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R. S. Stein,

The role of stress transfer in earthquake occurrence, Nature, 402, pp. 605-609, 1999.

[Online article][Printable article (885 kb)][Review of this paper in Complexity Digest]

 

 

The role of stress transfer in earthquake occurrence

Ross S. Stein

U.S. Geological Survey, MS 977, Menlo Park, CA 94025, USA. rstein@usgs.gov
 
 

An earthquake alters the shear and normal stress on surrounding faults. New evidence strengthens the hypothesis that such small, sudden stress changes cause large changes in seismicity rate. Rates climb where the stress increases (aftershocks) and fall where the stress drops. Both increases and decreases in seismicity rate are followed by a time-dependent recovery. When stress change is translated into probability change, seismic hazard is seen to be strongly influenced by earthquake interaction.

During the 75 years before the great 1906 earthquake on the San Andreas fault, the San Francisco Bay area suffered at least 14 M36 shocks on all major faults, including two M36.8 events; during the succeeding 75 years, there was but one M36 shock (Figure 1). Elsewhere, the occurrence of M36 extensional earthquakes seaward of subduction zones in the outer rise occur, with few exceptions, only in the years following great subduction events. Evidently, the rate of seismicity is not constant, and the rate -or probability- of earthquakes on one fault is not independent of another. Yet there is nothing in probabilistic seismic hazard assessment, the principal tool of the engineering, insurance, financial, and emergency response communities, that reflects or can reproduce such observations. Earthquake interaction is a fundamental feature of seismicity, leading to earthquake sequences, clustering, and aftershocks. One interaction criterion that promises a deeper understanding of earthquake occurrence, and a better description of probabilistic hazard, is Coulomb stress transfer.

Coulomb failure stress

An earthquake reduces the average value of the shear stress on the fault that slipped, but as Chinnery first showed in 1963, shear stress rises at sites in addition to the fault tips. This discovery lay in waiting for 20 years, when lobes of off-fault aftershocks were seen to correspond to small calculated increases in shear or Coulomb stress. In its simplest form, the Coulomb failure stress change,

where is the shear stress change on a fault (reckoned positive in the direction of fault slip), is the normal stress change (positive if the fault is unclamped), is the pore pressure change in the fault zone (positive in compression), and m is the friction coefficient (with range 0-1). Failure is encouraged if is positive and discouraged if negative; both increased shear and unclamping of faults promote failure. The tendency of to counteract is often incorporated into (1) by a reduced 'effective' friction coefficient, .

The calculated off-fault stress increases are rarely more than a few bars (1 bar = 0.1 MPa ~ atmospheric pressure at sea level), or just a few percent of the mean earthquake stress drop. In addition, the proximity to failure at any site is presumably variable but in any event unknown. So why would aftershocks concentrate at the site of such small stress increases? New studies find a surprisingly strong influence of stress change on seismicity, explain seek to explain it in terms of rupture nucleation phenomena observed in the laboratory.

Stress change and seismicity rate change

More than any other earthquake, the 1992 Mw=7.3 Landers, California, shock changed the landscape of stress-triggering investigations. Rich in aftershocks, the well-recorded event enabled detailed estimates of distribution of the fault slip needed to calculate the stress changes. Because in map view the strike-slip rupture is concave to the west, the earthquake was calculated to produce a 2-bar lobe of Coulomb stress increase 40 km west of the mainshock, where the M_w=6.5 Big Bear shock struck 2h30mn after Landers. While this association alone could result from chance, 67% of the 10,000 M>1 Landers-Big Bear aftershocks also occurred in regions calculated to have been brought >0.1 bar closer to failure (termed the stress-triggering zones), and few off-fault aftershocks occurred in regions discouraged >0.1 bar from failure (the stress shadows). Most of these comparisons of stress change to aftershocks rely on the assumption that small shocks occur on planes optimally oriented for failure as a result of the regional stress and the earthquake stress change. The association of calculated Coulomb stress increases with aftershocks is now widely reported (see and references therein).

Tantalizing as it may be, finding aftershocks in the stress trigger zones does not demonstrate that the stress imparted by the main shock had any effect on off-fault seismicity, since seismicity may have been as abundant in those zones before the mainshock. A stronger test of the Coulomb hypothesis is to compare the calculated stress change to the observed seismicity rate change. After a main shock, sites of both increased and decreased seismicity rate are seen; for the M_w=6.7 1994 Northridge, California, shock, 65% of the observed seismicity rate changes are correlated with the calculated Coulomb stress change (Figure 2).

