Rock Friction and Its Implications for Earthquake Prediction Examined Via Models of Parkfield Earthquakes (Constitutive Laws/Numerical Models) TERRY E

Proc. Natl. Acad. Sci. USA Vol. 93, pp. 3803-3810, April 1996 Colloquium Paper

This paper was presented at a colloquium entitled "Earthquake Prediction: The Scientific Challenge," organized by Leon Knopoff (Chair), Keiiti Aki, Clarence R. Alien, James R. Rice, and Lynn R. Sykes, held February 10 and 11, 1995, at the National Academy of Sciences in Irvine, CA. Rock friction and its implications for earthquake prediction examined via models of Parkfield earthquakes (constitutive laws/numerical models) TERRY E. TULLIS Department of Geological Sciences, Brown University, Providence, RI 02912-1846 ABSTRACT The friction of rocks in the laboratory is a implications of the observed time, velocity, and displacement function of time, velocity of sliding, and displacement. Al- dependence of rock friction in the laboratory for the possible though the processes responsible for these dependencies are behavior of natural faults, recognizing that this may represent unknown, constitutive equations have been developed that do only part of the story. This approach has the advantage that it a reasonable job of describing the laboratory behavior. These is clear from work already done that the laboratory-derived constitutive laws have been used to create a model of earth- constitutive laws offer at least one likely explanation for many quakes at Parkfield, CA, by using boundary conditions ap- phenomena involved in earthquakes (1-11). propriate for the section of the fault that slips in magnitude This success, combined with the fact that accelerating slip 6 earthquakes every 20-30 years. The behavior of this model precedes unstable sliding in the laboratory, leads one to prior to the earthquakes is investigated to determine whether investigate the relevance of rock friction data for predicting or not the model earthquakes could be predicted in the real earthquakes. One, debatably encouraging, fact is that there is world by using realistic instruments and instrument locations. general agreement between observations from the earth and Premonitory slip does occur in the model, but it is relatively predictions from the modeling on the small size of any restricted in time and space and detecting it from the surface premonitory signals. The fact that at least small signals may may be difficult. The magnitude of the strain rate at the exist suggests that it might eventually be possible to detect earth's surface due to this accelerating slip seems lower than them. the detectability limit of instruments in the presence of earth noise. Although not specifically modeled, microseismicity Review of Rock Friction Behavior related to the accelerating creep and to creep events in the model should be detectable. In fact the logarithm of the Observed Behavior. Generalities. Static friction increases moment rate on the hypocentral cell of the fault due to slip with the time ofbeing static and there is an associated tendency increases linearly with minus the logarithm of the time to the for dynamic friction to decrease with increasing slip velocity. earthquake. This could conceivably be used to determine when In addition, when the velocity is changed, the transition to the the earthquake was going to occur. An unresolved question is new frictional resistance requires sliding some distance. These whether this pattern of accelerating slip could be recognized features are illustrated in Fig. 1. Two competing effects are from the microseismicity, given the discrete nature of seismic seen in the behavior shown in Fig. 1, the evolution effect and events. Nevertheless, the model results suggest that the most the direct effect. likely solution to earthquake prediction is to look for a pattern Evolution effect. The most important single aspect of rock of acceleration in microseismicity and thereby identify the friction in terms of application to earthquakes is that surfaces microearthquakes as foreshocks. in stationary contact increase in strength with time. This effect is known as the evolution effect. It is demonstrated by the Frictional sliding of rocks on laboratory scale faults or on faults slide-hold-slide test, in which sliding is stopped for some in the earth are similar. Both systems involve an interaction period of time after having attained a steady-state frictional between the frictional response of the rocks in the fault zone resistance at constant velocity. When sliding is resumed, the and the elastically distorted surroundings that can either be resistance climbs to a peak value larger than the steady-state unstable or stable. value prior to the hold, subsequently decaying to the original It is not the coefficient of friction of rocks, but the way the steady-state value. As shown in Fig. 2, there is a nearly linear coefficient of friction depends on time, sliding velocity, and relationship between the magnitude of the peak value and the displacement that is important to whether sliding will be stable logarithm of the hold time (12, 13). This