Tectonoph?;sics,
199
169 (1989) 199206
Elsevier Science Publishers
B.V., Amsterdam
Independence,
 Printed
in The Netherlands
precursors hazard
and earthquake
b Q
IASPEI D.A.
RHOADES
Applied Mathematics Division, Department of Scientific and Industrial Research, P.O. Box 1335, Wellington (New Zealand) (Received
March
15,1988;
accepted
October
15, 1988)
Abstract Rhoades.
D.A., 1989. Independence,
Prediction:
TimeVariable
The problem when
conditional is presented particular occur
of combining
assumptions
the hazard
of independence
independence in which
at a constant
defined
for the time of occurrence when the precursors
of an earthquake
concerned
of different
Earthquake
Haza1.d and
the proportion
on precursors
and other
wide) of location following
have widely differing
of other
types has been found to be tractable
types
for continuoustime
are that
of earthquakes
by limits (however
In: F. Evison (Editor),
of several different
precursors
are not realistic
of information
hazard.
Tectonophysics, 169 (spec. sect.): 199206.
from precursors
assumptions
rate independently
and earthquake
Hazard.
between
assumptions the main
type is independent
within a domain
precursors
Earthquake
hazard
It is shown
estimation.
of earthquakes
that
An alternative
only
the usual approach
unrelated
to precursors
of a
types, and that false alarms
of a particular
type
alarms.
The method
and magnitude
a given alarm
are made.
applies
and to a general
of a particular
to hazard probability
type. The problem
estimation distrixition is simplified
timescales.
Introduction A central problem in timevariable hazard estimation is that of using information from several
ditional on a following to below as tion, leads,
different types of precursory observation, each of which, by itself, leads to an estimate of timevaria
plicative formula for combining the hazards 1981). A quite different form of independence
ble hazard,
proposed by Anderson (1982). Any such assumption, made for mathematical convenience, needs to be examined critically to see if it is physically and statistically reasonable in a particular applica
to arrive
at an estimate
of the com
bined hazard when some of the supposed precursors have been observed and others not. In practice the past experience of different types of precursor occurring together is generally insufficient to allow a direct estimate of their joint hazard, and it is well recognised that some form of independence assumption is necessary estimate the combined hazard. A been 1981; ferent
in order
form of independence assumption which suggested (Rhoades and Evison, 1979; Utsu, 1983; Rhoades, 1985) is that the types of precursor occur independently
00401951/89/$03.50
to
has Aki, difcon
0 1989 Elsevier Science Publishers
B.V.
the occurrence (or nonocc,_trrence) of earthquake. This assumption, referred the conditional independence assumpin some contexts, to a simple multi(Aki, was
tion, bearing in mind that it is impossible for precursors of different types to be mutually independent in the usual sense because they are related to each other through the ean.hquake or earthquakes which they precede. Theoretical approaches in the literature differ in many respects, and assumptions of apparently similar form are not necessarily comparable in their meaning. For instance, time is treated in
some studies as a discrete variable and in others as a continuous variable. The latter approach is adopted here as it allows for a clear distinction to be drawn between the instantaneous hazard, i.e. the estimated instantaneous rate of occurrence of events, and the probability of an event occurring in a short time interval. It also allows for a more satisfactory treatment of the uncertainty in time of occurrence of an earthquake which follows a precursor. Definitions
and terminology
Attention is restricted to earthquakes occurring within some domain defined by limits, however wide, on location and magnitude. The estimated average rate of occurrence of earthquakes in the domain is denoted X. Thus X is the hazard h(t) in the domain when no information on the occurrence or nonoccurrence of precursors is available. The problem is to estimate the timevarying hazard h( t 1I) when certain information, I, about precursory occurrences is available. This is defined by:
