Originally published as:

Originally published as: Kuehn, N. M., Hainzl, S., Scherbaum, F. (2008): Non-Poissonian earthquake occurrence in coupled stress release models and its effect on seismic hazard. - Geophysical Journal International, 174, 2, pp. 649 658. DOI: http://doi.org/10.1111/j.1365-246x.2008.03835.x

Geophys. J. Int. (2008) 174, 649 658 doi: 10.1111/j.1365-246X.2008.03835.x Non-Poissonian earthquake occurrence in coupled stress release models and its effect on seismic hazard N. M. Kuehn, 1 S. Hainzl 2 and F. Scherbaum 1 1 Institute of Geosciences, University of Potsdam, Potsdam, Germany. E-mail: nico@geo.uni-potsdam.de 2 GeoForschungsZentrum Potsdam, Potsdam, Germany Accepted 2008 April 28. Received 2008 April 28; in original form 2007 September 7 SUMMARY Most seismic hazard estimations are based on the assumption of a Poisson process for earthquake occurrence, even though both observations and models indicate a departure of real seismic sequences from this simplistic assumption. Instrumental earthquake catalogues show earthquake clustering on regional scales while the elastic rebound theory predicts a periodic recurrence of characteristic earthquakes on longer timescales for individual events. Recent implementations of time-dependent hazard calculations in California and Japan are based on quasi-periodic recurrences of fault ruptures according to renewal models such as the Brownian Passage Time model. However, these renewal models neglect earthquake interactions and the dependence on the stressing history which might destroy any regularity of earthquake recurrences in reality. To explore this, we investigate the (coupled) stress release model, a stochastic version of the elastic rebound hypothesis. In particular, we are interested in the time-variability of the occurrence of large earthquakes and its sensitivity to the occurrence of Gutenberg Richter type earthquake activity and fault interactions. Our results show that in general large earthquakes occur quasi-periodically in the model: the occurrence probability of large earthquakes is strongly decreased shortly after a strong event and becomes constant on longer timescales. Although possible stress-interaction between adjacent fault zones does not affect the recurrence time distributions in each zone significantly, it leads to a temporal clustering of events on larger regional scales. The non-random characteristics, especially the quasi-periodic behaviour of large earthquakes, are even more pronounced if stress changes due to small earthquakes are less important. The recurrence-time distribution for the largest events is characterized by a coefficient of variation from 0.6 to 0.84 depending on the relative importance of small earthquakes. Key words: Seismic cycle; Earthquake interaction, forecasting, and prediction; Statistical seismology. GJI Seismology 1 INTRODUCTION A key aspect in probabilistic seismic hazard analysis is the accurate modelling of the spatiotemporal distribution of earthquakes and in particular their recurrence times. These calculations are usually done with the Poisson process for distributed seismicity, making the assumption that seismicity in a specific time interval is independent of previous seismicity. Advantages of the Poissonian model are its simplicity and the fact that it arises naturally in a superposition of event sequences from independent sources (Vere-Jones 1995). Because the Poisson model has an exponential recurrence time distribution and thus a constant hazard function, this assumption leads to time-independent seismic hazard estimates. However, observations show that earthquake occurrence usually does not follow a Poisson process. While short- and intermediateterm earthquake clustering such as aftershock activity is evident (Kagan & Jackson 1991a; Rhoades & Evison 2004; Lombardi & Marzocchi 2007), the recurrence of large earthquakes on individual faults is rather speculative because seismic catalogues are short compared to the underlying processes (Parsons 2005). Even in regions with excellent seismic monitoring, like the Parkfield segment in California, the number of large earthquakes is below 10. Additional information may be gained from palaeoseismology, but a reliable statistical evaluation based on observational data is not feasible at the moment (see e.g. the dispute about the seismic gap theory Kagan & Jackson 1991b, 1995; Wesnousky 1994, 1996; Kagan 1996; Wyss & Wiemer 1999). Despite direct observational evidence, quasi-periodic recurrence of characteristic events is expected from the classical elastic rebound theory (Reid 1910, 1911) and has been observed for the proxy-data of small repeating earthquakes (Nadeau & McEvilly 1997) as well as in numerical simulations of fault dynamics (e.g. Zöller et al. 2006). Therefore, recent implementations of time-dependent hazard assessment in California and Japan (WGCEP 2003; Stein et al. 2006) C 2008 The Authors 649