An earthquake can thus enhance or suppress subsequent events, depending on their location and orientation. Viewed in this light, aftershocks are simply sites of seismicity rate increase, occurring where the stress has increased - whether on the fault rupture or off. Sites of seismicity rate decrease, or where the rate was higher before the earthquake than after, might logically be called 'antishocks' (in the sense of 'antipasto' - they precede rather than follow the main course). A spatial regression of stress change on seismicity rate change for the 1995 M_w=6.9 Kobe, Japan, earthquake (Figure 3a) reveals just how strong this effect is: a 1-bar stress increase corresponds to a 10-fold increase in rate of Ml32.6 shocks; a 5-bar stress change is associated with a 100-fold rate increase. But are such findings valid if small aftershocks, whose nodal planes are unknown, do not occur on faults optimally oriented for failure? One alternative is to consider only seismicity on (say, within 1 km of) major active faults and assume that these shocks share the known dip and rake of the faults on which they occur. Another approach is to calculate the Coulomb stress change on the nodal planes of the subset of shocks with focal mechanisms. Both approaches yield new insights.

Several researchers have examined stress changes and seismicity on major faults within 100 km of the 1989 M_w=6.9 Loma Prieta, California, shock. Parsons et al found that the seismicity rate change is associated with the calculated shear stress change for major faults (slip rates more than ~7 mm/yr and cumulative slip of more than ~50 km) (Figure 4a). For minor faults (with negligible cumulative slip and lower slip rates - typically thrust or oblique faults), seismicity is concentrated where the faults were unclamped (Figure 4b); both correlations are statistically significant. Restated in terms of eqn (1), the effective friction coefficient on the major faults is low (20.2), while for minor faults it is high (30.8). This inference accords with independent arguments that major faults such as the San Andreas develop thick, impermeable gouge zones that reduce sliding friction or trap pore fluids, both of which lower . The 1997 Umbria-Marche, Italy, sequence of eight 5>M6 shocks, on normal faults with low slip rates and modest net slip, also appears to have been promoted by unclamping. Seeber and Armbruster found that the ratio of encouraged to discouraged aftershocks of the Landers earthquake is greatest if =0.85, and minor faults surround Landers. Thus the high friction inferred for minor faults seems to be borne out by several Coulomb studies.

Using focal mechanisms to evaluate the Coulomb hypothesis affords additional insights but presents new problems, because the Coulomb stress change is different on the two nodal planes of the focal mechanism, except if =0. Hardebeck et al calculate the Coulomb stress change on both planes, and examined whether the percentage of encouraged planes increases after Landers and Northridge earthquakes (Figure 5a), the period before the earthquakes serving as a control group. The 20-25% increase in the percentage of encouraged shocks (Figure 5c and 5d) is statistically significant for stress increases 30.1 bar, and persists for the 5 years examined after both mainshocks. Seeber and Armbruster interpreted the fault plane for each of the 1900 aftershock focal mechanisms of Landers, and reached similar conclusions, although they find a larger initial increase in encouraged aftershocks that decays with time (Figure 5b). A test for aftershocks of the 1987 M_w=6.6 Superstition Hills, California, sequence, also shows significant correlations for stress increases 30.1 bar during 1.4-2.8 yr after the main shock.

Dynamic and tidal Coulomb stress change

The seismic waves excited by earthquakes produce dynamic Coulomb stress changes that, at distances more than about one source dimension from the fault, can be an order of magnitude larger than the static stress changes. How can one distinguish whether the static or dynamic stresses control seismicity? In other words, is it strong shaking or the weak permanent stress changes that promote seismicity? Because the dynamic stresses oscillate, they are everywhere positive at some point in time. All sites are shaken, and thus the dynamic stresses cast no stress shadows and should produce no antishocks (seismicity rate decreases), at odds with observations. Belardinelli et al calculated the dynamic stress evolution in the 1980 M_w=6.9 Irpinia, Italy, sequence in which nearby faults ruptured 20 s apart. The second event was not triggered at the time of the dynamic peak. Rather, a delayed triggering mechanism must be involved irrespective if static or dynamic stresses are responsible, because the second rupture nucleated 12 s after the dynamic peak and 6 s after the static value had been reached. Other evidence, however, suggests that at larger distances from the rupture, dynamic stresses may explain the distribution of seismicity rate changes better than the static stresses. Kilb et al find that the pattern of dynamic Coulomb stress changes bears similarities to that for static stress changes. While the peak dynamic stress field lacks shadows, it does exhibit lobes with small stress change in roughly the same positions as the static stress shadows. The key difference is that the dynamic stress increases are an order of magnitude larger in the direction of rupture propagation. The Landers rupture propagated unilaterally to the northwest, and produced more aftershocks in this direction. The observed seismicity rate may thus be influenced both by static and dynamic effects.