where P, denotes a probability evaluated for time t (assuming that no earthquakes occur in the domain between the time information I become avarlable and tune t) and Ec,,,+A,J denotes the event that an earthquake occurs in the domain during the time interval (t,t + At). It is assumed that there are n different kinds of precursory phenomena which have been demonstrated to be individually useful in forecasting earthquake hazard. For each kind, k, of phenomenon there is a set of criteria which, when satisfied, result in an alarm of type k being recognised. The proportion of earthquakes which fail to have an alarm of type k is assumed to be known and denoted fk. The proportion of alarms of type k which are related to a subsequent earthquake in the domain is likewise assumed known. This is called the type k valid alarm rate and is denoted uk. The event that an alarm of type k is valid is denoted V,. The nonoccurrence of an alarm of type k is denoted Nk. The information derived
from a particular occurrence of an alarm of type It is denoted A,. It is assumed that on the basis of this information, which includes its time of occurrence, t,,, and may include any other relevant observed features of the proposed precursor. gk(t 1V,), the probability density for the time of occurrence of the related earthquake given that the alarm is valid, is known. The corresponding cumulative distribution function is denoted Gk(t 1V,). Note that uk is the probability, immediately after an alarm has occurred, of its being valid, i.e.:
but this probability decreases as time passes without the related earthquake occurring. In fact for t>t,,: 1 1  Vk
P,(v, I’&) = ’ +
u,[l  G&l hk)]
The details were given by Rhoades (1985). I~ofthecoaditknrali sumption
In Rhoades and Evison (1979) an assumption of conditional independence between pieces of information given the validity or nonvalidity of a particular alarm was introduced. The caution was given that wrongly making such an assumption could lead to gross overestimation of the hazard. Aki (1981) and Utsu (1983) made assumptions of conditional independence between precursory anomalies of different types where the conditioning events were the occurrence or nonoccurrence of an earthquake in a future time interval. It seems to have been a commonly held view that precursors which have widelydiffering times scales would be likely to satisfy such assumptions whereas precursors with similar time scales would not. Rhoades (1985) adopted assumptions of a similar form in a continuous, rather than discrete, time context and stated that the assumptions were made for mathematical convenience and were likely to be a severe restriction in practice. Skitake (1987) applied Utsu’s formula to estimate the synthetic probability of occurrence, based on pre
INDEPENDENCE.
PRECURSORS
AND
EARTHQUAKE
201
HAZARD
cursory observations, for time periods of different lengths leading up to the IzuOshima Kinkai earthquake of 1978. His figs. 1115 indicate unrealistically high synthetic probabilities for a period of about 50 days preceding the earthquake. He sought to rectify this situation by drastically reducing the probabilities for the individual types of precursor (his fig. 16). Another possibility, of course, is that the conditional independence assumptions underlying Utsu’s formula are not applicable. The conditional independence assumption is of the form:
and:
1
P n:=r’,
~ i
~
I;:i.,::::i
1 o” b 1o*k 0
1= III:=iP [BiI
1
P nLB;lE~~,,f2)
loo 5 ?lO’
I 200
’ 400
600 Time
1 600
LOO0
1200
800
1000
1200
(Days)
IE;;(,_rZ) 1=n:=,P IBi(
denotes the complement of ECt,,12j, wherezc,,,12) and where B, stands for events similar to either Ai, y or N, in the above references. It was shown by Rhoades (1985) that if the events vi, i = I,. . , n are conditionally independent with respect to E C1,,12j and its complement, then: 1
0
200
400
000 Time
(Days)
lo’ T’
I
If ~nditional independence holds for all timeintervals (t, , t, > then, by considering the limit as the length of the interval tends to zero, it can be shown that:
1 3
I
__; Ii $ Fig. 1. Hazard alarm
h(t j Z,) calculated
in the scenario The dashed
for the individual
types
of
of Table 1. (a) i = 1; (b) i = 2; (c) i = 3. line shows the mean hazard
A.
.I
*o* i.__ii._i__ 0
: 200
400
600 Time
(Days)
800
1000
1200
Expressed ment
in words,
factor
mean hazard) ment factors
for individual
refineto the
of the hazard
refine
precursors.
