On the Role of Multiple Interactions in Remote Aftershock Triggering: The Landers and the Hector Mine Case Studies

Bulletin of the Seismological Society of America, Vol. 96, No. 1, pp. 80 89, February 2006, doi: 10.1785/0120050029 On the Role of Multiple Interactions in Remote Aftershock Triggering: The Landers and the Hector Mine Case Studies by A. Ziv Abstract The observed seismicity rate increase after large earthquakes in sites that are located several source lengths away from the mainshock centroid poses a major problem. This is because the static stress change induced by a mainshock in that region seems to be insignificant, and the dynamic stress changes can only enhance the seismicity during the passage of the seismic waves but not at later times. In quest for a physically viable triggering mechanism for delayed aftershocks in remote sites, we examine earthquake activities in remote sites that were triggered by two California earthquakes, the magnitude 7.1 Hector Mine earthquake and the magnitude 7.3 Landers earthquake. We introduce a new method for quantifying the degree to which the triggering effect of each aftershock is locally more important than the triggering effect of the mainshock. We apply this method to the Landers and the Hector Mine remote aftershock sequences. We show that multiple stress transfers from early aftershocks to later aftershocks played an important role in the enhancement of both the Landers and the Hector Mine aftershock activities in remote sites. We present a time-space diagram of the Hector Mine remote aftershock sequence in the Imperial Valley, which shows that this sequence is made up of several subsequences and that the onset of activity migrated southward. Introduction The M w 7.3 Landers earthquake ruptured a northwesttrending right-lateral fault in the eastern California shear zone on 28 June 1992. Less than 8 years later, on 16 October 1999, and only 50 km to the northeast, the M w 7.1 Hector Mine earthquake ruptured another northwest-trending rightlateral fault. Significant seismicity rate increase following the Landers earthquake has been observed as far as 600 km away from the mainshock rupture (Hill et al., 1993, 1995) and as far as 250 km away following the Hector Mine rupture (Gomberg et al., 2001). Observed seismicity rate increases following large earthquakes in sites that are located several source lengths away from the mainshock centroid pose a major problem. This is because the static stress change induced by a mainshock in that region seems to be insignificant. This led to the idea that dynamic stresses, because they decay at a slower rate with distance than static stresses, are the cause for remote triggering (e.g., Anderson et al., 1994; Gomberg and Bodin, 1994). Several mechanisms were invoked to explain remote triggering in volcanic regions. For example, Linde et al. (1994) suggested that shaking associated with the passage of the seismic waves can trigger degassing in the magma chamber, giving rise to inflation of the magma body. Hill et al. (1993) proposed that the seismic pulse can induce liquefaction of a partially crystallized magma body that may relax differential stresses stored in the solid phase, or it may trigger dike intrusions. Although these mechanisms may be viable for magmatic provinces, they are inadequate for nonvolcanic areas. In such cases, a question arises as to whether oscillatory stresses can directly induce slip on faults? Several workers have shown that oscillatory stresses of finite duration, applied on a fault that is governed by a rate- and state-dependent friction, may only trigger slip during the oscillatory phase (Gomberg et al., 1998; Belardinelli et al., 2003; Perfettini et al., 2003). Gomberg (2001) demonstrated that this result is not specific to the rate-and-state friction but holds for an entire class of failure models in which the physical property characterizing failure becomes self-accelerating. Thus, the persistence of remote aftershock activity after the passage of the seismic waves is not yet understood. It is constructive to distinguish between two types of remote aftershocks according to their timing. The first, immediate aftershocks are those quakes occurring during the passage of the seismic wave. Such aftershocks were identified after the Landers (Hill et al., 1993), the Izmit (Brodsky et al., 2000), and the Denali (Prejean et al., 2004) earthquakes. The second, but far more abundant type of aftershocks are referred to as delayed aftershocks. These aftershocks occur during the days and weeks after the passage of 80

