GEOPHYSICAL RESEARCH LETTERS, VOL.???, NO., PAGES 1 16, Does Aftershock Duration Scale With Mainshock Size? A. Ziv A. Ziv, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel. (e-mail: zival@bgu.ac.il)
2 ZIV: AFTERSHOCK DURATION Abstract. It has been claimed that aftershock duration scales with the mainshock recurrence interval [Dieterich, 1994]. This implies a scaling between aftershock duration and mainshock magnitude. To see if such a scaling exist, we compare properties of aftershock sequences associated with mainshocks whose size span several magnitude units. Earthquake properties were examined along two northern California fault segments, one is the Calaveras fault, another is a northern segment of the San Andreas fault. The first study area contains the rupture area of the M6.2, 1984 Morgan Hill, and the latter is adjacent the southern end of the M7.1, 1989 Loma Prieta rupture. Both the Morgan Hill and the Loma Prieta aftershock activities lasted 1 3 10 8 seconds. On the other hand, aftershock activities of small-to-moderate mainshocks lasted about 1 2 10 6 seconds. Nevertheless, there is no evidence for scaling between mainshock size and aftershock duration for moderate-to-small mainshocks. It thus seems that large earthquakes, because they break the entire seismogenic depth and interact with deeper parts of the crust, trigger a relaxation mechanism that is inactive during aftershock activity due to moderate-to-small quakes, and whose characteristic relaxation time is much longer than that of the brittle crust. Current seismicity models do not provide explanation for these observations.
ZIV: AFTERSHOCK DURATION 3 1. Introduction The effect of an earthquake is to perturb the stress field over adjacent areas. Depending on the local rheology, this stress perturbation may relax either seismically in the form of aftershocks or aseismically in the form of post-seismic creep. Being rheology-dependent, the rate and the duration of the relaxation processes that follow stress perturbations contain information that may help to unravel the mechanical behavior of the fault in question. Guided by this line of reasoning, we examine decay rate and duration of aftershock sequences along two Northern California faults, the Calaveras Fault and a segment of the Northern San Andreas Fault, and compare properties of aftershock sequences associated with mainshocks whose size span several magnitude units. We find that aftershock activities of moderate-to-small mainshocks (M < 6) last a few days, and show that the duration of such aftershock sequences is independent of the mainshock size. On the other hand, aftershock activity of large mainshocks (M > 6) last a few years. We suggest that large earthquakes, because they break the entire seismogenic depth and interact with deeper parts of the crust, trigger a relaxation mechanism that is inactive during aftershock activity due to moderate-to-small quakes, and whose relaxation time is several years long. Current seismicity models do not provide explanation for these observations. 2. Background and Motivation It is widely held that earthquakes are frictional instabilities [Brace and Byerlee, 1966]. Our understanding of friction is based mainly on the interpretation of laboratory experiments. Friction experiments show that the friction coefficient is a function of slip rate, δ,
4 ZIV: AFTERSHOCK DURATION and state, θ, as follows [Dieterich, 1979; Ruina, 1983]: µ = µ + A log δ δ + B log θ δ D c, (1) where A and B are unitless constitutive parameters, D c is a characteristic sliding distance from one steady state to another, δ is a reference velocity, and µ is the coefficient of friction when the contact surface slips at constant slip rate of δ. The state evolves with time, t, and slip according to [Ruina, 1980]: dθ dt = 1 θ δ. (2) D c From (2), a steady-state is reached when δ = D c /θ. Replacing δ with D c /θ in (1) yields a steady-state friction: µ ss = µ + (A B) log δ δ. (3) The constitutive parameters A and B are sensitive to the lithology [Marone and Scholz, 1988], the confining pressure [Shimamoto, 1986] and the temperature [Blanpied et al., 1998]. Depending on these factors the sign of B A may be either positive or negative. From (3), steady-state friction is velocity strengthening if A > B, and is velocity weakening if B > A. While the first favors stable sliding, the latter favors stick-slip. If slip on crustal faults is indeed governed by such a constitutive law, so are the properties (duration and decay rate) of aftershock sequences [Dieterich, 1972; Shaw, 1993; Dieterich, 1994; Schaff et al., 1998; Perfettini and Avouac, 2004]. Dieterich [1994] showed that a uniform stress step applied on a population of faults that are governed by a velocity weakening rate- and state-dependent friction results in an aftershock sequence that decays asymptotically according to 1/time, and returns to a background earthquake rate