650 N. M. Kuehn, S. Hainzl and F. Scherbaum assume quasi-periodic recurrence of characteristic events modelled by renewal processes such as the Brownian Passage Time model (Matthews et al. 2002; Zöller & Hainzl 2007). Although those renewal models account for some intrinsic variability, they neglect the dependence on the stressing history and the impact of earthquake interactions. However, stress transfer, especially from nearby faults and frequent smaller, Gutenberg Richter type earthquakes, can advance or delay the earthquake nucleation of large ruptures and it is an open question whether this will destroy any natural periodicity. In this work, we therefore, investigate numerical simulations of the stress release model which is a stochastic version of elastic rebound theory where the earthquake probability is a function of the stress level which is itself altered by each earthquake (Vere-Jones 1978). The stress release model is particularly interesting, since it is easily extendable into the spatial domain (the so-called coupled or linked stress release model) and can take stress interactions due to fault communication (Liu et al. 1999; Lu et al. 1999; Bebbington & Harte 2003). Several studies (for a review see Bebbington & Harte 2003) have applied the stress release model to reproduce the recurrence of large historical earthquakes in various seismically active regions showing that it is mostly superior to the Poissonian model (Zheng & Vere-Jones 1991, 1994; Zhuang & Ma 1998; Liu et al. 1999; Lu et al. 1999; Lu & Vere-Jones 2000; Lu 2005; Rotondi & Varini 2006, 2007; Imoto & Hurukawa 2006). Although this does not prove that the stress release model reflects the true evolution of seismicity on long timescales, it indicates that the stress release model can be regarded as a valid working hypothesis for studying the effect of stress interactions on recurrence time distributions. We investigate numerical simulations of the stress release model that consists of a large number of seismic cycles. Thereby, we explicitly take into account Gutenberg Richter distributed small and intermediate magnitude events because these might play an important role for stress redistribution and should not be neglected (Hanks 1992; Helmstetter et al. 2005; Marsan 2005). Additionally, we address the important question how coupling between different seismic zones affects the occurrence probability distributions on a local and regional scale. In the first sections, we will outline the model (Section 2) and the numerical setup (Section 3) in more detail. Subsequently, the resulting characteristics of the model are discussed (Section 4) and the main conclusions are summarized (Section 5). 2 THE MODEL The stress release model (Vere-Jones 1978) is a stochastic version of elastic rebound theory (Reid 1911), one of the classical models for earthquake occurrence. It assumes that the occurrence probability of an earthquake in a seismic active region depends on some quantity Z which may be interpreted as the mean stress level present in the region. The occurrence probability is assumed to increase with an increasing stress level and to drop when an earthquake occurs. Due to tectonic loading, the stress level is generally assumed to increase linearly between two earthquakes and to drop when an earthquake occurs, so its temporal evolution can be written as Z(t) = Z 0 + ρt t i <t Z i, (1) where Z 0 is the stress level at time t = 0 and ρ is the loading rate. t i <t Z i is the accumulated stress release of earthquakes that occurred prior to time t, with t i and Z i the occurrence time and stress release of the ith earthquake, respectively. It is important to note that the quantity Z, which is usually simply called stress, is a scalar and not the true stress tensor. Z is used to describe the underlying state of the system or region under study. It is assumed that the average stress release Z of an earthquake is related to some power of its energy release, Z E η. Combining the energy release with the magnitude using the relation m = 2 3 log 10 E + const (Kanamori & Anderson 1975), the following equation which connects the stress release with the magnitude is derived: Z = 10 h(m m 0), (2) where m 0 denotes the lower cut-off magnitude. For h = 0.75 and 1.5, the mean stress release is proportional to Benioff strain (Benioff 1951) and to seismic moment, respectively. Most applications use h = 0.75. The occurrence probability of an earthquake is controlled by a stress-dependent conditional intensity function λ(z). For small t, λ(z) t represents the probability of an event occurring in the time interval (t, t + t). Obviously, λ has to be a non-decreasing function of Z to agree with the physical understanding that higher stress levels result in a higher earthquake risk. λ(z) = const would result in a random (Poisson process) model, whereas { 0, Z Zc λ(z) =, Z > Z c corresponds to a