phenomenon is or unstable. These relatively small effects are more subtle and responsible for the restrengthening of surfaces between slip experimentally difficult to measure, but they are of primary episodes, without which repeated unstable slip is impossible, importance in determining whether a fault will slide by creep and for a tendency for frictional resistance to be lower at or by catastrophic earthquakes. There are other factors such as higher sliding velocities. This potential for decreasing strength changes in pore fluid pressures that may also play a role in with increasing velocity is the aspect of rock friction that allows earthquake generation, but some kind of dependence on for runaway instabilities (14, 15). velocity and sliding displacement is probably needed to allow After abrupt changes in velocity the frictional resistance a sustained runaway instability. In this paper I will explore the evolves to a new steady-state resistance over a characteristic slip distance (thus the term "evolution effect"). This has been The publication costs of this article were defrayed in part by page charge interpreted (16-18) as the slip required to destroy all the payment. This article must therefore be hereby marked "advertisement" in contacts established at one velocity and to create a new set of accordance with 18 U.S.C. §1734 solely to indicate this fact. contacts having an average age appropriate to the new velocity. 3803 Downloaded by guest on October 1, 2021 3804 Colloquium Paper: Tullis Proc. Natl. Acad. Sci. USA 93 (1996) inside the pressure vessel immediately adjacent to the sample. In Fig. 3 is shown a typical sequence of stick slip events V, V2 = eV, measured with internal stress and displacement transducers in our rotary shear apparatus (26). At first glance (Fig. 3A), the events look as if they occur without warning, but successively closer examinations (Fig. 3 B-D) show that there is acceler- /1 ating slip and an associated nonlinearity in the stress-time curve that foretells each unstable event. There is no reason to I believe that this precursory accelerating slip should not occur IAL in the earth. The real question is whether it will be large enough to be usefully detected and so provide the basis for a short-term earthquake prediction. In order to answer this we need to learn how to extrapolate the laboratory results to the FIG. 1. Response of friction to an abrupt increase in sliding earth. The constitutive laws discussed in a following section velocity. The parameters a, b, and Dc are those in Eqs. la, lb, and lc. form the present basis for doing this. The behavior given by the equations for a sudden velocity change is Constitutive Laws. constitutive laws have been illustrated graphically. The magnitude of the direct effect is measured Empirical a and of the evolution effect is measured b. used by many workers to fit the frictional behavior described by by (e.g., refs. 13, 17, 26-38). Although a variety of functions have However, a complete understanding of what occurs at the been presented, two laws are most commonly used, and even contacting asperities does not yet exist. these can be cast in slightly different ways. The form used in The other effect seen in all friction experiments is an initial ref. 13 is convenient because the state variable has dimensions increase in the resistance to sliding that occurs when the of time in both laws. Both laws represent friction as a function velocity of sliding is abruptly increased (Fig. 1). This is termed of velocity and a state variable by the same equation: the direct effect because the change of resistance occurs = + a + b instantaneously and in the same sense as the change in velocity. uL pto ln(V/Vo) ln(OVo/Dc). [la] Stability ofsliding. The absolute value of frictional resistance The direct effect is contained in the term a ln(V/V*) and is unimportant for controlling the stability of sliding. However, the evolution effect in the term b ln(0Dc/V*). The nature of the time, velocity, and displacement dependence of friction the evolution is what differs in the two laws. In the law interact with the elastic stiffness of the surroundings to pro- termed the slip law, because slip is required for evolution, the duce either stable or unstable sliding. The stability has been state variable evolves according to: analyzed by using the constitutive laws described in the following section (14, 16, 17, 19-24). d /dt = (OV/Dc)ln(OV/Dc). [lb] The stability of sliding is ultimately controlled by an interaction between the stiffness of the loading system and the In the law termed the slowness law, because the evolution dependence of frictional resistance on displacement. In sys- depends on the slowness (inverse of velocity) or on time, the tems for which the frictional behavior is as illustrated in Figs. evolution is given by: 1 and 2, the dependence of frictional resistance on displacement is itself a function ofhow friction depends on velocity (15, dO/dt = 1 - OV/Dc. [lc] pp. 566-569). If the frictional resistance decreases with in- Neither of these two commonly used laws fits all aspects of creasing