If static stress changes influence earthquake occurrence, then seismicity rates might be modulated by the solid earth tides. Unlike earthquakes, the tides produce no strong motion (shaking), but they do alter the stress on faults. The tidal Coulomb stress range, dominated by the normal-stress component, is only about ±0.01 bar, or 1/10 the threshold of detection in the most sensitive aftershock studies. Vidale et al calculated the tidal stresses on the fault planes of 13,000 earthquakes along the creeping portions of the San Andreas and Calaveras faults, finding that the seismicity rate is higher at times when the tidal stresses unclamped the fault, but not significantly so. Lockner and Beeler cycled stress in a laboratory sample to simulate the tides, and found that stress changes 30.1 bar caused strong correlations in the timing of stick-slip events, in accord with aftershock studies. They estimated that if detection increased with the square root of the sample size, >20,000 earthquakes would be needed to find a statistically significant association with tidal stresses, in which case ~1.5% of the seismicity would be correlated. Vidale et al repeated their experiment with 27,500 quakes, and found that the rate of seismicity during the peak tidal unclamping is 1.0% higher than average, a difference significant at the 95% confidence level. Thus the tides perceptibly alter the rate of seismicity, suggesting that the much larger off-fault stress changes associated with earthquakes are indeed one cause of seismicity rate changes.

Stress transfer in probabilistic hazard

The simplest way to incorporate stress transfer into probability models is to assume that a sudden stress change will alter the time until the next large earthquake by the ratio of the stress change on the fault to its long-term stressing rate. This is the ?time advance or delay? used in some consensus probability forecasts. Because stress changes on nearby faults are typically of the order 1 bar, and stressing rates are of the order 0.1 bar/yr, inter-event times are only changed by decades. Such a time change is inevitably much smaller than the uncertainty or variability of the earthquake inter-event time (typically assigned to be ±50%), and thus has little impact on the probability. But why, then, would earthquake stress changes exert such a strong influence on seismicity rates? The 1906 stress decrease on the faults in the San Francisco bay area, for example, is a few bars, but the rate of M6 shocks dropped by at least an order of magnitude during the ensuing 75 years (Figure 1), in a manner consistent with the Kobe results (Figure 3a).

A way out of this problem is suggested by the concept of state and rate friction. The observed dependence of seismicity rate on Coulomb stress change (Figure 3a) is well described by Dieterich?s earthquake-rate relation . In state and rate friction, seismicity is viewed as a sequence of independent nucleation events in which the ?state? depends on the fault slip, slip rate, and elapsed time since the last event. In the absence of a stress perturbation, the seismicity rate is constant. But under the assumption that there is a large number of earthquake nucleation sites on the fault, there is a nonlinear dependence of the time to instability on stress change. The ?time advance? causes a modest but permanent increase in earthquake rate and probability. The transient effect of the stress change strongly amplifies the permanent change, because the fault slips at a higher rate, causing a higher rate of earthquake nucleation (Figure 3b). The transient effect decays as the supply of nucleation sites is consumed; the duration of the transient is inversely proportional to the fault stressing rate. The seismicity rate equation in simplest form is

in which R is the seismicity rate as a function of time, t, following a Coulomb stress change, . A is a constitutive parameter, is the total normal stress, is the aftershock duration (equal to , where  is the stressing rate on the fault), and r is the seismicity rate before the stress perturbation. To evaluate (2), the Coulomb stress change is calculated and r, t_a and  are estimated from observations, permitting to be inferred (Figure 3a).