time-predictable model (Jaume & Bebbington 2004). We follow most of the articles in the literature and use λ(z) = exp [α + β Z]. (3) Our trials with a power law and a linear function for λ(z) indicate that the behaviour of the model does not depend strongly on its functional form. This is concordant with conclusions of Zheng & Vere-Jones (1994). Furthermore, it can be shown that with the exponential function, the stress release model is equivalent to the rate-state friction model of Dieterich (1994) as well as stress corrosion (Scholz 2002) in its instantaneous response to stress changes. Note that in eq. (3) the Poisson process (β = 0) is included. With eq. (1), the conditional intensity function becomes [ ( λ(t) = exp α + β Z 0 + ρt ) ] Z i. (4) t i <t Stress transfer and stress-triggering between distant faults and seismogenic zones can be included in a multivariate extension of the simple stress release model, leading to the so-called coupled (or linked ) stress release model (Liu et al. 1999; Lu et al. 1999; Bebbington & Harte 2003). Here, the local stress depends also on the earthquake activity in adjacent regions Z (i) (t) = Z (i) (0) + ρ (i) t j Z ( j) k θ ij S ( j) (t), (5) where S ( j) (t) = t k <t is the accumulated stress release of earthquakes in region j that occurred prior to time t. Superscript (i) denotes the ith region and θ ij measures the fraction of stress transferred from region j to region i. A positive or negative value of θ ij results in damping (stress shadowing) or excitation (stress triggering), respectively. The diagonal elements are θ ii = 1. The occurrence probability of the next event in each region is again modelled by a stress-dependent hazard function λ (i) (t) = exp [ α (i) + β (i) Z (i) (t) ]. (6)

Non-Poissonian earthquake occurrence 651 Note that in the simple model as described here, aftershocks that occur in the same region as the main shock are not taken into account. However, one might think of more elaborate models where one region is decomposed into a fault area describing the recurrence of main shocks which transfer stress to an aftershock zone (see, e.g. Borovkov & Bebbington 2003). Simulation of the stress release model requires specification of a probability distribution of the event sizes (i.e. magnitudes) to calculate the stress release for the events according to eq. (2). For that, we assume the well-known Gutenberg Richter distribution (Gutenberg & Richter 1944), empirically observed for regional seismicity. In particular, we use the double truncated Gutenberg Richter distribution where the maximum magnitude depends on the stress level. The probability density has the form ln 10 b 10 bm p(m) =, (7) 10 bm 0 10 bm max where m 0 and m max are the minimum and maximum magnitude, respectively. m max depends on the stress level in a way that the maximum possible earthquake just unloads the system. In the Appendix, it is shown that m max can be easily calculated by inverting eq. (2), m max = m 0 + log 10 Z. (8) h 3 EXPERIMENTAL DESIGN We investigate the (coupled) stress release model regarding its temporal properties and its consequences for seismic hazard. Therefore, we perform various simulations with different parameter sets. Parameters that can influence the temporal behaviour of the model are α, β and ρ in eq. (4) and the coupling strength θ in eq. (5). Furthermore, the parameters of the frequency magnitude distribution have to be specified. Since in seismic hazard analysis the frequency magnitude relation is often estimated independently, we assume that the a, b and m max value of the Gutenberg Richter relation are given. As a consequence, we can then fix the parameters α and ρ in eq. (4), as is shown in Appendix A. The temporal variability of the model is then only dependent on two free parameters, namely β and h. We perform simulations of both the uncoupled and the coupled model. Application of the coupled stress release model usually needs careful consideration concerning the number, shape and size of the regions taken into account (Bebbington & Harte 2003). However, to reduce complexity, we investigate the simple case of 10 10 isotropically coupled regions. Stress is only transferred to the four neighbouring regions, and the coupling strength θ between different regions is always the same in a single simulation, while θ ii = 1. We require that the total stress that is transferred to neighbouring regions must be smaller than the stress released in the host region, which leads to the constraint θ 0.25. To be as simple as possible, all regions of the model are assigned the same parameters. For the Gutenberg Richter relation, we use a maximum magnitude m max = 7.5 and its recurrence interval T r = 10 000a in all simulations. However, to study the parameter dependence of our results, we repeat the simulations with different values of θ, β and h. If values other than h = 0.75 are used, β needs to be adjusted