On the Role of Multiple Interactions in Remote Aftershock Triggering: The Landers and the Hector Mine Case Studies 81 the seismic waves emitted from the mainshock rupture. Although the triggering of immediate aftershocks is undoubtedly due to transient stress perturbation, the cause for the delayed aftershocks has remained uncertain (Freed, 2005). It is possible that different physical mechanisms are responsible for the triggering of immediate and delayed aftershock in remote sites. In quest for a physically viable triggering mechanism for delayed aftershocks in remote sites, we examine the importance of multiple interactions between aftershocks triggered by the Landers and the Hector Mine earthquakes. The main conclusion of this study is that that many of Landers and Hector Mine aftershocks in remote sites are not directly triggered by their mainshock but are instead aftershocks of previous aftershocks. First, to provide a useful context for the analysis that follows, we describe the spatial and temporal distribution of remote aftershocks triggered by the Landers and Hector Mine earthquakes. We then introduce a new method for quantifying the importance of multiple interactions in aftershock sequences. Next, we apply this method to the Landers and Hector Mine remote aftershock sequences. Finally, we discuss the implications of the results. The Data Figure 1. Maps of earthquake rate change following the Landers earthquake (a) and the Hector Mine earthquake (b). The quantity that is mapped is DN in equation (1) that is calculated for nonoverlapping spatial windows with dimensions of 0.5 0.5 deg. The time windows are Dt post 10 days, Dt pre 100 days, and Dt lt is between January 1985 and January 2002. The stars indicate the epicenters of the Landers and the Hector Mine earthquakes. Aftershock activities are examined in four regions that are indicated by solid rectangles and are labeled as North1, North2, LVC, and South. We use the composite catalog of the Advanced National Seismic System (ANSS, previously called the Council of the National Seismic System). This is a worldwide earthquake catalog that is created by merging earthquake catalogs from contributing ANSS institutions and then removing duplicate solutions for the same event. Seismic networks that contribute to ANSS in the study area are the Northern California Seismic Network, the Southern California Seismic Network, and the Nevada Seismic Network. Some of the analyzes presented in this study are sensitive to the catalog completeness. We have determined the smallest magnitude to which the data are complete from the Gutenberg-Richter relationship for various intervals and subregions within the studied area. For example, we compared earthquake size distribution during 1985 1990 and 1993 1998. Surprisingly, despite some post-landers network improvement, the minimum magnitude for completeness in regions North1, North2, and South (see Fig. 1) has not changed and is equal to 1.5. Earthquake size distribution in regions North1 and North2 during the 10 days after the Landers earthquake shows an increase in the detection threshold during aftershock activity by 0.5 magnitude units, from 1.5 to 2. Some of the analyses require precise relative location between adjacent earthquakes. In such cases we use, in addition to the conventional catalog of the ANSS, the relocated catalog of Hauksson et al. (2003). Location of earthquakes in this catalog were obtained by using either differential travel times from cross-correlation of waveforms or a 3D velocity model. Unfortunately, because the relocated catalog uses data only from the Southern California Seismic Network, it does not cover the entire studied area. Thus, unless otherwise stated, we use the ANSS catalog. The Spatiotemporal Distribution of Landers and Hector Mine Remote Aftershocks Next we report the spatial distribution and temporal decay of aftershock activity following the Landers and Hector Mine earthquakes. The objective of this section is to provide a context for the analysis that follows.

82 A. Ziv Spatial Distribution Previous investigators have mapped the seismicity following these earthquakes (Hill et al., 1993; Kilb et al., 2000; Wyss and Wiemer, 2000; Gomberg et al., 2001; Marsan, 2003; Voisin et al., 2004), and the results shown in the following text are in good agreement with what is already known from their studies. We compute the earthquake rate change for nonoverlapping spatial windows. The dimensions of the spatial windows are chosen to be 0.5 0.5 deg. Aftershock rate change, DṄ, is calculated according to: [N(Dt ) N(Dt post pre)] DN, (1) Ṅ(Dt lt) where N is the earthquake rate and Dt post, Dt pre, and Dt lt are the time windows for postmainshock, premainshock and long-term seismicity, respectively. The results presented in this study were obtained with Dt post 10 days, Dt pre 100 days, and Dt lt between January 1985 and January 2002. Only earthquakes with magnitude 2 were included in this calculation. Maps of the Landers and Hector Mine aftershock rates are shown in Figure 1. Although remote sites triggered by the Landers earthquake are located primarily to the north over a large area that extends up to 50 km north of Lake Tahoe, remote sites triggered by the Hector Mine earthquake are located to the south, occupying a much smaller area in the Salton Sea and the Imperial Valley. Gomberg et al. (2001) pointed