according to a characteristic time-scale, t a, that equals: where σ is the normal stress, and τ is the stressing rate. ZIV: AFTERSHOCK DURATION 5 t a = Aσ τ, (4) Schaff et al. [1998] suggested an alternative mechanism to account for the 1/time decay of aftershocks. According to their model, a permanent stress step that is applied on a segment governed by a velocity strengthening rate- and state-dependent friction causes an instantaneous increase in the creep rate, which relaxes proportionally to 1/time. Consequently, the stressing rate, and therefore the failure rate, of stick-slip patches that are surrounded by or are in the vicinity of the creeping segments is also proportional to 1/time. Perfettini and Avouac [2004] showed that the characteristic relaxation time of aftershock activity that arises in such a model is also equal to t a in (4), but with Aσ that is representative of the creeping segment. In reality, since parts of the fault may be slipping in stick-slip episodes while others may be creeping, both aftershock mechanisms may operate simultaneously and contribute to the total number of aftershocks. A non-trivial question is how to interpret the stressing rate in (4). Dieterich [1994] advocated for τ that is a function of the stress drop, τ, and the recurrence time, t r, as follows: τ = τ t r. (5) An alternative interpretation for the stressing rate is [Ziv and Rubin, 2003]: τ = G W δ tect, (6) where G is the shear modulus, W is the characteristic length scale for tectonic loading (e.g., the distance to the rigid plate boundary or the transition depth from seismic to
6 ZIV: AFTERSHOCK DURATION aseismic slip), and δ tect is the tectonic slip rate. What are the implications of (5) versus (6)? According to (5), because earthquake recurrence interval increases proportionally to its size, the duration of aftershock activity should increase with increasing mainshock magnitude as well [Dieterich, 1994]. In contrast, (6) implies that aftershock duration is independent of mainshock size. Quasi-static aftershock simulations on a velocity-weakening rate- and state-dependent fault with uniform distribution of Aσ suggest that the appropriate interpretation for the stressing rate in (4) is that of (6) [Ziv and Rubin, 2003], and that aftershock duration is independent of mainshock size [Figure 7 of Ziv et al., 2003]. 3. Properties of Aftershocks Sequences on Northern California Faults Does aftershock duration scale with the mainshock size? In order to address this question, we compare properties of aftershock sequences associated with mainshocks whose size span several magnitude units. 3.1. A Recipe for Analyzing Aftershocks of Micro-earthquakes The traditional approach is to consider as mainshocks only earthquakes that are large and infrequent. Recent studies show that small-to-moderate earthquakes also enhance the seismicity in their vicinity, and that just like aftershock sequences of large mainshocks, the increase in seismicity rate associated with these shocks decay according to an Omori law [Shaw, 1993; Ziv et al., 2003; Helmstetter, 2003]. When analyzing spatio-temporal clustering with respect to small earthquakes, it is useful to construct a composite catalog of stacked aftershock sequences. The construction of the composite aftershock catalog follows these steps: (a) We consider a given earthquake as a potential mainshock if it is larger than an arbitrary magnitude threshold that we shall detail below. For each such
ZIV: AFTERSHOCK DURATION 7 mainshock we compute the rupture dimensions using the moment-magnitude relationship of Abercrombie [1996] and the scaling of slip with length on a circular crack of Eshelby [1957], using stress drops of 10 MPa; (b) We calculate lag-times and distances between each potential mainshock and all later earthquakes that occurred within the study area; and (c) We stack mainshock-aftershock pairs with an inter-event distance that is less than twice the mainshock radius. This final step requires that relative location between mainshock-aftershock pairs be smaller than the rupture dimensions of the mainshock. Thus, for the smallest mainshocks we use relocated catalogs. 3.2. The Data and the Study Areas Earthquake properties were examined in two locations (see rectangles in Figure 1). The first is along the Calaveras fault. That study area contains the rupture surface of the M6.2 1984 Morgan Hill earthquake. The other is located along a segment of the San Andreas fault that is adjacent to the southern end of the M7.1 1989 Loma Prieta rupture. These areas were chosen for two reasons; the first is that both contain aftershock sequences of M > 6, and the second is that relocated seismicity catalogs are available for both areas [Rubin and Gillard, 2000; Rubin, 2002]. We use data from the Northern California Seismic Network for analyzing aftershocks of moderate and large mainshocks. From the magnitude distribution (not shown) we determined that this catalog is complete for M 1. Clearly there is some temporary worsening of the detection threshold shortly after major quakes, but in the next section we shall explain that this effect does not change the result of this study. Finally, we use relocated catalogs for analyzing properties of sequences triggered by small quakes.
