JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 96, NO. B7, PAGES 11,959-11,971, JULY 10, 1991
Prediction
Probabilities
DUNCAN
From Foreshocks
CARR AGNEW
Instituteof Geophysicsand Planetary Physics,
Universityof California, La Jolla
LUCILE
M. JONES
U.S. GeologicalSurvey, Pasadena, California
When any earthquakeoccurs,the possibilitythat it might be a foreshockincreasesthe probabilitythat a
largerearthquakewill occurnearbywithin the next few days. Clearly,the probabilityof a very largeearthquakeoughtto be higherif the candidateforeshockwere on or near a fault capableof producingthat very
largemainshock,
especiallyif the fault is towardsthe end of its seismiccycle. We derivean expression
for
the probabilityof a major earthquakecharacteristic
to a particularfault segment,given the occurrence
of a
potentialforeshocknearthe fault. To evaluatethis expression,
we need:(1) the rate of backgroundseismic
activity
in thearea,(2) thelong-term
probability
of a largeearthquake
onthefault,and(3) therateatwhich
foreshocksprecedelarge earthquakes,
as a functionof time, magnitude,and spatiallocation. For this last
functionwe assumethe averagepropertiesof foreshocks
to moderateearthquakes
in California:(1) the rate
of mainshock
occurrence
afterforeshocks
decays
roughlyas t-1, so thatmostforeshocks
arewithinthree
daysof their mainshock,
(2) foreshocks
and mainshocks
occurwithin 10 km of eachother,and (3) the fraction of mainshockswith foreshocksincreaseslinearly as the magnitudethresholdfor foreshocksdecreases,
with 50% of the mainshockshavingforeshockswith magnitudeswithin threeunits of the mainshockmagnitude (within three days). We apply our resultsto the San Andreas,Hayward, San Jacinto,and Imperial
faults,usingtlie probabilities
of largeearthquakes
from the reportof the WorkingGroupon CaliforniaEarthquakeProbabilities(1988). The magnitudeof candidateevent requiredto producea 1% probabilityof a
largeearthquake
on the SanAndreasfault withinthreedaysrangesfrom a highof 5.3 for the segmentin San
GorgonioPassto a low of 3.6 for the CarrizoPlain.
Probably the most evil feature of an earthquakeis its
suddenness.It is true that in the vast majority of casesa
severe shock is heralded by a series of preliminary
shocksof slight intensity.... [but] only after the havoc
has been wrought does the memory recall the sinister
warningsof hypogeneaction.
C. G. Knott (1908, p. 10)
1.
INTRODUCTION
Many damagingearthquakeshave been precededby smaller
earthquakesthat occur within a few days and a few kilometers
of the mainshock [e.g., Jones and Molnar, 1979]; these are
referred
to as immediate
foreshocks.
If such foreshocks
could
be recognized before the mainshock, they would be very
effectivefor short-termearthquakeprediction;but so far no way
has been found to distinguish them from other earthquakes.
Even without this, the mere existenceof foreshocksprovides
some usefulpredictivecapacity. When any earthquakeoccurs,
the possibilitythat it might be a foreshockincreasesthe proba-
bilitythata largerearthquake
wili soonhappen
nearby.For
these results,the U.S. Geological Survey has issuedfour shortterm earthquake advisories after moderate earthquakes[e.g.,
Goltz, 1985]. A more recent study by Kagan and Knopoff
[1987] developed a model for the clustering of earthquakes
which could indicate areas of space and time in which larger
events might follow smaller ones. The size of these areas
dependedon the probability gain, the ratio of probability of an
earthquakegiven the occurrenceof a possibleprecursor(suchas
a foreshock)to the probability in the absenceof such a precursor [Kagan and Knopoff, 1977; Vere-Jones,1978; Aki, 1981].
For low levels of probability gain, Kagan and Knopoff [1987]
found that one-third of all earthquakeswith magnitudes4 and
abovefell within their predictedregions.
These results are from studiesof earthquakecatalogs;Jones
[1985] used a catalog for southernCalifornia, and Kagan and
Knopoff [1987] used one for central California. As a consequence, both papers give generic results about pairs of earthquakes,without much regard for other factors. But it ought to
be possibleto do better: the probability of a very large earthquake shouldbe higher if the candidateforeshockwere to occur
near a fault capableof producingthat mainshockthan if it were
southernCalifornia, Jones [1985] showed that after any earthquake there is a 6% probability that a secondone equal to or located in an area where we believe such a mainshock to be
larger than the first will follow within five days and 10 km of very unlikely. Moreover, the chanceof a candidateearthquake
the first. The probabilityis much lower for a secondearthquake actually being a foreshockshouldbe higher if the rate of backmuch larger than the first; for example, the probability of an ground(nonforeshock)activity were low.
earthquaketwo units of magnitudelarger is only 0.2%. Using
In this study we derive an expressionfor the probability of a
major earthquakefollowing a possibleforeshocknear a major
fault from the basic tenetsof probabilitytheory. This probability tums out to depend on the long-term probability of the
Copyright1991 by the AmericanGeophysicalUnion.
mainshock,the rate of backgroundseismicityalong the fault,
Paper number91JB00191.
and
0148-0227/91/91JB-00191 $05.00
mainshocksand foreshocks. We then apply this expressionto
11,959
some
assumed
characteristics
of
the
relations
between
11,960
AGNEWAND JONES:PREDICTION
PROBABILITIES
FROMFORESHOCKS
the San Andreasfault systemto developshort-termprobabilities within a day of the mainshock);then we expect a foreshock
for possibleearthquakewarningsbasedon possibleforeshocks. every 1000 years. If a comparablebackgroundearthquake
occurs,on average,annually,we get 1000 backgroundearthquakesper foreshock. If an earthquakeoccursthat could be
2.
MODELS FOR PROBABILITIES FROM FORESHOCKS
eitherone,we thenwouldassume
theprobability
to be 10-3 that
Becauseof the natureof seismicityalong major fault systems it is a foreshock,and so will be followed by a mainshockwithin
such as the San Andreas fault, we have been led to address cer- a day. This is low, but still far abovethe backgroundone-day
tain fundamental issues about the relationship between
foreshocksand large earthquakes.These major faults illustrate
in an extreme form the "maximum magnitude" model introduced by Wesnouskyet al. [1983], in which the frequencyof
the largest earthquakeson a fault zone is much higher than
would be predicted by the extrapolation of the frequencymagnitudedistributionfor backgroundearthquakes.FOr many
parts of the San Andreas fault this is a straightforwardconsequenceof the low level of present-dayseismicity. For instance,
along the Coachella Valley segmentof the San Andreas fault
(Figure 4) an extrapolationof presentseismicityto highermagnitudespredictsa magnitude7.5 earthquakeevery 2900 years,
whereas the recurrence rate estimated from slip-rate data is
200-300 years.
This behaviorimpliesthat the large characteristic
earthquakes
on a fault zone are not simply the largestmembersof the total
populationof earthquakesthere, but are somehowderived from
a different population. Foreshocksto such eventscan thus reasonably be regardedas also being a separateclass of events
from the backgroundearthquakes.A physicalmodel that might
underlie this is that some special failure process takes place
before characteristicearthquakes,with an enhancedrate of small
earthquakesand eventual failure on a large scale both being a
result of it. It is of coursealso possiblethat no such process
occurs; a moderate shock might, dependingon the details of
stressnearby, trigger only smaller events(in which caseit is a
mainshock)or largerones(makingit a foreshock),as suggested
by Brune [1979]. There would then be no innate difference
betweenbackgroundeventsand foreshocks;but we believe that
it remainsfruitful (as will be shown) to make at least a concep-
probability
of 5.5 x 10--6.
For a formal treatmentwe begin by definingevents(in the
probability-theory
meaningof the term):
B:
A backgroundearthquakehas occurred.
F:
A foreshock has occurred.
C:
A large (characteristic)
earthquakewill occur.
