Bulletin of the Seismological Society of America, Vol. 79, No. 5, pp. 1439-1456, October 1989

T H E LONGER IT HAS BEEN SINCE T H E LAST EARTHQUAKE, THE LONGER T H E EXPECTED TIME TILL T H E NEXT?

BY PAUL M. DAVIS, DAVID D. JACKSON, AND YAN Y. KAGAN

ABSTRACT

We adopt a Iognormal distribution for earthquake interval times, and we use a locally determined rather than a generic coefficient of variation, to estimate the probability of occurrence of characteristic earthquakes. We extend previous methods in two ways. First, we account for the aseismic period since the last event (the "seismic drought") in updating the parameter estimates. Second, in calculating the earthquake probability we allow for uncertainties in the mean recurrence time and its variance by averaging over their likelihood. Both exten- sions can strongly influence the calculated earthquake probabilities, especially for long droughts in regions with few documented earthquakes. As time passes, the recurrence time and variance estimates increase if no additional events occur, leading eventually to an affirmative answer to the question in the title. The earthquake risk estimate begins to drop when the drought exceeds the estimated recurrence time. For the Parkfield area of California, the probability of a magnitude 6 event in the next 5 years is about 34 per cent, much lower than previous estimates. Furthermore, the estimated 5-year probability will decrease with every uneventful year after 1988. For the Coachella Valley segment of the San Andreas Fault, the uncertainties are large, and we estimate the probability of a large event in the next 30 years to be 9 per cent, again much smaller than previous estimates. On the Mojave (Pallett Creek) segment the catalog includes 10 events, and the present drought is just approaching the recurrence interval, so the estimated risk is revised very little by our methods.

INTRODUCTION

How can we estimate the probability of occurrence of "characteristic" earthquakes in a given time interval based only on the dates of past events? Existing methods assume a probability density function for the recurrence interval, with parameters to be determined by maximum likelihood. Nishenko and Buland (1987) find that characteristic earthquake recurrence intervals from various plate boundaries are fit well by a lognormal distribution. The parameters of this distribution are the geometric mean recurrence time r and the dimensionless coefficient of variation o. While ~ and r can both be estimated from a sufficiently long catalog for a given region, few regions have such an adequate catalog. Nishenko and Buland found that a single worldwide value of ~ = 0.21 was consistent with data for the separate fault segments. They used the estimated recurrence time for each region, with the worldwide value of o, to estimate probabilities of earthquakes within given time intervals on each of several fault segments.

In what follows, we add two important ingredients to the model. First, we account for the time since the last earthquake in estimating ~ and r. Second, we account for the uncertainty in the parameters ~ and r in estimating earthquake risk. Unlike Nishenko and Buland (1987), we do not assume a single worldwide value for the coefficient of variation in the lognormal process. Rather, we estimate r and independently for each fault segment from its earthquake history. We take this approach because ~ appears to vary significantly from place to place, and because

and its uncertainty strongly affect the estimated seismic risk.

1439

1440 P. M. DAVIS, D. D. JACKSON, AND Y. Y. KAGAN

We adopt the characteristic earthquake concept and the lognormal distribution for convenience and specificity, but we do not address the fundamental question of whether this model is appropriate. Since selection criteria for choosing sequences from earthquake catalogs to avoid bias have not been formalized, there is the possibility that the earthquake catalog used by Nishenko and Buland is biased towards quasi-periodic sequences. Also, the locations and magnitudes of most of the events are uncertain. In some areas, earthquakes are apparently clustered in time (Ambraseys, 1971; Kagan and Knopoff, 1976; Lee and Brillinger, 1979; Vere-Jones and Ozaki, 1982), a behavior antithetical to quasi-periodic occurrence. Our purpose here is not to endorse the lognormal model, but rather to show that seismic droughts, and uncertainties in the model parameters, have an important effect on earthquake risk calculations. Including the drought will affect earthquake risk calculation for any distribution. Parameter uncertainties affect seismic risk estimates strongly for some distributions such as the lognormal and weakly for the Poisson distribution.

MAXIMUM LIKELIHOOD WITH DROUGHTS

We assume that n + 1 "characteristic" earthquakes have occurred on a fault segment separated by n time intervals T~ and that the time elapsed since the last earthquake is Td. The lognormal probability density function for the interval time is given by

1 [ ( T ) - - - e -(l"(T/T))2/2~2 (1)

Tz ~2-~

The probability that no earthquake occurs in interval Td is

g P ( T d ) = / ( T ) dT. (2)

The likelihood function L is the probability of occurrence of intervals within dT~

of the Ti, and of a drought longer than Td, given z and r :

L =/ (Tk l ~, ~) = n / ( T , ) d T i P ( T d )

= \ a f ~ / IILT ~ JJT~ T (3)

where the product is over the index i, the dTi are arbitrarily small time intervals, and Tk is the sequence of observed time intervals Ti, including the open-ended interval Te. The factors dTi are included as a formality, so that each term in L represents a probability; dTi play no role in the calculations below. The log likelihood function, A, is then

A = c o n s t - n l n a - F ~ (ln Ti - In ~)2

2a ~ + lnerf(xo), (4)

where the constant contains terms that are independent of r and a, and Xo =

l n ( T J - r ) / ( ~ f 2 ) .