This result
to the main result of Aki (1981).
conditional
to all events
hazard
of the hazard
is the product
is closely related However,
the combined
(i.e. the ratio
independence
ECfl,12j is not
t, + t, the information
content
diminishes
and by taking
to nothing
with
plausible,
respect since
as
of the event ,$,,,12J the limit of
both sides of eqn. (2) it follows
that the events
must be mutually
i.e.:
P[n:=,B,]
independent,
= K,P(B,)
This cannot
be so because
each other through scenario
B,
further
the B, are related
the earthquakes.
illustrates
to
0
_____~i. ___ 200
~A
400
__L
600
~~
600
_~__I
..~
1000
1 1200
Time (Davs)
The following
Fig. 2. Combined hazard h( t 1I’) for the scenario of Table 1. assuming conditional independence (eqn. 3).
the contradiction.
A scenario with three types ojalarm Suppose magnitude
that n = 3 and that earthquakes, with and location within the adopted do
main, occur at an average rate of 1 in 30 yr. Then h = 9 x 10e5 events per day. Let the characteristics and times of occurrence of the three types of alarm be as in Table 1. In each case the distribution Gi( t ) q) is taken as lognormal on the interval of variation 0.5. Thus (t aI, co) with coefficient specifying the expected value of the precursor time E( T  to, 1y) is sufficient tribution Gi(t 1y). The timevariable
hazard
to identify resulting
the dis
from
each
type of alarm separately, i.e. h( t 1I,), i = 1, 2, 3, is shown in Fig. 1, where 1, is either A, or N, according to whether or not an alarm of type i is current at time t. It follows from formulae given in Rhoades
(1985) that:
h(t IN,) =.O
(4
TABLE 1 Individual alarm characteristics
i
f,
vi
EtTro,IVl
10,
“ longterm” “ mediumterm”
1 2
0.33 0.50
0.75 0.10
440 d 16 d
“shortterm”
3
0.15
0.05
4OOd 90 d. 650 d, 1OOOd 40 d, 560 d. 1020 d
1.5 d
and : h(tIA,)=uigi(tI~:)/[1oiG,(tI~)] Equation
(4) describes
+hX the constant
(5)
background
hazard due to the possibility of earthquakes uarelated to any alarm of type i occurring. Equation (5) describes a hazard which increases above the background level following the occurrence of an alarm and eventually subsides again to the background level (Rhoades and Evison, 1989). The combined alarm, shown
hazard
using
all three
types of
assuming conditional independence, is in Fig. 2. It can be seen that a maximum
hazard rate of approximately 30 events per day is reached when all three kinds of alarm are simultaneously current. To appreciate the impossibility of this result, it must be realised that the hazard existing during the currency of an alarm of type 3 depends primarily on only two things, viz., the estimated probability that the alarm is valid and the distribution for the time of occurrence of the earthquake. Given the relatively long time scales of alarms of types 1 and 2, the distribution for the time of occurrence of the earthquakes can hardly be affected at all by information on the occurrence of alarms of these types. Therefore, any increase in hazard resulting from the prior occurrence of alarms of types 1 and 2 must be due an increase in the probability that the alarm of type 3 is valid. However, Fig. 3 shows the hazard h( t 1V,)
INDEPENDENCE,
PRECURSORS
AND
EARTHQUAKE
203
HAZARD
Alternative
independence
Instead pendence
of any
of the abovementioned
conditions
the following
(1) The proportion
I t 1i
by an alarm
dent
information
of any
0
200
1
I 400
1000
1200
Time (Days)
Fig. 3. Hazard
h( t 1V3) calculated scenario
for alarms
that
would
exist possible
of type 3 in the
of type 2 than
assumed
value, one. The greatest
its
hazard
reached in Fig. 3 is only about 1.5 events per day, i.e. a factor of 20 below the greatest hazard in Fig. 2. The discrepancy can only be explained by concluding that the unrealistic conditional independence assumptions have led to an overestimation of the hazard in Fig. 2. It seems appropriate therefore to explore the possibility of adopting different independence assumptions which may be more realistic than conditional independence. Some alternative approaches Anderson dence” pressed
can already be found (1982) proposed a
of
earthquakes
to
which
of type k occur at a constant of earthquakes
rence of failures of type k is the average rate of occurrence
of Table 1.
