A semi-Markov model estimating the waiting times and magnitudes of large ... will vary from location to location, the probability of occurrence of a great earth-.
Bulletinofthe SeismologicalSocietyofAmerica,Vol.70,No. 1, pp. 323-347,February1980
A SEMI-MARKOV MODEL FOR CHARACTERIZING RECURRENCE OF GREAT EARTHQUAKES BY ASHOK S. PATWARDHAN,RAM B. KULKARNI,AND DON TOCHER* ABSTRACT
A semi-Markov model estimating the waiting times and magnitudes of large earthquakes is proposed. The model defines a discrete-time, discrete-state process in which successive state occupancies are governed by the transition probabilities of the Markov process. The stay in any state is described by an integer-valued random variable that depends on the presently occupied state and the state to which the next transition is made. Basic parameters of the model are the transition probabilities for successive states, the holding time distribution, and the initial conditions (the magnitude of the most recent earthquake and the time elapsed since then). The model was tested by examining compatibility with historical seismicity data for large earthquakes in the circum-Pacific belt. The examination showed reasonable agreement between the calculated and actual waiting times and earthquake magnitudes. The proposed procedure provides a more consistent model of the physical process of gradual accumulation of strain and its intermittent, nonuniform release through large earthquakes and can be applied in the evaluation of seismic risk.
INTRODUCTION The object of this paper is to describe an analytical mode] for characterizing the recurrence of great earthquakes (defined as earthquakes of magnitude M = 7.8) consistent with the general physical processes contributing to their occurrence. Available historical seismicity data suggest that great earthquakes exhibit patterns of nonrandomness in location, size, and time of occurrence (Mogi, 1968; Sykes, 1971; Kelleher et al., 1974). From a physical standpoint, the occurrence of great earthquakes can be represented by a continuous, gradual process of strain accumulation interrupted intermittently by episodes of sudden release. Several factors are believed to influence the size of great earthquakes in a given area; for example, accumulated strain, shearing resistance, slip rates, tectonic stress, and displacement over the interface area. Recurrence characterization includes estimation of sizes of and holding times between successive great earthquakes at a given location. Because of the uncertainties associated with the underlying physical processes, the characterization is probabilistic in nature. Several statistical models have been proposed to represent the process of earthquake occurrence. The most common model is the Poisson model, which assumes spatial and temporal independence of all earthquakes including great earthquakes; i.e., the occurrence of one earthquake does not affect the likelihood of a similar earthquake at the same location in the next unit of time. Other models such as those proposed by Shlien and Toksoz {1970) and Esteva {1976) consider the clustering of earthquakes in time. A few other probabilistic models have been used to represent earthquake sequences as strain energy release mechanisms. Hagiwara (1975) has proposed a Markov model to describe an earthquake mechanism simulated by a belt-conveyor model. A Weibull distribution is assumed by Rikitake * Deceased, July 6, 1979. See "Memorial",p. 400, this issue. 323
39,4
ASHOK S. PATWARDHAN, RAM B. KULKARNI, AND DON TOCHER
(1975) for the ultimate strain of the Earth's crust to estimate the probability of earthquake occurrences. Earthquake magnitudes, however, are not represented in this model. Knopoff and Kagan {1977) have used a stochastic branching process that considers a stationary rate of occurrence of main shocks and a distribution function for the space-time location of foreshocks and aftershocks. These models are useful in the broad context of predicting earthquake sequences over large tectonic regions. However, these models are not adequate to characterize the location-specific occurrences of great earthquakes. While a Poisson process does provide estimates of the probability of occurrence of great earthquakes of any size or the formation of a seismic gap which may be characteristic of a whole region, the estimates are independent of the size and time elapsed since the last great earthquake, invariant in time, and insensitive to location. The physical model outlined above would suggest, on the contrary, a dependence on at least two initial conditions--the size of and the time elapsed since the last great earthquake. Since both of these conditions will vary from location to location, the probability of occurrence of a great earthquake or continuation of a seismic gap can be expected to vary from location to location even within the same seismic region. A need exists, therefore, for establishing an analytical model that is more consistent with the underlying physical processes and that can characterize the recurrence of great earthquakes on a more location-specific basis. FORM OF THE SELECTED MODEL In this paper, a semi-Markov process has been utilized, which can model the spatial and temporal dependencies of great, main-sequence earthquakes. A semiMarkovian representation of earthquake sequences is consistent with the above generalized hnderstanding of earthquake generation consisting of gradual, uniform accumulation and periodic release of significant amounts of strain energy in the Earth's crust. Since the buildup of strain energy sufficient to generate another great earthquake would take some time, the occurrence of a great earthquake at the same location is less likely within short periods of time following an earthquake of similar size than within an area which has not experienced a similar earthquake for a long time. As the time elapsed without the occurrence of another great earthquak~ increases, so does the probability of its occurrence. It is reasonable to assume that both the size and waiting time to the next earthquake is influenced by the amount of strain energy released in the previous earthquake (related to the magnitude of that earthquake) and the length of time over which strain has been accumulating. For instance, in the simple case of a uniform strain rate, the strain buildup required to generate a magnitude 8.6 earthquake will take longer than the strain buildup to generate a magnitude 7.8 earthquake. These considerations are well modeled by a semi-Markovian representation of earthquake sequences. A semi-Markov process has the basic Markovian property of one-step memory (i.e., the probability that the next earthquake is of a given magnitude depends on the magnitude of the previous earthquake). However, an additional feature of a semi-Markov process is that it provides for the distribution of a holding time between successive earthquakes, which depends on the magnitudes of the previous and the next earthquake. Consideration of the holding time in effect provides a multi-step memory for the semi-Markov process. The following sections describe the development and application of the semiMarkov model.