As notedabove,if a small backgroundshockwere to happenby
coincidencejust before the characteristicearthquake,we would
certainly classit as a foreshock. Thus, B and C cannotoccur
together:they are disjoint. The sameholds true for B and F:
we can have a foreshockor a backgroundearthquake,but not
both.
The probability that we seek is the conditionalone of C,
given either F or B, becausewe do not know which has
occurred. This is, by the definitionof conditionalprobability,
P(C1F
•B) = P(C
c•(F
c9B
))
P(F•B )
(1)
BecauseF and B are disjoint, the probabilityof their union is
the sum of the individualprobabilities,allowing us to write the
numeratorof (1) as
P ((C •
)•(C rag )) = P (C c•F ) + P (C •
) = P (C c•F )
where the disjoinmessof C and B eliminatesthe P (C •
)
term. From the definitionof conditionalprobability,
P(Cc"•) = P(F 1C)P(C)
whereP (F IC) is the probabilitythata mainshock
is preceded
tual division.
That we make this division does not mean that there are any by a foreshock. Again using the disjoinmessof F and B, we
characteristicsthat can distinguishbetweenforeshocksand other can write the denominator as
earthquakes;indeed, if there were, we would not have had to
consider the second model above. We can only identify
P(F•B)
= P(F) + P(B)
(2)
foreshocks,like aftershocks,by virtue of their associationwith a
larger event; and, as our opening quotation suggests,for Because a foreshock cannot, by definition, occur without a
foreshockssuch identificationcan only be retrospective. Such mainshock,the intersectionof C and F is F, and therefore
classificationby associationmeans that any particular shock
might have been classified "incorrectly", and actually have
(3)
P (F) = P (F c•C) = P (F IC)P (C)
been a backgroundshock that just happenedto fall close to a
larger event. In our presentstateof knowledgethis is unavoid- We can use (2) and (3) to write (1) as
able, and it may always remain so.
2.1. Zero-Dimensional
Model
P(CIFuB) = P (F)+P
P(F)(B) = P (FPIC(C)P
(FIC) (4)
)P (C) + P (B)
Starting from the assumptionthat foreshocksare a separate
is small(the candiclass of earthquakefrom backgroundearthquakes,we can set For P (B) >>P (F Ic)P (C) this expression
out a formal probabilisticschemefor findingthe probabilityof a date event is probably a backgroundearthquake),while for
large shock,given the occurrenceof a possibleforeshock. For P (B)= 0, the expressionbecomesequal to one: any candidate
claritywe beginwith a "zero-dimensional"
model,ignoring earthquakemust be a foreshock.
The secondform of expressionin (4) is a function of three
spatial variations,magnitudedependence,and other complications, which will be added in later sections. With these quantities, which in practice we obtain from very different
simplifications,a numerical example will illustrate the reason- sources. P (B), the probability of a backgroundearthquake,
ing. Supposethat mainshocksoccur every 500 years (on aver- would be found from seismicity catalogs for the fault zone.
age), and that half of them have foreshocks(defined as being P (C), the probability of a characteristicearthquake,would be
AGNEWAND JONES:PREDICTIONPROBABILITIES
FROM FORESHOCKS
11,961
found from calculationsof the type presentedby the Working the fraction of mainshockswith foreshocks;this and equation
Group on California Earthquake Probabilities [1988]. If we (7) togetherconstrainthe normalizationof •Fc.
Next to the probability level itself, the socially most interesthad a record of the seismicitybefore many such characteristic
earthquakes,
we could evaluateP(FIC) (which we shall ing quantitieswould seemto be the chanceof an alert being a
hereafter call •FC) from it directly. (For this simple model, false alarm, and the rate at which false alarms occur for a given
•Fc is just the fraction of large earthquakespreceded by probabilitylevel. The probabilitythat an alert is a false alarm
foreshocks.)Of course, we do not have such a record, and so is P((•IFi•JBi), whichis just 1- P(CIFi•Bi): if we havea
are forced to make a kind of reverse ergodic assumption, 10% chanceof having a mainshock,we have a 90% chanceof
namely that the time averageof •Fc over many earthquakeson not having one. The rate of false alarms is equivalent to the
one fault is equal to the spatialaverageover many faults. This probability of a false alarm happeningin some given time, and
this is just the probabilitythat an alert is a false alarm times the
may not be true, but it is for now the best we can do.
probabilityof the event that triggersit, namely
2.2. One-Dimensional
Model
N
As a simple extension to the previous discussion,suppose
•_•[1 - P (C IFi [.JBi
)] [P(Bi) + P (Fi)]
i=1
that we have N "regions" and that Ci, Bi, and Fi denotethe
occurrenceof an event in the i th region, with C (for example)
As will be shown in section4, we would in practice usually
now being the occurrenceof a large earthquakein any possible
choose
the probability of a mainshock given a small event,
region. Theseregionscan be sectionsof the fault or (as we will
see below) volumes in a multidimensionalspace of all relevant P (C IFicJBi)to havea fixedvalue(e.g., 1%), whichwe denote
variables.The quantityof interestis now ?(C [FiuBi): we by S, for all regions. This value of S then sets the value of
have a candidateforeshockin one region, and want the proba- P (Bi ) for the i th region; from (6), P (Bi ) = P (Fi )[(1 - S )/S ],
bility of a large earthquakestartinganywhere. Assumingthat which makesthe probabilityof a false alarm
the occurrences
Ci are disjoint(the epicente•r
can only be in one
place), we then have that the probabilityof a foreshockin the
i th regioncan be written as
N
P (Fi) = • •FC(i, j)P (Cj)
(5)
j=l
where•Fc(i ,j)= P (FiICj). We mayregard•Fc astheprobability of a foreshockin region i given a large earthquakein
region j. We call this the precurrentprobability becauseit
refers to the probability of an event precedinga secondone
(not, it shouldbe noted, with an implication of violated causal-
ity). As a simpleexample,we couldtake •Fc(i,j)=
P (Fi) + P (C)P (Bi)
P (Bi) + P (Fi)
where we have used (5). For fixed S and •Fc this expression
is proportionalto P (Ci) only: the rate of false alarmsfor a
given probability dependsonly on the rate of mainshocksand
not on the rate of backgroundactivity. In terms of the simple
example at the beginning of section 2.1, fixing a probability
level of 0.1% means that we would set the magnitudelevel of
candidate events such that there would be 1000 background
events for each actual foreshock; but the absolute rate of such
mined only by the rate of foreshocks,and thus of mainshocks.
3.
at all.
We can then easilyrevise(4) aboveto get the probabilitywe
seek;simply addingsubscriptsto the candidateevent yields
P (C IFiLJBi)=
(1-S) N N
ct•ii•, backgroundearthquakes(and thus of false alarms) is then deter-
which would imply that large earthquakesare preceded by
foreshocksonly in the same region, and even then only a fraction ot of them have foreshocks
N
S S)
i=1
S /•lj•l(I)F½(i
-= .=
' j)P(Cj)
(1ZP(Fi)=
(6)
A MULTIDIMENSIONAL
MODEL FOR FORESHOCKS
We now develop an expandedversion of (5), which contains
more variables. The first step is to define our events more
thoroughly:
B:
A backgroundearthquakehas occurredat coordinates
(xo+eo,Yo+eo), duringthe time period[t,t +150],with
magnitudeM +g. (All of the quantitiese0, •50,and g
Equations(5) and (6) are the basic ones we shall use in the
are small and are included becausewe will be dealing
more generalcase. Equation(5) showsus how to computethe
with probability density functions; as will be seen
probabilityof a foreshockhappeningin the locationof our canbelow, they cancel from the final expression).
didate earthquake,by summingover all possiblemainshocks.