8 ZIV: AFTERSHOCK DURATION 3.3. The Calaveras Fault The diagram in Figure 2a shows aftershock rate as a function of time since the Morgan Hill earthquake for M 1 (black) and M 2 (gray). The straightness of the seismicity curve for M > 1 between 10 3 and 10 8 seconds indicates that if there is a temporary worsening of the detectability threshold, it may only be for intervals that are less than 10 3 seconds following the mainshocks (but this does not affects the conclusions of this study). Aftershocks duration may be determined in two ways; one is from the intersection between the seismicity curves and the long-term average curves (dashed lines), and the other is from the flattening of the seismicity curves. From these we infer that aftershock activity returned to the background earthquake rate after 2 10 8 seconds, corresponding to about 6 years (indicated by the circles). Additionally, the sequence decayed according to a modified Omori law [Utsu, 1961] with an Omori exponent that is smaller than 1 (but that is larger than 0.5). Differences in the M 1 and M 2 slopes suggest that large aftershocks decay faster than small ones. Similar dependence of decay rate on aftershock size has been observed in other data sets [Wiemer and Katsumata, 1999; Hosono and Yoshida, 2002; Ziv et al., 2003], and was observed in quasi-static simulations of seismicity on rate-and-state dependent faults [Ziv and Rubin, 2003]. Visual inspection of relocated micro-seismicity in the study area by other investigators revealed that most of the seismicity in that area is organized in sub-horizontal streaks and in isolated patches that fail repeatedly [Rubin, 2002; Schaff et al., 1998; Schaff et al., 2002]. Peng et al. [2005] found that the lag-times of repeating quakes after the Morgan Hill earthquake decayed according to Omori law with an Omori exponent that varies in space, but that
ZIV: AFTERSHOCK DURATION 9 is generally less than 0.7. Thus, the rate of aftershock decay in Figure 2a is the result of spatial averaging of many sequences each of which decayed at a different rate. We now show that the duration of aftershock sequences triggered by smaller quakes along the very same fault segment is by nearly two orders of magnitude shorter than that of the Morgan Hill aftershock sequence. The diagram in Figure 2b shows aftershock rate as a function of time for composite mainshocks with 3.5 M 4.5. In order to reduce the possible effect of the Morgan Hill earthquake, earthquakes that occurred prior to 1996 were excluded from this analysis. Note that the decay rate of aftershock sequences triggered by these mainshocks follows a power-law with a power exponent that is very close to 1. The duration of the sequence is only 10 6 seconds, corresponding to about 10 days. The diagram in Figure 2c shows aftershock rate as a function of time for composite mainshocks with M 3.2. Here too pre-1996 were excluded from the analysis. Since here relative location should be very accurate, we used relocated catalog [Rubin, 2002]. We find that the duration of aftershock activity associated with 3.2 is the same as for aftershock sequences triggered by 3.5 M 4.5. 3.4. The Northern San-Andreas Fault Here we perform the same analysis for earthquakes that occurred within the San- Andreas study area. The diagram in Figure 3a shows aftershock rate as a function of time since the Loma Prieta earthquake for M 1 (black) and M 2 (gray). The jump in earthquake rate near 10 7 seconds is due to a M5.8 followed shortly by a M5.4 that ruptured on April 18, 1990, near the northern end of the study area. Note that the sequence returned to the long-term average rate after 10 8 seconds.