in order to have comparable impacts of large earthquakes on the conditional intensity function. Therefore, we recalculate β by requiring that an event of magnitude m max has the same effect on the occurrence probability as before, namely β10 h(mmax m0) = β 0.75 10 0.75(mmax m0). (9) After neglecting the first 4 000 000 events in each simulation to ensure that the system has evolved towards the equilibrium state, two catalogues are obtained. One contains 1 000 000 events of any magnitude. Since we are particularly interested in the behaviour of the large magnitudes, the other catalogue consists of 1000 events per region with a magnitude m > 6.5. The simulations themselves are carried out as follows. (1) Using the conditional intensity function λ(t) (eq. 4), the next event time t i in each region (index i) is simulated using the transform method (Devroye 1986), namely by inverting u i = exp [ ti τ i e α+β Z i (s) ds ], (10) where u i is uniformly distributed within [0, 1] and τ i is the time of the last stress jump in this region (for the derivation, see Appendix B). (2) In the region with the smallest value, the next event occurs at time t i. (3) The magnitude of this event is simulated from distribution (7) and converted into a stress release Z by eq. (2). (4) The stress transfer is calculated by θ Z. In all affected regions, the stress level is updated and the subsequent event times are recomputed with eq. (10) according to the altered intensity functions. Then step 2 is again repeated and so on. 4 SIMULATION RESULTS We analyse the characteristics of large earthquakes occurring in this model. At first we investigate the uncoupled model describing an isolated seismic zone. Subsequently, we turn to the effects of interactions between different seismic regions. Our standard simulations are carried out with the parameters β = 0.1, h = 0.75 and b = 1in the Gutenberg Richter relation. 4.1 The uncoupled model In the case of the uncoupled model, a key result of our simulations is that the occurrence of small and intermediate earthquakes does not lead to a random recurrence of the largest events. Even if earthquakes occur with a fractal size distribution, related to a Gutenberg Richter distribution, the evolution of the stress level shows some regularity. A weak seismic cycle occurs which is illustrated for one example in Fig. 1. A systematic analysis of the interevent-time distribution is shown in Fig. 2 for a much longer simulation. While this distribution is monotonically decaying (with a power-law tail) if all events are considered, it is peaked if only larger magnitude earthquakes (m > 6.5) are taken into account. The peak corresponds to a preferred interevent-time, indicating quasi-periodicity. We have also fitted the exponential, gamma, log-normal, Weibull and Brownian passage time distributions to the interevent-time distribution, since these are widely used in seismic hazard calculations (Matthews et al. 2002; Yakovlev et al. 2006), using the maximum likelihood method. The highest likelihood value was achieved by the gamma distribution, with log-likelihood values of LL = 7961 for the gamma, LL = 7966 for the Weibull, LL = 7973 for the log-normal, LL = 7979 for the exponential and LL = 8008 for the BPT distribution. The right part of Fig. 2 shows the interevent time histogram of the larger magnitude events together with the best-fitting gamma distribution. A general characteristic of the interevent-times can be quantified by the coefficient of variation, which is defined by the ratio of

652 N. M. Kuehn, S. Hainzl and F. Scherbaum magnitude 5.5 5 4.5 4 3.5 3 100 110 120 130 140 150 time [a] Figure 1. Example of evolving state of a 50-yr-long simulation with magnitudes (bars) and conditional intensity (dashed line). 10 0.0010 1 0.0008 density 0.1 0.01 0.001 10 4 density 0.0006 0.0004 0.0002 10 5 0.001 0.01 0.1 1 10 100 0.0000 0 500 1000 1500 2000 interevent time [a] interevent time [a] Figure 2. Left-hand side: Interevent-time distribution for all events with m > 3 (dashed) and fitted exponential distribution (solid); right-hand side: Intereventtime histogram for the large events (m > 6.5) and fitted gamma distribution. The parameters of the simulation are β = 0.1, h = 0.75, b = 1. standard deviation to mean, C V = SD/mean. For a Poisson (completely random) process the coefficient of variation has a value C V = 1, whereas values C V > 1 and C V < 1 correspond to a clustered and quasi-periodic process, respectively (Telesca et al. 2002). Fig. 3 illustrates how the gamma distribution the best fitting distribution for the interevent times of the larger magnitudes changes for the different C V -values of 0.5, 1 and 2, if the mean value is constant. For C V = 1, the gamma distribution corresponds to the exponential distribution. In Fig. 4, the resulting coefficient of variation is shown as a function of the lower magnitude cut-off for different values of