A SEMI-MARKOV MODEL FOR RECURRENCE
OF GREAT EARTHQUAKES
325
D E V E L O P M E N T OF THE M O D E L
The theoretical development of a semi-Markov process is discussed in the literature (Howard, 1971). The model is described by two parameters, state, i, and holding time, v. A state is defined by the magnitude of a great earthquake. The continuous magnitude scale can be divided into appropriate intervals to specify discrete states of the system. Figure 1 is a schematic representation of the semi-Markov process. It shows the present conditions at a given location given by the magnitude of the last great earthquake, Mo, and the time elapsed since its occurrence, to. In the next unit of time, the system may either experience no great earthquake or make a transition to any of the other discrete states, M~, M2, or 2143.The representation of earthquake
Q
G Present
2 Time Units
Fro. 1. Schematic representation of the trajectory of a semi-Markov process.
occurrences by a semi-Markov process implies that the likelihood of the next great earthquake being of a particular magnitude (i.e., transition to state j), depends on the magnitude of the previous great earthquake {present state i). The holding time, r, represents the time period for which the system holds in a given state, i. As discussed by Howard (1971), the successive state occupancies (earthquake magnitude} will be governed by the transition probabilities of a Markov process, but the stay in any state (holding time) will be described by an integer-valued random variable that depends on the state presently occupied and on the state to which the next transition will be made. The formal model. Let pij be the probability that a semi-Markov process which
326
ASHOK S. P A T W A R D H A N , RAM B. K U L K A R N I , AND DON TOCHER
entered state i on its last transition will enter state j on its next transition. The transition probabilities must satisfy the following properties pij>=0
i=1,2,-..,N;
j=I,
2,...,N
(1)
and N j=l
P~i = i
(2)
where N is the total number of states in the system. Whenever the process enters a state i, the likelihood t h a t it will go to state j at some future time is determined by the transition probability Pii. However, after j has been selected, but before making the transition from state i to state j , the process "holds" for a time T~j in state i. The holding times ~ii are positive, integervalued random variables each governed by a probability mass function h~j (m) called the holding time mass function for a transition from state i to state j. Thus, P(~"ij = m) = hij(m)
m=
1, 2, 3 , . . .
i = 1,2, . . . , N j=I,
2,...,N.
(3)
We assume t h a t a system entering a state i at time 0 will not make another transition at time 0; i.e., hq(0) = 0.
(4)
After holding in state i for Tij, the process makes the transition to state j and then immediately selects a new destination state k using the transition probabilities pj 1, pj2, . . . , pjN. It next chooses a holding time Tjk in state j according to the probability function hjh (m) and makes its transition at time ¢/k after entering statej. The process continues developing its trajectory in this way indefinitely. A possible trajectory of such a process is shown in Figure 1. The time a semi-Markov process spends in state i given t h a t it enters i at time 0 without knowing the destination state is called the waiting time Ti in state i. Let Wi(. ) be the probability mass function of ~i; i.e., N
Wi(m) = P(¢i = m) = Y~ Pij(m).
