For twenty years there has been a dilemma in earthquake physics, because the observed scaling law for large earthquakes did not appear to be consistent with the stress-drop invariance of small earthquake scaling. Surprisingly, slip was seen to continue to increase with rupture length L even for events with lengths much longer than the event widths W (the brittle crust down-dip depth), whereas it might have been expected to saturate for lengths much beyond the width. If this implies that the physics of great earthquakes is somehow different from that of their smaller counterparts, this casts serious doubts on predicting the effects of the rare and damaging great events from observations of the more common smaller events. Here we bring together recently compiled observations of very large aspect ratio earthquakes with results of a 3 dimensional dynamic earthquake model to show that slip-length scaling observations are, in fact, consistent with a scale-invariant physics. Further, we discuss the origin of the large earthquake scaling in the model

Scholz, Christopher H.

Shaw, Bruce E.

English

Columbia University Academic Commons

GEOPHYSICAL RESEARCH LETTERS, VOL. 28, NO. 15, PAGES 2995-2998, AUGUST 1, 2001 Slip-length scaling in large earthquakes' Observations and theory and implications for earthquake physics Bruce E. Shaw and Christopher H. Scholz Lamont-Doherty Earth Observatory, Columbia University, New York Abstract. For twenty years there has been a dilemma in earthquake physics, because the observed scaling law for large earthquakes did not appear to be consistent with the stress-drop invariance of small earthquake scaling. Surpris- ingly, slip was seen to continue to increase with rupture length L even for events with lengths much longer than the event widths W (the brittle crust down-dip depth), whereas it might have been expected to saturate for lengths much beyond the width. If this implies that the physics of great earthquakes is somehow different from that of their smaller counterparts, this casts serious doubts on predicting the ef- fects of the rare and damaging great events from observa- tions of the more common smaller events. Here we bring together recently compiled observations of very large aspect ratio earthquakes with results of a 3 dimensional dynamic earthquake model to show that slip-length scaling observa- tions are, in fact, consistent with a scale-invariant physics. Further, we discuss the origin of the large earthquake scaling in the model. Introduction Earthquakes have long been expected to scale like cracks with scale-invariant stress drops. Thus for small earth- quakes, with dimensions less than the seismogenic thickness, displacement should scale with radius, and for long earth- quakes, with lengths L greater than the seismogenic thick- ness W, displacement D should scale with W [Kanamori and Anderson, 1975]. Although this is the observed scaling for small earthquakes [Hanks, 1977], it was pointed out in 1982 [$cholz, 1982] that the observations indicate that displace- 12 ment scales with L rather than W for large earthquakes. There have been continued discussions about the data and their interpretation [Romanowicz, 1992; $cholz, 1994a; Pe- 10' gler and Das, 1996; Mai and Beroza, 2000], with little resolu- tion, and various interpretations have been suggested for the • s meaning ofthis finding, such as the suggestion that higher • stress drop earthquakes imply propagate farther [Heaton, • 1990]. There was always the unsettling possibility that new ..a 6 physics, uch as frictional melting [McKenzie and Brune, • 1972], may accompany the large earthquakes but notthe • small ones. If that were so, the study of small events may 4 not be helpful in predicting the effects of the much rarer but destructive great events, which would undercut a key 2 assumption of earthquake hazard analysis. Copyright 2001 by the American Geophysical Union. Paper number 2000GL012762. 0094-8276/01/2000GL012762505.00 Observations The scaling question hinges on observations of slip in very large aspect ratio earthquakes, which are very rare and were few in the early data sets. Gradually, however, such data has accumulated, and the most recent compilation does suggest a rollover to constant slip for very long earthquakes [$cholz, 1994b]. These data, shown in Figure 1, are for large earth- quakes, in which L > W, where W has a constant value of 15-20 kin. The different scales for interplate and intraplate earthquakes reflect the systematically greater stress drops of the latter events, here shown to be about a factor of three. The difference in the two types of earthquakes is that the in- traplate earthquakes occur on faults with geologic slip rates one or two orders of magnitude slower than the interplate ones, and their larger stress drops are thought to result from greater fault healing due to their longer recurrence times [Marone, 1998]. The linear trend in D vs. L is clearly