Y. Y. Kagan and L. Knopoff Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California 90024, USA

Geophys. J. R. astr. SOC. (1987) 88, 723-731 Random stress and earthquake statistics: time dependence Y. Y. Kagan and L. Knopoff Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California 90024, USA Accepted 1986 August 22. Received 1986 August 22; in original form April 24. Summary. The interearthquake time distribution is analysed on the assumption that those stresses that are not observed directly, change in a way that is describable by a random walk, i.e. as a Brownian motion. In this case, the time intervals between earthquake pairs has a power-law distribution with exponent -3/2. If tectonic stress loading is added to the Brownian motion, the interearthquake time distribution changes from a power-law to an almost Gaussian renewal distribution. The actual distribution depends on the ratio of the size of the random component to that of the tectonic component. We find that after about two days, tectonic stresses influence the temporal distribution of aftershocks, for main shocks with ML = 1.5. In the random regime the Omori law holds, while in the tectonic regime, an exponential distribution holds. The time of transition between random and tectonic effects increases with the size of the main shock. Key words: random stress, earthquake statistics, Omori law 1 Introduction In earlier papers (Kagan & Knopoff 1981; Kagan 1982) we have described a kinematic stochastic model of an earthquake process, which generates most, if not all, of the statistical features of earthquake occurrence identified thus far. Since the model is based on a small number of arbitrary assumptions, it follows that the model is phenomenological, even though the number of such assumptions is small. For the purposes of this paper, only one of these assumptions is important and may be summarized as follows: the probability of occurrence of one infinitesimal earthquake in the wake of another such infinitesimal earthquake is a power-law function of time with an exponent for shallow seismicity that is close to -312. The major item of seismological evidence which supports this assumption is the wellknown Omori law of aftershock occurrence. We have shown that the above assumption leads to the Omori law directly (Kagan & Knopoff 1980, 1986a); this law is generally valid, not 24

724 Y. Y. Kagan and L. Knopoff only for the time distribution of aftershocks, but also for the distribution of foreshocks; indeed it explains almost all of the temporal clustering of shallow earthquakes that is reliably known at present. Because of its phenomenoloacal nature, the model cannot be extended significantly beyond what has already been accomplished. In particular, we have no convenient way to evaluate the influences of the steady stress increase due to plate tectonics or other long-term effects on the statistical output from the model; while the present model adequately reproduces short-term statistical phenomena, it cannot reproduce long-term behaviour. In this paper we rectify this defect by introducing a physical basis for the original phenomenological model. We show that a rational modelling of the fluctuations in the stresses near a crack due to all the other cracks in the neighbourhood will allow us to duplicate the phenomenology of the original model. This interpretation will permit us to add tectonic stresses to the model as well. An inversion of the procedure will then allow us to identify the statistical consequences of the superposition of both the steady and the random processes. We show that the exponent -3/2 in the phenomenological model arises as a natural outcome of the conventional assumption that an earthquake occurs when the stress in the vicinity of a fault edge exceeds some critical value. In the absence of any information about tectonic stresses, we can assume that the stress is controlled by random factors. Our aim is to show that the value of the exponent is not an adjustable parameter, but is the consequence of a well-defined physically based assumption. Once we have established the physical basis, we can proceed to the discussion of the other factors of earthquake occurrence, such as the size distribution and spatial statistics which have been discussed phenomenologically in the above papers. We consider the problems of the relationship between the physical and stochastic modelling of the spatial earthquake parameters in a companion paper (Kagan & Knopoff 1986b). 2 Interearthquake time statistics As we have indicated, in our model (Kagan & Knopoff 1981) the time intervals between infinitesimal events that constitute our earthquake source-time function, are distributed according to a power-law distribution with exponent -3/2. Because of the infinity in the distribution at t = 0, we arbitrarily impose a dead-time to before which no new earthquakes can occur. A physical argument also suggests that a short characteristic time be introduced. In our model the dead-time arises because we bypass consideration of the dynamical nature of fracturing. If we consider a fracture to be an accelerating quasistatic process, then the seismic event itself is the culmination of the quasistatic growth phase at the moment when the rate of fracture growth approaches seismic wave velocities. Because we bypass the problems of dynamical rupture theory, the radiation from catastrophic fractures is taken to be instantaneous in the model; we introduce a very small time constant to describe the finiteness of the rupture process. We have identified to roughly with the coda time of an earthquake (Kagan & Knopoff, 1981), and have scaled this time as the square root of the scalar seismic moment. The exponent -3/2 also arises in studies of Brownian motions. Thus, heuristically, we can argue for a model of fracture as follows. At the moment that any given elementary earthquake rupture stops, the stress near the edge of the crack is lower than the critical breaking stress for extension of the fracture. Let us assume the subsequent stress history near the crack tip is due to a set of random factors due to other fractures in the neighbourhood. In this case, the time history of the stress might resemble a Brownian random walk. The stress-