velocity, a behavior termed velocity weakening, then experiments data (39), and a better law is needed. Both laws unstable sliding is possible. This unstable sliding generally is do a reasonable job of fitting data until one looks carefully at not possible for a velocity strengthening material, one in which the details. If the processes that cause the observed behavior the resistance increases with slip speed. This generality must be can be understood, then the correct form of this law may be modified if the resistance does not show a monotonic change found, and we can have more confidence in extrapolating its after the direct peak (26). behavior outside the range of existing laboratory data. Predictability of unstable sliding. The onset of an unstable Lack ofData and Constitutive Laws for Dynamic Slip Under sliding event in the laboratory is typically preceded by some Realistic Conditions. No laboratory experiments combine the nonlinearity in the loading curve and by accelerating slip. large displacement, high slip rate, high normal stress, and Whether this is a universal feature of all laboratory experi- presence of pressurized pore fluids that characterize dynamic ments is not known, since detecting this behavior may require earthquake slip. Most experiments involving high slip velocity high precision measurement of the stress and perhaps mea- (e.g., refs. 40-42) have been done at low total displacement surement of the displacement using a transducer mounted and low normal stress, without the presence of pore fluids. These failings mean that processes that may occur during 0.10- Initially bare granite dynamic slip in earthquakes have not been explored experimentally. Chief among these are shear heating and associated 0.08- possible melting or increase in pore fluid pressures. Some experiments at low normal stress at relatively high velocity and 0.06- large total displacement have produced shear melting (43-45). Experiments on foam rubber suggest the importance of re- 0.04- 8 duction of normal stress during dynamic slip (46). Thus, although we have considerable data that may be 0.02- relevant to the accelerating slip that occurs prior to an earthquake, we have no experimental data that is really useful for the or 101 102 103 104 105 characterizing velocity displacement dependence of hold time (s) frictional resistance that might cause self-healing slip pulses (47-49), the lack of a heat flow anomaly in area of faults that FIG. 2. Increase of static friction as a function of the time period typically slip in seismic events (50-52), or other evidence for of holding static. the absolute and relative weakness of major faults (53-55). Downloaded by guest on October 1, 2021 Colloquium Paper: Tullis Proc. Natl. Acad. Sci. USA 93 (1996) 3805

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0 6 100426 0 u 0.68 UTL 100450 0.70 0.67 z ui. 100425 2 ,/ 2 ° /'8 % of Dynamic 0.66 100440 S SipT Z 25 MPaznormal stress at 1 ,um/s. zI- : 100424 g g-,,L0.65 0.69 'U 100430 u.I 0 0.634 E 5asi 100420 Z' 0 0.63 Elastiq 100420 ° -- Elast 5 849940 850000 850060 0 ...... TIME, SEC 0.6850030 850040 850050 850060 850070 850080 00422 TIME, SEC FIG. 3. Four successively closer looks (A-D) at the behavior of stick slip events found in the laboratory, showing that some premonitory slip occurs. The jagged line is the friction and the one that increases in steps is the displacement. The data is for a bare quartzite sample that slid at 25 MPa normal stress at 1 gxm/s. Implications of Instability Model of Parkfield Earthquakes designated L1 in ref. 4. In the Parkfield section itself, the fault for Earthquake Prediction is considered to have the properties of granite, which is velocity-strengthening at the surface, velocity-weakening from A three-dimensional model of magnitude -6 earthquakes on a depth of 1.8-8.2 km, and velocity-strengthening below that. the San Andreas fault at Parkfield, CA, has been developed, To the north of the Parkfield section, the material is assumed by using the laboratory-based constitutive behavior described to act like serpentine at low velocity; namely, it is velocity- above (3, 4). This model takes the boundary conditions strengthening and shows only the direct effect. Because ser- relevant to the Parkfield section of the San Andreas fault, and pentine is velocity strengthening and is weak (30-33), its by using constitutive parameters consistent with laboratory presence on the San Andreas fault in the creeping section experiments, generates a spontaneous sequence of character- could simultaneously explain the lack of significant earth- istic earthquakes. The magnitude and spatial and temporal quakes there and the low absolute and relative strength of this extent of any premonitory creep in this model can be studied part of the fault (53-55). A smooth transition over 6 km along to see whether it could be detected and form the basis for a strike occurs between the granite-like and the serpentine-like prediction of the earthquake. Although differences will exist sections. between the model and real Parkfield earthquakes, the model The slip on the fault