THE LONGER IT HAS BEEN SINCE THE LAST EARTHQUAKE 1441

If the likelihood funct ion is cont inuously differentiable in • and o, the maximum likelihood values of r and o satisfy

(ln Ti - In r ) c e -x°2 OA/Or = 0 = + (5a)

~" o z or~,f~ erfc(xo)

O A / 0 o = 0 = - n / o + ~( ln Ti - In ~)2 CXoe-Xo ~

+ (5b) 0 3 o e r f c ( x o ) '

where c = 2/~f~. We solve these equations using a Newton Raphson iterative scheme in which the

functions are expanded in a Taylor ' s series about chosen start ing values and increments are found to improve the fit. Because of nonlinear constraints such as positivity of 7 and o, the likelihood function might a t ta in its maximum at a boundary such as o = 0 or T = 0, where its derivatives may not vanish. For this reason, it is useful to plot the likelihood function vs. a and 7, to verify tha t the maximum actually occurs at a s ta t ionary point. We discuss below a case where the maximum likelihood occurs on o = 0, so tha t equations (5a and b) are not satisfied.

COMPARISON WITH OTHER METHODS

Nishenko and Buland (1987) also used the maximum likelihood method to est imate the parameters of the lognormal distribution, but their procedure differed from ours in at least two respects. First, they neglected the last terms in each of equations (5a and b). These terms include the effects of the drought, and they are negligible only if the drought t ime Td is much less than the recurrence interval, or if there are many characterist ic events in the sequence. Second, they assumed tha t o is the same everywhere, so they coupled together many equations of the form 5a (one for each region) with one equation like 5b. The assumption tha t o is a global constant is ra ther arbitrary; it nei ther violates, nor can it be tested against, available data because there are so few known characterist ic events on any one fault segment. Simulat ion studies (Rundle, 1988; Knopoff , personal comm., 1988) show tha t the regularity of ear thquake occurrence depends strongly on the mechanical coupling between fault segments, with independent segments being more regular than strongly coupled segments. As the coupling between various fault segments is poorly understood, we assume tha t the coefficient of variat ion a must be evaluated separately for each fault segment. The Working Group on California Ear thquake Probabili t ies (1988; hereaf ter called WGCEP ) used a very similar method to tha t of Nishenko and Buland, except tha t they modified the estimate of a to account for uncertaint ies in the data for each fault segment.

CONDITIONAL EARTHQUAKE RISK

If r and o are known, the probabil i ty of an ear thquake in a t ime interval AT given tha t no ear thquake has occurred in Td is given by

P ( E [Tk, "r, o) = (err(x1) - er f ( xo ) ) / e r f c ( xo ) , (6)

where xl = ln((Td + A T ) / T ) / o ~ / 2 and Xo is given above. In 'practice, the maximum likelihood method is often used to evaluate ~ and ~, which are then subst i tuted into (6). Note tha t xo and xl depend on r and ~. But the maximum likelihood method is

1442 P. M. DAVIS, D. D. JACKSON, AND Y. Y. KAGAN

deficient in some respects. First, the maximum likelihood values are assumed to occur where the likelihood function is stationary with respect to T and ~, which may not be the case if ~ or a is subject to positivity or other constraints. Second, and most important, the maximum likelihood estimates may be biased when there are only a few samples (that is, few earthquakes). In this case, the maximum of the likelihood function may be only broadly defined, with a wide range of values being consistent with the observations. Furthermore, the maximum likelihood estimates may be biased if the product of the likelihood function and the conditional proba- bility is asymmetrical about its maximum.

An improvement over the maximum likelihood technique is the "mean likelihood" method, in which one accounts for the uncertainty in parameters (such as T and ~) by integrating over all possible values, weighting by the likelihood function (Bar- nard, 1949; Jenkins and Watts, 1968, p. 124). Jenkins and Watts (1968, p. 193) give an illustration of how the method works when the likelihood function is not stationary at its maximum because of a constraint on one of the parameters. In the present case, we account for the uncertainties by integrating (6) over all acceptable (that is, positive) values of r and 0.

The probability of an earthquake in some time interval, given the catalog Tk (including the drought) is

P ( E I T ~ ) = f f f ( E , ~ , a l T ~ ) d ~ d ~ , (7)

where f is the joint density function for r, o, and an earthquake occurrence, given the catalog Tk. But

[(E, T, ~ I Tk) = P(EI Tk, -r, a ) f ( r , a I Tk), (8)

where [ is the density for ~ and a given Tk. We estimate the latter using Bayes' rule,

(9)

Here f(Tk) is the unconditional density function for the catalog Tk. Being an integral over all possible values of ~ and a, it is independent of these two parameters, and it serves as a normalization constant:

We further assume that f(r , a), the unconditional density function for T and a, is a "noninformative prior" (Tarantola and Valette, 1982); that is, it may be assumed constant for positive values of r and o. This means that before we observe any interval times, no particular positive value of r or a is any more likely than any other. Then we may write (7) as

PiE' Th) = f f PiE' Tk, r, a)fiTk,r, a)f(T, cr)/fiTk) dT d~. (11)

THE LONGER IT HAS BEEN SINCE THE LAST EARTHQUAKE 1443

Using (10) gives

P(EI Th) = f f P(EI Tk, r, ~)/(T~ I ~, ,,)f(~, ~) dr da (12) f f f(Tkl r, a)[(r, a) dr dz

P ( E I Tk) = f f P ( E I r, a)f(T~ I ~, ~) dr d z $ $ f(T~I~, ~) d r da ' (13)

i.e.,

P ( E I Tk) =' f ~ {{er f (x l ) -er f (x°)} /er fc(x°)a-ne-=ln(Ti / ' )~ /2~er fc(x°)}d 'rdz (14) { f f 6-ne - 21n(Ti/r)2/2z2 erfc (Xo) d r da }

The probability is an integrated average of the conditional probability for an assumed r and a multiplied by the likelihood of obtaining that (r, a) given the earthquake catalog and the drought time interval. This accounts properly for the spread in the acceptable values of r and ~ which arises from the small data set. As we show below, the likelihood function is non-negligible for relatively large values of r and ~, for which the conditional probability is low; thus the mean likelihood risk estimate is often smaller than the maximum likelihood estimate, which selects only a single, relatively high," conditional probability.