The rate equation 2 has the form of Omori's law, which describes the observed temporal decay of aftershocks on the mainshock rupture surface. Thus the decay of seismicity following a stress change may instead be a general property of earthquakes, restricted neither to aftershocks nor to the rupture surface. Such an interpretation is borne out by observations: Even at distances more than 40 km from the Loma Prieta fault - well outside the traditional aftershock zone - the response to a stress change of either sign is a sudden seismicity rate change followed by a recovery roughly to the former rate (Figure 6). The close resemblance between the observed behavior of seismicity following such a stress increase (Figure 6b) and that modeled with equation 2 (Figure 3b) is evident.

Although for small shocks seismicity rate changes and probabilities can be tested against observations, the probabilities for large earthquakes are more difficult to validate because there are so few of them. Probability calculations are also fraught with uncertainties associated with the prospective earthquake location and magnitude, the variation of the earthquake inter-event time, and the probability density function (how the probability grows with time). Monte Carlo simulation can capture the range of behavior, but this range can be frustratingly large. The probability changes associated with earthquake transfer, however, are less sensitive to these assumptions, enabling inferences about how the probability on one fault is affected by a nearby large earthquake on another.

Several attempts to estimate earthquake probabilities using stress transfer and state and rate friction have been made. The seven M_w36.8 'falling-domino' shocks on the North Anatolia fault during 1939-1967 are obvious candidates, because it is difficult to explain such a rapid and progressive sequence unless transient probability gains associated with stress transfer are included. In 1997, my colleagues and I calculated that there was an average 3-fold probability gain at the site of each successive event caused by the preceding earthquakes. We also identified two segments -two dominoes still standing- where the stress rise since 1939 was greatest. We calculated a 15% 30-year probability for a large earthquake near Erzincan, a 12% probability near Izmit, and < 1% probability on the remaining 750 km of the fault. The 1999 M_w=7.4 Izmit shock occurred in one of these stressed sites.

Japan and California have proven to be key sites for testing such interaction-based probabilities. Toda et al calculated a 10-fold probability drop during the next 30 years on the section of the major fault most discouraged from failure by the 1995 M_w=6.9 Kobe shock, and a 5-fold probability gain on the section most stressed, which lies near Kyoto. Three studies have focused on the San Francisco Bay area where, as in the other cases, the consequences of an urban M_w=7 shock loom large, but the interaction of several sub-parallel faults is simpler than in Japan. Building on the stress analysis of Jaumé and Sykes, Harris and Simpson retrospectively evaluated the suppression of M_w6 shocks in the Bay area after 1906, finding the set of stress and state/rate constitutive parameters consistent with the observed rate change. Prospective studies of the probability of a M_w=6.8 shock on the Hayward fault during 2000-2030, such as occurred in 1868, find that the probability is 15-25% lower if the effect of the 1906 shock is included.

These nascent probability calculations are faithful to the presence of aftershocks, the characteristics of large earthquake sequences, the change in location and style of upper-plate earthquakes following subduction events , and the cessation of large shocks in the Bay area after 1906. Perhaps most surprising, one finds that the stress trigger or shadow cast by a great earthquake can exert an influence earthquake occurrence for more than a century. What lies ahead is further incorporation of viscoelastic and poroelastic effects that modify the Coulomb stress with time, real-time tests of earthquake rate and probability calculations, and further exploration of seismicity rate changes and the dynamic Coulomb stresses.
 
 

The author is at the U.S Geological Survey, Menlo Park, CA

Acknowledgements. I thank the many generous colleagues who shared their preliminary research; and R. Dmowska, J. Lin, R. Madariaga, T. Parsons, F. Pollitz, J. Rice, and S. Toda for thoughtful reviews. The article was written while at Ecole Normale Supérieure and the Institut de Physique du Globe de Paris. The support of Pacific Gas & Electric Co. is gratefully acknowledged.