evident for the shorter events in Figure 1, but now one can also see, for earthquakes with W/L > 10 a tendency for slip to attain a constant value, as expected from a 2-D crack model. What is surprising is that this crossover is so gradual and occurs at such a high L/W ratio [Bodin and Brune, 1996]. The large scatter in the data also makes one wonder if the observed trend is real [Bodin and Brune, 1996]. We explore both of these issues now with a 3-D dynamic earthquake model. Aspect ratio, L/W , • 10 i , 20 , • , ß ß [] [] intraplate ß interplate 0 100 200 300 400 Length, km I 500 Figure 1. Compilation of mean slip v$ length for large crustal earthquakes (modified after [Scholz, 1994b]). The aspect ratio is based on an assumed value of W - 15 krn. 2995 2996 SHAW AND SCHOLZ: LARGE EARTHQUAKE SCALING surface slip 0.4 0.2 o• • 5 70 75 80 S5 90 95 distance along fault / W vertical exaggeration 8:1 slip unstable sliding stable sliding peak velocity initial displacement initial stress Figure 2. Five different views of an example event. From top to bottom, we see the slip at the surface, and then four different views of the fault plane: the slip in the event, the maximum velocity, the initial net displacement, and the initial stress. Theory Our model both retains a number of complications of the Earth, while simplifying others. The model is three dimen- sional, retaining the large scale geometry of faults. We use a planar fault, neglecting the complication of geometric irreg- ularity. Our model uses fully inertial Newtonian dynamics, so that waves mediate interactions. We consider just one scalar elastic mode, and thus the bulk satisfies the wave equation. The fault is loaded from a steady, slowly moving, distant boundary. The upper surface is a free surface, as in the Earth. Friction on the fault plays a central role in the dynam- ics. When friction increases with slip or slip rate, the fault slides stably, creeping along at the plate loading rate. In contrast, when friction decreases with slip or slip rate, the fault slides unstably, rupturing in sudden stick-slip events [Brace and Byeflee, 1966; Carlson and Langer, 1989]. In the Earth, a seismogenic unstably sliding brittle crust of roughly 15 km depth overlays a stably sliding ductile lower crust [Blanpied et al., 1987; Tse and Rice, 1986]. We model this using friction with two layers, with frictional weakening in the top layer and frictional strengthening in the bottom layer. As with the geometry, we simplify the friction in our model, taking a friction which is uniform along the fault, and with constant properties with depth within each of the two layers. The friction we use has been described in detail and studied in lower dimensional models [Shaw, 1995; Shaw and Rice, 2000]. It combines a mixture of slip and velocity weakening effects. While it differs from the more elaborate rate and state laboratory derived friction [Dieterich, 1979; Ruina, 1983], the scaling results we present appear to be insensitive to many of the details of the frictional instability. It has a crucial advantage of allowing for a faster numerical scheme, which is useful since we are at the limits of our computational resources. SHAW AND SCHOLZ: LARGE EARTHQUAKE SCALING 2997 0.4 0.2 [] [] 0.0 ' ' ' ' ' •0 0 ]0 20[] [] [] [] Figure 3. Average surface slip as a function of surface rupture in the model. Each point corresponds to an individual event. Note the remarkable similarities with Figure 1, both in terms of the mean behavior, and the variation about the mean. The solid line is the scaling expected from a simple constant stress drop estimate. Beginning from any nonuniform initial condition, the sys- tem self-organizes into a sequence of events, on an attractor which appears to be independent of the initial conditions. Two types of attractors are seen. When there is insufficient weakening or when there is only slip weakening without any velocity weakening in the upper layer, throughgoing events which break the entire fault length develop. Alternatively, for sufficient weakening and when there are some velocity weakening effects, a nonperiodic, chaotic sequence of events ensues. This chaotic regime, which most resembles nature, is our focus here. Figure 2 shows a few different views of the fault surface for one example event. This figure illustrates what large surface breaking events typically look like in the model, and some aspects of the complex attractor that develops. It also illuminates an important aspect of the behavior: when the fault is long enough, there are attractors on which events die out without propagating the whole length of the fault. They die out because neighboring regions have slipped in a large event in the not too distant past. Figure 3 shows, as in Figure 1, a plot of the average surface slip as a function of the length of the surface rupture for the model events. The events plotted are the events which break the surface, taken from