h. Here we see that for small cut-off magnitudes the coefficient of variation takes values C V > 1, indicating that small events occur clustered. For larger cut-off magnitudes, the values of the coefficient of variation decrease, reaching a minimum at C V = 0.85 at m = 5.1 for h = 0.75 and C V = 0.61 at m = 7 for h = 1.5. Hence, earthquakes with a magnitude greater than these are characterized by quasi-periodic occurrence. The quasi-periodic behaviour stems from the fact that large earthquakes decrease the stress level by a significant amount, leaving it far from its equilibrium state, whereas small earthquakes cause only slight deviations in the stress level. However, the quasiperiodic behaviour applies only if all large events are considered together (i.e. m > 5.1 for h = 0.75 or m > 7 for h = 1.5), as they all produce a stress release high enough to cause significant density 1.4 1.2 0.8 0.6 C V = 0.5 C V = 1 C V = 2 0.4 0.2 0.0 0.0 0.5 1.5 2.0 Figure 3. Gamma distribution with parameter values corresponding to C V = 0.5, = 1 and = 2, where the mean value is fixed to 1. x

Non-Poissonian earthquake occurrence 653 CV 2.0 1.8 1.6 1.4 1.2 0.8 0.6 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 cutoff magnitude CV 2.0 1.5 0.5 0.0 4.0 4.5 5.0 5.5 6.0 6.5 cutoff magnitude Figure 4. Left-hand side: Coefficient of variation for a simulation with β = 0.1 and b = 1 as a function of the lower magnitude cut-off for h = 0.75 (stars), h = 1 (crosses) and h = 1.5 (rhombs); right-hand side: Coefficient of variation for seismicity of a single fault, taken from Zöller et al. (2006). stress release per magnitude 0.8 0.6 0.4 0.2 h = 0.75 h = h = 1.5 0.0 3 4 5 6 7 magnitude Figure 5. Fraction of stress released per magnitude for Benioff strain (h = 0.75), seismic moment (h = 1.5) and a value h = 1. perturbations in the stress level. The largest earthquakes, if considered separately, show an increasing trend of the coefficient of variation (right-hand part of Fig. 4) and hence a more random occurrence since slightly smaller events frequently occurring in between also significantly perturb the stress level. The result of a quasi-periodic behaviour of large events is in general agreement with the results for the fault model by Zöller et al. (2006) which simulates the seismicity on a single fault plane by spontaneous quasi-dynamic rupture propagation and stopping on a discretized fault embedded in the elastic crust. For this physically based fault model, the corresponding dependence of the C V -value on the lower magnitude cut-off is shown in Fig. 4. The general trend the decrease of C V for larger magnitudes cut-offs is very similar to the stochastic stress release model, although some differences occur for very large magnitude cut-offs: This discrepancy is probably related to the fact that the Gutenberg Richter frequency magnitude distribution breaks down in the physical model at a magnitude of approximately 6 due to the occurrence of so-called runaway ruptures. These are events which will rupture, after growing to a certain size, the whole fault because the stress concentration at the rupture tip becomes large enough to overcome all stress barriers. Thus independent magnitude 6 and 7 earthquakes do not occur in this physical model in contrast to our simulations of the stress release model. Despite this specific difference, the general similarity of the clustering properties between the stochastic stress release model and the computational expensive quasi-dynamic fault simulations indicates that the stress release model is able to mimic the fundamental fault dynamics in a first approximation. While it is not yet computationally feasible to study the effects of fault interactions in these more realistic models, this can be done with the linked stress release model (see Section 4.2). For the stress release model, the dependence of the recurrence time statistics on the h-value can be seen in Fig. 4. The dependence is caused by differences in the amount of stress that is released in different magnitude ranges. This is shown in Fig. 5. If Benioff strain is used (h = 0.75), the greater portion of stress is released by the smaller events, whereas for seismic moment (h = 1.5) the main stress release results from the large magnitude events. Hence, in the latter case, the magnitude level that can cause a large deviation from the normal stress level is enhanced. Together with the survivor function (Daley & Vere-Jones 2003), the interevent-time distribution can be used to calculate an empirical hazard function, which is defined as the event rate at time t given survival up to t. The hazard function is calculated as the probability density function divided by the survivor function. The Poisson process has a constant hazard function (Daley & Vere- Jones 2003). Fig. 6 shows the empirical hazard function (event rate) for all (left-hand side) and the large events (right-hand side).