DATA ERRORS

Prehistoric earthquakes are generally dated by radiocarbon dating of displaced sediments. These dates, and thus the interval times between events, are subject to experimental error. Let ai be the standard deviation of the i th interval estimate. The above analysis must now be repeated with ~ replaced with ai' = ~/(a 2 + ai ~) and ~,- = ~/(STi 2 + 5T~:12)/T~, where 5T~ is the standard deviation of the ith earthquake date and Ti is the i th time interval between earthquakes. We have assumed that the data consist of independent intervals. However, they are linked across common boundaries. We have not solved the more complicated problem that includes this covariance. If we assume that the intervals with errors are independent, equation (3) for the likelihood function becomes

L = f ' ( T k l r , a) = 1-I Ti e-~l"(Ti/T)2/2~i'2 erfc(xo'), (15)

where

Xo' = l n ( T j r ) / ( ~-ff'(~7),

and 5T,~+~ is the standard deviation of the time of the last earthquake. Equations (5a and b) become

2;(ln Ti - In T) 2 ce -x°'2 OA/Or = 0 = ,2 + (16a)

r~i aer~]2 erfc(xo')

Z(ln Ti - In r)2a aCXo' e -x°'2 OA/O~ = 0 = -:~ - - ~ + ,4 + (16b)

ai ai ad erfc (Xo' ) '

1444 P. M. DAVIS, D. D. JACKSON, AND Y. Y. KAGAN

where c = 2/~/-~. Equations (13) and (14) can then be rewritten with f(Tk[ r, ~r) replaced with f ' (Th [ r, a) from equation (15).

POISSON PROCESS

We can compare the results above, which assume a quasi-periodic process, to those for a Poisson process. For a Poisson process, the probability of an earthquake in the time interval AT is hAT, and the probability of a drought longer than Td is P(E) = exp(--XTd), where X is known as the rate parameter. The maximum likelihood estimate of the rate parameter, given the catalog Tk, is simply X = N/T , where N is the number of events in the catalog and T is the total time span of the catalog. Because X must be estimated from data, it is uncertain, and we should include its uncertainty in estimating seismic risk. By analogy with equation (13), we have

P(EI Tk) = f P(EI X)f(Th [ X) dX f f(Thl X) dX

Where

P(E[ X) = 1 - P(E[ X) = 1 - exp(--XAT) (17)

and

f(Tkl k) = hNexp(-XT)

where P(E[ X) is the probability that no event occurs in AT, given X. Then

(18)

P(EI Tk) -- 1 - f XNexp(--X(T + AT)) dX f XNexp(--XT) dX

= I - ( T / ( T + A T ) ) x . (19)

The probability depends only on the number of events and the time span covered by the catalog, independent of the specific event times. In this special case, the mean likelihood probability (equation 19) is nearly equal to the maximum likelihood estimate (equation 17 with X = N /T) .

PARKFIELD

Bakun and Lindh (1985) suggest that characteristic M = 6 earthquakes occurred in 1857, 1881, 1901, 1922, 1934, and 1966. For discussion purposes, we assume this catalog, although the 1857 and 1901 events could have been as far as 50 km from Parkfield (Toppozada, personal comm., 1987) and additional characteristic events might have occurred at Parkfield since 1857 (Toppozada, 1987).

Ignoring the 22 year drought at the end, the mean recurrence time is 21.8 years with a coefficient of variation a = 0.33 (Nishenko and Buland, 1987). Including the drought, we estimate r = 21.9 years and ¢ = 0.32. Using this maximum likelihood estimate, the probability of another event in the next 5 years is P = 0.47, compared with Nishenko and Buland's average estimate of P = 0.53 to 0.84 and the value of P = 0.20 assuming a Poisson process.

Figure 1 shows how the maximum likelihood estimates of r and ¢ would evolve with time in the absence of further earthquakes. Figure 2 illustrates the mean

THE L O N G E R IT HAS BEEN SINCE T H E LAST EARTHQUAKE 1445

0.9

0.7

I0

0.5

0.3

0.1

UPDATED MAXIMUM LIKELIHOOD PARAMETER ESTIMATES, PARKFIELD

M

m

m

I I I I I I I I I I J

MEAN RECURRENCE TIME - . . / " " " "

-"---- COEFFICIENT OF VARIATION

,60 I I I I I 1 I I I I

1980 2 0 0 0 2 0 2 0 2 0 4 0 2 0 6 0 YEAR

30

co - f E

w >-

15

FIG. 1. Maximum likelihood estimates of the mean recurrence interval r and coefficient of variation a vs. time for Parkfield segment of the San Andreas fault, assuming no earthquakes occur there.

likelihood risk estimates as of 1988; part (a) shows the likelihood function f(Tk I r, ~); (b) shows the conditional probability P(EI r, ~), and (c) shows their product, the integrand of equation (13). The mean likelihood probability, 0.34, is less than the maximum likelihood estimate because the likelihood function has a long tail where the conditional probability is small. Figure 3 shows how the 5-year probability of one or more characteristic events would evolve with time, assuming continued drought, calculated by each of four methods: lognormal distribution assuming ~ = 0.21 (Nishenko and Buland, 1987) with no updating to account for the drought; lognormal with updated parameters estimated by maximum likelihood; lognormal with updated parameters determined by mean likelihood; and Poisson process (that is, exponential distribution of interval times) with updated parameters determined by mean likelihood. Among the three lognormal models, we feel that the third method employing mean likelihood estimation is most proper because it accounts for the seismic drought and the uncertainties in r and a. The Poisson model is shown for comparison. Note that curves (B) and (C) both drop below the Poisson curve (D) after sufficiently long times. This occurs because the conditional proba- bility for lognormal models decreases with increasing recurrence time and coefficient of variation, and these parameters both increase as the drought continues. In the Poisson model, by contrast, the conditional probability depends only on the recur- rence interval.