10 ZIV: AFTERSHOCK DURATION Earthquake rate as a function of time since the M5.8 of April 18, 1990, is shown in Figure 3b for earthquakes that occurred within two mainshock rupture radii from that earthquake. This sequence returned to a near constant rate after 1 2 10 6 seconds. Finally, we used a relocated catalog [Rubin and Gillard, 2000] to construct a composite catalog. Earthquakes that occurred during the first two years after the Loma Prieta earthquake were excluded from that catalog. Aftershock rates as a function of time for composite mainshocks whose magnitude satisfy 2.5 < M 3.7 (gray curve) and M 2.5 (black curve) are shown in Figure 3c. Note that here the rate curve falls below the background rate near 10 6 seconds and returns to it at later times. Similar intervals of quiescence following aftershock activity were observed on other fault segments in Northern California [Ziv et al., 2003], and were reproduced by the quasi-static rate- and state-dependent aftershock simulations of Ziv and Rubin [2003]. Similar to the activity triggered by the M = 5.8, these sequences too lasted about 10 6 seconds. 4. Conclusions Aftershock rates in the study areas decay according to the modified Omori function. Aftershock duration data is summarized in Figure 4. Note that while both the Morgan Hill and the Loma Prieta aftershock activities lasted 1 3 10 8 seconds, aftershock activities of M 5.8 lasted about 1 2 10 6 seconds. There is no evidence for scaling between mainshock size and aftershock duration for moderate-to-small mainshocks. That the Loma Prieta is larger than the Morgan Hill by more than a magnitude unit, and yet the duration of aftershock sequences that were triggered by the two mainshocks were about the same, also suggests a luck of scaling between aftershock duration and mainshock
ZIV: AFTERSHOCK DURATION 11 size of large earthquakes. Nevertheless, since the data set contains only two M > 6, the result concerning the large mainshocks is inconclusive. Can these observations be explained in the context of rate- and state-dependent friction? Simple rate-and-state aftershock models [Dieterich, 1994; Schaff et al., 1998; Perfettini and Avouac, 2004] reproduce the decay of aftershock rate according to the modified Omori law, but are unable to reproduce the aftershock duration data of Figure 4. It may be possible that more complex rate-and-state models, which incorporate both velocityweakening and velocity-strengthening segments coupled through an elastic medium may explain the data. We have carried out a preliminary exploration of this possibility using the quasi-dynamic rate-and-state seismicity simulator of Ziv and Cochard [in press], employing various distributions of constitutive parameters. These tests failed to achieve the desired result, namely that aftershock duration of sequences triggered by large mainshocks is by two orders of magnitude longer than the duration of sequences triggered by small-to-moderate mainshocks. We suggest that large earthquakes, because they interact with deeper portions of the lithosphere, trigger a relaxation mechanism that is inactive during aftershock activity due to moderate-to-small quakes. This relaxation mechanism may either occur in the form of creep along the downward aseismic extension of the fault, or it may be due to interaction with visco-elastic layer underlying the seismogenic depth, or it may be a combination of the two. Regardless of the exact nature of that mechanism, the data presented here suggest that the characteristic relaxation time of this process is a few years long. Because aftershock duration of small-to-moderate earthquakes does not scale with mainshock magnitude, we conclude that the correct interpretation for the stressing rate in (4)
12 ZIV: AFTERSHOCK DURATION is that of (6). That the durations of aftershock sequences occurring on the Calaveras and San Andreas fault segments are similar further implies that the tectonic stressing rate on the two fault segments is of similar magnitude. Acknowledgments. I thank Allan Rubin for discussions. Comments from the associate editor Aldo Zollo, and two anonymous reviewers helped to improve the manuscript.