(5)
j=l
The cumulative distribution function (CDF) and complementary CDF of rij are denoted by hi j ( . ), respectively. The same functions of vi are denoted by _-_Wi (.) and > Wi (.), respectively. If the system has already spent some time (say, to) in a particular state (say, i), its first transition out of state i will not be governed by the functions pij and hij (.). These functions apply for a system which has just entered state i; t h e y do not apply for a system which is observed in state i at the present time. Let the transition and holding time probability functions for the first transition out of state i given t h a t it
A S E M I - M A R K O V MODEL FOR R E C U R R E N C E OF GREAT E A R T H Q U A K E S
327
has stayed in state i for the previous to time periods b epij1 and h ij1 (.), respectively. By using Bayes' theorem, the following expression is found forp~i p~j = P(transitions to j[i, to) P(holding time at least to[i, j ) P { t r a n s i t i o n t o j I i) iF, P(holding time at least to [ i, j ) P ( t r a n s i t i o n to j[ i) -- >h~i (to)p~
(6)
j~ >hii(to)pij An expression for h~j(. ) can be derived as follows h}i(m) = P(holding time in i is m[ holding time in i is at least to and the next transition is to j)
= P(z~j = m + to) = hij{m + to) P(¢ij > to)
>hij(to).
(7)
A superscript I is used to distinguish the probability functions for the first transition from those for the subsequent transitions; e.g., =7.8) will occur in the next 40 yr is 0.19, while the probability of occurrence of exactly one 8.4 (+_0.2) magnitude earthquake is 0.056. The holding time and transition probabilities used in obtaining these results are based on the assumption of a constant magnitude-rupture length relationship for all the areas included in the analysis (data shown in Tables 2 through 5). The basic output of the semi-Markov model (i.e., the probabilities such as those shown in Table 6) provide one of the inputs necessary for seismic risk calculation at a given location. UTILIZATION OF RESULTS
The set of probabilities of different magnitudes of great earthquakes within a given time period yielded by this model can be readily applied for a number of purposes. The model is particularly helpful in characterization of a N ( M ) relationship, in the magnitude range where the Poisson model has difficulty due to lack of
338
ASHOK S. PATWARDHAN, RAM B. KULKARNI, AND DON TOCHER
s u f f i c i e n t d a t a , i.e., n e a r t h e t a i l o f a d i s t r i b u t i o n . I t c a n a l s o b e u t i l i z e d t o a s s e s s t h e p r o b a b i l i t y o f l a c k o f e a r t h q u a k e s ; i.e., t h e p r o b a b i l i t y o f c o n t i n u a t i o n o f a " s e i s m i c gap." Both determinations can be location-specific and on a "real-time" basis. D i s c u s s e d b e l o w is t h e m e t h o d o l o g y f o r e v a l u a t i o n o f " s e i s m i c g a p s " a n d c h a r a c t e r ization of tails of magnitude distributions.
Characterization of seismic gaps. T h e r e is n o w e l l - a c c e p t e d d e f i n i t i o n o f a "seismic gap." The term "gap" has been applied both in a spatial and/or temporal TABLE 6 EARTHQUAKE PROBABILITIES FOR THE NEXT 40 YEARS CACULATED FROM THE MODEL (M0 -- 8 + 0.2; to = 5 yr)
Probability
0.1933 0.1691 0.9840 × 10 ~ 0.9539 x 10-~ 0.5852 × 10-~ 0.5627 x 10-~ 0.5593 × 10 ~ 0.4684 × 10-1 0.3966 x 10-~ 0.2371 × 10-~ 0.1981 × 10-~ 0.1849 x 10-~ 0.1770 x 10-~ 0.1724 x 10 ~ 0.9683 x 10-2 0.8271 x 10-z 0.7256 x 10-2 0.7229 x 10-2 0.6807 x 10-2 0.5505 x 10-2 0.2764 x 10-2 0.2490 x 10-2 0.2451 x 10-2 0.1828 × 10-2 0.1081 x 10-2 0.1067 x 10-z 0.7314 x 10-~ 0.6698 x 10-z 0.3025 x 10-3 0.2575 x 10-3 0.1035 x 10-3 0.5808 × 10-4
Number of Great Earthquakes of Each Magnitude 8 -+ 0.2
8.4 ± 0.2
&75 +_0.15
0 1 2 0 1 0 1 3 2 2 3 4 0 1 1 2 3 2 0 0 1 1 0 0 1 1 2 0 0 0 0 0