causes changes in strain on the surface has the advantage of forming a concrete example of the of the earth where it is possible to make measurements in the implications of the laboratory results. Its success can in part be real world. Since detection of these changes in strain is one of evaluated when the next Parkfield earthquake occurs. the more likely ways that the slip at depth might be detected, Description of the Model. More details of the model can be calculation of these changes has been done for this model. found in ref. 4. The fault is assumed to be vertical and planar Because the rate of slip on the fault best characterizes any and shows only strike slip motion. To the south of the Parkfield particular stage in the model earthquake, the associated rate section, the fault is assumed to be locked at the surface down of strain at the earth's surface has been calculated and to a depth of 9 km as a result of the reduction in stress after displayed. the great 1857 Fort Tejon earthquake. The fault is assumed to The detailed behavior of the model is affected by the size of be creeping at 35 mm/year at the surface to the northwest of the cells on the fault plane used to approximate a more the Parkfield section along the central creeping section of the continuous distribution ofslip. A larger number of smaller cells San Andreas. This boundary condition of 35 mm/year is also does a betterjob of approximating the behavior of a continuum applied at depth under the Parkfield section and under the (5), but it uses considerably more computer time. Because of locked 1857 section (Fig. 4A). In the Parkfield section itself to this time constraint, the behavior has been studied most a depth of 39 km and to 45 km northwest of Middle Mountain extensively for a model in which the smallest cells are squares, (the north end of the Parkfield section) the behavior of the 1.0 km on a side. A few examples of the behavior when the fault is a computed result of the boundary conditions, the smallest cells are 1080 m, 360 m, and 120 m on a side have been constitutive properties, and the initial conditions. briefly studied, but this work is still in progress. Preliminary The constitutive properties of the fault have been assigned results are shown comparing the behavior as the cell size is in several different ways in the model examples studied in ref. decreased. Because larger cells are more compliant than 4. The properties used in the present paper are taken from the smaller ones, cells are more able to slip independently from behavior of granite (26-29) and serpentine (30-33) and are their neighbors when they are larger. The discreteness intro- Downloaded by guest on October 1, 2021 3806 Colloquium Paper: Tullis Proc. Natl. Acad. Sci. USA 93 (1996) A B

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FIG. 4. Behavior at selected times in the model with the smallest grid size h = 1 km. In A the boundary condition of 35 mm/year is applied over the brown area and the behavior is calculated over the grey area with the grid. In the interseismic period (B), the Parkfield patch is essentially locked and the rest of the fault is slipping at 35 mm/year. A rapid creep event (D) would represent a false alarm. At other times (C and E-H), the only unusual behavior is that a variety of cells at the north end of the Parkfield section are creeping rapidly, due to the boundary condition of slip extending to the surface in the north. Finally, within an hour of the earthquake (I-K), the pattern of strain rate on the earth's surface is developed, but the magnitudes are too small to detect. The earthquake (L) in this model does not deal properly with the dynamics of rupture. duced by oversized cells introduces a spatial and temporal center to nearby slightly larger cells and so successive models heterogeneity that in a crude way proxies for the heterogeneity were run in each case to find a distribution of cells so that the that variations in rock type and geometric irregularities on the hypocenter occurred within the smallest cells. fault may cause in the real world. As models are studied with From the point of view of earthquake prediction, it is the smaller grid sizes, it will be possible to introduce heterogene- pattern in space and time of the accelerating slip leading up to ities in material properties that can represent real world the main shock that might be used to predict it. variations in a more controlled way. The cell size used (5) Slip in the model at velocities higher than the 35 mm/year should be substantially less than h* = 2GDc/[vr(B - A)m] boundary condition might be expected to correspond to in- where G is the shear modulus (30 GPa) and (B - A)m is the creased microearthquake activity. The coarseness of the grid maximum value of B - A, where B - A = an(b - a). For the precludes explicit modeling of small earthquakes. It is reason- parameter values used here [model Li of ref. 4], (B - A)m is able to imagine that the more rapid the slip event, the more 0.575 MPa and Dc is 13 mm, giving h * = 431 m. Our cell sizes numerous the small earthquakes or the larger their