A simulation of earthquake occurrence

1972, Phys. Earth Planet. Interiors 6, 311—3 15. North-Holland Publishing Company, Amsterdam

A SIMULATION OF EARTHQUAKE OCCURRENCE

MICHIO OTSUKA Department of Physics, Faculty of Science, Kumamoto University, Kuniamoto, Japan

Using an electronic computer, a two dimensional model of blocks connected to one another by springs, is used to simulate earthquakes. The frequency of occurrence of events (slips of one or more blocks) as a function of the number of blocks that slip (size of the event) is found to obey a relation similar to Gutenberg and Richter’s frequency—magnitude relation for earthquakes. In-

tervals of time between events obey an exponential distribution. Large events often occur in gaps between areas where other events have already occurred. Thus, the model simulates well earthquake activity in space and time and may contribute to our understanding of strain accummulation and release in the Earth.

The purpose of the present paper is to propose an earthquake simulator built in an electronic computer and to demonstrate that its operation is helpful in understanding the physical principles on which seismic phenomena are based. To get an understanding of the simulator, it is helpful to imagine a hypothetical mechanical device as shown in fig. 1. The construction is similar to a model proposed by BURRIDGE and KN0POFF (1967). The main differences are the expansion of the model to two dimensions and an increase in the number of elements, thus enabling us to simulate not only the temporal character, but also its spatial patterns. Fig. 1 shows a perspective view of the simulator. It is constructed with a number of blocks which are suspended from a fixed ceiling board by leaf springs M with regular spacing. The blocks are in frictional contact with a floor G which is moving slowly in the direc-

tion indicated by the arrows. Moreover, they are interconnected with the neighboring blocks by leaf and coil springs crosswise and lengthwise. The spring constants and frictional parameters are assigned appropriate values with random stochastic fluctations so that one can construct models with heterogeneity of an arbitrary degree in the computer. For details of the numerical values assigned to the model,see OTSUKA (1971). As the computer step proceeds, elastic energy is built up in the springs M by a steady motion of the floor G until a critical state is reached for any of the blocks. Then the restitutive force exerted by spring M surpasses the frictional force which had caused the block to be dragged forward. The block slips back abruptly. Fig. 2 shows a top view of a constituent single block and related spring system. It is suspended by a spring M and interconnected with the neighboring blocks by springs K and L. Suppose a critical state is reached by

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a block in the center of the figure. It slips back by some amount. The frictional character between the block B and the floor G is such that once a critical state is reached for any block and the block starts moving, the frictional force becomes zero for that block until motion terminates. Hence, the new position of the block is determined only by the equilibrium condition of the relating spring system. Following the slippage of any block, readjustment of energy in the relating springs as well as the reduction of stored energy in the main spring M will occur. This process gives rise to some amount of additional restitutive force to the neighboring blocks. If any of these blocks is in a state close to critcai, slippage may be propagated and the situation is repeated for the new block. Thus, by a kind of chain reaction, a cluster of slippages may grow. The extent to which a cluster of slippages will spread is controlled by many factors such as spring constants, frictional parameters, and above

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all, by the past history of slippages. Thus it is hard to predict the extent of the slipped region until all disturbances are brought to complete termination. Whether this concept of seismic energy liberation is acceptable or not is open to question. In any case, this is the way the mechanism of seismic energy release is interpreted throughout this paper. In fig. 3, examples of simulated seismic activity in the three successive computer steps are shown. The simulator is constructed with 2000 blocks in all 100 blocks crosswise and 20 blocks lengthwise. All 2000 coordinates of the blocks are typed out by small dots and the slipped blocks at the corresponding step are marked by solid circles. Clusters of slippages of various size can each be considered as corresponding to individual earthquakes with their respective magnitudes. It is particularly interesting that smaller clusters of slippages occur more frequently than the larger ones suggesting that some parallel relation exists with real seis—

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Fig. 5. Simulated seismic energy release. Actual seismic energy release from Japan Islands and surrounding area is shown as an inset for comparison.

mic activity. However, there is no guarantee so far that this simulation is successful in simulating the magnitude—frequency relation in a quantitative way because we have not established that the release of seismic energy is determined by a probability controlled chain reaction. Fig. 4 corresponds to the magnitude—frequency relation of the simulated earthquakes. The horizontal axis shows the number of slippages in each cluster and the vertical axis shows the frequency of their occurrence. The left figure corresponds to a homogeneous model with a standard deviation of the assigned parameters of 0.1, and the right figure to an inhomogeneous model with a standard deviation of 0.5 of the respective mean values. If the assumption is made that the energy liberated from a single slippage is constant over the whole simulator, the horizontal axis is equivalent to the earthquake magnitude. The regular decline of frequency with increasing magnitude is clearly observed. The trend is almost linear suggesting that the Gutenberg—Richter relation holds, Of particular interest is that the larger slope is obtamed for the more heterogeneous case, which is in harmony with MOGI’s (1962) finding in the laboratory experiments of brittle fractures, To go into more detail of this problem, however, we have to discuss a wider range of earthquake magnitudes. In principle, this is possible if we keep the cornputer calculating for a long time. However, it is so time consuming and laborious that the author has construct-