0 0 0 o 1 1 0 0 1 0 1 0 1 1 2 1 0 2 2 0 2 0 2 1 1 3 0 3 2 3 0 1
0 0 0 1 0 0 1 0 0 1 0 0 1 1 0 1 1 0 0 2 1 2 1 2 2 0 2 0 2 1 3 3
sense. When used only in a spatial sense, it refers to a zone in which no great earthquake has occurred for sometime since the last great earthquake {usually for m o r e t h a n 2 0 t o 30 y r ) , w h i l e t h e a d j a c e n t z o n e s h a v e e x p e r i e n c e d g r e a t e a r t h q u a k e s within the same time period. When used in a spatial-temporal sense, it refers to the nonoccurrence of a great earthquake in a given zone or part of it for a period of time s i n c e t h e l a s t g r e a t e a r t h q u a k e . I n t h i s p a p e r , t h e t e r m is u s e d i n t h e l a t t e r s e n s e (i.e., s p a t i a l - t e m p o r a l s e n s e ) . T h e m i n i m u m t i m e p e r i o d f o r t h e i d e n t i f i c a t i o n o f a g a p is t a k e n a s o n e u n i t t i m e , i.e., 5 y r . T h e m o d e l c a n b e u t i l i z e d t o e s t i m a t e t h e
A SEMI-MARKOV
MODEL
FOR
RECURRENCE
OF
GREAT
EARTHQUAKES
339
probabilities that there will be no earthquake within a period of interest (say, 40 yr) for different initial conditions. Figure 5a shows an example in which the relationship between the probability of continuation of a gap for the next 40 yr is plotted against the number of years for which there have been no earthquakes following earthquakes of M = 8 +_ 0.2 and M = 8.75 _ 0.15. Figure 5b shows similar relationships for area B. The relationships in Figure 5a indicate that the probability that a gap will be formed following a 0.5
r
..
I
(a) For all areas except Area 8
t 0.4
1
Mo= 8.75 +- 0.15 - -
- -
== o .~
0.3 Mo = magnitude of last great earthquake
•~
\
0.2
o 0.1
0
0
0.3
==
20
40
60
80
100
I (b) For Area B
£9 E = ._~ ~ 0.2
.~-z 0.1 ~.~
Mo= 8.78 ± 0.15
o 0
0
20
40
60
80
100
Number of Years Since the Last Great Earthquake, t o
FIG. 5. Probabilityof continuationof a seismic gap for differentinitial conditions. magnitude 8 _ 0.2 earthquake is approximately 20 per cent in the first 5 yr. If no great earthquake occurs for 20 yr, the probability of continuation of the gap for another 40 yr decreases to approximately 10 per cent; this result differs from a Poisson model, which indicates a constant probability of formation or continuation of a gap. Figure 5a would seem to indicate that for a previous earthquake of magnitude 8.75 _ 0.15, the probability of continuation of the gap would show little change for elapsed times of up to 60 yr. This result is primarily due to relatively large estimated holding times for earthquakes following an 8.75 +_ 0.15 magnitude earthquake used in present analysis. It can be expected that for larger elapsed times (>100 yr) the probability of continuation of the gap would decrease, thus conforming to the
340
ASHOK
S. P A T W A R D H A N ,
RAM
B. K U L K A R N I ~
AND
DON
TOCHER
generally non-Poissonian character. A similar trend can also be observed for area B (Figure 5b) which has relatively shorter holding times. In a different seismic e n v i r o n m e n t such as area B, the corresponding probabilities for M = 8 + 0.2 for the continuation of a gap for periods of 5 and 20 yr are 10 and 4 per cent, respectively; i.e., considerably smaller t h a n the probabilities for the continuation of gaps in other areas. In other words, gaps in area B h a v e a greater likelihood to be filled t h a n the gaps in other areas. T h e relationships in Figure 5a also indicate t h a t the probability of continuation of a gap after the lapse of any given t i m e period increases with the m a g n i t u d e of the last earthquake; this result is to be anticipated. T h e trends for higher m a g n i t u d e s show s o m e interesting differences especially w h e n the elapsed t i m e is greater t h a n a p p r o x i m a t e l y 50 yr. As seen TABLE