654 N. M. Kuehn, S. Hainzl and F. Scherbaum 5 0.0010 4 0.0008 hazard rate 3 2 hazard rate 0.0006 0.0004 1 0.0002 0 0 2 4 6 8 10 0.0000 0 200 400 600 800 1000 time [a] time [a] Figure 6. Empirical hazard function calculated from the interevent-time histogram with β = 0.1, h = 0.75 and b = 1. Left-hand side: All events with m > 3; right-hand side: All events with m > 6.5. 5 4 CV 3 2 1 3 4 5 6 7 magnitude cutoff Figure 7. Coefficient of variation for a single region with (θ = 0.23, crosses) and without (θ = 0, stars) coupling (h = 0.75 and β = 0.1). The hazard function for all events decreases from high values at small interevent-times to an almost constant level. This corresponds to the clustering observed for the small magnitudes. By contrast, the hazard level for the large events (m > 6.5) is almost zero immediately after the last large earthquake and then starts to rise for increasing waiting times. This is due to the fact that the system needs to be reloaded before another large earthquake can happen again. On long timescales, the hazard function reaches a constant level. Changing the values of β does not alter the general characteristics of the model: events with a small magnitude occur clustered (C V > 1), whereas large ones occur quasi-periodically (C V < 1). However, for β 0, eq. (3) reduces to λ(t) = e α = const, corresponding to a Poisson process (C V = 1). Hence, the C V -values for low cutoff magnitudes decrease with decreasing β, but increase for higher values of β. However, the minimum of the coefficient of variation is quite stable: For 30 simulations (h = 0.75) with β-values varying between 0.01 and 0.3, the mean minimum takes place at C V = 0.841 ± 0.009. This indicates that the observed seismic cycle is mainly due to the seismic gap after large earthquakes which results from the time needed to reload the seismic zone such that it can produce the next large event. This reloading time depends on ρ but not on β. 4.2 The effect of coupling In the previous section, we examined the simple (uncoupled) stress release model concerning its temporal properties. Now we turn to the effects of stress interaction between different seismic regions. We find that the presence of coupling does not significantly affect the general temporal variability observed in each seismic zone, as can be seen in Fig. 7. Here, the coefficient of variation is shown for one specific region of the uncoupled (θ = 0, stars) and a strongly coupled (θ = 0.23, crosses) model, respectively. Parameters h = 0.75 and β = 0.1 are used in both cases. Since stress is transferred into the four neighbouring regions, a coupling strength θ = 0.23 means that 92 per cent of the stress released by an earthquake is redistributed in the system and 8 per cent is lost. As one can see in Fig. 7, the values of the coefficient of variation are very similar for both the uncoupled and coupled case. In the presence of coupling, the C V -values are just slightly increased, and the minimum of the curve is shifted a little to higher magnitudes. However, the general appearance of the graph is unchanged. Hence, equivalently to the uncoupled case, large earthquakes occur quasi-periodically in each individual seismic region. On the other hand, earthquakes in different regions start to correlate in the presence of stress interactions. This effect can be seen in

Non-Poissonian earthquake occurrence 655 1.6 3.0 1.4 2.5 CV 1.2 0.8 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 magnitude cutoff CV 2.0 1.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 magnitude cutoff Figure 8. Dependence of the coefficient of variation on the coupling strength for the stacked activity, for θ = 0 (crosses), θ = 0.1 (stars) and θ = 0.23 (rhombs). Left-hand side: Benioff strain (h = 0.75); right-hand side: Seismic moment (h = 1.5). 0.5 0.4 hazard rate 0.3 0.2 0.1 0. 0.001 0.01 0.1 1 normalized time Figure 9. Hazard function for large events (m > 6.5) in the uncoupled (solid) and strongly coupled (dashed) model. The long dashed line represents the hazard in the same region, the short dashed line in regions adjacent to that where the earthquake occurred. The times are normalized by the mean recurrence time. Fig. 8. Here, all events from all 100 regions are considered together as one catalogue for which the coefficient of variation is shown. This corresponds to a superposition of the individual stress release models for the different regions. The left-hand side of Fig. 8 shows simulations with h = 0.75, the right-hand part with h = 1.5. In the case of an uncoupled model (θ = 0), the different regions are independent, so their superposition leads to a Poisson process (Vere-Jones 1995) with C V = 1 (crosses in Fig. 8). However, if coupling is present, the processes in the different regions are not independent any more, so deviations from C V = 1 emerge. Coupling leads to a clustering of the events, independent of the magnitude, since each earthquake leads to an increase of the occurrence probability in adjacent zones. The clustering is also observed for small magnitudes because of triggered aftershock activity in neighbouring zones. Hence, on the global scale seismic cycling cannot be observed. One can see in Fig. 8 that the strongest clustering occurs for intermediate to large cut-off magnitudes. We explain the tendency that the largest events are again more random in the same way as before for the isolated region: The intermediate to large earthquakes can already perturb the stress