The Nishenko and Buland method does not account for the fact that the parameters r and a must be continually evaluated as time goes by (with or without additional earthquakes). If the drought continues, those methods which provide for updating of the parameters predict a much steeper drop in the conditional proba- bility than predicted by the Nishenko and Buland method. This drop in conditional

1446 P. M. DAVIS, D. D. JACKSON, AND Y. Y. KAGAN

CONDITIONAL PROBABILITY OF EVENT AT PARKFIELD IN 5 YRS

j A

LIKELIHOOD FUNCTION FOR PARKFIELD, 1988

3%

B ~o

FIG. 2. (a) Conditional 5-year earthquake probability, (b) normalized likelihood function for observed catalog, and (c) their product, the integrand of equation (13), for Parkfield.

THE LONGER IT HAS BEEN SINCE THE LAST EARTHQUAKE

C O N D I T I O N A L P R O B A B I L I T Y x L I K E L I H O O D , P A R K F I E L D

1447

X

q

.,(9,

FIG. 2. C

o.gL FIVE YEAR PROBABILITY, PARKFIELD

I I I I I I I I I I

(A) LOGNORMAL, NO UPDATE

0.7

P

0.5 RMAL, UPDATED MAXIMUM LIKELIHOOD

0.3 ~ O D

o . , ,D, PO,SSO. UPDATEO -- . _

1960 1980 2000 2020 2040 2060 YEAR

FIG. 3. Five-year earthquake probabilities vs. time for Parkfield, assuming continued drought, as calculated by (a) Lognormal distribution with a = 0.21, (b) Lognormal distribution with updated maximum likelihood estimation, (c) Lognormal distribution with mean likelihood estimation using equation (13), and (d) Poisson distribution with mean likelihood estimation using equation (19).

1448 P. M. DAVIS, D. D. J A C K S O N , AND Y. Y. KAGAN

T A B L E 1

PARKFIELD 5-YEAR PROBABILITY

Distribution Parameters Update? Method P robab i l i t y Reference (%)

Lognormal 0 < r < ~, 0 < a < ~ Yes M E L E 34 th is work

Lognormal r = 21.9 yr, a = 0.32 Yes M L E 47 th is work Lognormal r = 21.8 yr, a = 0.33 No M L E 53-84 NB

Poisson 0 < ~ < co Yes M E L E 20 th is work

Poisson h = 1/21.8 yr Yes M L E 20 th is work

Notes: M L E = M a x i m u m Likel ihood Es t imate ; M E L E = Mean Likel ihood Es t imate ; NB = Nishenko and Bu land (1987); Probab i l i ty = condi t ional ea r thquake probabi l i ty for specified t ime interval .

probability for the updated methods is completely reasonable; after a long drought the estimated recurrence interval and coefficient of variation would both be larger than before the drought. In fact, it would be quite unreasonable to use estimates based on incomplete data, as one would do by neglecting the drought. The effect of the drought is especially important for catalogs with only a few events and when the drought exceeds the estimated recurrence time.

Table 1 summarizes the various estimates of the probability of a characteristic earthquake at Parkfield in the next 5 years. The estimates range from 20 per cent for the Poisson process, to a range of 53 to 84 per cent by Nishenko and Buland.

COACHELLA VALLEY SEGMENT OF THE SAN ANDREAS FAULT

Here there are fewer documented events than at Parkfield, and the earthquake dates are noisy. Sieh (1986) gives the following dates: 1020 _+ 20, 1300 _ 30, 1450 _+ 150, and 1680 _ 40. The ranges implied by the uncertainties represent 95 per cent confidence intervals; because of a highly nonlinear and in some cases multiple valued relationship between radiocarbon date and calendar date, the errors are strongly nongaussian. Nevertheless, we will halve the confidence interval to estimate the standard deviation, and thus the variance, of the dating errors. Neglecting the drought, but including the dating errors, we estimate r -- 220 years and ~ = 0.32, using the method of Nishenko and Buland. The fact that the 308-year drought exceeds 220 years has inspired some official concern about the Coachella segment; WGCEP (1988) estimates a recurrence probability of 0.4 for the next 30 years. However, when the drought is included, the estimated recurrence time is 305 years.

The maximum likelihood estimate of the coefficient of variation ~ is zero, because the variation in interevent times is more than adequately explained by the dating errors. Thus the maximum likelihood solution for this case satisfies only the first of equations (5a and b). This is illustrated in Figure 4 which shows that the likelihood function would peak at negative a were it not for the positivity constraint on ~. The variation in observed event times could be due entirely to measurement errors, even if the earthquakes were perfectly periodic.

The reported uncertainties of the dates may be overestimated; smaller dating uncertainties would lead to a nonzero maximum likelihood estimate for ~. Figure 5 illustrates how earthquake risk would evolve with time, barring further earthquakes; the format is the same as that of Figure 3, except that the maximum likelihood curve is omitted. We have omitted the maximum likelihood curve for Coachella for reasons cited above: the likelihood function is nonstationary at its maximum, so the maximum likelihood estimates of r and a do not exist in the conventional sense. We calculate probabilities for a prediction interval of 30 years.

THE LONGER IT HAS BEEN SINCE THE LAST EARTHQUAKE

L I K E L I H O O D F U N C T I O N F O R C O A C H E L L A , 1 9 8 8

1449

Fro. 4. Likelihood for Coachella Valley showing that the likelihood function is not stationary with respect to a at its maximum.