F:
A foreshockhas occurred,with the same parametersas
The use of the precurrentprobability •Fc is the key to this
in event B.
approach;we can (and in the next sectionshall) designit to
C:
A major earthquakewill occursomewherein the region
embody our knowledge and assumptionsabout the relation
of concern, which we denote by Ac (also using this
betweenforeshocksand the earthquakesthey precede. Having
variable for the area of this region). This earthquake
found the foreshockprobability,we then use (6) to find the conwill happenduring the time period [t + A, t + A + •5•],
ditionalprobabilityof a large earthquake.
with magnitudebetweenMe and Mc + gc.
An importantconsequence
of (5) is that we may sum over all We assumethat we are computingthe probability at some time
possibleforeshocks(again assumingdisjoinmess)to get
in the interval (t + •50,t + A); the possibleforeshockhas happened,but the predictedmainshockis yet to come.
N
N
P (F) = •_••_••FC(i, j)P (Cj)
i=1 j=l
(7) 3.1. Rate Densitiesof EarthquakeOccurrence
giving us the overall probabilityof a foreshocksomewherein
the total region. This must satisfyP (F)= otP (C), where ct is
We beginby defininga rate of occurrencefor the background
seismicity(in the literature on point processesthis would be
11,962
AGNEWANDJONES:
PREDICTION
PROBABILITIES
FROMFORESHOCKS
called an intensity, a term we avoid becauseof existingseismo- regions),we may in fact makethemthree-dimensional
or onelogical usage). This rate (or, strictly speaking,rate density)we dimensionalif we so choose,making sure that we adjust the
call A(x ,y ,M); it is suchthat the probabilityof B is
numbersof the integralsin (8) and (13) accordingly. The onedimensionalmodel is easiestto develop analytical expressions
x0+e0 Y0+e0
M+g
for, and may be an adequate approximationfor the case of a
long fault zone. In this case,of course,we needto projectthe
P(B)=i50
I dx I dyl dmA(x,y,M) (8) background
seismicity (out to some distance away) onto the
xo-e0
Y0-e 0
M-g
fault zone.
By not making A dependent on the time t we make the
occurrenceof backgroundearthquakesinto a Poissonprocess. 3.2. Computationof the ForeshockProbability
If we assume that at any location the Gutenberg-Richter
We are now in a position to write the formal expressionfor
frequency-magnitude
relationholds,we may write
the foreshockprobabilityP (F) in the sameway as was done in
(5) for the discrete one-dimensional case. In this case, •rc
(9)
A(x,y, M) -- As(x,y ) e-•(•c'y)M
becomesa density function over all the variables involved, its
value indicating with what relative frequency foreshockswith
where [• is 2.3 timesthe usualb value. (While commonrather differentparameters
occurbeforemainshocks
With particular
Ones.•Insteadof a single sum, as in (5), we have a multiple
than natural logarithmsare conventionalin this area, they lead
to messier expressions,and we have thereforenot used them).
integral:
If [• is constantover a regionof areaA, andduringa time interval T the cumulativenumber of earthquakesof magnitudeM or
greateris given by the usualformula
t+50 x0+e
0
M+g
t+A+•
1
MC+gC
P(F)=
l dtl dxl dyl dM
I a,'JIax'ay'
I
t
N(M) = 10a-bM
Y0+e0
(10)
xo-e0
Y0-e0
M-g
t+A
MC
'•ec(t, t',x ,y ,x',y',M ,M') fs (x',y', t')e-•'(x"y')•'
(•5)
then, since the expectedvalue of N (M) is
Of these eight integrals,the last four are the integrationof the
precurrentprobability density times the density of mainshock
occurrenceover the space of possiblemainshocksand are the
(11)
M
A
equivalentof the sum in (5). But this gives only the rate density for foreshocks,which must in turn be integratedover the
we have that As = (10a•)/(AT) for As constantwithin the spaceof the candidateevent (the first four integrals)to produce
the actualprobabilityP (F).
region.
Equation (15) is clearly quite intractable as it stands. To
Similarly, we can define a rate density for the occurrenceof
renderit less so, we assumethat we can separatethe behaviors
large earthquakes,
of P (F) in time, magnitude,and location. This implies the folf(x, y ,M,t) = fs(X, y ,t)e -•'(x'y)M
(12) lowing assumptions:
E[N(M)]
=Tl dMIldxdyAs(x,y)e
-•M
1'
where[•' is 2.3 timesthe b value for theseevents. In this case, 2:
we introducea dependenceon time t becausethe occurrenceof 3'
large earthquakes
is oftenformulatedas a renewalprocess[e.g.,
Nishenkoand Buland, 1987], with time being measuredrelative
to the last earthquake. The probabilityof C is then
•' doesnotdepend
onx' ory'.
Over the rangeof integration,fs doesnot dependon t'.
The functionalforms of the precurrentprobabilitydensity
for time, space, and magnitude are independent,so that
we can write the functionas the productof the marginal
distributions:
MC+gC
= Ilex
I
AC
MC
•rc = •s(X, y ,x', y')•t(t,t')•m(M,M')
f(x,y,M,t+A) (13)
Of these assumptions,the third seemsthe least likely to be
valid,
sincethe dependenceon both distanceand time might be
whereAc is the areaof concern,i.e., the particularsegmentof a
correlatedwith the magnitudeof either the mainshockor the
fault.
For lack of better informationwe would usuallytake fix to candidate foreshock. The most likely correlation, with
be a constant,but we could chooseto make it spatiallyvarying. mainshockmagnitude, does not matter very much, since our
Suchvariationcouldincludeincreasesnear fault jogs and termi- rangeof integrationof this variableis small.
These assumptions
made, we can divide the integral in (15)
nationsif we think that rupturenucleationis more likely there,
or a proportionalityto Ax if we suspectthat backgroundearth- into a productof threeintegrals(in space,time, andmagnitude):
quakes are (on the average) the likely triggersof large ones
t+50
(both issues are discussedin section 3.2.2). For fix constant,
P(F)
we have that
fs =
P(C)[•'
Active
-•'•c(1- e-•'•c)
(14)
Note that while we have regarded both A and Ac as twodimensional regions (and hence also as the areas of such
=
t+A+fi
1
M+g
MC+gC
I at I att(I),
(t,tt)I am I dm
t(I)
m(m,m
')e-•'M'
t
x o+e0
t+A
M-g
MC
Y0+e0
I dx I * If dx'*'*x(x'y'x"y')nx(x"y')
XO-e0
YO-eO
AC
(16)
AGNEWAND JONES:PREDICTION
PROBABILITIES
FROM FORESHOCKS
3.3. Functional Formsfor the ForeshockDensity
To evaluatethe integralsin (16), we need to know the three
precurrentprobabilitydensities(I)t, (I)s, and (I)m. Our expressions for these incorporateour knowledge and assumptions
about foreshocks. In the following sections,we describein
the candidateearthquake)is small. The normalizationis determined by the requirementthat
t+•o+tw
dt' •t(t,t')
functions for the relevant (I); these functions must include both
= 1
(19)
t+b0
somedetail what is known aboutthe temporal,spatial,and mag-
nitude dependences
of foreshocks.From these data, we find
11,963
where tw is the total time window within which we admit
to be foreshocks.This then gives
the actual dependenceon the variablesand a normalization. precedingearthquakes
The nature of the normalization can be seen if we imagine
t+{3
0
extendingthe rangeof the firstfourintegralsin (15) to coverall
possibleforeshocks(howeverwe choseto define them); the
resulting? (F) mustthenbe equalto ctP(C), where(x is, as for
the one-dimensionalmodel, the fraction of mainshockspreceded
t+A+•51
ln[1 + b•/(A + c)]
dt ' (I)t (t, t') = bo
ln[1 + tw/(•5o+ c )]
= bob(/X,8•)
(20)
by foreshocks.In derivingour expressions
we have aimedfor
simplicityratherthanattempting
to find a functionthat can be where, with an eye to future simplifications,we have separated
shownto be statisticallyoptimal.