one long sequence of events. Notice the striking similarity to Figure 1, not only in the mean properties of the scaling, but in the scatter about the mean as well. For the mean, we see a rapid initial increase, followed by a rollover to roughly constant, or at least slowly increasing slip with rupture length, with the rollover occurring at roughly an aspect ratio 10. An assumption of constant stress drop, taking into account the finite brittle depth, captures some, but not all, of the scaling. Roughly, we can estimate it as follows. For an average slip D, the strain is given by D/LI q- D/Lz where LI is the rupture length along the fault and Lz is the rupture length in the depth direction. Equating this to a constant stress change and taking into account the finite depth W gives • L<2W' D .,• -•+-} - ' • L>2W. (1) where the factor of 2 comes from the effect of the free sur- face. With this estimate there is only one free parameter, the overall amplitude, which is fit to the slip of the largest events. The solid line in Figure 3 shows this scaling, for comparison to the model. It captures some of the mean be- havior, including the rapid increase and rollover at several times the crust depth. This result indicates a constant stress drop of 4 MPa for the interplate events and 12 MPa for the intraplate events. While the scaling argument captures the basic first order effect, there seem to be some systematic second order effects in the results which appear to go beyond this simple scaling. We see, in particular, a slower increase in the slip with length in the measurements. Interestingly, the model data fit the observed data, in this respect, even better than either fit the simple scaling argument! What appears to be underlying the surprisingly large lengthscales in the problem are dynamic effects. Slip pulses carrying along potential and kinetic energy concentrations are seen to take very long distances, both to get going, and to die away. Though the complex attractor contains a very heterogeneous stress field, as Figure 2 illustrates, these ef- fects are easiest to see by looking at pulses propagating over faults with constant initial stress conditions. In Figure 4, we plot the slip that results from a kick to a fault hav- ing an initially uniform stress. Here we see long transients in the initiation of ruptures through constant stress fields. The lengthscales of these transients are only weakly depen- dent on friction parameters, and weakly dependent on initial stress. What we see is that, in general, only over very long lengthscales, of order 5 to 10 W (10 to 20 W bilaterally), do the slip pulses saturate in slip. These long transients give us basic insight into our scaling problem. It suggests that it takes some quite long distance to reach the maxi- mum amount of slip, and that for very long ruptures we can 1.5 0.0 40 .... I .... I .... I .... I .... I,,,,I .... I,,,, 30 20 10 0 10 20 30 40 distance of rupture propagation / W Figure 4. Surface slip for kicks into initially uniform stress. Dif- ferent lines are for different initial stresses, and different friction parameters (thicker lines for friction parameter corresponding to less weakening). Note the large distances it takes for pulses to saturate in slip: lengths of order 10 W. 2998 SHAW AND SCHOLZ: LARGE EARTHQUAKE SCALING expect to see peak slip not necessarily at the epicenter, but away from it. It is these long transient lengths which stretch the saturation of the scaling out to such large aspect ratios. Dynamic energies are central to the long saturation lengths in the model, and, we expect, in the Earth. Implications and Conclusions Thus far, we have focussed our attention on the mean behavior of the scaling. The scatter, however, also contains important information. Here again, there is a remarkable similarity of the model to the observations, with both show- ing roughly a factor of 2 variation between events in the average slip for a given rupture length. The scatter in the model data comes from the intrinsic roughness of the slip which has developed along the fault. That is, it arises from naturally developed variations of initial conditions, and not from intrinsic strength variations, which are absent in the model. This has significant implications for the source of heterogeneities in earthquake behavior, where a debate has raged about the relative importance of geometrical and ma- terial heterogeneities versus stress heterogeneities. The fact that the model scatter is already of order the scatter in the observations suggests that stress heterogeneities cannot be neglected compared with geometrical and material hetero- geneities. Earthquake scaling laws provide constraints on the phy- sics of the earthquake source. Here we have shown that a longstanding mystery, why slip in earthquakes continues to increase with rupture length even for lengths many times the crust depth, can be reproduced by dynamic models hav- ing