T H I R T Y Y E A R P R O B A B I L I T Y , C O A C H E L L A

I I I I I I I I I I

0 . 9 -

0 .7 - (A) LOGNORMAL, NO UPDATE _

P

0 .5 --

0 .3 --

EAN LIKELIHOOD

1 6 0 0 1 8 0 0 2 0 0 0 2 2 0 0 2 4 0 0 2 6 0 0 Y E A R

FIG. 5. Thirty-year earthquake probabilities vs. time for Coachella Valley, assuming continued drought, as calculated by (a) Lognormal distribution with a = 0.21, (b) Lognormal distribution with mean likelihood estimation using equation (13), and (c) Poisson distribution with mean likelihood estimation using equation (19).

1450 P. M. DAVIS, D. D. JACKSON, AND Y. Y. KAGAN

TABLE 2

COACHELLA VALLEY 30-YEAR PROBABILITY

Distribution Parameters Update? Method Probability Reference (%)

Lognormal 0 < ~ < 0% 0 < a < ~ Yes MELE 9 This work Lognormal T = 220 yr, a = 0.30 No MLE 40 WGCEP Poisson 0 < X < co Yes MELE 9 This work

Notes: WGCEP = Working Group for California Earthquake Probabilities (1988); See also Table 1 Notes.

TABLE 3

MOJAVE SEGMENT 30-YEAR PROBABILITY

Distribution Parameters Update? Method Probability Reference (%}

Lognormal 0 < T < ~, 0 < ~ < ~ Yes MELE 21 This work Lognormal ~ = 125 yr, ~ = 0.66 Yes MLE 26 This work Lognormal ~ = 131 yr, a = 0.21 No MLE <50 WGCEP Lognormal T = 150 yr, ~ = 0.3 No TP 30 WGCEP Lognormal r = 125 yr, ~ = 0.21 Yes MLE 69 tw/NB Weibull X = 1/166 yr, B = 1.5 Yes MLE 21 SEA Poisson 0 < ), < ~ Yes MELE 19 This work Poisson ), = 1/146 yr Yes MLE 19 This work

Notes: TP = time-predictable method; tw/NB = this work, using method of Nishenko and Buland and fixing a = 0.21; SEA = Sieh et al. (1989); See also notes to Tables 1 and 2.

T a b l e 2 s u m m a r i z e s our e s t ima te s of the 30-year p robab i l i t y for the Coachel la

segment , a long wi th p rev ious ly p u b l i s h e d es t imates . T h e c u r r e n t m e a n l ike l ihood l ogno rma l p robab i l i ty , P = 0.09, is nea r ly equal to the m e a n l ike l ihood Po i s son e s t ima te (P = 0.087) a n d m u c h smal le r t h a n some prev ious es t imates . W G C E P

(1988) o b t a i n P = 0.4 u s ing the same m e t h o d as N i s h e n k o a n d B u l a n d (1977), wi th a = 0.30. U s i n g the N i s h e n k o a n d B u l a n d m e t h o d wi th the global a of 0.21, we

e s t i m a t e d P = 0.58. A m o n g the l ogno rma l models , the m e a n l ike l ihood e s t ima te is

p re fe rab le because the m e t h o d of c o m p u t a t i o n yields be t t e r es t imates . T h e r e is no c o n v i n c i n g way to choose one d i s t r i b u t i o n over a n o t h e r wi th so few data , b u t for

th i s s e g m e n t the re is no p rac t i ca l d i f ference b e t w e e n the l ogno r ma l a n d the P o i s son

d i s t r i bu t i on .

PALLETT CREEK

Sieh e t al. (1989) repor t da tes for ma jo r e a r t h q u a k e s a t P a l l e t t Creek as 671 _ 13, 797 ± 22 ,997 _ 16, 1048 ± 33, 1100 _+ 65, 1346 ± 17, 1480 ± 15, 1812, a n d 1857,

where the u n c e r t a i n t i e s def ine the 95 per cen t conf idence in te rva l . T h e m a x i m u m l ike l ihood r ecu r r ence t ime , 125 years , is s l ight ly less t h a n the m e a n r ecu r rence t ime of 131 years. T h e c o r r e s p o n d i n g a = 0.661 is m u c h larger t h a n the worldwide 0.21 va lue of N i s h e n k o a n d B u l a n d (1987) a n d gives rise to a lower p robabi l i ty .

T h e e s t ima t e s of c o n d i t i o n a l p robab i l i t y of a n e a r t h q u a k e in the n e x t 30 years are s u m m a r i z e d in T a b l e 3. T h e m e a n l ike l ihood es t ima tes are 0.21 a s s u m i n g a

l o g n o r m a l d i s t r i b u t i o n a n d 0.19 a s s u m i n g a P o i s son d i s t r ibu t ion . N i s h e n k o a n d B u l a n d give a r ange of 1.4 to 38.4 per cen t for the 20-year p robab i l i ty . U s i n g the m a x i m u m l ike l ihood t e c h n i q u e wi th P a l l e t t Creek da ta only, we e s t ima te ~ = 125 yr, ~ = 0.66, a n d the c o n d i t i o n a l p robab i l i t y = 0.26. T h e l a t t e r va lue is qui te close to t h a t e s t i m a t e d by S ieh e t al., who a s s u m e d a W e i b u l l d i s t r i b u t i o n a n d a p p a r e n t l y i n c l u d e d the d r o u g h t in t he i r ca lcu la t ions . All e s t ima te s are s imi la r because the

PREDICTING EARTHQUAKES USING INTERVAL TIMES 1451

1.3

1 . 1 - -

0 . 9 -

0 . 7 -

0 .5 -

1 8 0 0

UPDATED MAXIMUM L IKELIHOOD PARAMETER ESTIMATES, MOJAVE SEGMENT

I I I I I I I I I

MEAN RECURRENCE TIME ~

- * - C O E F F I C I E N T OF VARIATION

2 5 0

- - 2 0 0

- - 1 5 0 (D ,'r"

- - UJ >-

t -

- - 1 0 0

- - 5 0

2 8 0 0 I I I I I I [ I I

2 0 0 0 2 2 0 0 2 4 0 0 2 6 0 0

Y E A R

FIG. 6. Maximum ]ikelihoodest imatesofthe mean recurrenceintervalrandcoeff ic ientofvar ia t ion avs. time ~ r P a l l e t t Creek even t son the San Andreas ~ult, assumingcontinueddrought.

long catalog defines a and r rather well and because the drought time does not greatly exceed the estimated recurrence time.