3.3.1 Time. Most foreshocks occur just before the
mainshock. An increasein earthquakeoccurrenceabove the
backgroundrate has only been seen for a few days [Jones,
1984; 1985; Reasenberg,1985] to a week [Jonesand Molnar,
1979] before mainshocks.For 26% of Californian mainshocks,
the foreshocksare most likely to occur within 1 hour of the
mainshock;the rate of foreshockoccurrencebefore mainshocks
(Figure1) varieswiththet-• typebehavior
alsoseenin Omori's
law for aftershocks [Jones, 1985; Jones and Molnar, 1979].
This variation can be well fit by the function that Reasenberg
and Jones [1989] found for California aftershocksequences:
(I)t(t,t') =
Nt
t'-t
(17)
+c
where t is the foreshock time and t' the mainshock time; c is a
out the •50term. Note that (17) predicts a finite rate for all
times, whereas the assumptionof a limited time window
automaticallyforces the rate to fall to zero beyondsome time;
we can easily modify (I)t to allow for this.
3.3.2. Location. Foreshocksnot only occur close in time to
the mainshock,but are also nearby in space. Jonesand Molnar
[1979] found that epicentersof mainshocks(M > 7) and their
foreshocks in the National Earthquake Information Center
(NEIC) catalog were almost all within 30 km of each other,
approximatelythe location error for the NEIC catalog. Jones
[1985], with the more accuratelocationsof the California Institute of Technology(Caltech) catalog,found that epicentersof
mainshocks(M _>3) and their foreshockswere almost all within
10 km of each other; this result also held for foreshocks of
M > 5 mainshocks within the San Andreas system [Jones,
1984] if relative relocations were used. Even the largest
foreshocks(M > 6 at Mammoth Lakes and SuperstitionHills)
constant,foundby Reasenberg
and Jones[1989] to be 200 s for have had epicenterswithin 10 km of the epicentersof their
mainshocks.
aftershocks.The relevantintegralfrom (16) is then
We have assembleda data set of sequenceswith high-quality
t+50 t+A+$1
locationsto examine the dependenceof the distancebetween
dt' (I)t(t, t') = •5oNtIn[ 1 + tS•/(A+ c )]
(18) foreshocksand mainshockson the magnitudesof the eartht
t+A
quakes. This data set includesall foreshock-mainshock
pairs
with mfore_>2.5 andMmain
->3.0 recordedin southern
Califorwhere we have assumedthat •50(the uncertaintyof the time of
3OO
• 25O
o
'½ 200
nia since 1977 (the start of digital seismic recording), and
severalsequences
relocatedin specialstudies,with relative location accuracyof at least 1 km. Figure 2 showsthe distance
between foreshock and mainshock versus magnitude of the
mainshock(2a) and magnitudeof the foreshock(2b). The epicentral separationbetween foreshockand mainshockdoes not
correlatestronglywith either magnitude. Rather, the data seem
to groupinto two classes:foreshocksthat are essentiallyat the
same site as their mainshock (<3 km) and foreshocks that are
-• 150
clearly separatedfrom their mainshocks. Only foreshocksto
larger mainshocks(mmain
->5.0) OCCur
at greaterepicentraldistances(5-10 km). Of these spatially separateforeshockssome
(but not all) rupturedtowardsthe epicenterof the mainshock
(the rupture zones are shownby the ovals in Figure 2). The
greatestreporteddistancebetweenforeshockandmainshockepicenters is 8.5 km; the greatest reported distance between
foreshockrupturezone and mainshockepicenteris 6.5 km. It
o
,- 100
z
50
0
0
24
48
72
96
120
144
168
Time between Foreshock and Mainshock, Hours
would therefore seem that, whatever other behavior (I)s may
have, it can be taken to be zero for distancesgreater than 10
Fig. 1. The number of foreshock-mainshock
pairs recordedin southern km.
California
versus the time between
for foreshocks M
1932 and 1987.
foreshock
> 2.0 and mainshocks M
and mainshock
in hours
> 3.0 recorded between
It is possible(and allowedfor in our choiceof variablesfor
(I)s) for foreshocksto be preferentiallylocatedin somesections
11,964
AGNEWAND JONES:PREDICTIONPROBABILITIES
FROM FORESHOCKS
(a)
city. For example, the Calaverasfault in central California has
a relativelyhigh rate of backgroundactivity and no foreshocks.
15
Foreshocks
and mainshocks
thusclearlyoccurclosetogether
o
o
EP10o
1987
ß,•
-
1972
•
1968
I 970
0
3.5
foreshock
4.5
5
5.5
1986_1I
6
6.5
possible
AC
I l dx'dy
' f•s(X',
y')
7
Magnitude
o
E•e10
1
198
0
;•o 5
•
•
0
Pw(1 - Pw2/4Ac
)
1972
• 1968
•
•
••,•••
2.•98•3
0
1970
4
4.5
Foreshock
5
5.5
6
if p <_Pw
if p > Pw
(22)
We use Pw = 10 km to agree with the data presentedabove.
Then, providedthat the locationx0 of the candidateearthquake
•s more than a distancePw from an end of the fault zone and
1986
0•1975
• 198
0 01979
0,•,I0,•--1•85
,m,,
........ •1966,
II .... I
3.5
(21)
AC
which in generalcan be done only numerically,even for •s
constantand •s havinga simpledependence
on p. If, however,
we makethe simplification,
mentionedin section3.1, of making
our spatialintegralsone-dimensional
(with Ac then being the
length of the fault), assume•s constant,and make •s constant
for p <_Pw and zero for largerp, we find that •s is
o
•0
epicenters
I laxay
I ldx'dy'rbs(X,
y,x',y')f•s(X',
y')=
AC
(b)
u.
mainshock
erly normalizedis
01979
4
and
(p = [(x-x') 2+ (y _y,)2],/2).Thecondition
for •s to beprop-
0 1975- ,"•1981
(•I
....,,;
,,,,
?,,
Mainshock
ß,•
in space,within 10 km of each other in all resolvablecases-but
showno other clear dependenceon location. We thereforehave
made •s depend only on p, the distancebetween candidate
that •s (x') is constantover a distance2pw, the integralneeded
6.5
in (16) is
x 0+e0
Magnitude
Fig. 2. Distancebetweenforeshockand mainshockepicentersversusthe
(a) magnitudeof the mainshockand (b) magnitudeof the foreshockfor
foreshock-mainshock
sequences(foreshocksM > 2.5 and mainshocks
M _>3.0) recordedin the Caltechcatalogbetween1977 and 1987.
Sequencesthat have been relocatedin specialstudiesare also plotted
and include 1966 Parkfield, 1968 BorregoMountain,1970 Lytle Creek,
1972 Bear Valley, 1975 Haicheng(M = 7.3), 1975 Galway Lakes
(M = 5.2), 1979 Homestead,1980 Livermore, 1981 Westmoreland,
1985 Kettleman Hills, 1986 Chalfant Valley, and 1987 Superstition
Hills. For the three foreshocksequenceswith known rupturezonesthe
distancerangeof foreshockrupturezone to the mainshockepicenteris
shown by the elongatedovals; the circles inside theseshow the distance
for the foreshockepicenter.
of major faults. Jones [1984] suggestedthat foreshocksare
more commonat areasof complicationalong faults; this would
require that either P(C) or •s (or both) be larger at such
places. An increaseof P (C) would be in accordancewith the
notionthat epicentersof mainshocks
are mostlyat suchpoints
I
I
XO-e0
AC
2e0
•s(Xo)
-- 2eols(xo)
1-pw2/4Ac
(23)
where we have definedIs in a parallel way to It; the dependence on x0 comes throughthe dependenceon the value of •s
near the candidateearthquake.
