Independence, precursors and earthquake hazard

Independence, precursors and earthquake hazard

Tectonoph?;sics, 199 169 (1989) 199-206 Elsevier Science Publishers B.V., Amsterdam Independence, - Printed in The Netherlands precursors haza...

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Tectonoph?;sics,

199

169 (1989) 199-206

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:

Time-Variable

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 continuous-time

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.): 199-206.

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

time-scales.

Introduction A central problem in time-variable 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 time-varia-

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

0040-1951/89/$03.50

to

has Aki, difcon-

0 1989 Elsevier Science Publishers

B.V.

the occurrence (or non-occ,_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 non-occurrence of precursors is available. The problem is to estimate the time-varying 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 non-occurrence 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 non-validity 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 non-occurrence of an earthquake in a future time interval. It seems to have been a commonly held view that precursors which have widely-differing 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 Izu-Oshima Kinkai earthquake of 1978. His figs. 11-15 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 time-intervals (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 time-variable

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

EtT-ro,IVl

10,

“ long-term” “ medium-term”

1 2

0.33 0.50

0.75 0.10

440 d 16 d

“short-term”

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~:)/[1-oiG,(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 above-mentioned

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, long-term. medium-term and short-term 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 non-occt.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 non-occurrence 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=l-fk 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 non-occurrence of alarms of the-first 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“

L-Ad

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 longest-term

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 well-known relationships between time and earthquake magnitude (Riki-

1979) and between

tude indicate

source

Prediction,

566-574.

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: 879-888. Assigning

of four large Chinese

probability

earthquakes.

gain

for

J. Geophys.

Res.. 88: 2185-2190. 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. Long-range

earthquake

cal Press, Beijing, pp. 374-382. Rhoades,

D.A. and Evison,

forecasting Astron. Rhoades.

based

on

a single

predictor.

Geophys.

F.F.,

1984. Method

J.R.

Sot., 59: 43-56. D.A. and Evison,

long range earthquake ion, Proceedings quake

forecasting. Terrapub.

assessment

In: Earthquake

of the International

Prediction.

in

Predict-

Symposium

on Earth-

Tokyo/Unesco.

Paris,

pp.

497-504. 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:

201-210. Rikitake,

T.,

1979.

Tectonophysics,

Classification

105-114.

approach

on the empirical

their

of earthquake

precursors.

54: 293-309.

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. Vere-Jones 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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