out that the asymmetry in the location of remotely triggered sites is similar to the asymmetry in the distribution of the peak dynamic stress because of the effect of rupture directivity. This similarity has been interpreted as a fingerprint of the dynamic role in remote aftershock triggering. Note that Landers aftershock rate map shows significant triggering over a small region to the east of the Imperial Valley (dashed rectangle in Fig. 1a). A more careful inspection of the seismicity in that region revealed that the increase of earthquake rate is the result of four events that occurred during the first 10 days after the Landers earthquake against a very low background seismicity rate. Because the aftershock statistics in that area are insignificant, properties of that sequence cannot be examined. Finally, Figure 1a suggests that Landers enhanced the seismicity in the future location of the Hector Mine hypocenter. Note, however, that much of the activity in that area has been triggered by the M w 5.4 Pisga earthquake that occurred 7 days after the Landers earthquake (Felzer et al., 2002). Aftershock Decay Rate Remote aftershock activities are examined in four regions (Fig. 1): (1) the northernmost region, labeled as North1, includes Lake Tahoe, Mono Basin, White Mountains, and Death Valley; (2) a small area that includes the Long Valley Caldera and the Mono-Inyo craters is labeled as LVC; (3) a region to the north of Garlock fault, labeled as North2, includes the Little Skull Mountain, Indian Wells Valley, and the Coso Hot Springs; and (4) the southernmost region labeled as South includes the Salton Sea and the Imperial Valley. The rationale for splitting the northern regions into two is that the relocated catalog, which we use in the Evidence for Intense Interaction among Remote Aftershocks section, is available for North2 but not for North1. Histograms of event count as a function of time before and after each mainshock (Fig. 2) confirm that both North1 and North2 regions were triggered by the Landers earthquake and not by the Hector Mine earthquake. Some triggering in LVC by Hector Mine is also apparent (see also Fig. 1b), but the size of this aftershock sequence is very small compared with that triggered by the Landers earthquake in the same area. Landers aftershock activity in LVC has already been studied in depth by Hill et al. (1995), and it is believed that activity there is coupled with magmatic processes. For these reasons, in the remainder of this article we focus attention on aftershock activities in regions North1, North2, and South. It is a common practice to fit the decay rate of large aftershock sequences with the modified Omori law (Utsu, 1961): K Ṅ(t) p, (2) (c t) where N is the aftershock rate, t is time since the mainshock, and K, c, and p are fitting coefficients. The aftershock sequences studied here are small. In such cases, the calculation of the seismicity rate is inaccurate, and the fitting coefficients are poorly constrained. Thus, here, the fitting is done to a cumulative form of (2), with a p-value that is fixed to 1: 0 t N(t) N(t)dt K ln(t/c 1). (3) Cumulative earthquake plots in Figure 3 are limited to earthquakes of magnitude greater than 2 (this magnitude threshold tends to lessen the affect of temporal changes in the detection threshold). We use equation (3) to fit cumulative plots and find a good fit between the modified Omori law with a p-value that is equal to 1 (dashed lines) and aftershock sequences in North1 and North2 (best fits were obtained with K 131 and c 3 days in North1 and with K 173 and c 4 days in North2). In contrast, the Hector Mine aftershock sequence in South cannot be fitted with such a decay law. Inspection of the histograms and the cumulative curves indicates that the duration of Landers aftershock activity in areas North1 and North2 is much longer than the duration of Hector Mine aftershock sequence in area South. The Hector Mine aftershock sequence in area South is further examined in the Hector Mine Sequence section, where we

On the Role of Multiple Interactions in Remote Aftershock Triggering: The Landers and the Hector Mine Case Studies 83 Figure 2. Histograms of event counts as a function of time with respect to (a d) Landers and (e h) Hector Mine origin times. A vertical line indicates the time of the mainshock in question. The width of the time bins is 10 days. show that this sequence is made up of several subsequences and that the onset of activity migrated southward. A New Method for Quantifying Multiple Interactions in an Aftershock Sequence We introduce a new method for quantifying the degree to which the triggering effect of each aftershock is locally more important than the triggering effect of the mainshock and the previous aftershocks. In the next section we apply this method to investigate multiple interactions among Landers and Hector Mine remote aftershocks. Mainshock Index and Secondary Mainshocks We compare the number of earthquakes