3.3.3. Magnitude. The functional form for •m(m,m') is
probablythe least certainpart of •ec. Plots of the differencein
foreshockand mainshockmagnitudes
with a uniformmagnitude
thresholdfor foreshocksand mainshocks[e.g., Jones, 1985]
showthe magnitudedifferenceto be a negativeexponentialdistribution. However, to considerall possibleforeshocksto a
given mainshock,the completeness
thresholdfor the foreshocks
shouldbe muchlower thanfor the mainshocks.A bivariateplot
of foreshock and mainshock magnitudes for all recorded
foreshocksin southernCalifornia (Figure 3) suggeststhat for
any given narrow range of mainshockmagnitude,foreshock
magnitudesclose to that of the mainshockare more common;
however, for the larger mainshockmagnitudesof interesthere,
the (admittedlysparse)data suggestthat all foreshockmagnitudesare equallylikely for givenmainshockmagnitude.
Becauseof the simplicity of this last assumption,we have
used it here by making (I)m constant;we set (I)m(m,m t)• Nm,
a normalizingfactor. The normalizationof (I)m is in generalset
[King and Na7•elek, 1985; Bakun et al., 1986). While this
seemslike a valid refinement,in practicedifferentiating
between
the many possiblecomplexsitesand the "smooth" partsof the
fault would requiringgriddingat the kilometerscale,a level of
detail that does not seem justified by our presentlevel of
knowledge. One furtherchoicewould be to make •s proportionalto the local rate of background
activityAs, thusasserting
that mostmainshocks
with foreshocks
occurin areaswith high by
backgroundseismicity. The data on foreshocksto moderate
earthquakes
in California [Jones,1984] doesnot supportthis:
while the fractionof earthquakes
with foreshocks
doesvary by
region, it doesnot appearto be relatedto backgroundseismi-
oo /•
oo
I I (I)m
(m,m')dmrim'
=(•I e-•'V•'dM'(24)
AGNEWAND JONES:PREDICTIONPROBABILITIES
FROM FORESHOCKS
Southern
California
M+p
o
11,965
MC+PC
dM'(I)m(M, M')e-lY'•'=
o
M -g
MC
0
o
o
0
o
o
0
o
o
0
o
o
0
o
o
0
o
o
3.4. Mainshock Probability
0
o
o
We now can combine the integrals in (18), (23), and (26)
into (16) to get the foreshockprobability:
0
o
o
-13'PC
g
2
1118
16
38
10
72
19
6
5
1
1
0
o
o
15230
9
2
0
0
0
0
o
o
7123
6
1
0
0
0
0
o
o
d.oo
7'.00
a'.oo
60
"%.oo
•'.oo
•'.oo
2pNm
e-•'•c1- e
= 2pIm(Mc,Pc)
(26)
where we have assumed!-t small, and again separatedit out
from the rest of the expression.
P(F) = 45opeolthlm
Solving the integralin (8) for the backgroundevent gives
P (B) = 4 50}.te0As(x0)e-•a4
9.00
M(Main)
We substitutethesevaluesof the backgroundand foreshockproFig. 3. The numberof foreshock-mainshock
pairsin half unit of magni- babilities into (6) to obtain:
tude bins for the magnitudes of foreshock and mainshock. Data
included all M
_>2.0 foreshocks and M _> 3.0 mainshocks recorded
between 1932 and 1987 in southern California.
P (C IF t•B ) -
Is/tim
IsIt Im + As(xo)e-•M
(27)
The candidateearthquake
errors80, e0, and}.thavecanceledout.
Equation (24) says that if we look before all mainshockswith
magnitudesgreaterthan MB for foreshocksabove a cutoff magnitude of Mo, we find that a fraction 0• of the mainshocks have
foreshocks. Note that we have chosen to normalize (I)t and •s
to integrateto 1, so (I)m containsthe informationabout the total
fraction
of mainshocks
with foreshocks.
For making calculations,it is also useful to set It equal to 1
(solve for the probability in a fixed time interval) and (for the
case of a linear fault) take Is in (23) to be equal to g2s(x0). If
we take g2s to be constant and combine (14) and (26), we find
that the dependenceon Mc and Itc cancelsout, and we are left
with
Making (I)m constantimplies that the fraction of mainshocks
precededby foreshockswill increaseas the magnitudethreshold
for foreshocks decreases. This is consistent with reported
foreshock activity, since the data suggestthat foreshocksare
relatively common before major strike-slip earthquakes. Jones
and Molnar [1979] found that 30% of the M > 7.0 earthquakes
occurring outside of subduction zones were preceded by
foreshocksin the NEIC catalogue (M > 4.5-5.0) and almost
50% had foreshocksM > 2 reported in the literature. Jones
[1984] showed that half of the M > 5.0 strike-slip earthquakes
in California were preceded by M > 2.0 foreshocks.
(Foreshockswere less common on thrust faults.)
For (I)m constantand equal to Nm, (24) implies that
P (C IF twB) =
4.
(NmP (C )/Ac
(NmP(C)/Ac5•)+ As(x0)e-•a4
(28)
APPLICATION TO THE SAN ANDREAS
FAULT SYSTEM, CALIFORNIA
We now have an expressionfor the conditionalprobabilityof
a characteristicearthquake on a fault segment given the
occurrenceof an earthquakethat is either a backgroundevent or
a foreshock. To evaluatethis, we need the long-termprobability of the characteristicmainshock(the terms involving the
actualmagnitudeof the characteristic
earthquakehave canceled
out), the length of the fault segment,and the rate density of
Nm=
(25) backgroundseismicity for that segment. To show how this
1+ [•'(MB-Mo )
works, we now apply this to the San Andreasfault systemin
California, becausethe long-termprobabilitiesfor characteristic
The data presentedby Jones [1984], with MB = 5.0 and Mo =
earthquakesthat we need have been estimatedfor the major
2.0, gave 0• equal to 0.5 for strike-slipearthquakes. Adopting faults of this system,the San Andreas,Hayward, San Jacinto
this value,with a [•' of 2.3, givesNm = 0.15. A consequence
of and Imperial faults. This was first done by Lindh [1983] and
taking (I)m constant is then that all earthquakesshould have Sykesand Nishenko[1984], and more recentlyby the Working
foreshockswithin 6.5 units of magnitude of the mainshock. Group on California Earthquake Probabilities [1988], hereafter
Holding (I)m constant for all M would of course lead to the
referred
absurd
Our division of the fault into segmentsand our values of
P (C) for each segmentcome largely from WGCEP-88. One
exception is that the lengths of the SouthernSanta Cruz Mountains and the San Francisco Peninsula segmentshave been
alteredto matchthe rupturezone of the 1989 Loma Prieta earth-
result
that
more
than
100%
of
mainshocks
have
foreshockswithin, say, 8 magnitude units. For the smaller
range of magnitudesconsideredhere a constant(I)m does not
presentany difficulties.
The integralneededfor (16) is then
to as WGCEP-88.
11,966
AGNEWAND JONES:PREDICTIONPROBABILITIES
FROM FORESHOCKS
Southern
San
1977-1987
36
ø
Andreas
M>1.8
Fault
Declustered
Parkfield
Cholame
MAGS
¸
Carrizo
0.0+
•
4.0+
MAGNITUDES
35
ß
0.0+
o
2.0+
o
3.0+
Mojave
•
Sail
.o
4.0+
• 5.0+
6.0+
34 ø
aim
Springs
ca
20
KM
33 ø
120 ø
119 ø
118 ø
117 ø
116 ø
Fig. 4. A mapof M _>1.8 declustered
earthquakes
locatedwithin10 km of the southern
SanAndreasfaultrecorded
in the
Caltech
catalog
between
1977and1987and(forParkfield)
theCALNETcatalog
between
1975and1989.
quake (A. Lindh, personal communication, 1990). We took
P (C) to be constantalong each segment;as noted in section
3.2.1, we have not tried to includethe possibilitythat nucleation
points (and higher values of P (C)) are more likely at points of
complication. We have also not alteredthe distributionof P (C)
to account for any possible relationship between nucleation
point and level of backgroundactivity.