field significantly and thus randomize the occurrence times of the largest events. Similar to the case of a single region, the clustering is enhanced for h = 1.5 when the stress is mainly released by the largest earthquakes (see Fig. 5). Fig. 9 compares the hazard functions in coupled and uncoupled seismic regions. On the one hand, the probability for the recurrence of large earthquakes in the same seismic zone is found to be almost identical in both cases. This indicates again that coupling does not significantly alter the seismic cycles in each region. On the other hand, the hazard level of a large earthquake is greatly enhanced in adjacent regions immediately after the event, if both regions are positively coupled. Thus correlated occurrence of major events in neighbouring regions will occur. However, the hazard function decays in adjacent regions quickly with time and, on long timescales, it reaches in all cases a constant level after approximately 20 per cent of the mean recurrence time of the large events. 5 SUMMARY AND CONCLUSIONS In probabilistic seismic hazard assessment (Cornell 1968), most studies assume a Poissonian occurrence of earthquakes, that is, a random occurrence in time and space. However, earthquake observations show that earthquakes cannot be treated as independent random events, because the occurrence of earthquakes is characterized by long- and short-term patterns (Scholz 2002). In seismically highly active regions where active faults have been identified and their characteristics are well known, often a quasi-periodic

656 N. M. Kuehn, S. Hainzl and F. Scherbaum recurrence of characteristic events combined with Poisson distributed background activity is deemed applicable (Matthews et al. 2002; WGCEP 2003). However, the validation of this assumption is difficult and it is thus controversially debated (see e.g. Kagan 1996). In particular, it is questionable whether or not the self-organized stress evolution due to fault communication destroy any regularity of the recurrence of large earthquakes. In particular, also small earthquakes seem to play an important role for the stress redistribution and should not be neglected (Hainzl et al. 2000; Helmstetter et al. 2005; Marsan 2005). In general, observational data could be used to directly estimate the hazard function, e.g. by non-parametric methods (Faenza et al. 2003). However, this usually cannot solve the problem because catalogues are mostly short compared to the underlying processes (Parsons 2005). In this work, we investigate the impact of stress evolution and interactions for seismic hazard estimations by means of the stress release model. In this model the earthquake activity depends on a scalar stress-variable denoting the mean stress level in the seismic region under consideration (Vere-Jones 1978). Tectonic loading increases this level steadily whereas it is decreased by earthquakes (dependent on the earthquake magnitude) based on the assumptions of classical models of earthquake mechanism (Reid 1911). The stress release model can be considered as a mean field model describing the net effect of earthquakes in a seismogenic volume. In previous studies, it has been shown that the stress release model achieves a fit to main shock occurrence that is usually superior to the Poisson process (e.g. Zheng & Vere-Jones 1994; Bebbington & Harte 2001; Imoto 2001; Imoto & Hurukawa 2006; Rotondi & Varini 2006). Hence, it seems worthwhile thinking about its application to seismic hazard analysis, although several issues pertaining the fitting of the model (e.g. the spatial zonation) have to be carefully considered (Bebbington & Harte 2003) and the model has not yet been rigorously tested (as opposed to other models, see e.g. Schorlemmer et al. (2007) and other articles in the same volume). Nevertheless, given its physical basis in elastic rebound theory and the number of good fits achieved, the stress release model can be regarded as a viable hypothesis for earthquake occurrence in seismically active regions. The aim of this work is to investigate the predictions of the stress release model in the general case of Gutenberg Richter type seismicity and fault interactions. Therefore, we perform numerical simulations of this model based on the well-known Gutenberg Richter relation for the frequency magnitude distribution. In practice, the parameters of the magnitude distribution can be determined independently, and thus most of the model parameters of the stress release model can be fixed (see Appendix). The remaining two parameters are describing the strength of the time dependence (β) and the scaling between stress release and magnitude (h). Both parameters are found to influence the strength but not the general characteristics of the observed non-poissonian behaviour. Our simulations show that the occurrence of earthquakes is generally correlated, in particular small earthquakes tend to cluster, while large ones occur quasiperiodically. Large earthquakes decrease the occurrence probability of the next event significantly within the first approximately 20 per cent of the mean recurrence time (see Fig. 9). On longer timescales, the hazard function reaches a constant level. The corresponding peaked recurrence-time distribution for the largest events is characterized by coefficients of variation from 0.61 (for h = 1.5) to 0.84 (for h = 0.75). This is similar to results from simulations of a physically based model of a single fault (Zöller et al. 