if this probability
types
which have
in the literature. novel “indepen
condition for combining hazards, exin terms of the maximum magnitude M,,,
to be experienced within a fixed time interval, where there are discrete magnitude classes m,, k = 1,. . , n. The condition can be written
An unacceptable feature of this condition is that splitting or amalgamating any of the magnitude classes may change the probabilities on the left hand side for all magnitude classes.
and of all
other alarms. Since the average rate of occurrence of earthquakes is h and the average rats of occur
type
maximum
earthquakes
do not have an alarm of type 1. rate and are independent
600
of type k is indepen
of type 1 are no more or less likely
have an alarm
_,I__1 600
which are
on the other
alarm. Thus, for instance, an alarm
inde
are proposed.
f,, of earthquakes
not preceded
(2) False alarms lo’
assumptions
k
is (1 
fk)A
and
f,h.
it follows that of valid alarms of
therefore
the
rate
of
occurrence of false alarms is (1  uL )(l  fk)X/o,. (3) The precursor time for alarms 01‘ type k is very alarms
much
shorter
than
the precursor
of type (k  1); k = 2..
time
, n. This
for
condi
tion would apply if the types of alarm were, say, longterm. mediumterm and shortterm as in example 1. It means first, that if an earthquake is to have alarms of types j and k, where ,i < k, then to, < to, and secondly, that the hazard based on information from alarm types 1, . _. k  1 is, to a good approximation, constant throughout the time during which an alarm of type k is current. The first two conditions are fundamental to the approach.
The third
condition
cause it would be satisfied and because its inclusion mathematical
is made
here be
in many applications greatly sitrplifies the
development.
Although the perceived rate of occurrence of false alarms of type k is constant by condition 2. the perceived rate of occurrence of valid alarms of type k may vary according to what information is available on occurrences or nonocct.rrences of alarms of other types. If an alarm occms at a time when, based on prior information from other types of alarm, the rate of occurrence of alar,ms of that type is considered to be relatively high. then the new alarm is reinforced by the prior information. Because of condition 3 it is only necessary to take account of current hazard at time t,,,, based on information from alarms of types 1,. . , k  1. in
estimating that rate of occurrence. Without condition 3 it would be necessary to take explicit account of the hazard for all future times estimated at time tok and also of the probability density function of the precursor time for an alarm of type k. A result of condition 3 is that at the time when an alarm of type k occurs it is known whether or not an earthquake related to that alarm would also be related to alarms of types 1, . . . , k  1. Information on whether such an earthquake would have alarms of types k + 1, . . , n does not become available until some time later. There is thus a natural order for processing the information from the different types of alarm. When making an estimate of instantaneous hazard one has, in theory, information on the occurrence or nonoccurrence of all n types of alarm right up to, but not including, the time to which the estimate applies. However, at the time just after an alarm of type k has been observed it is possible to evaluate the hazard for all future times given only the information on alarms of types 1,. . . , k (assuming that no earthquakes intervene) without making any use of information on alarms of types k + 1, . . . , n. Such estimates might be useful for some purposes. Consistency
of assumptions