Random stress and earthquake statistics 725 time function is thus given by the solution to the diffusion equation. When the stress reaches the critical threshold level, a new earthquake begins. The distribution density of the time intervals of first passage (Feller 1966) is the function where D is the diffusion coefficient, t is the time, and u is the threshold or barrier stress, i.e. the difference between the initial breaking stress and the stress at the time of cessation of extension. This distribution as a function of stress is the Rayleigh distribution; as a function of time it is the Ldvy distribution (cf. Zolotarev 1983, p. 79). In this model the stress is taken to be a scalar, which corresponds to the addition of perfectly aligned stress tensors; the interaction of misaligned tensors will be discussed in a second contribution. The Le vy distribution (1) has been used by Kagan (1982) [see equations (8) and (9)] to model the time dependent part of the space-time sequence of these infinitesimal events. It is only the time sequence that concerns us here. The average recurrence time Tis infinite for the model of (1). This model has been used to describe aftershocks, and in the aftershock case, as noted, it yields the familiar Omori t- aftershock frequency law. However, the usual time-scale of aftershock series is of the order of months; this interval is too short to observe tectonic effects. In fact, we can easily assume without penalty that the amount of stored energy decreasesmonotonically during an aftershock sequence. On the other hand, the motion of the plates restores the energy lost in the main shock and its subsequent aftershocks, in anticipation of the next major earthquake. Thus, over a long time span, the statistics of earthquakes niust have a different interval-between-earthquakes law than is appropriate for aftershocks. In this note, we discuss the appropriate pairwise interval law for a model in which a steadily increasing source of stress, which we assume is due to plate tectonics, is added to a random or diffusion component, where the distribution (1) describes the density of earthquake recurrence times in the absence of tectonic loading. Under the model of Brownian motions in the presence of an external steady field, the frequency of occurrence law will be asymptotic to the Omori law for short time intervals. There is no guarantee that the familiar exponential law for the time intervals between Poisson random events will be the result. If the rate of tectonic loading is a constant C, the distribution densityf(t) is modified to (Feller 1966, p. 368) The average recurrence time in this case is no longer infinite, but becomes 112 where we have also given the standard deviation. If we set f = o/jm and q = C /ad we obtain the distribution density in terms of only two parameters,

726 Y. Y. Kagan and L. Knopoff 1.000. l 0BB A H.0010 U. m 0 a! a 0001.0000 3.a v) Fl.I0 I. 00 10.00 100.00 TInE &=L C fi ~ ~+CRmCAL l)=o -=+- TECTONC -TME LOADING 1 0 Figure 1. Time interval distributions. The probability densities for equations (1) and (4) are displayed for various values of the dimensionless random 6 and external stress q fields. Time is measured in arbitrary units which depend on the values of the quantities D and C. In the lower part of the figure we show schematically Brownian motions in the absence and presence of an external stress field. B It is a trivial matter to rewrite (3) in these variables. If we pass to the limit 77 + 0, we get (1) which can be written as a function of I and t. The time dependence of the stress is governed by the Fokker-Planck equation (Feller 1966). Depending on the relative sizes of and 77, i.e. on the scaled value of the tectonic loading rate, the resulting distribution may vary from the power-law or scale-invariant law to a Gaussian renewal distribution (Fig. 1). The curve with I = 1, 77 = 0, for example, corresponds to equation (l), and has a power-law long-time tail; as already noted, this is appropriate for the case of aftershocks. As the value of the normalized drift velocity 77 increases, the resulting distribution of earthquake intervals becomes asymptotically Gaussian. Similarly, if the normalized stress drop I increases for non-zero 77, the distribution of earthquake intervals also becomes almost Gaussian (see Fig. 1). The plots at the bottom of Fig. 1 illustrate schematically several realizations of the time histories of randomly changing stresses for the cases of absence (left) and presence (right) of tectonic loading. The stress trajectories start from a stress level which is less than the critical threshold stress; under the influence of random factors it subsequently increases or decreases. An earthquake starts at the time that it reaches the critical breaking stress.