References

1. Bakun, W. H. Seismic activity of the San Francisco Bay region. Bull. Seism. Soc. Amer. 89, 764-784 (1999).
2. Lay, T., Astiz, L., Kanamori, H. & Christensen, D. H. Temporal variation of large interplate earthquakes in coupled subduction zones. Phys. Earth Planet. Inter. 54, 258-312 (1989).
3. Chinnery, M. A. The stress changes that accompany strike-slip faulting. Bull. Seismol. Soc. Amer. 53, 921-932 (1963).
4. Das, S. & Scholz, C. H. Off-fault aftershock clusters caused by shear stress increase? Bull. Seismol. Soc. Amer. 71, 1669-1675 (1981).
5. Stein, R. S. & Lisowski, M. The 1979 Homestead Valley earthquake sequence, California: Control of aftershocks and postseismic deformation. J. Geophys. Res. 88, 6477-6490 (1983).
6. Oppenheimer, D. H., Reasenberg, P. A. & Simpson, R. W. Fault plane solutions for the 1984 Morgan Hill, California, earthquake sequence: Evidence for the state of stress on the Calaveras fault. J. Geophys. Res. 93, 9007-9026 (1988).
7. King, G. C. P., Stein, R. S. & Lin, J. Static stress changes and the triggering of earthquakes. Bull. Seismol. Soc. Amer. 84, 935-953 (1994).
8. Stein, R. S., King, G. C. P. & Lin, J. Change in failure stress on the southern San Andreas fault system caused by the 1992 Magnitude=7.4 Landers earthquake. Science 258, 1328-1332 (1992).
9. Harris, R. A. & Simpson, R. W. Changes in static stress on southern California faults after the 1992 Landers earthquake. Nature 360, 251-254 (1992).
10. Jaumé, S. C. & Sykes, L. R. Change in the state of stress on the southern San Andreas fault resulting from the California earthquake sequence of April to June 1992. Science 258, 1325-1328 (1992).
11. Gross, S. & Kisslinger, C. Estimating tectonic stress rate and state with Landers aftershocks. J. Geophys. Res.102, 7603-7612 (1997).
12. Harris, R. A. Introduction to special session: Stress triggers, stress shadows, and implications for seismic hazard. J. Geophys. Res.103, 24,347-24,358 (1998).
13. Reasenberg, P. A. & Simpson, R. W. Response of regional seismicity to the static stress change produced by the Loma Prieta earthquake. Science 255, 1687-1690 (1992).
14. Simpson, R. W. & Reasenberg, P. A. in The Loma Prieta, California, earthquake of October 17, 1989-Tectonic processes and models: (ed. Simpson, R. W.) F55-F89 (U.S. Geological Survey Professional Paper 1550-F, 1994).
15. Parsons, T., Stein, R. S., Simpson, R. W. & Reasenberg, P. A. Stress sensitivity of fault seismicity: A comparison between limited-offset oblique and major strike-slip faults. J. Geophys. Res. 104, 20,183-20,202 (1999).
16. Zoback, M. D. et al. New evidence on the state of stress of the San Andreas fault system. Science 238, 1105-1111 (1987).
17. Rice, J. R. in Fault Mechanics and Transport Properties of Rock (eds. Evans, B. & Wong, T.-f.) 475-504 (Academic Press, London, 1992).
18. Cocco, M., Nostro, C. & Ekström, G. Static stress changes and fault interaction during the 1997 Umbria-Marche earthquake sequence. J. Geophys. Res., in press (1999).
19. Seeber, L. & Armbruster, J. G. Earthquakes as beacons of stress change. Nature submitted (1999).
20. Hardebeck, J. L., Nazareth, J. J. & Hauksson, E. The static stress change triggering model: Constraints from two southern California aftershocks sequences. J. Geophys. Res. 103, 24,427-24,437 (1998).
21. Anderson, G. & Johnson, H. A new statistical test for static stress triggering: Application to the 1987 Superstition Hills earthquake sequence. J. Geophys. Res. 104, 20,153-20,168 (1999).
22. Belardinelli, M. E., Cocco, M., Coutant, O. & Cotton, F. Redistribution of dynamic stress during coseismic ruptures: Evidence for fault interaction and earthquake triggering. J. Geophys. Res. 104, 14,925-14,946 (1999).
23. Kilb, D., Gomberg, J. & Bodin, P. Aftershock triggering by dynamic and static Coulomb stress changes. Eos, Trans. AGU, supplement 79, F647 (1998).