Figure 6 shows the variation of r and a with hypothetical drought time and Figure 7 shows the corresponding evolution of the various probabilities. Comparing the lognormal curve with a = 0.21 with the other estimates reveals striking differences in risk estimates as time progresses.

DISCUSSION

It seems obvious that earthquakes are inevitable in some regions; the conventional wisdom is that "the longer it has been since the last one, the closer it is to the next one" (widely attributed to Professor C. F. Richter of Caltech, but also credited to Stanford Professor R. Jahns by Gere and Shah, 1984, p. 91). It follows that the seismic hazard would increase with each passing moment. Why then should our results show a decrease in the perceived hazard? The answer is that the perceived hazard depends on the estimated recurrence time and its variability, both of which increase with time since the last event. It is important to distinguish between the actual risk, which does increase with time in the quasi-periodic model, and the estimated risk, which may not.

The conventional intuition is based on the assumption that the recurrence time is known, a condition that could lead to increasing hazard with time. The recurrence time might be determined, for example, by plate tectonic considerations. If all of the plate motion is accomplished by "characteristic" earthquakes with similar displacements, and if the relative plate velocity is known, then the mean recurrence time is just the ratio of these quantities. But the characteristic earthquake model may not hold in specific cases, the mean displacement estimated from only a few

1452 P . M . DAVIS, D. D. JACKSON, AND Y. Y. KAGAN

0.9 -

THIRTY YEAR PROBABILITY, MOJAVE SEGMENT

1 1 1 - - I l I ..... i I I

0.7 --

p - -

0 . 5 - -

(A) LOGNORMAL, NO UPDATE _ _

0 . 3 - - (B) LOGNORMAL, UPDATED MAXIMUM LIKELIHOOD

-- / 1 " ~ / ~ ~ UPDATED MEAN LIKELIHOOD--

//,/ o . , . . . . .

(C) LOGNORMAL, UPDATED MEAN LIKELIHO

! i _ i I .... I L__ , ,i I 1800 2 0 0 0 2 2 0 0 2 4 0 0 2 6 0 0 2 8 0 0

YEAR

FIG. 7. Thirty-year earthquake probabilities vs. time for Pallett Creek assuming continued drought, as calculated by (a) Lognormal distribution with ~ = 0.21, (b) Lognormal distribution with updated maximum likelihood estimation, (c) Lognormal distribution with mean likelihood estimation using equation (13), and (d) Poisson distribution with mean likelihood estimation using equation (19).

events may be seriously in error, and aseismic slip may contribute to plate motion. Thus the historic record usually provides a much better estimate of the recurrence time than such "independent" methods as plate tectonics. The Parkfield segment lies just at the transition between a creeping and a locked section of the San Andreas, and a substantial contribution from aseismic slip should be expected. Furthermore, the seismic displacement can only be estimated directly for the last (1966) event, and that estimate is uncertain by as much as a factor of 2. At Coachella, there is no direct evidence of a characteristic displacement. Displace- ments in the last two events (A.D. 1450 and 1680) are estimated to exceed 3.1 m and 2.0 m respectively, and the slip rate on the fault is estimated at 30 mm/yr during the last millenium (Sieh, 1986). These data suggest a mean recurrence time exceeding 85 years, not a very helpful estimate. At Mojave the displacement is known well only for the last event, but the dates of the last 10 events are reasonably well known. Thus at all three of the segments considered in this paper, the dates of the past earthquakes provide a more precise estimate of the recurrence time than does an accounting of seismic slip. And as long as the recurrence interval must be estimated from the dates of past events, the perceived risk will decrease until the next earthquake in each of the three segments we have studied.

The conditional probabilities for the Parkfield, Coachella Valley, and Mojave segments using the lognormal distribution with the mean likelihood method differ only marginally from those of the Poisson distribution. We have adopted the lognormal distribution to illustrate our improvements in methodology and to facil- itate comparison with previous results. In so doing, we do not mean to suggest that the data are better fit by the lognormal distribution than by some others. In fact,

PREDICTING EARTHQUAKES USING INTERVAL TIMES 1453

the choice of distribution is a crucial question deserving future study. For a given mean recurrence time, simple point-process distributions can be roughly grouped into three categories: clustering models, Poisson models, and quasi-periodic models. For clustering models, both short and long repeat times are relatively more likely than in other models. For quasi-periodic models, repeat times near the mean are favored more than in other distributions. The lognormal distribution is most appropriate for quasi-periodic processes, but data from Poisson and clustering processes can be fit to a lognormal distribution. In the former case, the estimated coefficient of variation would be near 1.0, while in the latter it may exceed 1. At Mojave, the segment with the most complete record, our estimate for a is not significantly different from 1.0, suggesting that the catalog could be treated as Poisson. Sieh et al. (1989), using a Weibull distribution, found maximum likelihood parameters that strongly suggested clustering, although they could not reject a Poisson process either. At Coachella, the a cannot be estimated meaningfully from the data because the earthquake dating errors account for all of the variability of observed repeat times. Parkfield is the only segment we examined for which a Poisson process can apparently be rejected in favor of a quasi-periodic process. We have not actually made a rigorous test of the Poisson model, but the ~ of 0.32 for Parkfield (Table 1) is much lower than 1.0, and the likelihood function (Fig. 2b) appears to exclude the a = 1.0, at a very high confidence level. The Parkfield example of quasi-periodic behavior cannot be generalized to other fault segments, however, because the Parkfield segment was selected for its apparent regularity.