Random stress and earthquake statistics 727 Figure 2. Numbers of dependent earthquakes in several time intervals for the USGS catalogue of earthquakes for each of four (numbered) fixed distance intervals from the main shock (see equation 5). Successive time intervals increase by a factor of m. We find partial confirmation of the above model from the time distribution of the numbers of dependent shocks occurring in certain time and distance intervals from other shocks (Kagan & Knopoff 1986a) for the USGS Central California catalogue (Fig. 2). The distribution density p(t) is the numbpr of dependent earthquakes that occur in the magnitude interval between M and M - 0.1, where M is the magnitude of a main shock. (In practice, the number of aftershocks, or dependent shocks with magnitudes near M is close to zero. To determine this quantity we smooth the distribution of the number of aftershocks as a function of magnitude to determine this asymptotic value. The smoothing makes use of the function (1 O/O) - p exp [p(m- Md)] analogous to the magnitude-frequency law, where Md is the magnitude of the dependent shock, and is an adjustable parameter which is close to In 10 for most catalogues.) Successive time intervals in Fig. 2 increase logarithmically; the p values are per unit logarithmic time interval, instead of per unit linear time interval as in Fig. 1, The unit time in Fig. 2 is 2.22 x days, which is our estimate of the coda length for a main earthquake with ML = 1.5. Because of the logarithmic time scale, a flat curve in Fig. 2 implies a power-law distribution. Since the roll-off in Fig. 2 is about lo4 time units. we infer that for times less than about 2 days, the temporal properties of aftershock sequences are dominated by local random effects and for times greater than 2 days, tectonic influences begin to be felt, for main earthquakes with ML = 1.5. The unit time is scaled according to the magnitude of the main shock as T, = 2.22 x 1 0-4 - 3 M-1.5. Details of smoothing and scaling procedures are to be found in Kagan & Knopoff (1 980, 1986a). The constancy of any of the curves in Fig. 2 indicates that the number of dependent shocks falls off as ljt. However, this roughly uniform rate drops drastically for large time intervals. We have used a fairly complicated spatial scaling relation because of the presence