2006). While the effect of fault interactions cannot be analysed appropriately by latter models due to computational limits, the linked stress release model can serve as a first approximation. We find that the above characteristics are robust if stress coupling between adjacent seismic zones is considered. Even very strong coupling does not significantly change the recurrence-time distributions of each zone, and major earthquakes occur quasi-periodically in each region. This is in agreement with conclusions drawn from the analysis of a number of real, but very short sequences of recurrent sequences on specific faults (Ellsworth et al. 1999). However, this does not imply that seismicity in adjacent seismic zones are independent in our simulations. Large events in different zones tend to cluster which is reflected in a strongly increased hazard level after a major failure in neighbouring regions (cf. Fig. 9). The role of small earthquakes can be studied in this model by varying the parameter h for a fixed Richter b-value. If Benioff strain is used (h = 0.75), a greater portion of stress is released by small magnitude events than by large earthquakes, whereas it is vice versa if seismic moment scaling (h = 1.5) is assumed (see Fig. 5). The non-poissonian behaviour mentioned above is found to be strongly enhanced for larger h-values when small earthquakes play only a minor role. Thus small earthquakes randomize the stress state evolution. However, even if they are dominating the stress release (in the case of h = 0.75), the seismicity shows a clear time-dependence. 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APPENDIX A Here we derive how the parameters α and ρ in the conditional intensity function (eq. 5) can be fixed if a Gutenberg Richter relation for the frequency magnitude relationship is assumed. In the stress release model, two processes oppose each other which both affect the stress level: stress gain due to tectonic loading and

658 N. M. Kuehn, S. Hainzl and F. Scherbaum stress release due to earthquakes. The system reaches an equilibrium stress state Z eq when these two processes equal each other. Tectonical loading is expressed by the loading rate ρ, which is the time-derivative of the stress function (eq. 1). The average stress release due to earthquakes is the product of the average stress loss of an earthquake Z times the conditional intensity function λ(t). Since the stress level trends to the equilibrium state Z eq, the value of the conditional intensity function at Z eq can serve as an approximation of λ(t). Hence, we get as a condition for the stable state the equation ρ = λ(z eq ) Z. (A1) However, the equilibrium state is also a good approximation for the maximum stress level, Z eq = Z max, since larger values of Z are unstable. The maximum stress level can be calculated by Z max = 10 h(mmax m0). On the other hand, the average stress loss Z can be calculated by summing up the average stress loss per magnitude for all magnitudes. Thus, Z is given by Z = mmax m 0 p(m) Z(m)dm, (A2) ln 10 b 10 bm where p(m) = is the probability density function 10 bm 0 10 bmmax of magnitudes and Z = 10 h(m m0) is the stress loss due to an earthquake of magnitude m. The solution of the integral gives b Z = (h b) 10(h b)(mmax m0) 1. (A3) 1 10 b (mmax m 0) Hence, the condition for the stable state (eq. A1) can be written as α+β b Zmax ρ = e (h b) 10(h b)(mmax m0) 1. (A4) 1 10 b (mmax m 0) Assuming that the stress level is near its stable state, the earthquake rate for all events with a magnitude m > m 0 can be constrained by λ(z max ) = 10 a bm 0. (A5) Thus, using λ(z) = e α+β Z as the conditional intensity function, this becomes e α+β Zmax = 10 a bm 0. leading to α = ln 10(a bm 0 ) β Z max. (A6) (A7) Thus, assuming a maximum magnitude as well as the a- and b- value of the Gutenberg Richter relation for a seismic active region, eqs (A7) and (A4) can be used to fix the parameters α and ρ, leaving β free. APPENDIX B In this section, the inverse transform method for calculating the next event time of the stress release model is explained. Therefore, first the conditional survivor and the conditional intensity function of a point process are defined. The conditional survivor function S n (t H τn 1 ) represents the probability that no event occurs in the interval [τ n 1, t], given the history H τn 1 of the process. It is related to the conditional probability density p n (t H τn 1 ), representing the probability of an event occurring at time t given H τn 1,via t S n (t H τn 1 ) = 1 p n (s H τn 1 )ds. (B1) τ n 1 For t (τ n 1, τ n ] the conditional intensity function can be expressed by λ(t) = p n(t H τn 1 ) S n (t H τn 1 ) = d dt log [ S n (t H τn 1 ) ], (B2) where λ(t) is also called the risk or hazard function. The inverse transform method employs that the survivor function S n (t H τn 1 ) of a point process is uniformly distributed in [0, 1]. Hence, if u is a sample from a uniform distribution on [0, 1], t = S 1 n (u) is a simulated point of the process, where S 1 n denotes the inverse of the survivor function. The hazard function for the stress release model is [ ] λ(t) = exp α + β(z 0 + ρt t i <t Z i ). (B3) Hence, using eq. (B2), the survivor function for t >τcan be calculated: [ t ] S(t H τ ) = exp e α+β Z(s) ds = exp { A[e βρ(t τ) 1] }, τ (B4) with A = 1 βρ eα+β Z(τ). So, if u is a sample from a uniform distribution on [0, 1], solving u = S(t H τ ) (B5) gives for the next event time t = τ + 1 [ βρ ln 1 ln u ]. (B6) A