24. Hill, D. P. et al. Seismicity remotely triggered by the Magnitude 7.3 Landers, California, earthquake. Science260, 1617-1623 (1993).
25. Vidale, J. E., Agnew, D. C., Johnston, M. S. J. & Oppenheimer, D. H. Absence of earthquake correlation with earth tides: An indication of high preseismic fault stress rate. J. Geophys. Res.103 (1998).
26. Lockner, D. A. & Beeler, N. M. Premonitory slip and tidal triggering of earthquakes. J. Geophys. Res. 104, 20,133-20,152 (1999).
27. Vidale, J., Agnew, D., Oppenheimer, D., Rodriquez, C. & Houston, H. A weak correlation between earthquakes and extensional normal stress and stress rate from lunar tides. Eos, Trans. AGU, supplement 79, F641 (1998).
28. Working Group on the Probabilities of future large earthquakes in southern California. Future Seismic Hazards in Southern California, Phase I: Implications of the 1992 Landers Earthquake Sequence, Calif. Div. Mines & Geol., 42 p. (1992).
29. Working Group on California Earthquake Probabilities. Seismic hazards in southern California: Probable earthquakes, 1994-2014. Bull. Seismol. Soc. Amer. 85, 379-439 (1995).
30. Dieterich, J. A constitutive law for rate of earthquake production and its application to earthquake clustering. J. Geophys. Res. 99, 2601-2618 (1994).
31. Dieterich, J. H. & Kilgore, B. Implications of fault constitutive properties for earthquake prediction. Proc. Nat. Acad. of Sci. USA 93, 3787-3794 (1996).
32. Parsons, T. & Stein, R. Earthquake probabilities in the San Francisco Bay area after 1906: A rate/state approach to fault interactions. J. Geophys. Res., in prep. (2000).
33. Stein, R. S., Barka, A. A. & Dieterich, J. H. Progressive failure on the North Anatolian fault since 1939 by earthquake stress triggering. Geophys. J. Int. 128, 594-604 (1997).
34. Toda, S., Stein, R. S., Reasenberg, P. A. & Dieterich, J. H. Stress transferred by the Mw=6.5 Kobe, Japan, shock: Effect on aftershocks and future earthquake probabilities. J. Geophys. Res. 103, 24,543-24,565 (1998).
35. Jaumé, S. C. & Sykes, L. R. Evolution of moderate seismicity in the San Francisco Bay region, 1850 to 1993: Seismicity changes related to the occurrence of large and great earthquakes. J. Geophys. Res.101, 765-789 (1996).
36. Harris, R. A. & Simpson, R. W. Suppression of large earthquakes by stress shadows: A comparison of Coulomb and rate-and-state. J. Geophys. Res. 103, 24,439-24,451 (1998).
37. Reasenberg, P. A., Simpson, R. W., Matthews, M. V., Ellsworth, W. L. & Parsons, T. Recurrence models and fault interactions: an example based on the 1868 Hayward, California, earthquake. Bull. Seismol. Soc. Amer. submitted (1999).
38. Taylor, M. A., Dmowska, R. & Rice, J. R. Upper plate stressing and seismicity in the subduction earthquake cycle. J. Geophys. Res. 103, 24,523-24,542 (1998).
39. Pollitz, F. F. & Sacks, I. S. The 1995 Kobe, Japan, earthquake: A long-delayed aftershock of the offshore 1944 Tonankai and 1946 Nankaido earthquakes. Bull. Seismol. Soc. Amer. 87, 1-10 (1997).
40. Freed, A. M. & Lin, J. Time-dependent changes in failure stress following thrust earthquakes. J. Geophys. Res. 103, 24,393-24,410 (1998).
41. Kenner, S. & Segall, P. Time-dependence of the stress shadowing effect and its relation to the structure of the lower crust. Geology 27, 119-122 (1999).
42. Bakun, W. H. Scenarios for historic San Francisco Bay region earthquakes. U.S. Geol. Surv. Open File Rep. 98-785 (1998).
43. Stein, R. S., King, G. C. P. & Lin, J. Stress triggering of the 1994 M=6.7 Northridge, California, earthquake by its predecessors. Science 265, 1432-1435 (1994).
44. Reasenberg, P. A. Seismicity patterns in southern California before and after the 1994 Northridge earthquake: A preliminary report. U.S. Geol. Surv. Open-File Rep. 95-484 (1995).
45. Hardebeck, J. L. & Hauksson, E. Background Stress State Plays a Role in Earthquake Triggering. Eos, Trans. AGU, supplement, in press (1999).