ed an independent model for this particular purpose extracting only the chain reaction mechanisms and has obtained a promising result (OTSUKA, 1972). Briefly stated, the magnitude—frequency relation of earthquakes is clearly understood by the introduction of a chain reaction hypothesis. Fig. 5 shows the simulated seismic energy release. The lower plots correspond to the energy liberation from the whole simulator, while the upper ones correspond to that from a small subregion composed of 100 blocks. The actual seismic energy release from the Japan Islands and surrounding area compiled by TSUBOt (1964) is shown on the top of the figure for comparison. The simulation can be said to be quite satisfactory in expressing the validity of seismic energy release and its uniformity as well. While in some cases a large amount of energy is liberated by a single event, in other cases the same amount of energy is released little by little by a number of minor events. As the area under consideration is increased, a uniformity in the pattern prevails that is similar to the patterns for natural seismic events. The statistics of the time intervals between successive earthquakes tells us that if the earthquake occurrence is a random process, the duration of the time intervals between events is expressed by an exponential distribution. In other words, if we plot the logarithm of the number of intervals between events versus the length of the time interval (step interval), the result should be a straight line. The step intervals between successive

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earthquakes in the simulator were counted and such a plot was made for two cases of homogeneous and inhomogeneous distribution of parameters (fig. 6). The results yield a satisfactory straight line suggesting that the simulated earthquakes do occur as a random process. The slope is slightly larger for the homogeneous case. No corresponding study of earthquakes, however, is available for comparison, as far as the author knows. It is frequently reported that the seismic activity is low in the period preceding the occurrence of large earthquakes in the region from which a large amount of seismic energy is to be liberated. In other words, great earthquakes occur as a complement to the deficiency of seismic energy release in the preceding stages (see for example M0GI, 1968). Fig. 7 shows a simulation of this phenomenon. The shaded area in the lower figure shows the extent over which energy was released at a particular computer step by a single event. The upper figure shows the region where energy had been released in the preceding stages by a number of minor events. Comparing these two figures, it is clearly observed that in the focal area where seismic energy will be liberated by a large event, seismic activity is anomalously low before the occurrence of the large event,

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A simulated diastrophism. Time increases to the right, and following a slow accummulation of strain, a large slip occurs relieving the strain. Fig. 8.

Fig. 8 shows a simulated example of diastrophism. It has been remarked, especially in Japan, that the inhabitants of the coastal area have repeatedly suffered from large scale subsidence and upheaval of the ground

A SIMULATION

OF EARTHQUAKE

that are related to the occurrence of large earthquakes. The geodetic surveys have disclosed that the typical pattern of diastrophism in time is a saw-tooth wave form of the ground deformation; that is, a period of gradual accummulation of strain energy is followed by a sudder~release of it by a large event, and the period of energy accummulation is repeated again. TAKEUCHE and KANAMORI (1968) have interpreted this phenomenon as due to the elastic rebound of the Earth’s crust from the drag of the lithosphere by mantle convection. Whatever the reason is, the above mentioned surface deformation should be a reflection of some kind of strain accummulation and release. The averages of displacements of blocks from two subregions each composed of 100 blocks are shown in fig. 8. The typical pattern of diastrophism is nicely simulated although the vertical movements cannot yet be realized in the simulator because of its constructional simplicity. We have seen in this paper that the simulator proposed herein works satisfactorily in modeling various

315

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aspects of seismic phenomena. Although many alterations and improvements should be made for perfect simulation to be achieved, it certifies that we are approaching a basic understanding of what is occurring in the seismic region and how seismic energy is stored and released. Acknowledgment The author wishes to thank P. Molnar who has made the helpful critical reading of the manuscript. References BURRIDGE,

R. and L. KNOPOFF (1967), Bull. Seismol. Soc. Am.

57, 341—347. M0GI, K. (1962), Bull. Earthquake Res. Inst., Tokyo Univ. 40,

83 1-853. Moo!, K. (1968), Bull. Earthquake Res. Inst. Tokyo Univ. 46, 1225—1236. M., (1917), J. Seismol. Soc. Japan Ser. 2, 24 (in Japanese) 13—25.

OTSUKA, OTSUKA,

M., (1972), J. Phys. Earth 20, 35—45. (1968), J. Seismol. Soc. Japan

TAKEUCHI, H. and H. KANAMORI 5cr. 2, 21, 317 (in Japanese). TsuBoi, C. (1964), J. Phys. Earth

12, 25—36.