7
COMPARISON OF RELATIVE FREQUENCY OF GAP FROM HISTORICAL SEISMICITY DATA WITH PROBABILITIES CALCULATED FROM MODEL (M0 -- 8 - 0.2)
M~)
t~, yr
Period of Interest, tyr
8 +- 0.2 8 _+0.2 8 +_0.2 8 +_-0.2
5 5 30 5
5 20 20 40
Number of Events Number of Events with at Least One with No Earthquake Earthquake in Time in Time t t
5 12 2 18
Relative Frequency of Gap from Data
Probability of Gap from Model
0.81 0.50 0.60 0.18
0.74 0.42 0.42 0.19
22 12 3 4
TABLE 8 COMPARISON OF RELATIVE FREQUENCY OF GAP FROM HISTORICAL SEISMICITY DATA WITH
PROBABILITIESCALCULATEDFROMMODEL(Mo = M(~
t~, yr
Period of Interest, tyr
8.4 +- 0.2 8.4 + 0.2 8.4 +--0.2 8.4 -+ 0.2 8.4 +- 0.2 8.4 -+ 0.2
5 5 30 5 30 30
5 20 20 40 40 5
Number of Events Number of Events with at Least One with No Earthquake Earthquake in Time in Time t t
4 10 2 16 2 1
24 16 3 4 2 6
8 + 0.2)
Relative Frequency of Gap from Date
Probability of Gap from Model
0.86 0.62 0.60 0.20 0.50 0.86
0.77 0.40 0.58 0.24 0.31 0.88
f r o m the posterior probability distribution for zij > 40 yr, the probability of having a m a g n i t u d e 8 +__0.2 or 8.4 +__0.2 e a r t h q u a k e s is very small, and the probability of continuation of a gap is primarily the probability of having no e a r t h q u a k e s of m a g n i t u d e 8.75 +_ 0.15. T h i s result is discussed further in the next p a r a g r a p h . T h e probabilities of continuation of a gap for different initial conditions (M0, to) a n d different t i m e periods of interest (t) are shown in T a b l e s 7 and 8 for all areas except area B. T a b l e 7 gives values of probabilities (M0 = 8 +_ 0.2) e s t i m a t e d f r o m the m o d e l and corresponding relative frequency values e s t i m a t e d by directly using the sample d a t a and d a t a on the events with no e a r t h q u a k e s for a n u m b e r of years following M / s h o w n in T a b l e 3. As seen from the table, the values calculated f r o m the m o d e l show very good a g r e e m e n t with the data. F o r example, to e s t i m a t e the probabilities f r o m the d a t a in T a b l e 3 for M0 = 8, to --- 5, and t -- 5, the sample d a t a and the d a t a on n u m b e r of years for different events with no e a r t h q u a k e following Mi = 8 ± 0.2 were examined to count the n u m b e r of events with at least one e a r t h q u a k e in 5 yr (5 events) and the n u m b e r of events with no e a r t h q u a k e s between 5 a n d 10 yr (22 events), which yield a relative frequency of 22/27 -- 0.81 for the
A
SEMI-MARKOV
MODEL
FOR
RECURRENCE
OF
GREAT
EARTHQUAKES
341
continuation of a gap. This value compares favorably with the probability of 0.74 estimated from the model. Similar estimates for other initial conditions show good agreement with the model except where the data are too scanty (e.g., Mo = 8 + 0.2, to = 30 yr, t = 20 yr). Table 8 shows a similar comparison for Mo = 8.4 + 0.2. As in the previous case, the agreement between estimates of probability of continuation of gaps from the data and the probabilities estimated from the model is reasonable. This agreement between the probabilities of formation of gaps also suggests that the prior distributions based on subjective assessment are reasonable. 1.2
m
t s
D; Semi-Markov Model Mo = 8 -+ 0.2, t o = 80 years
1.0
.E
s A ;
0.8
~ ~ " ' , ~ ' ~
A
2,
Poisson Model, b = 1.0
0.6
C; Semi-Markov Model ~1o = 8 -+ 0.2, to = 30 years
" \ . \,~
2
0.4
Mo ~ 8 -+ 0.2, t o = 5 years ~
*~.~.
0.2
0
7.8
I
I
I
8.0
8.2
8.4
I .
8.6
8.8
Earthquake Magnitude. M Mo, t o initial conditions of semi-Markov model Mo = magnitude of last great earthquake t o = waiting time since last great earthquake
Fro. 6. Comparisonof magnitudedistributionrelationshipsderivedfromthe modelfor differentinitial conditionswith relationshipsbased on the Poissonmodel.