preceding and following each aftershock in temporal windows that are equal to the lag time between the mainshock and the aftershock in question, Dt, and in a radial region surrounding the aftershock hypocenter with dimensions that scale with the size of the aftershock rupture. Mainshock index of event i is defined as: N(Dti t 2Dt i, r 2R i) ki, (4) N(0 t Dt, r 2R ) where t is the time measured from the mainshock time, r is the horizontal component of the interevent distance, N is the number of earthquakes satisfying the time and distance criteria, and R is the rupture radius. Explanatory notes are as follows. Interevent distances are projected onto a map view, because uncertainties in hypocenter depth are much larger than uncertainties in epicenter location. The size of the spatial window is set proportionally to the rupture radius, because the stress change that an earlier aftershock induces on the site of a later aftershock is proportional to the ratio between the dimension of the early rupture and the interevent distance. The choice of r 2R is somewhat arbitrary. We want the spatial window to be not too large, so that very distant aftershocks that were not triggered by the mainshock in question will be left out. On the other hand, we want the spatial window to be not too small, so that nearby aftershocks that were directly triggered by the mainshock in question will be included. i i

84 A. Ziv transfer) of that aftershock in that region is stronger than the triggering effect of the mainshock and the previous aftershocks. Hereafter, aftershocks with mainshock index that is greater than 1 are referred to as secondary mainshocks. This is not to say that aftershocks with mainshock index smaller than 1 did not promote later aftershocks, only that their effect was smaller than the combined effect of the mainshock and the previous aftershocks. Comparison with a Mainshock Index of a Noninteracting Sequence It is possible to compare the observed mainshock index, k, with a theoretical mainshock index, k th, of a sequence with noninteracting aftershocks. The assumption here is that the theoretical aftershock sequence decays uniformly according to a modified Omori s law. The theoretical mainshock index is then: 2Dt i Dti p K/(c t i) dt k th. (5) i Dt i p 0 K/(c t ) dt i Integration for p 1 gives: and for p 1: 2Dti c ln Dt c th i ki, (6) Dti c ln c 1 p 1 p th (c 2Dt) (c Dt) i 1 p 1 p k. (7) (c Dt) c Figure 3. Plots of cumulative earthquake counts as a function of time. Cutoff magnitude is 2. The mainshocks and the studied areas are indicated in each panel. Dashed lines are best fits to equation (3). The dotted line in (a) is the best fit to the first 60 days. Rupture radius is estimated based on the moment-magnitude relation of Abercrombie (1996) and the scaling of slip with length on a circular crack (Eshelby, 1957), with a stress drop of 10 MPa. Premainshock time windows are defined such that the mainshock in question is included, ensuring that the denominator is never equal to zero. A mainshock index greater than 1 is indicative of seismicity rate increase in the vicinity of the aftershock in question, suggesting that the triggering effect (but not the stress Both solutions have the properties that k th r 1asDt r 0, and that k th r 0asDt r. This shows that our definition of a secondary mainshock is a strict one, in the sense that it provides the most conservative measure of aftershock triggering. On the other hand, the least conservative approach would be to examine the fraction of aftershocks for which ki k th i. Several researchers view aftershock sequences as being made up of direct and secondary aftershocks (e.g., Ogata, 1988). Although the first are triggered directly by the mainshock, the latter are triggered by previous aftershocks that are either direct or not. Felzer et al. (2002, 2003) proposed a method for estimating the fraction of secondary aftershocks of a natural aftershock sequence. This is done by numerically reproducing the sequence in question with a stochastic cascade model. That model utilizes the Gutenberg- Richter relationship, the Omori law, Bath s law, and several input parameters. Our method has the advantage that it only utilizes the Omori law (but not the definition of secondary