scale invariant physics. This provides evidence for the notion that there is nothing essentially different about the physics of great earthquakes relative to large earthquakes, and provides support for the use of measurements of the more numerous magnitude 7•s in preparing for the rarer and more devastating magnitude 8•s. Acknowledgments. We thank C. Marone and an anon- ymous referee for their comments. This work was supported by NSF grant EAR-99-09287 and USGS grant 1434-HQ-97- GR-O3074. References Blanpied, M.L., Tullis, T.E. and Weeks, J.D, Frictional be- havior of granite at low and high sliding velocities, Geo- phys. Res. Lett., 14, 554, 1987. Bodin, P. and J.N. Brune, On the scaling of slip with rupture length for shallow strike-slip earthquakes: Quasistatic models and dynamic rupture propagation, Bull. $eis. $oc. Am., 86, 1292, 1996. Brace, W.F. and Byerlee, J.D, Stick-slip as a mechanism for earth- quakes, Science, 153, 990, 1966. Carlson, J.M. and Langer, J.S, Mechanical model of an earth- quake fault, Phys. Rev. A, 60, 6470, 1989. Dieterich, J, Modeling of rock friction: 1. Experimental results and constitutive equations, J. Geophys. Res., 84, 2161, 1979. Hanks, T.C, Earthquake stress-drops, ambient tectonic stresses, and the stresses that drive plates, Pure Appl. Geophys., 115, 441, 1977. Heaton, T, Evidence for and implications of self-healing pulses of slip in earthquake rupture, Plan. Earth Int., 64, 1, 1990. Kanamori, H. and D.L. Anderson, Theoretical basis for some em- pirical relations in seismology, Bull. $eis. $oc. Am., 65, 1073, 1975. Mai, P.M., and G. C. Beroza, Source scaling properties from finite rupture models, Bull. Seis. $oc. Am., 90, 604, 2000. Marone, C, The effect of loading rate on static friction and the rate of fault healing during the earthquake cycle, Nature, 391, 69, 1998. McKenzie, D.P. and J.N. Brune, Melting on fault planes during large earthquakes, Geophys. J. Roy. Astron. Soc., 29, 65, 1972. Pegler, G. and S. Das, Analysis of the relationship between seis- mic moment and fault length for large crustal strike-slip earth- quakes between 1977-1992, Geophys. Res. Lett., 23, 905, 1996. Romanowicz, B., Strike-slip earthquakes on quasi-vertical tran- scurrent fault: inferences for general scaling relations, Geo- phys. Res. Lett., 19, 481, 1992. Ruina, A.L., Slip instability and state variable friction laws, J. Geophys. Res., 88, 10359, 1983. Scholz, C.H., Scaling laws for large earthquakes: consequences for physical models, Bull. Seis. $oc. Am., 72, 1, 1982. Scholz, C.H., A reappraisal of large earthquake scaling physical models, Bull. Seis. Soc. Am., 84, 215, 1994a. Scholz, C.H., Reply to comments on 'A reappraisal of large earth- quake scaling', Bull. Seis. Soc. Am., 84, 1677, 1994b. Shaw, B.E., Frictional weakening and slip complexity on earth- quake faults, J. Geophys. Res., 100, 18239, 1995. Shaw, B.E. and J.R. Rice, Existence of continuum complexity in the elastodynamics of repeated fault ruptures, J. Geophys. Res. 105, 23791, 2000. Tse, S. and J.R. Rice, Crustal earthquake instability in relation to the depth variations of frictional slip properties, J. Geo- phys. Res., 91, 9452, 1986. B. E. Shaw and C. H. Scholz, Lamont-Doherty Earth Obs., Palisades, NY 10964. (e-mail: shaw@ldeo.columbia.edu, scholz@ldeo.columbia. edu) (Received December 12, 2000; accepted March 14, 2001) 

Slip-length scaling in large earthquakes' Observations and theory and implications for earthquake physics

Abstract. For twenty years there has been a dilemma in earthquake physics, because the observed scaling law for large earthquakes did not appear to be consistent with the stress-drop invariance of small earthquake scaling. Surpris-ingly, slip was seen to continue to increase with rupture length L even for events with lengths much longer than the event widths W (the brittle crust down-dip depth), whereas it might have been expected to saturate for lengths much beyond the width. If this implies that the physics of great earthquakes is somehow different from that of their smaller counterparts, this casts serious doubts on predicting the ef-fects of the rare and damaging great events from observa-tions of the more common smaller events. Here we bring together recently compiled observations of very large aspect ratio earthquakes with results of a 3 dimensional dynamic earthquake model to show that slip-length scaling observa-tions are, in fact, consistent with a scale-invariant physics. Further, we discuss the origin of the large earthquake scaling in the model

Bruce E. Shaw

Christopher H. Scholz

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Slip-length scaling in large earthquakes: Observations and theory and implications for earthquake physics

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Slip-length scaling in large earthquakes' Observations and theory and implications for earthquake physics

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