728 Y. Y. Kagan and L. Knopoff of location errors in the catalogue. The formula for the distance boundaries is where rj = 0.35, 0.70, 1.1, 1.6 km, and pi = 0.50, 1.25, 2.5, 5.0 km. (For the more spatially distant curves there is a weak maximum that is shifted towards later times; this result suggests an outward 'migration' of seismicity.) For comparison, the smooth curve of Fig. 2 is drawn for the case of a normalized version of equation (4) with = 0, 7 = 1 O-2.25. We observe that a small but non-zero value of the drift component 7 causes the power-law distribution density of (I) to transform to an exponential distribution for large time intervals and is similar to the results for real aftershock sequences as well. The time rate of occurrence of aftershocks in other earthquake catalogues is similar (Kagan & Knopoff 1980). For P q, a good approximation to equation (4) is the gamma distribution (cf. Kagan & Knopoff 1986a, section 2) f'(t)= 0 t< to f(t) = c-l t-3/2. exp (-t/tx) f a to, (6) where tx = 7-2 is a parameter that controls the distribution for large values of time and fo is the characteristic dead time of the earlier model, which we identify as fo = 4C;*/n. The coefficient c is where Erfc (x) is the complementary error-function. For time intervals close to to, formula (6) is not a particularly good approximation to (4); for large time intervals the agreement between the formulas is very good. The above discussion is valid for sequences of the strongest earthquakes in a region. The source volume of a weak earthquake will most likely be imbedded in the focal zone of a future stronger event; thus the renewal chain for the specific volume will be destroyed. For the strongest earthquakes, the stress levels in formulas (l)-(4) are to be scaled according to the size of the focal region. In effect, we interpret u in the above equations as an appropriately scaled integral of the stress-drop over spatial extent of the earthquake focal region. Thus u is rather close to the scalar moment of the earthquake; hence, the recurrence time in (3) will be dependent on the size of event. The precise way to scale this relation is not obvious. It is, of course, exactly this scaling that yields the seismic moment-frequency law. We have discussed the time scaling of the seismic moment in Kagan & Knopoff (1981); the spatial scaling of a focal region has been discussed by Kagan (1982). The distribution (2) depends on two parameters, so its scaling should be more complicated than that for (I), discussed in Kagan & Knopoff (198 1). As noted, another limitation of the discussion in this paper is the quasistatic nature of the model; we do not consider here dynamical rupture effects which, most probably, control the final size of an earthquake focal zone. The model we have used is a rather crude approximation to the real picture. In reality, stress is a tensor, rather than a scalar. The deviation of the seismic moment tensor from complete coherence during the process of rupture may be the cause of intermittency of rupture (Kagan & Knopoff 1985; we will discuss this point extensively in our companion paper, see Kagan & Knopoff 1986b); most probably, these disorientations of the moment tensor are due to asperities or barriers in the propagation of a fault. To study these problems we need to consider the properties of the stress tensor, and the spatial distribution of the elementary events.

3 Discussion Random stress and earthquake statistics 729 These demonstrations show that we can obtain distributions of interearthquake time intervals that vary from a power-law to an almost Gaussian renewal type distribution. Both of these limiting distributions are the result of rather simple assumptions concerning the behaviour of the stresses in the lithosphere. The second of these limiting distributions is of the repulsive type and has been used by many advocates of the characteristic earthquake hypothesis (see, for instance, Wesnousky et al. 1983; Schwarz & Coppersmith 1984) to model the distribution of main sequence earthquakes. Unfortunately, these repulsive distributions of the interoccurrence time, which are not of cluster type, are not readily adaptable to the multidimensional generalization which is necessary to describe spatial distributions as well. It is no accident that all models based on repulsive-type renewal distribution are 1-D. To reduce the dimensionality of the earthquake process to one time dimension only, we need to isolate the region under consideration, a procedure which is difficult to formalize. Unfortunately, opinions regarding methods of spatial isolation are usually divergent. Put simply, we do not know how to minimize edge effects or the effects of earthquakes in adjacent or other more distant regions, on the statistics of an extended region considered as a point in space. The (g, q) distributions cannot be directly compared to the known distributions of interearthquake time intervals or to the distributions of numbers of aftershocks following a main earthquake, because the clustering of earthquakes must be taken into account. In simulations of earthquake occurrence (Kagan & Knopoff 198 l), we model this clustering by a hypothesis that earthquakes are composed of infinitesimal events which multiply according to a critical branching process. We show that the power-law distribution of interevent time with the exponent value of -3/2, combined with the above branching assumption, reproduces aftershock-foreshock properties of shallow seismicity. These properties are studied by an application of the same statistical analysis technique to both real and synthetic catalogues (Kagan & Knopoff 1981). We can also offer the following heuristic basis for the above agreement. As mentioned in the introduction, the best known of the statistical distributions for temporal occurrence is the Omori law of aftershock occurrence. This law is usually formulated as N(r, At) 0: tka - At, where N(t) is the number of aftershocks in a small interval of time (t, t + At), t is the elapsed time after a main event, and ci is an exponent close to one. The distributions (1) and (2) have been calculated for interevent time intervals; we need to compare them with the distribution of the numbers of earthquakes. We use the fractal model of Mandelbrot (1982) to make the transition from the former distribution to the latter. If we assume that earthquakes form a temporal, 1-D, renewal chain of events, then our results (equation 1) suggest that the exponent ci should be 1/2; in other words the fractal dimension of an earthquake sequence should be 1/2. The fractal dimension is defined for our purposes as the exponent in the dependence of the cumulative number of events on the time, which is one more than the exponent on the time in (I), for sufficiently large timeintervals (Mandlebrot 1982). The critical branching process which we use to approximate earthquake ruptures (Kagan & Knopoff 1981), has a stationary first moment; thus, in principle, it can be regarded as a renewal chain of events. However, two considerations must be taken into account before comparing our results with the Omori law: (1) higher moments of the critical branching process are not stationary, and (2) the numbers in (7) are not