The rate densityfor the backgroundseismicityis determined
from the microearthquakes
recordedbetween 1977 and 1987 by
the Caltech/U.S. GeologicalSurvey SouthernCalifornia Seismic
Network [Given et al.,
1988] for southern California and
between 1975 and 1989 by CALNET, the U.S. GeologicalSurvey Central California Seismic Network (P. Reasenberg,personal communication, 1990), for northern California.
Back-
groundseismicitycan be definedin many ways; it is important
in this application that it be defined in the same way as the
foreshocks will be. Because foreshocks can be up to 10 km
from their mainshock (Figure 2), backgroundseismicityup to
10 km
from
the
surface
trace
of
the
San Andreas
fault
is
includedin the backgroundrate.
Another issue is how to handle temporal clustering in the
catalog. We assume that if an earthquake of M = 6 (for
instance) were to occur on the southernSan Andreas fault with
an aftershock sequence,we will only evaluate the probability
that the M = 6 earthquakeis a foreshock,and not individually
determinethe probabilitiesthat the M = 6 and each of its aftershocksis a foreshockand then sum them. For consistencywe
therefore want to determine the backgroundseismicityusing a
catalogfrom which aftershocksequencesand swarmshave been
removed. In sucha declusteredcatalogue,sequencesare recognized by somealgorithmand replacedin the cataloguewith one
event at the time of the largest earthquakein the sequence,
which is given a magnitudeequivalentto the summedmoment
of all the earthquakes in the sequence. To produce our
declusteredcatalogs, we used the algorithm of Reasenberg
[1985].
The resulting backgroundseismicity within 10 km of the
faults is shown in Figures 4-6. It is clear from these that the
rate of backgroundseismicitycan vary significantlywithin the
fault segments defined by WGCEP-88. For example, the
Coachella Valley segment of the San Andreas includes the
active region aroundDesert Hot Springs(includinga M = 6.5
event in 1948) and a very quiet region (near the Salton Sea)
where the largestearthquakein 55 yearshas beenM = 3.5. To
account for this variation, we have divided some of the
WGCEP-88 segmentsinto smaller regions,which are shownin
Figures4-6 and listed in Table 1.
Table 1 providesthe data neededfor each segment. To use
(28) we also need the time period 81, which we set to 3 days
(1.09x 105s), to matchtherecentusageof theU.S.Geological
Survey and the Califomia's Governor's Office of Emergency
Servicesin issuingearthquakeadvisories. Alert levels for such
advisoriesare definedto correspondto certainprobabilities;the
magnitudesof earthquakesneeded to trigger those alert levels
can then be computedfrom (28), and are also given in Table 1.
Figure 7 showsthe probabilityas a functionof the magnitudeof
AGNEWAND JONES:PREDICTIONPROBABILITIESFROM FORESHOCKS
Northern
11,967
San Andreas Fault System
Deelustered
1975-1989
40 ø
Poi
MAGS
39 ø
O
0.0+
•
4.0+
MAGNITUDES
Coast
ß
0.0+
o
2.0+
o
3.0+
[•)
38 ø
5.0+
N•rthHayward
South
4.0+
6.0+
Hayward
Peninsula
37 ø
Loma
P
50 KM
I,,,.l,,,d..,.&,d .... 1
124 ø
123 ø
122 ø
121 o
Fig. 5. A mapof M > 1.8 declustered
earthquakes
locatedwithin10 km of the SanJacintofaultrecordedin the Caltechcatalog between1977 and 1987.
the candidateearthquakefor each segment.
We have treatedthe Parkfieldsegmentin two different ways.
In the Table 1 listing for Parkfield, we treat it in the same way
as the other segments,regardingthe foreshockas equally likely
anywhere along the segment,and taking P (C) from WGCEP88. Theseassumptionsgive short-termprobabilitiesmuch lower
than those estimated by Bakun et al. [1987] for the Parkfield
earthquakeprediction experiment. Bakun et al. [1987] used a
somewhat different methodology and also used different
assumptionsin two areas:their value of P (C) is 1.5 times that
of WGCEP-88, and they assume that the foreshock will be
located in a small region under Middle Mountain, making a
smallerarea for definingbackgroundseismicity. (Their assumption that 50% of Parkfield mainshockswill be preceded by
foreshocksagreeswith our choice in section3.2.3). For a better
comparisonwe have usedthe Bakun et al. assumptionsto deter-
mine short-termprobabilitieswith our methodologyand given
these in Table 1 as Middle Mountain probabilities. These
remain lower than the Bakun et al. results; for example, a magnitude 1.5 shock gives a probability of 0.1% from our methodology and 0.68% (Level D alert) accordingto Bakun et al.
As with the long-term probabilities of major earthquakes,
these short-termforeshock-basedprobabilitiesare better seen as
a meansof ranking the relative hazardfrom different sectionsof
the faults than as highly accurateabsoluteestimates. The probabilities are as uncertain as the data used to calculate them,
which in some cases are uncertain indeed. For example, the
values of P (C) found by WGCEP-88 are up to a factor of 4
larger than thosefound by Davis et al. [1989]; this would lead
to similarlylarge differencesin the short-termprobabilities.
The relative short-term probabilities for different segments
shownin Table 1 and Figure 7 are stronglyaffectedby both the
San
Jacinto
and
1977-1966
an
Imperial
M>I.6
Faults
Declustered
Bernardino
34 ø
MAGS
¸
0.0+
•
4.0+
MAGNITUDES
0.0+
2.0+
3.0+
4.0+
5.0+
33 ø
6.0+
San Diego
2O
Imperial
KM
h,, ....,I.........I
117 ø
116 ø
115 ø
Fig. 6. A map of M > 1.5 declustered
earthquakes
locatedwithin 10 km of the northernSanAndreasandHaywardfaults
recordedin the CALNET catalogbetween 1975 and 1989.
TABLE 1. Parametersand MagnitudeLevels
Segment
Length,
Pc
km
/3 days
a
b
Mecca
60
1.1 x 10-4
3.67
PalmSprings
SanGorgonio
50
60
1.1x 10-4
5.5x 10-5
4.00 0.97 1.29x 10-6
4.46 0.94 2.99x 10-6
San Andreas
San Bernardino
As,
[3
events/kms
Magnitudefor
0.1%
1%
10%
2.18
3.1
4.2
5.3
2.23
2.16
3.5
4.2
4.5
5.3
5.6
6.4
Fault
0.95
40
5.5 x 10-5
3.95
2.12
4.0
5.1
6.2
Mojave
Tejon
100
100
8.2x 10-5
2.7x 10-5
3.85 0.90 4.22x 10-7
3.49 0.88 1.80x 10-7
2.07
2.02
3.3
3.7
4.4
4.9
5.6
6.1
Carrizo
Cholame
Parkfield
Middle Mountain
Loma Prieta
Peninsula
North Coast
Point Arena
60
50
35
20
50
100
150
100
2.7 x
8.2 x
8.2 x
1.2 x
8.2 x
5.5 x
1.4 x
1.4 x
2.58
2.87
4.17
3.40
4.41
4.57
3.26
2.95
10-8
10-8
10-6
10-7
10-6
10-6
10-8
10-8
2.37
1.91
2.00
1.70
2.32
2.64
2.02
1.59
2.6
2.3
2.5
1.5
3.4
3.3
3.7
3.8
3.6
3.6
3.6
2.9
4.4
4.2
4.9
5.3
4.6
4.8
4.8
4.3
5.4
5.1
6.1
6.8
4.94 x 10-6
3.18 x 10-6
4.68 x 10-6
2.25
2.32
2.18
4.0
4.1
3.9
5.0
5.1
5.0
6.1
6.1
6.1
1.84x 10-6
2.28
4.0
5.0
6.1
0.99 1.39x 10-6
1.01 2.52x 10-6
2.28
2.32
3.5
3.6
4.5
4.6
5.5
5.6
2.21
3.6
4.7
5.8
10-5
10-5
10-4
10-3
10-5
10-5
10-5
10-5
0.92
4.91 x 10-7
1.03
0.83
0.87
0.74
1.01
1.15
0.88
0.69
1.36 x 10-6
4.32 x
8.15 x
1.79 x
4.52 x
2.52 x
2.08 x
6.09 x
1.40 x
San Jacinto Fault
San Bernardino
San Jacinto
Anza
50
65
50
5.5 x 10-5
2.7 x 10-5
8.2 x 10-5
4.58
4.49
4.57
0.98
1.01
0.95
Borrego
40
1.4x 10-5
4.05 0.99
Hayward Fault
NorthHayward
SouthHayward
60
50
5.5x 10-5
5.5x 10-5
4.24
4.41
Imperial Fault
Imperial
50
1.4x 10--4 4.59 0.96 4.95x 10-6
AGNEWANDJONES:
PREDICTION
PROBABILITIES
FROMFORESHOCKS
Short-term
10 2
Probabilities:
Southern
Son
i
11,969
Andreas
i
i
I-1
1ø1
I-I Mecca
A
' •' Palm Springs
0
OSan Gorgonio
San Bernardino
-i---
100
,,,+ Mojave
'•
•' Tejon
x
x Carrizo
I-I
I-I Cholame
10-1
2
3
Shod-term
4
5
6
Magnitude of Candldote Event
Probabilities:
Northern
San
7
Andreas
10 2
A
101
•
A Parkfield
OMiddle
0
I
10
0: ////
10-1
0 Lama
Mountain
Prieta
I Peninsula
•
• North
Coast
x
x Point
Arena
,..