In view of the contradiction derived above in the case of the conditional independence assumptions, it is pertinent to ask whether the alternative independence assumptions are consistent, i.e., is it possible to construct a stochastic process of earthquakes and alarms which satisfies conditions 1 and 2? To see that it is possible, observe that such a process could be simulated in the following way. Let the earthquake process be the superposition of 2” independent stationary Poisson processes or 1; k=l,...,n) where an (P+...,LX~: a,=0 individual process Pa,....,“” represents the earthquakes which have valid alarms of those types, and only those types k for which (Ye= 1. Let Pa,....,a. have rate parameter X X II;=,r,, where r, =fk if (Ye= 0 and rk=lfk if +=l. It is readily confirmed that the total earthquake process is itself a stationary Poisson process and has rate parameter h. Furthermore it can be seen that the proportion of earthquakes which fail to have
an alarm of type k is .fk and that the same proportion holds for any subcollection of the (PI+,,” ) processes chosen on the basis that some or all of the subscripts (y, : j f k take on particular values (0 or 1). It is clear therefore that a process of earthquakes constructed according to the above scheme would satisfy condition 1. A process R,, representing valid alarms of type k, could be constructed from the processes ) by independently simulatR,. .. . . a,_,.l.ai+ I,..., U. ing the time interval between each alarm and its related earthquake to conform to a given probability distribution for the precursor time. Further, let ( Qk : k = 1, . . . , n ) be stationary Poisson processes which are independent of each other and of (P,,,,_,,,J. Let Qk have rate parameter (1  uk)(l  fk )h/uk. Qk represents the false alarms of type k and is, by construction, independent of earthquakes and of all other alarms. Condition 2 is therefore satisfied. A stepwise procedure to evaluate the hazard
Assuming that conditions 1, 2 and 3 are satisfied, attention is now given to the procedure for evaluating the hazard in the light of certain precursory information. Let Ik denote the information on the occurrence or nonoccurrence of alarms of thefirst k types. Then Ik is an event of the form:
Ik = ll;“,lIi where, for each i, I, represents either Aj or Ni. The hazard h( t 1I’) has already been dealt with (eqns. 4 and 5 above). Consider the step of evaluating h(t 1Ik) given that h(t Irk‘) has already been evaluated. The simplest case is if Ik = Nk. Then:
by condition 1 above. In the case where 1, = A, it is necessary to consider the conditional rate of occurrence of alarms of type k, based on the prior information Ike’, at the time t,, when the alarm occurred. It follows from conditions 1 and 3 above that this rate is approximately (1 fk)h(tOk I Zk+‘). On the other hand the conditional rate of occur
INDEPENDENCE,
PRECURSORS
rence of false alarms
AND
EARTHQUAKE
at time t,,
the prior information
Ike’
(l  uk)(l fk)x/uk. the information that the alarm
The Ik and of type
HAZARD
is independent
by condition probability,
based
of valid alarms
conditional
valid and invalid) &(I/,
=
rate
of type k (based
on
I k‘) divided by the correrate of occurrence of all (i.e.
the prior information sponding
on
evaluated for time tOk, k is valid, is denoted
P,,,,(V, 1I”), and is given by the conditional of occurrence
of
2, and is
alarms
of type k. Thus:
1 I”)
q,(
vk
( Ik‘nAk)
10“
LAd
0
I
400
200
600 Time
c1 (I
fk)h(tOk
fkb(hk 1 Ik‘)
1 Ik‘) +
c1

Fig. 4. Combined
uk)(l
using
fk)x/uk
alternative
hazard
Whereas the probability that an alarm of type k is valid is affected by the prior information Ik‘, it is a consequence of condition 3 that the probability density gk(t ) V,) is not. The updated hazard for t > t,, is given by:
1000
1200
(Days)
h( t1I') for the scenario
independence formulae
~__/
600
assumptions
of Table
1
and the stepwise
(eqns. 6, 7 and 8).