On the Role of Multiple Interactions in Remote Aftershock Triggering: The Landers and the Hector Mine Case Studies 85 mainshocks). Additionally, our analysis is carried out directly on the data, without resorting to Monte Carlo techniques and stochastic models. Finally, whereas the secondary aftershocks in Felzer et al. (2002, 2003) are identified in a statistical sense (i.e., the fraction of the population), our secondary mainshocks are identified individually. Evidence for Intense Interaction among Remote Aftershocks Figure 4. Mainshock indexes, k, as a function of event magnitude for earthquakes that occurred during the 100 days after the Landers earthquake within regions North1 (a) and North2 (b). Landers Sequence The analysis described previously requires precise relative location between adjacent ruptures. Relocated catalog is available for North2 (Hauksson, 2003) but not for North1. Thus, we use the ANSS to calculate mainshock indexes in region North1 and use Hauksson s relocated catalog to compute mainshock indexes in region North2. Mainshock indexes as a function of event magnitude for earthquakes that occurred within regions North1 and North2 during the 100 days after the Landers earthquake are shown in Figure 4. Note that many of the mainshock indexes are greater than 1. A better view is provided in Figure 5, which shows the percentage of secondary mainshocks for aftershocks with magnitude greater than the magnitude threshold (dashed curve). Because this analysis may be affected by the worsening of the magnitude of completeness shortly after the Landers earthquake, we have recalculated the fraction of secondary mainshocks after excluding events that occurred during the first 24 hr following Landers (solid curve). We find that the fraction of secondary mainshocks increases due to excluding the first 24 hr. Note that the fraction of secondary mainshocks increases with increasing magnitude. For example, the fraction of Landers aftershocks with mainshock index greater than 1 in North2 is 45% for aftershocks of magnitude greater than 3, but 60% for aftershocks with magnitude greater than 4. The percentage of k 1 in North1 is much smaller than in North2. Whether this reflects a true difference or an artifact of poor earthquake locations in North1 is difficult to determine. Nevertheless, in the Appendix we present the results of Monte Carlo tests, which show that the result presented in Figure 5a is very unlikely to arise by chance. In the Aftershock Decay Rate section we showed that aftershock decay in areas North1 and North2 is fittable by a modified Omori law with an Omori exponent that is equal to 1, and c 3 4 days. We substitute c 4 in equation (6) (because c 4 gives a larger k th than c 3), and compute k th for each earthquake. In Figure 6 we show the percentage of aftershocks with a mainshock index greater than the theoretical (i.e., noninteracting) index as a function of the magnitude threshold. For example, we find that the fraction of Landers aftershocks in region North2 with mainshock index greater than the theoretical index is 30% for all magnitudes and 70% for aftershocks with magnitude greater than 3. We thus conclude that multiple interactions play an important role in Landers remote aftershock sequence and that many aftershocks are not directly triggered by the Landers earthquakes but are aftershocks of previous aftershocks. Hector Mine Sequence Do multiple stress transfers play an important role in the Hector Mine aftershock sequence as well? Here too we used the relocated catalog (Hauksson, 2003) and calculated mainshock indexes for aftershocks that occurred within region South during the 10 days after the Hector Mine earthquake. In Figure 7 we show the percentage of aftershocks with a mainshock index greater than 1 as a function of the magnitude threshold. Similarly to the Landers aftershock sequence in regions North1 and North2, we find that the fraction of secondary mainshocks increases with increasing magnitude. Additionally, we show a time-space diagram of Hector Mine aftershocks in area South, which reveals a complex internal structure for that sequence (Fig. 8). The sequence consists of several subsequences, and the onset of activity

86 A. Ziv Figure 6. Percentage of k k th (p 1) as a function of the threshold magnitude for earthquakes that occurred during the 100 days after the Landers earthquake within regions North1 (solid) and North2 (dashed). Earthquakes that occurred during the first 24 hr were excluded from this analysis. Figure 5. Percentage of k 1 as a function of the threshold magnitude for earthquakes that occurred during the 100 days after the Landers earthquake within regions North1 (a) and North2 (b). Dashed and solid curves are for earthquakes with Dt 0 and Dt 1 day, respectively. migrated southward, that is, away from the mainshock. Many of the earthquakes that occurred between latitude 33 N and 33.5 N are aftershocks of a M w 4.3 earthquake that ruptured about 10 min after the Hector Mine earthquake. Additionally, a M w 4.37 earthquake that occurred 2.4 days after the Hector Mine earthquake, near the southern end of the studied area, triggered the burst of seismicity to the north, near latitude 33 N. Thus, also in the case of Hector Mine aftershocks in the Imperial Valley, there is clear evidence that many aftershocks were aftershocks of previous aftershocks. Can Multiple Stress Transfers Explain Remote Aftershocks? Indeed, the contribution of small earthquakes to the total slip, the sum of the seismic moment, and the seismic energy budget are negligible. For these reasons, there has been a tendency among seismologists to downplay the importance of small earthquakes. More recently, however, there is