A Semi-markov Model for Characterizing Recurrence of Great Earthquakes by Ashok

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 3 2 5 DEVELOPMENT OF THE MODEL 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