Characterization of tails of earthquake magnitude distribution. A Poisson model of earthquake occurrences based on a typical N ( M ) curve {i.e., a and b values) generally does not give reasonable estimates of probabilities of great earthquakes (e.g., Ms >- 7.8). We believe that the semi-Markov model developed in this study provides a better characterization of the tails of the probability distribution of earthquake magnitudes since the model takes into account the interaction between the length of holding time and the magnitude of the next great earthquake, and the influence of recent energy releases on the magnitude and time of the next great earthquake. Figure 6 shows the probability of at least one earthquake of magnitude M or greater in the next 40 yr for different initial conditions. For comparison, the probabilities calculated from a Poisson model using an average number of 2.44 earthquakes = 7.8 in 40 yr have a b value of 1.0. These values approximately
342
ASHOKS. PATWARDHAN, RAM B. KULKARNI, AND DON TOCHER
represent the level of seismic activity for great earthquakes in zone 4 of the AlaskaAleutian area. It is seen that the results of a Poisson model are close to those of the semi-Markov model with initial conditions Mo = 8 and to = 5 yr; i.e., the case in which a magnitude 8 + 0.2 earthquake occurred 5 yr ago. For areas where a great earthquake has not occurred for a relatively long period of time (e.g., 11//o= 8, to = 8 yr), the probability of occurrence of another great earthquake in, say, 40 yr is significantly higher than that estimated by a Poisson~model. The Poisson model seems to give higher probabilities for the occurrences of multiple earthquakes (n => 3) in 40 yr than the semi-Markov model for the initial conditions of Mo --- 8 and to in 5 yr. For the latter model, the probabilities of occurrences of a given number of earthquakes are influenced by the initial conditions (recent releases of strain energy); these probabilities provide inputs to a seismic risk model that are more consistent with the postulated earthquake mechanism of strain accumulation and release. For example, the results would indicate that to achieve the same level of risk, a facility in a zone with a seismic "gap" of, say, 50 yr after the occurrence of a magnitude 8 earthquake should be designed for a higher seismic loading than a facility in a zone where a magnitude 8 earthquake occurred, say, 5 yr ago; this inference appears to be reasonable. Figure 7a indicates that the probability of a continuation of a seismic gap in the next 40 yr increases for elapsed times greater than approximately 50 yr. This result would appear to stem from the fact that for the number of years elapsed since the last earthquake exceeds a certain value, an earthquake of magnitude 8 + 0.2 is less likely to occur than an earthquake of a higher magnitude. In other words, under the given circumstances, the system is likely to wait a little longer and produce a larger magnitude earthquake. As the elapsed time continues to grow, the probability of a higher magnitude earthquake would increase further and, consequently, the probability of continuation of a seismic gap would decrease. Additional data and interpretations are necessary to examine the trends for larger earthquake magnitudes and longer elapsed times. PARAMETRIC ANALYSES To assess the effect of variation in input parameters on the results provided by the model, two analyses were made. In one, the holding times in the prior distributions were increased by factors of 1.5 and 6. The probabilities of continuation of a seismic gap for the next 40 yr for both cases of increased holding time for Mo = 8 + 0.2 and 8.75 + 0.15 and different time periods (to) after the previous great earthquakes are shown in Figure 7. When the holding times are increased by a factor 1.5, the probability of continuation of a seismic gap followinu a magnitude Mo = 8 + 0.2 earthquake increases by a factor of 2 to 3; while for an increase by a factor 6, the corresponding probability increases by a factor 10. In the case of higher magnitudes (Mo = 8.75 _+ 0.15), a different trend is observed. If the holding time is increased by a factor 1.5, the probability of continuation of a gap for 40 yr increases by a factor 7. If the holding time is increased by a factor 6, the probability of a gap increases by a factor 14. This is so because, as the average holding times increase, the probability of having earthquakes of magnitudes 8 +_. 0.2 or greater are not insignificant even after 40 yr and the probability of continuation of a gap is not small. The historical seismicity data indicate a similar trend. The second parametric analysis of the zones was defined, based on variable rupture lengths. Considerable uncertainty exists in the rupture lengths appropriate for different areas. Kelleher and McCann (1976) give estimates of maximum rupture
A SEMI-MARKOV MODEL FOR RECURRENCE OF GREAT EARTHQUAKES 0.9
I
343
T
(a) For all areas except Area B (holding times increased by a factor 6)
0.8 ~ E =~ ~
]
/
Mo = 8.75 ± 0.15
0.7
r~Q
. = z 0.6
~._= o .Q
~
Mo = 8 ± 0.2
J
0.5
4
J
0,4
0.5
20
40
I
I
60
80
100
(b) For all areas except Area B (holding times increased by a factor 1.5) 0.4 M o = magnitude of last great earthquake E ~ .~ ~ 0.3
o~ .~z
~=._=
0.2
t~
0.1
0
0
0.3
20
40
60
80
100
I
(c) For Area B (holding times increased by a factor 1.5)
~~
o.~ =
.
+
.