2
3
4
5
6
7
Magnitude of Candidate Event
Fig. 7. The probability
thatan earthquake
of givenmagnitude
is a foreshock
to a characteristic
mainshock,
plottedagainst
magnitudefor eachfault segmentlistedin Table 1. Shownare resultsfor the (a) southernSan Andreas,(b) northernSan
Andreas,and (c) otherfaultsof the San Andreassystem.
long-termprobabilityP (C) and the rate of backgroundseismicity. Outside of the Parkfield and "Middle Mountain" segments,which have very high P (C), the highestshort-termprobabilities are from the Carrizo and Cholame segments;even
thoughthe 30-year probabilityis only 10% on the Carrizo segment, the background seismicity there is almost nonexistent.
The possibilityof the next Parkfieldearthquaketriggeringa
larger earthquakeon the Cholamesegmenthas been much discussed. Our proceduregives a magnitude6 in Cholamea 52%
chance of being a foreshockto a characteristicmainshock on
that segment;but this result comes from the low background
rate for the Cholamesegmentitself. Since this rate predictsa
The San Francisco Peninsula and the San Bernardino Mountain
magnitude6 shockevery 1400 years, not every 22 years as at
segments
both have a 30-yearprobabilityof 20%, but the proba- Parkfield, this high probability does not apply to a possible
bilities in the San Gorgonio subregionare much lower than Parkfield trigger. We can, however, use (3) of our zerothosenear San Franciscobecauseof higher backgroundseismi- dimensionalmodel to roughly estimatethe probabilitythat a
city. At high magnitudes,the lowest probabilities are for the Parkfieldearthquakewill be a foreshockto a larger earthquake
PointArenasegment
because
of its very low [5,whichmay be a at Cholame. The WGCEP-88 probabilityof a Cholameearthresultof catalogincompleteness
at low magnitudes.
quake is 30% in 30 years, while the backgroundrate for
11,970
AGNEWANDJONES:
PREDICTION
PROBABILITIES
FROMFORESHOCKS
Shorf-ferrn
Probabilities:Son Jacinfo/Hayward/Irnperial
lO 2
I-I Son
D
Son
1ø1
0---
Bernardino
dacinto
0 Anza
<•........... -(>Borrego
I
lOo
•
x
I N. Hayward
---•
S. Hayward
x Imperial
lO-1
3.0
4.0
5.0
6.0
7.0
Magnitude of Candidate Event
Fig. 7. (continued)
Parkfieldmainshocks
(andhencecandidate
Cholameforeshocks) mainshock[Jonesand Lindh, 1987]. If that relationship
were
is oneevery21.7years.To determine
theshort-term
probabil- parameterized,•ec and the integrationin (15) could include
ity with (3) we needto assume
a valuefor ? (F IC), therateat variablesdescribingthe difference in focal mechanisms;thus
which Cholame mainshocks have Parkfield mainshocks as normal- or thrust-faulting
earthquakes
would be given a lower
foreshocks.
If we assume
that,as for an average
magnitude
7 probability of being a foreshockto a San Andreasmainshock.
earthquake,
15%of Cholame
earthquakes
arepreceded
by mag- If any othercharacteristics
are recognized
as beingmorecomnitude6 foreshocks,
thenthe short-term
probability
of a Cho- mon in foreshocksthan backgroundearthquakes(such as
lameearthquake
aftera Parkfield
earthquak•
is 3%. At the numberof aftershocks),
we can rigorouslyincludethis informaotherextreme,
if we assume
that50% of Cholame
earthquakestion in our computationof the conditionalprobabilities.
are precededby Parkfield earthquakes
(the only type of
Anotherdirectionto go is in improvingour estimatesfor the
foreshockit has is Parkfieldearthquakes),
the probabilityprecurrentprobabilitybeyondthe rathersimpleformsdescribed
becomes 10%.
above. Considerable
work hasbeendonein the lastfew years
on howto estimatemultivariate
densityfunctions,
whichis predependson the background
probabilityfor the characteristicciselytheproblemat hand[Silverman
1986]. An obviousquesearthquake. A cumulative false-alarm rate for the whole San tion is whetherthe estimateddensitiesdiffer significantly
Andreas fault is thus dominatedby the contributionfrom betweenregions;if so, this couldreflectsignificantdifferences
Parkfield,for whicha 10% probability
leveloccursevery8.4 in thenucleation
andtriggeringof largeearthquakes.
years. By comparison,,
for the Coachella
Valleysegment
of the
Of course,nothingin the derivations
of section2 is specific
SanAndreasfaultthe falsealarmratefor a 10%probability
is to foreshocks;
this procedurecan be used for any potential
onceevery63 years. For 0.1% it is onceevery5.5 months,but earthquake
precursor.Equation(6) showsthat whatis neededis
thisprobability
levelis only9 timesthebackground
one.
a long-termmainshock
probabilityP (C), a ratefor background
As discussedin section 2.2 above, the rate of false alarms
5.
DISCUSSION
eventsP (B), and a precurrentprobability•ec, which would in
many casesjust be the fractionof mainshocks
with precursors.
At present,thesedata are not availablefor any precursorbut
The procedure
developedhere can be mademore general foreshocks.For instance,the background
rate of creepevents
thanhasbeenappropriate
for the aboveapplication
to the San can be determined for some sections of the San Andreas fault
Andreas fault. As discussedin sections3 and 4, we could system, but we have almost no data on the fraction of
includea differentdependence
of •t on time or make? (C) mainshocks
precededby suchevents.
includeinformation
aboutthe mostlikely epicenters
for the
Therehavebeena numberof earlierpapersonestimating
the
mainshock(suchas fault jogs or terminations).Anotherexten- probabilitiesof earthquakes
in the presenceof precursors
sionwouldbe to set P(C) from an extrapolation
of the [Kagan and Knopoff 1977; Vere-Jones1978; Guagentiand
frequency-magnituderelation; while a violation of the Scirocco1980;Aki 1981;Anderson1982;Grandoriet al. 1984].