ZIV: AFTERSHOCK DURATION 13 References Abercrombie, R. E., The magnitude-frequency distribution of earthquakes recorded with deep seismometers at Cajon Pass, southern California, Tectonophysics, 261, 1 7, 1996. Blanpied, M. L., C. J. Marone, D. A. Lockner, J. D. Byerlee, and D. P. King, Quantitative measure of the variation in fault rheology due to fluid-rock interactions, J. Geophys. Res., 103, 9691 9712, 1998. Brace, W. F., and J. D. Byerlee, Stick slip as a mechanism for earthquakes, Science, 153, 990 992, 1966. Dieterich, J. H., Time-dependent friction as a possible mechanism for aftershocks, J. Geophys. Res., 77, 3771 3781, 1972. Dieterich, J. H., Modeling of rock friction, 1. Experimental results and constitutive equations, J. Geophys. Res., 84, 2161 2168, 1979. Dieterich, J., A constitutive law for rate of earthquake production and its application to earthquake clustering, J. Geophys. Res., 99, 2601 2618, 1994. Eshelby, J., The determination of elastic field of an ellipsoidal inclusion and related problems, Proc. Roy. Soc. London, A241, 379-396, 1957. Helmstetter A., Is earthquake triggering driven by small earthquakes? Phys. Rev. Lett., 91, 058501, 2003. Hosono, K., and Yoshida, A., Do large aftershocks decrease similarly to smaller ones? Geophys. Res. Lett., 29, 1482, 2002. Marone, C., and C. Scholz, The depth of seismic faulting and the upper transition from stable to unstable slip regimes, Geophys. Res. Lett., 15, 621 624, 1988.
14 ZIV: AFTERSHOCK DURATION Peng, Z., J.E. Vidale, C. Marone and A. Rubin, Systemic variations in recurrence interval and moment of repeating aftershocks, Geophys. Res. Lett., 32, L15301, doi:10.1029/2005gl022626, 2005. Perfettini H, Avouac JP, Postseismic relaxation driven by brittle creep: A possible mechanism to reconcile geodetic measurements and the decay rate of aftershocks, application to the Chi-Chi earthquake, Taiwan, J. Geophys. Res., 109, B02304, doi:10.1029/2003jb002488, 2004. Rubin, A. M., Aftershocks of microearthquakes as probes of the mechanics of rupture, J. Geophys. Res., 107, doi:10.1029/2001jb000496, 2002. Rubin A. M., and D. Gillard, Aftershock asymmetry/rupture directivity among central San Andreas fault microearthquakes, J. Geophys. Res., 105, B8, 19095 19109, 2000. Ruina, A. L., Friction laws and instabilities: A quasi-static analysis of some dry frictional behavior, Ph.D. thesis, Brown Univ., Providence, R. I., 1980. Ruina, A., Slip instability and state variable friction laws, J. Geopys. Res., 88, 10359 10370, 1983. Schaff, D. P., G. C. Beroza, and B. E. Shaw, Postseismic response of repeating aftershocks, Geophys. Res. Lett., 25, 4549 4552, 1998. Schaff, D. P., G. H. R. Bokelmann, G. Beroza, F. Waldhauser, and W. L. Ellsworth, High-resolution images of the Calaveras fault seismicity, J. Geopys. Res., 107, 2186, doi:10.1029/2001jb000633, 2002. Shaw, B. E., Generalized Omori law for foreshocks and aftershocks from a simple dynamics, Geophys. Res. Lett., 20, 907 910, 1993.
ZIV: AFTERSHOCK DURATION 15 Shimamoto, T., A transition between frictional slip and ductile flow undergoing large shearing deformation at room temperature, Science, 231, 711 714, 1986. Utsu, T., A statistical study on the occurrence of aftershocks, Geophys. Mag, 30, 521 605, 1961. Wiemer, S., and K. Katsumata, Spatial variability of seismicity parameters in aftershock zones, J. Geophys., Res., 104, 13,135 13,151, 1999. Ziv, A., and A. M. Rubin, Implications of rate-and-state friction for properties of aftershock sequence: quasi-static inherently discrete simulations, J. Geopys. Res., 108, 2051, doi:10.1029/2001jb001219, 2003. Ziv, A., A. M. Rubin, and D. Kilb, Spatio-temporal analyses of earthquake productivity and size distribution: Observations and simulations, Bull. Seismol. Soc. Am., 93(5), 2069 2081, 2003. Ziv, A., and A. Cochard, Quasi-dynamic modeling of seismicity on a fault with depthvariable rate- and state-dependent friction, in press in J. Geopys. Res. (pre-prints may be downloaded from www.bgu.ac.il/geol/ziv/ziv-cochard.pdf).