Our seismic risk estimates, like those of Bakun and Lindh (1985), Nishenko and Buland (1987), WGCEP (1988), and Sieh et al. (1989), are subject to important limitations:

(1) We have not allowed for systematic errors in the catalog, which are quite likely in some cases. For example, the historic data for Parkfield may be incomplete, and they do not distinguish well between "characteristic" earthquakes and other events before 1922. Another source of systematic errors is failure to resolve two or more closely timed earthquakes in the geological record. In the case of earthquake recurrence, the greatest danger from missed events during the dead time is that we may fail to recognize the proper form of the distribution; clustered events may appear to be quasi-periodic if each cluster is perceived as a single event. But if the process is actually quasi-periodic, then the dead time would be a minor problem because earthquake occurrence would be extremely unlikely in the short dead time just after a major event. Sieh et al. (1989) suggest that the temporal resolution of paleoseismic data is about 2 years. But if sediment deposition is very slow, this "dead-interval" time could be significantly larger. Statistically this situation is similar to that of Geiger-Mueller counters, which also experience dead-interval periods after each particle registration. The statistical theory of such counters is well-developed (see, for example, Smith, 1958); for a Poisson process, the dead interval simply decreases the active observation time; that is,

T - Td

(Smith, 1958; p. 274); where Td is the average dead-interval time, Ti is the corrected interval time, and T is the measured time. More complicated cases can be solved by methods discussed in the above-referenced review.

(2) We chose a distribution function without examining whether it really de- scribes earthquake occurrence well in specific cases. At Parkfield and Mojave, where

1454 P. M. DAVIS, D. D. JACKSON, AND Y. Y. KAGAN

the current droughts approximate the mean recurrence time, we expect that a clustering model would predict smaller conditional probabilities than either the Poisson or lognormal models. This expectation follows from the tendency of clustering models to favor short and long intervals at the expense of average ones. But at Mojave, the Poisson and lognormal distributions give similar results (cf. Table 2, lines 1 and 3), so the choice of distribution may not matter much. At Parkfield, the distribution does seem to matter (Table 1) but the data strongly favor a quasi-periodic distribution. For the Coachella segment, a clustering model might give a higher conditional probability than the lognormal or Poisson, because the drought is long. But the Poisson and lognormal give nearly identical results (Table 3), so again the choice of distribution may not matter. In the regions we have studied, the choice of distribution is either clear from the data, as at Parkfield, or not important to the final result, as at Mojave and Coachella. But the choice of distribution is not, in general, moot; choice of the wrong distribution could cause serious errors, especially for a fault segment with an inadequate earthquake record.

(3) While our results seem rather different from previous results, there is not a powerful test that would readily distinguish between the various ideas. The best opportunity for a test is at Park field, where we calculate a 34 per cent chance of a characteristic earthquake within 5 years (1988 to 1993), while Nishenko and Buland (1987) calculate a range of 53 to 84 per cent. Let H0 be the hypothesis that earthquakes there are lognormal with T = 22 yr and ~ = 0.21 (essentially the Nishenko and Buland model), and let/-/1 be the hypothesis corresponding to our mean likelihood estimate with updating (line 1 of Table 1). Let us start the clock on 22 September 1988, the date we submitted the first draft of the paper. Take the critical date 5 years later, i.e., 22 September 1993. According to Ho, the probability of at least one event before 22 September 1993 is 69 per cent, while for H~ the probability is only 34 per cent. Suppose we accept/to if the predicted earthquake occurs before that date, and reject H0 if it occurs therafter (see, for example, Martz and Waller, 1982, p. 60). Then the probability of a type I error (reject/to, when it is true) is 31 per cent, whereas the probability of a type II error (accept/to, when it is false) is 34 per cent. The likelihood ratio, i.e., the ratio of the two probabilities of nonoccurrence, is 1 : 2. These error values indicate that it is difficult to distinguish between/to and H~ on the basis of the early occurrence of an earthquake. However, if no earthquake occurs before a critical date of February 1997, this will favor H1, since according to Ho the probability of this is less than 5 per cent, whereas we calculate that according to H~ it is 35 per cent. Then the probability of a type I error is 5 per cent, whereas for a type II error it is 65 per cent. The likelihood ratio is 7 : 1. We would reject Ho with 95 per cent confidence. Although this test becomes more definitive as the drought continues after 1997, eventually the adoption of the lognormal distribution would be rejected in favor, for example, of the Poisson model.

(4) Estimates of earthquake probabilities based on the assumption of quasi- periodic characteristic earthquake sequences neglect the possibility of interaction between an earthquake fault with characteristic earthquakes and other faults. Earthquake simulations derived from mechanical models of rupture (Rundle, 1988) indicate that when interaction is taken into account, earthquakes on the fault cease to be periodic or characteristic.

(5) We treated with one-dimensional techniques a problem that involves at least five dimensions: time and magnitude as well as three space dimensions. Until we incorporate these variables, our models only predict "characteristic" earthquakes, which we have defined only by example. The danger of circular reasoning should be readily apparent. There is no appropriate two-dimensional generalization of inter-