Figure 2. Correlation between calculated Coulomb stress change and seismicity rate change for the 1994 M_w=6.7 Northridge earthquake. a. The largest Coulomb stress change on optimally-oriented thrust or strike-slip faults at depths of 3-10 km; the compressive axis of the regional stress is oriented N4°E, and m = 0.4. Active surface faults, and M_L31.5 shocks during 3-6 months after the mainshock, are superimposed in black. b. The seismicity rate change (new/old), comparing the rate during the 78 months before the Northridge earthquake to that 3-6 months afterward (the first 3 months after Northridge have a poorer level of completeness and are excluded). The rate is calculated in 10 km cells on a grid with 1 km spacing and then smoothed with a Gaussian filter. The rate change in the white areas is unresolved. Some 65% of the resolved area is positively correlated. Such correlations extend 40 km (or 4 fault lengths) southeast of the mainshock to Los Angeles (L.A.). Observed seismicity rate decreases in the Santa Monica Bay (S.M. Bay) and along parts of the San Andreas fault are correlated with the calculated stress decrease.

Figure 3. Observed seismicity rate change as a function of calculated Coulomb stress change for the 1995 Mw=6.9 Kobe earthquake. a. Seismicity rate change for M_L 32.6 earthquakes (the completeness magnitude) during 8 years before the Kobe shock compared to the following 1.5 years; the seismicity rate climbed after Kobe where R/r>1 and fell where R/r<1. The maximum Coulomb stress change on optimally oriented strike-slip and thrust faults is calculated at depths of 0-20 km, for m=0.4. b. Schematic illustration of the time-dependence of seismicity rate (graphically similar to conditional earthquake probability), following a sudden stress increase; the transient effect of the stress increase decays to the permanent rate change over the aftershock duration, t_a.

Figure 4. Seismicity and stress changes associated with the 1989 M_w=6.9 Loma Prieta earthquake resolved on nearby faults (faults are shown in Figure 6a). M_L 31.5 seismicity during 7 years before and after Loma Prieta are plotted with size proportional to magnitude. Distance increases toward the southeast. a. The fault is assumed to dip 70°NE with a 160° rake, consistent with focal mechanisms and marine terrace deformation; the thin line is the coast. Earthquakes within 2 km of the fault are shown. The change in the seismicity distribution is associated with the calculated shear stress change, suggesting m is low (no correlation is seen for normal stress change). b. The fault dips 55±5° SW and has a 135° assumed rake; the thin line is the surface projection of the San Andreas fault. Earthquakes within 1 km of the fault are shown. The post-1989 seismicity is concentrated where the fault was unclamped, suggesting a high m (no correlation is seen for shear stress change).
 
 Figure 5. Stress changes associated with the 1992 Mw=7.2 Landers and 1994 Mw=6.7 Northridge ruptures resolved on nodal planes of earthquakes with focal mechanisms. The percentage of encouraged planes is ~50% by chance before the main ruptures, but increases significantly afterward, indicating that aftershocks tend to occur where the main rupture has brought faults brought closer to Coulomb failure. a, Focal mechanisms of the main shock and largest aftershock in both sequences; LA is Los Angeles. b, M_l 1.5 shocks 327 km from the Landers rupture are used, at sites where the calculated stress change is 30.2 bar, assuming = 0.85 . c, M_l 2.5 shocks within a box defined by 32.5-36.5°N/115.5-119.5°W are included, assuming =0.4 ; 95% confidence bounds are shown. Events with both planes encouraged receive more weight than those with one encouraged plane. d, M_l2.5 shocks within a box defined by 33.8-34.9-36.5°N/117.8-119.5°W, assuming =0.4 ; 95% confidence bounds shown.

Figure 6. The influence of stress changes associated with the Loma Prieta earthquake on M_L31.5 seismicity rates well outside what is traditionally regarded as the aftershock zone. a, Map of Loma Prieta source (rectangle) and fault sections (bold) on which stress change is calculated and seismicity rates are measured; S.F. is San Francisco; S.J. is San Jose. b, Seismicity rate jumps on the southern San Gregorio fault, 40 km west of Loma Prieta. The 95% confidence limits for the pre-1989 rate and the post-1989 decay are shown by the thin gray lines. The decay obeys Omori?s law; the aftershock duration, t_a = 5-28 yr at 95% confidence. c, Seismicity rate drops on the southern Hayward fault, 40-80 km north of Loma Prieta, and then recovers during the following ~8 years. (Some 30 km southeast of the south Hayward fault, the 1984 M_w=6.2 Morgan Hill shock disrupts the seismicity during 1984-88, so this period is not used to estimate the pre-1989 seismicity rate. The highest values exceed 20 shocks/yr during this period.)