16 ZIV: AFTERSHOCK DURATION 38 00' HF CALIFORNIA 37 30' SAF 1984 Morgan Hill 37 00' 1989 Loma Prieta 25 km CF 237 30' 238 00' 238 30' 239 00' Figure 1. Map showing the study areas and the main faults. The Calaveras fault study area contains the 1994 Morgan Hill rupture plane. The San-Andreas study area is located south of the southern end of the 1989 Loma Prieta rupture. The stars indicate the location of the Morgan Hill and the Loma Prieta epicenters. The inset shows a map of California. Abbreviations are CF, Calaveras Fault; HF, Hayward Fault; and SAF, San Andreas Fault.
ZIV: AFTERSHOCK DURATION 17 10-2 rate [quakes/s] 10-3 10-4 10-5 slope=-0.5 slope=-1 10-6 10-7 10-2 (a) NCEDC, M=6.2 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 time since Morgan Hill [s] rate [quakes/s] 10-3 10-4 10-5 10-6 slope=-1 (b) NCEDC composite, 3.5<=M<=4.5 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 lag-time [s] rate [quakes/s] 10-1 10-2 10-3 10-4 slope=-1 10-5 (c) relocated composite, M<=3.2 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 lag-time [s] Figure 2. Aftershock rate as a function of time for earthquakes along the Calaveras fault. (a) Time is measured with respect to the Morgan Hill earthquake. The black and gray solid curves are for M 1 and M 2, respectively. The black and gray dashed curves indicate average earthquake rate between Jan. 1980 and Jan. 1984 for earthquakes whose magnitude is greater than or equal to 1 and 2, respectively. (b) Aftershock rate is calculated for composite catalogs with mainshock magnitude between 4.5 and 3.5. (c) Aftershock rate is calculated for relocated composite catalogs with mainshock magnitude that are less than 3.2. The circles in each panels indicate the return of seismicity rate to the background rate.
18 ZIV: AFTERSHOCK DURATION rate [quakes/s] 10-2 10-3 10-4 10-5 18 April 1990 M=5.8+5.4 rate [quakes/s] 10-6 10-2 10-3 10-4 10-5 10-1 10-2 (a) NCEDC, Loma Prieta (M=7.1) 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 time since Loma Prieta [s] (b) NCEDC, M=5.8 slope=-1 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 time since M5.8 [s] rate [quakes/s] 10-3 10-4 10-5 slope=-1 10-6 (c) relocated composite 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 lag-time [s] Figure 3. Aftershock rate as a function of time for earthquakes along a segment of the San Andreas fault (Figure 1). (a) Time is measured with respect to the Loma Prieta earthquake. The black and gray solid curves are for M 1 and M 2, respectively. The black and gray dashed curves indicate average earthquake rate between Jan. 1980 and Jan. 1984 for earthquakes whose magnitude is greater than or equal to 1 and 2, respectively. (b) Aftershock rate is calculated for earthquakes that occurred within two mainshock radii from the M5.8 of April 18, 1990, with time measured since that earthquake. (c) Aftershock rate is calculated for relocated composite catalogs with mainshock magnitude that are less than 3.7. The gray and black curves are for mainshocks whose magnitude satisfy 2.5 < M 3.7 and M 2.5, respectively. The circles in each panels indicate the approximate return of seismicity rate to the background rate.
ZIV: AFTERSHOCK DURATION 19 aftershock duration [s] 10 9 10 8 10 7 10 6 San-Andreas Calaveras Calaveras Morgan Hill Loma Prieta 18 April 1990 M=5.8 10 5 1 2 3 4 5 6 7 8 mainshock magnitude Figure 4. Summary diagram showing aftershock duration as a function of mainshock magnitude. The vertical dashed line indicates the transition from one regime to another. While aftershock sequences triggered by mainshocks whose magnitude was less than 6 lasted a few days, those triggered by mainshocks whose magnitude was more than 6 lasted a few years.