hazard existing at the time of the earthquake occurrence. Other matters relating to the physics of precursors
and earthquakes
may aho
vant. These are not pursued here. Figure 4 shows the result of applying
be relethe step
wise formulae in the scenario above (Ta4le 1). The hazard graphed is h(t I 13). A comparison of Fig. 4 with Fig. 2 shows that the peak hazard evaluated at the high level of 30 events per day under the conditional independence assumptions in Fig. 2 is only a little more than alternative independence
+f,/z(
t 11k‘)
(8)
being the sum of the hazard due to possible earthquakes related to the alarm of type k and that due to possible
earthquakes
unrelated
to the alarm
of
type k. By successively employing either eqn. (6) or eqns. (7) and (8) while stepping through the information on alarm types 1, 2,. . . , k it is possible to evaluate the hazard h (t ) Ik). The above formulae apply until such time as an earthquake happens. After an earthquake occurs in the domain it is necessary to determine whether the alarms have been completely validated or whether some other large earthquake may still be expected. This may involve evaluating the probability that the earthquake is related to each alarm in the light of the contributions each made to the
1 event per day under the assumptions in Fig. 4, a
result which is more consistent with F’ig. 3. The previous occurrence of precursors of types 1 and 2 affects the hazard resulting from the occurrence of a precursor of type 3 by increasing the \ralid alarm probability I’,,,( V, 1I’) to a value close to 1. The alternative independence assumptions thus lead to a more credible estimate of the combined hazard in this example. Conclusion A stepwise procedure has been given for evaluating the hazard in a domain when information is available on the occurrence of several different types of precursor. The independence assumptions applied here may be more realistic in some situations than those found in the literature.
However, it is always necessary to examine any such assumptions critically in the light of a particular
application
be approximately mance
to see whether
tests on individual
not be overlooked Although
the basis
spatial
extent
would
and Evison, strength,
of the longestterm
quake magnitude be desirable
to allow
al
of earthis nevstudy.
for a general
It
joint
location following
a valid alarm of a particular type. Treating the uncertainty in all of these three variables simultaneously, however, greater complexity
would have introduced much into the discussion. Some re
strictions on interdependence between these three variables may be necessary to make the problem tractable. It should be noted that interdependenties between the distributions for magnitude, location and time of occurrence are, in general, to be expected. precursor take,
The wellknown relationships between time and earthquake magnitude (Riki
1979) and between
tude indicate
source
Prediction,
566574.
uncertainty
probability density for the magnitude, and time of occurrence of an earthquake
quake
of interand
area and magni
cursors. Cao,
Aki,
precursors
and making
several
valuable
D.C..
pp.
for the probabilities of
(of
unreliable
pre
Sot. Am., 72: 879888. Assigning
of four large Chinese
probability
earthquakes.
gain
for
J. Geophys.
Res.. 88: 21852190. Rhoades,
D.A.,
multiple
1985.
Earthquake
precursors.
hazard
in the presence
of
In: Feng Deyi and Liu Xihui (Editors),
Fuzzy Mathematics
in Earthquake
Researches.
Seismologi
F.F.. 1979. Longrange
earthquake
cal Press, Beijing, pp. 374382. Rhoades,
D.A. and Evison,
forecasting Astron. Rhoades.
based
on
a single
predictor.
Geophys.
F.F.,
1984. Method
J.R.
Sot., 59: 4356. D.A. and Evison,
long range earthquake ion, Proceedings quake
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assessment
In: Earthquake
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in
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Paris,
pp.
497504. Rhoades,
D.A. and Evison,
earthquake Hazard
Hazard.
Assessment
F.F., 1989. Time variable
In:
M.J.
Berry
and
Prediction.
(Editor),
factors
in
Earthquake
Tectonophysics,
167:
201210. Rikitake,
T.,
1979.
Tectonophysics,
Classification
105114.
approach
on the empirical
their
of earthquake
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54: 293309.
T.. 1987. A practical
tion and
of this paper
Ewing
Washington
estimates
1983.
Utsu, T., 1983. Probabilities
The author is indebted to Professor F.F. Evison and Professor D. VereJones for reading earlier
Earth
Review. Maurice
observations
rence time. J. Seismol.
Acknowledgements
suggestions.
K.,
phenom
(Editors),
Union,
following
Bull. Seismol.
T. and
ion basing
drafts
Geophys.
J.G., 1982. Revised
earthquakes
of precursory
and P.G. Richards
an International
IV. Am.
Anderson,
Rikitake.
this.
synthesis
ena. In: D.W. Simpson Ser.,
precursors
in the present
Aki. K., 1981. A probabilistic
1989).
location
the treatment
and location restrictive
types should
to select domains
of the
lows for some flexibility, quite
precursor
(Rhoades
the ability
est on
ertheless
they are likely to
true. Also, the need for perfor
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