a Figure 7. Percentage of k 1 as a function of the threshold magnitude for earthquakes that occurred during the 10 days after the Hector Mine earthquake within region South. growing awareness of the role that small earthquakes play in the enhancement of spatiotemporal clustering and the redistribution of stresses in the crust. It turned out that the smallness of small ruptures is exactly compensated for by their greater abundance, such that small earthquakes are just as important as large ones in redistributing the driving forces along active faults (Hanks, 1992). In addition, time-space analyses of various catalogs show that, similar to large earthquakes, small earthquakes too induce aftershock activity that decays according to the Omori formula (Shaw, 1993; Rubin, 2002; Helmstetter, 2003; Ziv et al., 2003). Felzer (2003) examined the spatiotemporal clustering of Landers aftershocks that occurred in the vicinity of the Landers hypocen-

On the Role of Multiple Interactions in Remote Aftershock Triggering: The Landers and the Hector Mine Case Studies 87 Figure 8. Time-space diagram for the Hector Mine aftershocks in area South. The size of the circles is proportional to the earthquake magnitude. The vertical dashed lines indicate the timing of the three largest earthquakes. ter and concluded that many of these aftershocks were not directly triggered by the Landers earthquake but were instead secondary aftershocks triggered by earlier aftershocks. The results of the previous section indicate that multiple interactions between aftershocks also play an important role in areas that are located far from the mainshock. We would like to take it a step further and suggest that all delayed aftershocks (i.e., aftershock occurring after the passage of the seismic waves) that occurred in remote sites could be triggered by secondary mainshocks. Below we summarize the results of previous studies that prove the viability of this mechanism. Ziv (2003) studied earthquake clustering in a synthetic catalog generated by the fault model of Ziv and Rubin (2003), which is a quasistatic inherently discrete model governed by rate- and state-dependent friction. Ziv (2003) found that the increase in the postmainshock earthquake production rate covers an area surrounding the mainshock rupture with radial dimensions that are several times larger than the mainshock dimensions. By comparing the spatial distribution of aftershock activity with and without elasto-static stress transfers, he showed that the increase in seismicity rate far from the mainshock is entirely due to the effect of multiple stress transfers. Furthermore, it has been suggested that the remote seismicity rate increase could be triggered by the passage of seismic waves. Remote triggering in a quasistatic model indicates that it is not necessary to invoke a dynamic effect to explain distant aftershocks. The idea that multiple interactions can explain remote aftershocks is also consistent with the results of a stochastic cascade model that utilizes Omori s law, the Gutenberg- Richter law, and a power-law distribution of distances between triggering and triggered quakes. It turned out that the effect of multiple triggering in this model is to increase the aftershock region with time (e.g., Helmstetter and Sornette, 2002; Helmstetter et al., 2003). Thus, this model too predicts aftershock occurrence in areas where the triggering effect of the mainshock alone is negligible. Summary and Conclusions We examine the decay of remote aftershock sequences. We find that Landers aftershock rate is fittable with the modified Omori law, with a p-value that is equal to 1. In contrast, the Hector Mine aftershock sequence cannot be fitted with such a decay law. We define a new parameter, a mainshock index, that quantifies the degree to which each aftershock is acting locally as a mainshock. A mainshock index greater than 1 indicates a seismicity rate increase in the vicinity of the aftershock in question, suggesting that the triggering effect of that aftershock in that region is more important than the triggering effect of the mainshock and earlier aftershocks. We show that many of the Landers and Hector Mine aftershocks have a mainshock index greater than 1 and that the fraction of aftershocks with mainshock index greater than 1 increases with increasing aftershock magnitude. Additionally, we present a time-space diagram for the Hector Mine aftershock sequence, which shows that this sequence is made up of several subsequences and that the onset of activity migrated southward. The main conclusion of this study is that that many of the Landers and Hector Mine aftershocks in remote sites are not directly triggered by their mainshock but are instead aftershocks of previous aftershocks. Acknowledgments This study benefited from discussions with M. Chaljub, R. Madariaga, and L. Margerin. I thank S. Wiemer, Y. Y. Kagan, and an anonymous reviewer for constructive remarks that helped to improve the manuscript.