maximum-magnitude
model,thiswouldallowapplication
of this Most of thesetake a slightlydifferentdefinitionof eventsfrom
technique
to manymoreregions.The greatest
flexibilitycomes the oneswe have used. Ratherthan distinguishing
between
from the precurrentprobabilitydensity•ec, sincewe can, as background
events(independent
of largeearthquakes)
andprethesepapers
thedatawarrant,
alterthisfunction
to include
•ddi[ional
data cursors(alwaysfollowedby a largeearthquake),
types. For example,there is evidenceto suggest
'that most assumethat all possibleprecursors
fall into one classof events,
foreshockshave focal mechanisms
similar to that of their with someprobability
of a possible
precursor
notbeingfollowed
AGNEW
ANDJONES:
PREDICTION
PROBABILITIES
FROMFORESHOCKS
11,971
Thatcher, Parkfield, California, earthquakeprediction scenariosand
by an earthquake. (For example, Anderson [1982] computesthe
responseplans, U.S. Geolog. Surv. Open-fileRep. 86-365, 1987.
probabilitiesof a precursorbeing useful or useless). For seismiBakun, W. H., G. C. P. King, and R. S. Cockerham,Seismicslip, asecity, a division into backgroundand precursoryevents appears
ismic slip, and the mechanics of repeating earthquakes on the
to be a better approximation to the likely physics. Most of
Calaverasfault, California, in Earthquake Source Mechanics (Geophythesepapersalso deal with the case (not discussedhere) of how
sical Monograph 37), edited by C. Scholz, pp. 195-207, American
GeophysicalUnion, WashingtonD.C., 1986.
possiblemultiple precursorscould increasethe conditional probability above that for a singleprecursor. The discussionabove Brune, J. N., Implicationsof earthquaketriggeringand rupturepropagation for earthquakepredictionbased on premonitoryphenomena,J.
suggeststhat this will usually be a moot point, since only rarely
Geophys.Res., 84, 2195-2197, 1979.
do we have the information
needed to estimate
the conditional
probabilities. With the exception of the work of Kagan and
Knopoff [1987] and (in part) Anderson [1982], there does not
seem to have been much considerationof any multidimensional
cases of the kind described in section 3. The Kagan and
Knopoff study is closestto the approachpresentedhere, though
the functionalform employedby them is derived from a fracture
mechanicsmodel, whereas ours is more purely empirical. The
models also differ considerablyin their specificationof longterm probability. In the Kagan and Knopoff model, this is
given by a Poissonrate derived from the frequency-magnitude
relation (10), whereas here it can be independentof that. As
noted in section 2, such independenceappears to be a more
satisfactoryrepresentationof the seismicity of an active fault
Seismol. Soc. Amer., 79, 1439-1456, 1989.
Given, D. D., L. K. Hutton, L. Stach, and L. M. Jones, The Southern
California Network Bulletin, January-June, 1987, U.S. Geol. Surv.
Open-file Rep. 88-408, 1988.
Goltz, J., The Parkfield and San Diego earthquakepredictions:a chronology, Special Report by the Southern California Earthquake
PreparednessProject, Los Angeles,Calif, 1985.
Grandori, G., E. Guagenti,and F. Perotti, Some observationson the probabilistic interpretationof short-termearthquakeprecursors,Earthq.
Eng. Struct.Dynam., 12, 749-760, 1984.
Guagenti,E.G. and F. Scirocco,A discussionof seismicrisk including
precursors,
Bull. Seismol.Soc.Amer., 70, 2245-2251, 1980.
Jones,L. M., Foreshocks(1966-1980) in the San Andreas System, California, Bull. Seismol. Soc. Amer., 74, 1361-1380, 1984.
Jones,L. M., Foreshocksand time-dependentearthquakehazard assessment in southern California,
1669-1680, 1985.
zone.
6.
For the San Andreas
fault the two extremes
Bull.
Seismol. Soc. Amer.,
75,
Jones,L. M. and A. G. Lindh, Foreshocksand time-dependentconditional probabilitiesof damagingearthquakeson major faults in Cali-
CONCLUSIONS
We have shown that the probability that an earthquakethat
occursnear a major fault will be a foreshockto the characteristic mainshock depends on the rate of backgroundearthquake
activity on that segment, the long-term probability of the
mainshock,and the rate at which the mainshocksare preceded
by foreshocks, which we call the precurrent probability.
Assuming certain reasonableforms for the density function of
this probability (as a function of time, location, and magnitude)
we have found an expressionfor the short-termprobability that
an earthquakeis a foreshock,and applied it to the faults of the
San Andreas system. Because the rate of foreshocks before
mainshocksis assumedto be the same for all segments,the
differences in short-term probabilitiesbetween segmentsarise
from differencesin backgroundrate of seismicityand in longterm probabilities. The backgroundrates are more variable
betweenregions and lead to larger variationsin short-termprobabilities.
Davis, P.M., D. D. Jackson,and Y. Kagan, The longerit has been since
the last earthquake,the longer the expectedtime till the next?, Bull.
are the
nearly aseismic Carrizo Plain, where a 1% probability for a
characteristicearthquakewould be found for a magnitude 3.6
candidateevent, and the highly seismic San Gorgonio region,
where it would take a magnitude5.3 to reachthis level.
fornia, Seismol. Res. Letters, 58, 21, 1987.
Jones, L. M. and P. Molnar, Some characteristicsof foreshocks and their
possiblerelationshipto earthquakepredictionand premonitoryslip on
faults, J. Geophys.Res., 84, 3596-3608, 1979.
Kagan, Y. and L. Knopoff, Earthquakerisk prediction as a stochastic
process,Phys.Earth Planet. Int., 14, 97-108, 1977.
Kagan, Y. and L. Knopoff, Statisticalshort-termearthquakeprediction,
Science, 236, 1563-1567, 1987.
King, G. C. P. and J. Na'belek,Role of fault bendsin the initiation and
terminationof rupture,Science,228, 984-987, 1985.
Knott, C. G., The Physics of Earthquake Phenomena, 278 pp., Clarendon Press, Oxford, 1908.
Lindh, A. G., Preliminaryassessment
of long-termprobabilitiesfor large
earthquakesalong selectedsegmentsof the San Andreasfault system
in California, U.S. Geolog. Surv. Open-File Rep. 83-63, pp. 1-15,
1983.
Nishenko,S. P. and R. Buland,A genericrecurrenceintervaldistribution
for earthquakeforecasting,Bull. Seismol.Soc.Amer., 77, 1382-1399,
1987.
Reasenberg,P., Second-ordermoment of Central California seismicity,
1969-1982, J. Geophys.Res., 90, 5479-5496, 1985.
Reasenberg,P. A. and L. M. Jones,Earthquakehazardafter a mainshock
in California, Science, 243, 1173-1176, 1989.
Silverman,B., DensityEstimation, 175 pp., Chapmanand Hall, London,
1986.
Sykes, L. and S. P. Nishenko,Probabilitiesof occurrenceof large platerupturing earthquakesfor the San Andreas, San Jacinto,and Imperial
Acknowledgements.We thank the membersof the Working Group
faults, California 1983-2003, J. Geophys.Res., 89, 5905-5927, 1984.
on Short-term Earthquake Alerts for the Southern San Andreas Fault,
especiallyBrad Hager and Dave Jackson,for raising someof the issues Vere-Jones, D., Earthquake prediction-a statistician'sview, J. Phys.
Earth, 26, 129-146, 1978.
that led to this paper. We have benefitedgreatly from reviews by Andy
Michael, Dave Jackson(again), A1 Lindh, and especiallyMark Mathews. Wesnousky, S. G., C. H. Scholz, K. Shimazaki, and T. Matsuda, Earthquake frequencydistributionand faultingmechanics,J. Geophys.Res.,
We also thank Paul Reasenbergfor commentsand for providing the
88, 9331-9340, 1983.
declusteredCALNET data. Preparationof this paper was in part supWorking Group on California EarthquakeProbabilities,Probabilitiesof
portedby U.S. GeologicalSurvey grant 14-08-0001-G1763.
large earthquakesoccurring in California on the San Andreas fault,
U.S. Geol. Surv. Open-File Rep. 88-398, 1988.
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(Received August 17, 1990;
revised January 14, 1991;
acceptedJanuary 18, 1991.)