PREDICTING EARTHQUAKES USING INTERVAL TIMES 1455

earthquake interval, since we cannot define the nearest-neighbor distance in the time-magnitude space. Thus, if an earthquake with magnitude 5.4 occurs on a segment where the characteristic event in a magnitude range 5.5 to 6.5 is predicted, there will always be a confusion, whether (taking into account the accuracy of magnitude or seismic moment determination) it is a forecasted event or not. The reduction of seismicity to a one-dimensional temporal process is difficult to formal- ize because, even for a simple earthquake fault, two spatial boundaries have to be drawn, involving the loss of two degrees of freedom. In the case of a complex spatial seismicity pattern, complicated polygons may be needed as the boundaries of a region. Selecting such region boundaries based on catalog data invites bias; one will find those features (artifacts) in the subcatalogs that were used in the selection process. For example, using various spatial and magnitude limits, it is easy to find quasi-periodic sequences in a sufficiently large synthetic random earthquake catalog. Such "data-based" sampling has no predictive value. The dangers of such biased selection have been discussed, for example, by Savage and Cockerham (1987). We note that in the case of Parkfield the catalog data have been used in the selection, whereas the Pallett Creek data selection has been controlled by considerations other than seismic data. This might explain why these regions have significantly different coefficients of variation: in the former case, it is about 0.3 (see Fig. 1), hence strongly periodic; in the latter case, the coefficient is close to one (Fig. 7). Sieh et al. (1989) report that even the hypothesis of Poisson occurrence cannot be rejected for Pallett Creek.

CONCLUSIONS

We have devised a likelihood function for parameters in a probabilistic recurrence model that accounts for an open-ended interval (a drought) at the end. The method could be generalized easily to include an open-ended period at the beginning of a catalog (a time when earthquakes were known not to occur). The likelihood function can be used to obtain maximum likelihood estimates of the parameters, or, better, it can be used as a weighting function in the "mean likelihood" method to average seismic risk over the parameters. Assuming the lognormal distribution for interevent times, the mean likelihood earthquake probability for Parkfield is 34 per cent for the next 5 years, considerably less than the 53 to 84 per cent estimated by Nishenko and Buland (1987). For the Coachella segment of the San Andreas Fault, the mean likelihood risk is 9 per cent, much lower than the rate of 40 per cent estimated by WGCEP (1988). For Coachella Valley, the coefficient of variation a is poorly known, partly because of large uncertainties in the dates of prehistoric events. More accurate dates, or even a better estimate of dating uncertainties, could improve our knowledge of seismic risk there. We estimate a probability of 21 per cent for a large earthquake on the Mojave segment of the San Andreas during the next 30 years. Our value agrees with that of Sieh et al. (1989), although WGCEP calculate a higher rate.

ACKNOWLEDGMENTS

We appreciate support from the National Science Foundation through grant EAR 88-04883. We thank Alan Lindh for suggesting the problem of accounting for seismic droughts in estimating statistical parameters. We thank Kerry Sieh for providing prehistoric earthquake dates for Pallett Creek prior to publication. We thank D. Vere-Jones and D. Perkins for helpful reviews.

REFERENCES

Ambraseys, N. N. (1971). Value of historical records of earthquakes. Nature 232,375-379. Bakun, W. H. and A. G. Lindh (1985). The Parkfield, California, earthquake prediction experiment,

Science 229, 619-624.

1456 P. M. DAVIS, D. D. JACKSON, AND Y. Y. KAGAN

Barnard, G. A. (1949). Statistical Inference, J. Royal Stat. Soc. B l l , 115-149. Gere, J. M. and H. C. Shah (1984). Terra Non Firma, Understanding and Preparing for Earthquakes, W.

H. Freeman and Company, New York, pp. 203. Jenkins, G. M. and D. G. Watts (1968). Spectral Analysis and its Applications, Holden-Day, Oakland,

California, pp. 525. Kagan, Y. and L. Knopoff (1976). Statistical search for non-random features of the seismicity of strong

earthquakes, Phys. Earth Planet. Inter. 12,291-318. Lee, W. H. K. and D. R. Brillinger (1979). On Chinese earthquake history--an attempt to model

incomplete data set by point process analysis, Pageoph 117, 1229-1257. Martz, H. F. and R. A. Waller (1982). Bayesian Reliability Analysis, Wiley, New York, 745 pp. Nishenko, S. P. and R. Buland (1987). A generic recurrence interval distribution for earthquake

forecasting, Bull. Seism. Soc. Am. 77, 1382-1399. Rundle, J. B. (1988). A physical model for earthquakes, 1. Fluctuations and interactions, J. Geophys.

Res. 93, 6237-6254. Savage, J. C. and R. S. Cockerham (1987). Quasi-periodic occurrence of earthquakes in the 1978-1986

Bishop-Mammoth lake sequence, Eastern California, Bull. Seism. Soc. Am. 77, 1347-1358. Sieh, K. (1986). Slip rate across the San Andreas fault and prehistoric earthquakes at Indio, California

(abstract), EOS 67, 1200. Sieh, K., M. Stuiver, and D. Brillinger (1989). A more precise chronology of earthquakes produced by

the San Andreas fault in Southern California, J. Geophys. Res. 94,603-623. Smith, W. L. (1958). Renewal theory and its ramifications, J. Roy. Stat. Soc. B20, 243-302. Tarantola, A. and B. Valette (1982). Inverse problems = quest for information, J. Geophys. 50, 159-170. Toppozada, T. R. (1987). Earthquake History of Parkfield and Surroundings (abstract), EOS 68, 1345. Vere-Jones, D. and T. Ozaki (1982). Some examples of statistical estimation applied to earthquake data.

I. Cyclic Poisson and self-exciting models, Ann. Inst. Statist. Math. B34, 189-207. Working Group on California Earthquake Probabilities (WGCEP) (1988). Probabilities of large earth-

quakes occurring in California on the San Andreas fault, U.S. Geol. Surv., Open-File Rept. 88-398.

DEPARTMENT OF EARTH AND SPACE SCIENCES

UNIVERSITY OF CALIFORNIA LOS ANGELES, CALIFORNIA

(P.M.D., D.D.J.)

INSTITUTE OF GEOPHYSICS AND PLANETARY PHYSICS UNIVERSITY OF CALIFORNIA

LOS ANGELES, CALIFORNIA 90024 (Y.Y.K.)

Manuscript received 22 September 1988