88 A. Ziv References Abercrombie, R. E. (1996). The magnitude-frequency distribution of earthquakes recorded with deep seismometers at Cajon Pass, southern California, Tectonophysics 261, 1 7. Anderson, J. G., J. N. Brune, J. Louie, Y. Zeng, M. Savage, G. Yu, Q. Chen, and D. depolo (1994). Seismicity in the western Great Basin apparently triggered by the Landers, California, earthquake, June 28, 1992, Bull. Seism. Soc. Am. 84, 863 891. Belardinelli, M. E., A. Bizzarri, and M. Cocco (2003). Earthquake triggering by static and dynamic stress changes, J. Geophys. Res. 108, 2135, doi 10.1029/2002JB001779. Brodsky, E. E., V. Karakostas, and H. Kanamori (2000). A new observation of dynamically triggered regional seismicity: earthquakes in Greece following the August, 1999 Izmit, Turkey earthquake, Geophys. Res. Lett. 27, 2741 2744. Eshelby, J. D. (1957). The determination of elastic field of an ellipsoid inclusion and related problems, Proc. R. Soc. London Ser. A 241, 376 396. Felzer, K. R., T. W. Becker, R. E. Abercrombie, G. Ekström, and J. R. Rice (2002). Triggering of the 1999 Mw 7.1 Hector Mine earthquake by aftershocks of the 1992 Mw 7.3 Landers earthquake, J. Geophys. Res. 107, 2190, doi 10.1029/2001JB000911. Felzer, K. R., R. E. Abercrombie, and G. Ekström (2003). Secondary aftershocks and their importance for aftershock forecasting, Bull. Seism. Soc. Am. 93, 1433 1448. Freed, A. M. (2005). Earthquake triggering by static, dynamic, and postseismic stress transfer, Annu. Rev. Earth Planet. Sci. 33, 335 367. Gomberg, J. (2001). The failure of the earthquake failure models, J. Geophys. Res. 106, 16,253 16,263. Gomberg, J., and P. Bodin (1994). Triggering of the M s 5.4 Little Skull Mountain, Nevada, earthquake with dynamic strains, Bull. Seism. Soc. Am. 84, 844 853. Gomberg, J., N. M. Beeler, M. L. Blanpied, and P. Bodin (1998). Earthquake triggering by transient and static deformations, J. Geophys. Res. 103, 24,411 24,426. Gomberg, J., P. Reasenberg, P. Bodin, and R. Harris (2001). Earthquake triggering by transient seismic waves following the Landers and Hector Mine, California earthquakes, Nature 411, 462 466. Hanks, T. C. (1992). Small earthquakes, tectonic forces, Science 256, 1430 1432. Hauksson, E., W-C. Chi, and P. Shearer (2003). Comprehensive waveform cross-correlation of southern California seismograms: part 1. Refined hypocenters obtained using the double-difference method and tectonic implications (abstract), in Proc. Fall Ann. Meeting, American Geophysical Union, 8 12 December 2003, San Francisco, California. Helmstetter, A. (2003). Is earthquake triggering driven by small earthquakes? Phys. Rev. Lett. 91, 058501. Helmstetter, A., and D. Sornette (2002). Diffusion of epicenters of earthquake aftershock, Omori law and generalized continuous-time random walk model, Phys. Rev. E. 66, 061104. Helmstetter, A., G. Ouillon, and D. Sornette (2003). Are aftershocks of large Californian earthquakes diffusing? J. Geophys. Res. 108, 2483, doi 10.1029/2003JB002503. Hill, D. P., P. A. Reasenberg, A. Michael, W. Arabaz, G. C. Beroza, J. N. Brune, D. Brumbaugh, S. Davis, D. DePolo, W. L. Ellsworth, J. Gomberg, S. Harmsen, L. House, S. M. Jackson, M. Johnston, L. Jones, R. Keller, S. Malone, S. Nava, J. C. Pechmann, A. Sanford, R. W. Simpson, R. S. Smith, M. Stark, M. Stickney, S. Walter, and J. Zollweg (1993). Seismicity in the western United States remotely triggered by the M 7.4 Landers, California, earthquake of June 28, 1992, Science 260, 1617 1623. Hill, D. P., M. J. S. Johnston, J. O. Langbein, and R. Bilham (1995). Response of Long Valley caldera to the M w 7.3 Landers, California, earthquake, J. Geophys. Res. 100, 12,985 13,005. Kilb, D., J. Gomberg, and P. Bodin (2000). Triggering of earthquake aftershocks by dynamic stresses, Nature 408, 570 574. Linde, A., I. S. Sacks, M. J. S. Johnston, D. P. Hill, and R. G. Bilham (1994). Increased pressure from rising bubbles as a mechanism for remotely triggered seismicity, Nature 371, 408 410. Marsan, D. (2003). Triggering of seismicity at short timescales following Californian earthquakes, J. Geophys. Res. 108, 2266, doi 10.1029/ 2003JB002886. Ogata, Y. (1988). Statistical models for earthquake occurrence and residual analysis for point processes, J. Am. Stat. Assoc. 