730 Y. Y. Kagan and L. Knopoff counted from any arbitrary event, as suggested by Mandelbrot's (1982) definition of the dimension, but instead are conditional on the occurrence of the especially strong cluster of elementary events that comprise a main earthquake. The total number of events at the beginning stages of development of a critical branching process is proportional to n2 (Athreya I% Ney 1972, p. 20), where n is the number of generations in the process. The number of new events is proportional to n. Thus, the fractal dimension of the set of events at the beginning of the sequence should be (1/2) x 2 = 1, (see Mandelbrot 1982, ch. 32), i.e. there should be a continuous stream of events at the beginning of any earthquake sequence. This stream of events is what we call an earthquake: the exact definition of the time of termination of an earthquake depends on the frequency response of the seismographic network, as well as other properties of the network (see more in Kagan & Knopoff 1986a). New episodes of energetic branching in the same sequence should be interpreted as new, separate earthquakes (aftershocks). When the sequence of events starts to die out, the numbers of new events should first stabilize at an interval rate -3/2, and then fall off to zero. Thus, the fractal dimension of the set should fall first to 1/2, and then to zero. The zero value of the dimension corresponds to a= 1 in (7), i.e. the cumulative number of events in the time interval T increases only as log (7'). 4 Conclusions The Ornori law of aftershock occurrence and, in general, the time clustering of earthquake events, has been shown to be easily derived from a simple assumption of the behaviour of the stress prior to an earthquake and from elementary probability theory. If tectonic stress loading is present, then the interearthquake time distribution may change from a powerlaw to an almost Gaussian renewal distribution, depending on the ratio of the random component to the drift component. Acknowledgments This research was supported in part by Grant CEE-84-07553 of the National Science Foundation. Publication Number 2893, Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA 90024. References Athreya, K. B. & Ney, P, E., 1972. Branching Processes, Springer-Verlag, New York. Feller, W., 1966. An Introduction to Probability Theory and its Applications, 2, J. Wiley, New York. Kagan, Y. Y., 1982. Stochastic model of earthquake fault geometry, Geophys. J. R. astr. Soc., 71, 659-6 91. Kagan, Y. Y. & Knopoff, L., 1980. Dependence of seismicity on depth, Bull. saism. Soc. Am., 70, 1811-1822. Kagan, Y. Y. & Knopoff. L.. 1981. Stochastic synthesis of earthquake catalogs, J. zaophys. RES., 86, 2853-2862. Kagan, Y. Y. & Knopoff, L., 1985. The two-point correlation function of the seismic moment tensor, Geophys. J. R. astr. SOC., 83,636-651. Kagan, Y. Y. & Knopoff, L., 19863. Statistical and stochastic models of earthquake occurrence and earthquake prediction (in preparation). Kagan, Y. Y. & Knopoff, L., 1986b. Random stress and earthquake statistics: spatial dependence (in preparation). Mandelbrot, B. B., 1982. The Fractal Geometry of Nature, W. H. Freeman, San Francisco.

Random stress and earthquake statistics 731 Schwarz, D. P. & Coppersmith, K. J., 1984. Fault behaviour and characteristic earthquakes: Examples from Wasatch and San Andreas fault zones, J. geophys. Res., 89,5681-5698. Wesnousky, S. G., Scholz, C. H., Shimazaki, K. & Matsuda, T., 1983. Earthquake frequency distribution and the mechanics of faulting, J. geophys. Res., 87,9331-9340. Zolotarev, V. M., 1983. Odnomernye ustoichivye raspredeleniya (One-Dimensional Stable Distributions), Nauka, Moscow, (in Russian).