. ~ z 0.1
0
0
20 40 60 80 100 Number of Years Since the Last Great Earthquake, t o
Fro. 7. Probability of continuation of a seismic gap for different initial conditions (holding t ~ e increased approximately by a factor 1.5 and 6).
344
ASHOK S. P A T W A R D H A N ~
RAM B. K U L K A R N I ,
•- d r -
c~
~-
~
cq
c~
c~
¢~
cq
A N D D O N TOCI-IER
c~
z
C) t~
c~
+1
c'q
+1 +~ +1 o~
+1
+1 ~d
+~
A SEMI-MARKOV MODEL FOR R E C U R R E N C E OF GREAT EARTHQUAKES
345
T A B L E 10 SUMMARY OF TRANSITION STATES USED IN PARAMETRIC ANALYSIS Prior Fractiles (Mag) Initial State Mi
8 -+ 0.2
8.4 _ 0.2
8.75 ___0.15
Posterior Fractiles (Mag)
Sample Data (Mag)
8.4, 8, 8.7, 8.8, 7.9, 7.9, 7.9, 8.3, 7.8, 8.3, 8.8, 8.1, 8.4, 8.4, 8.4, 8.3, 7.9, 8.3, 7.9 7.9, 7.8, 8.1, 7.9, 7.9, 8.1, 7.8, 7.9, 8.4, 8, 7.9, 8.3, 8.3, 8.3, 8.6, 8.3, 8.2 7.8, 8.3
0.25
0.50
0.75
1.0
0.25
0.50
0,75
1.0
8.4, 7.8,
7.0
8
8.4
8.8
7.9
8.1
8.5
8.8
7.8, 8.6,
7.9
8
8.4
8.8
7.9
8
8.4
8.8
7.9
8
8.4
8.8
7.8
8
8.6
8.8
Note: Zones were defined by using variable magnitude-rupture length relationships (all areas except area B, see Figure 2).
0.6
I
(a) For all areas except Area B
0.5
8.75 z 0.15
0.4
~>-
o.3
o~ '~._~
M o = magnitude of last great earthquake
\
0.2
e 0.1
0
0
0.3
~
-+ 0.2
20
40
60
80
100
80
100
l (b) For Area B
._
0.2 +- 0.15
. ~ z 0.1
~._~ e~
+
0
0
20
40
60
Number of Years Since the Last Great
Earthquake, t o
FIC. 8. Probability of continuation of a seismic gap for different initial conditions using variable magnitude-rupture length relationships.
346
AS~.tOK S. PATWARDHAN, RAM B. KULKARNI, AND DON TOCHER
lengths but do not suggest magnitude-length relationships for different areas. Assuming that these lengths are associated with the largest magnitude from historical data in a given area, the rupture lengths of lesser magnitudes were selected by judgment. Thus, zones in the Alaska-Aleutian area are based on a rupture length of 800+_ km for M0 = 8 _+0.2, while those in the Honshu area are based on rupture lengths of 150 to 200 kin. Table 9 shows the sample data, prior and posterior estimates of holding times, and lengths of gaps in the various areas. Table 10 shows the sample data and prior and posterior estimates of transition states. A comparison of Table 9 with Table 3, and Table 10 with Table 2, is instructive. Use of variable rupture lengths decreases the size of sample data, increases the number of gaps, and suggests generally longer holding times. The probability of continuation of seismic gaps for various initial conditions are expected to be higher, as the case in Figure 8 illustrates. The parametric analyses provide a useful insight into the effect of rupture sizes and holding times on the formation and continuation of gaps. Thus, in areas where the rupture lengths are higher or holding times are longer, the probability of formation and continuation of a gap is higher. The results illustrated in Figures 5 and 7 provide an approximate quantitative assessment of the degree of variation. These results can be applied for differentiating between the characteristics of gaps in different areas, e.g., between Alaska-Aleutians area and Japan, or between Central America and New Guinea. SUMMARY AND CONCLUSIONS A semi-Markov model is developed to estimate the likelihoods of occurrences of great earthquakes (magnitude >7.8) at a given location during a specified period of interest. The model takes into account the influence of the length of time over which strain energy is accumulating since the most recent great earthquake in a zone on the magnitude and time of the next great earthquake in the zone. The basic parameters of the model are (1) probability distribution of holding times between earthquakes of magnitudes Mi and Mj, (2) transition probabilities (i.e., the probabilities that the next earthquake will be of specified magnitudes following an earthquake of magnitude Mi), and (3) initial seismicity conditions of a zone (i.e., magnitude Mo of the most recent great earthquake and time to since that earthquake). These parameters were obtained by combining historical seismicity data and expert judgments through the use of a Bayesian procedure. This procedure provided better reliability in the estimation of the parameters than using only the limited historical seismicity data. The values of probabilities