83, 9 27. Perfettini, H., J. Schmittbuhl, and A. Cochard (2003). Shear and normal load perturbations on a two-dimensional continuous fault: 2. Dynamic triggering, J. Geophys. Res. 108, 2409, doi 10.1029/2002JB001805. Prejean, S. G., D. P. Hill, E. E. Brodsky, S. E. Hough, M. J. S. Johnston, S. D. Malone, D. H. Oppenheimer, A. M. Pitt, and K. B. Richards- Dinger (2004). Remotely triggered seismicity on the United States west coast following the M W 7.9 Denali fault earthquake, Bull. Seism. Soc. Am. 94, S348 S359. Rubin, A. M. (2002). Aftershocks of microearthquakes as probes of the mechanics of rupture, J. Geophys. Res. 107, no. 87, doi 10.1029/2001 JB000496. Shaw, B. E. (1993). Generalized Omori law for aftershocks and aftershocks from a simple dynamics, Geophys. Res. Lett. 20, 907 910. Utsu, T. (1961). A statistical study on the occurrence of aftershocks, Geophys. Mag. 30, 521 605. Voisin, C., C. Cotton, and S. Di Carli (2004). A unified model for dynamic and static stress triggering of aftershocks, antishocks, remote seismicity, creep events, and multisegmented rupture, J. Geophys. Res. 109, B06304, doi 10.1029/2003JB002886. Wyss, M., and S. Wiemer (2000). Change in the probabilities for earthquakes in Southern California due to Landers M 7.3 earthquake, Science 290, 1334 1338. Ziv, A. (2003). Foreshocks, aftershocks and remote triggering in quasistatic fault models, J. Geophys Res. 108, 2498, doi 10.1029/2002 JB002318. Ziv, A., and A. M. Rubin (2003). Implications of rate-and-state friction for properties of aftershock sequence: quasi-static inherently discrete simulations, J. Geophys Res. 108, 2051, doi 10.1029/2001JB001219. Ziv, A., A. M. Rubin, and D. Kilb (2003). Spatio-temporal analyzes of earthquake productivity and size distribution: observations and simulations, Bull. Seism. Soc. Am. 93, no. 5, 2069 2081. Appendix Monte Carlo Tests In the Landers Sequence section we show that the fraction of aftershocks with a mainshock index greater than 1 increases with increasing aftershock magnitude. For example, the fraction of Landers aftershocks in North1 with mainshock index greater than 1 is 12% for aftershocks of magnitude greater than 3 and more than 16% for aftershocks with magnitude greater than 4 (Fig. 5a). We wish to test the null hypothesis that this result may arise by chance. Specifically, we would like to know what the chances are of getting a fraction of k 1 in a random catalog, that is, equal to or greater than what is observed in North1 following the Landers earthquake. To that end we calculated mainshock indexes for synthetic catalogs produced by shuffling the times in the original catalog. Following randomization with respect to times, the synthetic catalog is still clustered. Yet, although the temporal and spatial clusterings are correlated in the original catalog they are not correlated in the randomized catalog. The histograms in Figure A1 show the distributions

On the Role of Multiple Interactions in Remote Aftershock Triggering: The Landers and the Hector Mine Case Studies 89 of percentage of k 1 for aftershocks with magnitude greater than 3 and 4, calculated for 1000 synthetic catalogs. The dashed lines indicate the observed fraction of k 1. Note that only 9 of 1000 synthetic catalogs exceed the observed fraction of k 1 for magnitude threshold equal to 3, and only 47 exceed the observed fraction for magnitude threshold equal to 4. On the basis of this analysis, we conclude that the null hypothesis that the observed mainshock index versus magnitude in Figure 5a could arise by chance may be rejected at the 95% confidence level. Ben-Gurion University of the Negev Department of Geological and Environmental Sciences Beer-Sheva, 84105, Israel zival@bgu.ac.il Manuscript received 15 February 2005. Figure A1. A histograms showing the distributions of percentage of k 1 for aftershocks with magnitude greater than 3(a) and 4(b), calculated for 1000 synthetic catalogs. The dashed lines indicate the observed fraction of k 1. The synthetic catalogs were produced by shuffling the times in the original catalog.