit´edeNice-SophiaAntipolis,ParcValrose,06108NiceCedex2,
Helmstetter, A., and D. Sornette (2002), Subcritical and super-
France. ([email protected])
criticalregimesinepidemicmodelsofearthquakeaftershocks,
J.Geophys.Res.,107(B10),2237,doi:10.1029/2001JB001580. M. J. Werner, Department of Earth and Space Sciences,
Helmstetter,A.andD.Sornette(2003),Foreshocksexplainedby and Institute of Geophysics and Planetary Physics, Uni-
cascades of triggered seismicity, J. Geophys. Res. 108 (B10), versity of California, Los Angeles, California 90095, USA.
245710.1029/2003JB002409 01. ([email protected])
X - 8 SORNETTEANDWERNER: APPARENT CLUSTERINGANDBACKGROUNDEVENTS
Figure 1. Schematic representations of the branching
structureoftherealETASmodel(left)andtheapparent
ETAS model (right). The initial mainshock is circled.
Only events above the detection threshold m are ob-
d
served. The apparent branching ratio does not take into
account unobserved triggered events (dashed lines). An
observed event triggered by a mother below m appears
d
as an untriggered background source event (circled).
1
0.9
0.8
a
n
s 0.7
k
c
o
h
s
er 0.6
aft
n of 0.5
o
cti
nt fra 0.4
e
ar 0.3
p
p
A
0.2
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
fraction of aftershocks n
Figure 2. The apparent fraction of aftershocks (ap-
parent branching ratio) na varies linearly with the real
fraction of aftershocks (real branching ratio) n with a
slopefixedbythesmallest triggering earthquakem . As
0
m decreases,theapparentfractionofaftershockssignifi-
0
cantlyunderestimatestherealfraction. Asexamples,we
chose m0 = md = 3 (solid), i.e. na = n and no events
are missed; m = 0 (dashed); m = −5 (dotted); and
0 0
m = −10 (dash-dotted). We further assumed parame-
0
ters md = 3, mmax = 8, b = 1, and α = 1.0. A small
valueof α amplifies this effect (see Figure 3).
SORNETTEANDWERNER: APPARENTCLUSTERINGANDBACKGROUNDEVENTS X - 9
1
0.9
0.8
0.7
0.6
n
/ a
o n0.5
ati
R
0.4
0.3
0.2
0.1
0
−10 −8 −6 −4 −2 0 2 4
Magnitude m of the smallest triggering earthquake
0
Figure 3. The ratio of the apparent fraction of after-
shocks (apparent branching ratio) na over the real frac-
tion of aftershocks (real branching ratio) n varies as a
function of the smallest triggering earthquake m . For
0
m0 =md, na = n and all events are detected above the
threshold. For a small value of m , the ratio becomes
0
small, indicating that na significantly underestimates n.
Decreasing α amplifies this effect. We used parameters
md = 3 (vertical reference line), mmax = 8, b = 1. We
varied α=0.5 (dash-dotted), α=0.8 (dashed), α=1.0
(solid).
X - 10 SORNETTEANDWERNER: APPARENT CLUSTERINGANDBACKGROUNDEVENTS
1
0.9
0.8
s n0.7
nt
e
v
e0.6
d
e
er
g0.5
g
of tri
n 0.4
o
cti
a
Fr0.3
0.2
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Apparent fraction of triggered events n
a
Figure 4. The fraction of aftershocks (branching ra-
tio) n can be estimated from the apparent fraction of
aftershocks (apparent branching ratio) na by using four
estimates of the smallest triggering earthquake m as
0
a function of n as determined in Sornette and Werner
[2004] (see text). Theestimates of m asa function of n
0
wereobtainedfromcomparisonsoftheETASmodelpre-
diction ofthenumberofobserved aftershocksandfitsto
observed aftershock sequences performed by Helmstetter
et al. [2004] (solid), Felzer et al. [2002] (dash-dotted),
Reasenberg and Jones [1989] (dotted) and from B˚ath’s
law (dashed). The additional diagonal solid line na = n
corresponds to m =m (no undetected events). Along
0 d
any of the four lines, m varies from minus infinity to
0
mmax. Given that we can rule out m0 ≥ md, we can
restrict the physical range to the left side of the refer-
ence curve na = n. Given an estimate of the apparent
(measured) fraction of aftershocks na, we can estimate
the real fraction from one of thefour lines. Because n is
linearly proportional to na, an estimate of na not only
determines n from one of the curves, but also estimates
m0fromtheslopeofthelineconnectingthepoint(na,n)
totheorigin. Forexample,theestimateofHelmstetteret
al. [2004] of 55 percent of observed aftershocks approx-
imately gives a real fraction n of 75 percent according
to their own fits to aftershocks and thereby determines
m0 =1.2 for mmax =8 and md =3.