of different magnitudes and holding times are influenced in part by the accuracy and completeness of the historical seismicity record with respect to location and magnitude. Careful reevaluation of the data should be made before applying the model to a specific area. The model is based on a qualitative assessment of strain accumulation and intermittent release. The possibility of making quantitative assessments in terms of seismic moments should be explored. The application of the model was discussed for high seismicity areas in the circumPacific belt in which the primary process of earthquake generation is that of subduction. The basic output of the model is the set of probabilities of occurrence of a different number of various magnitude earthquakes (~7.8) in a given zone during selected periods of interest. This output can be used for a variety of purposes in seismicity evaluation problems: (1) characterization of a seismic gap, (2) definition
A SEMI-MARKOV MODEL FOR RECURRENCE OF GREAT EARTHQUAKES
347
of real time inputs to the seismic risk model, (3) characterization of tails of earthquake magnitude distributions. The semi-Markov model provides several advantages over other models in that it is location specific, can take into account initial conditions, and is flexible enough so that its parameters can be adjusted to represent any regimen of great earthquake occurrences such as predominance of certain magnitudes and variations in holding times. The probabilities of earthquake occurrence (and gaps) estimated from the model show reasonably good agreement with the values obtained from available data. ACKNOWLEDGMENTS This work is part of an ongoing study supported by the Professional Development Program of Woodward-Clyde Consultants. The authors gratefully acknowledge this support from WCC, especially Dr. I. M. Idriss, Mr. Douglas C. Moorhouse, and Dr. Keshavan Nair. Dr. Chhote Saraf assisted in the computer analyses, and Mr. Robert L. Nowack assisted in the compilation of earthquake data. Dr. William U. Savage reviewed the manuscript and made several useful suggestions. REFERENCES Benioff, H. (1951). Global strain accumulation and release as revealed by great earthquakes, Bull. Geol. Soc. Am. 62, 331-338. Esteva, L (1976). Developments in geotechnical engineering ser.: 15, in Seismic Risk and Engineering Decisions, Elsevier, New York. Hagiwara, Y, (1975). A stochastic model of earthquake occurrence and the accompanying horizontal land deformations, Tectonophysics 26, 91-101. Howard, R. A. {1971). Dynamic Probabilistic Systems, vol. 2, John Wiley & Sons, New York. Kelleher, J. and W. McCann (1976). Buoyant zones, great earthquakes, and unstable boundaries of subduction, J. Geophys. Res. 81, 4885-4896. Kelleher, J., J. Savino, H. Rowlett, and W. McCann (1974). Why and where great thrust earthquakes occur along island arcs, J. Geophys. Res. 79, 4889-4899. Knopoff, L. and Y. Kagan (1977). Analysis of the theory of extremes as applied to earthquake problems, J. Geophys. Res. 82. Mogi, K. (1968). Some features of recent seismic activity in and near Japan, Bull. Earthquake Res. Inst., (Tokyo Univ.) 46, 1225-1236. Nair, K. and L. S. Cluff (1977). An approach to establishing design surface displacements for active faults, Proc. World Conf. Earthquake Eng., 6th, New Delhi. Patwardhan, A. S., D. Tocher, and E. D. Savage (1975). Relationship between earthquake magnitude and length of rupture surface based on aftershock zones, Bull. Geol. Soc. Am. (Abstracts with Programs), 7. Raiffa, H. (1968). Decision Analysis, Addison-Wesley, Reading, Massachusetts. Raiffa, H. and R. Schlaifer (1960). Applied Statistical Decision Theory, Harvard University, Cambridge. Rikitake, T. (1975). Statistics of ultimate strain of the earth's crust and probability of earthquake occurrence, Tectonophysics 26, 1-21. Sykes, L. R. (1971). Aftershock zones of great earthquakes, seismicity gaps, and earthquake prediction for Alaska and the Aleutians, J. Geophys. Res. 76, 8921-8041. Tocher, D. (1958). Earthquake energy and ground breakage, Bull. Seism. Soc. Am. 48, 147-153. U. S. Geological Survey (1977). National Oceanic and Atmospheric Administration Hypocenter Data File. WOODWARD-CLYDE
CONSULTANTS
3 EMBARCADEROCENTER SAN FRANCISCO,CALIFORNIA94111 Manuscript received March 13, 1979