size, since h of 1080, 360, and 120 m represent h/h * of 2.51, 0.84, and 0.28. these more rapid creep events correspond to larger moment Behavior of the Model. In the model the velocity weakening rates. Thus the spatial and temporal variation in moment rate part of the Parkfield segment slips very slowly in the inter- in the model might be expected to be detected in the earth by seismic period. Its northwest end is eaten into by episodic creep changes in seismicity. The model predicts that foreshocks events until one of them, at a depth where the velocity should occur and that they are more numerous as the main- weakening is the greatest, accelerates to a slip velocity of 1 m/s, shock approaches. The model also shows what might be the definition used for dynamic rupture, and becomes the considered to be false alarms, for example, a rapid creep event hypocenter. Representative typical images from an earthquake at 2.8 years prior to the mainshock (Fig. 4D) that can be cycle are shown in Fig. 4 B-L. A complete set of such images regarded as a foreshock. exists in a video showing the behavior of the model in more Some of the creep events that occur in the few years prior detail. to the mainshock are large enough that they are reflected in Results of a preliminary examination of the effect of cell size the pattern of strain rate on the earth's surface. This is on the behavior of the model is shown in Fig. 5. In this figure, invariably true for the creep event that accelerates to become models with three sizes for the smallest cells are shown, all for the mainshock, and it is true for the larger of the "foreshocks." 0.1 s before the earthquake to show the final stage of accel- Predictability of the Model Earthquake. The premonitory eration of the eventual hypocentral cell. The general location increases in slip in the hypocentral region that precede the of the hypocentral cell is similar in these models. However, this main shock are expected to correspond in the earth to in- is a complex issue, since the model tends to move the hypo- creases in microseismicity and in foreshocks that could be Downloaded by guest on October 1, 2021 3808 Colloquium Paper: Tullis Proc. Natl. Acad. Sci. USA 93 (1996) in perhaps an hour, a strain of 2 x 10 -I would have to be detectable. Given the noise spectrum for strain (57-59), strains of 10-9 are detectable at that period, suggesting that the increase in strain rate is about two orders of magnitude below the detectable level. Similarly at 1 s before the mainshock the maximum strain rates are 2 x 10-11 s-1, requiring a detectability of better than 10-11 at a 1-s period. This is about an order of magnitude below the detectability limit at a 1-s period. Fig. 6 shows plots of the maximum strain rates observed on the Earth's surface in the model versus logarithm of time before the earthquake, together with an estimate of the detectability limit for sampling at appropriate rates for that period. This is analogous to the plots shown in ref. 2 for the two-dimensional earthquake model of ref. 1. As expected (2), the signals for the three-dimensional model are lower than for the two- dimensional model. The difference is large enough that whereas the two-dimensional model strain rates might have been detectable, the three-dimensional ones seem too small. Note that in the model for which h/h * is the smallest, a large number of creep events occur that might be observed. How- ever, it is important to note that the duration of these creep events is very small and so the fact that they seem to rise above the detectability limit is misleading since the time axis is time before the eventual earthquake, not the time before the particular creep event. On the other hand, as discussed below, these events could be associated with microseismicity and so be detected on seismometers. Note that the detectability arguments just presented correspond to one's ability to detect the pattern of the rate of strain at any instant prior to the mainshock, without regard for prior history. If a pattern of strain develops that is consistently growing in the same area over some interval of time (4), then that pattern might be detected, even if it could not be with only a brief measurement. Thus, based on the analysis done to date, it is too early to be sure whether or not the strain at the earth's surface might be useful in predicting the earthquake. Microseismicity. It is also an open question as to whether the expected increase in slip rate could be detected from increases in microseismicity. In the model, there is an increase in the moment release in the vicinity of the hypocenter in the few years late in the seismic cycle prior to the earthquake. As far as we know, even if much of this increase in slip occurred aseismically, it should be accompanied by an increase in microseismicity. This should be detectable seismically and might serve to give a general intermediate-term prediction of the earthquake. The question is whether or not such an increase could be used for a short-term earthquake prediction. If it were to occur as smoothly as in the model, it might be quite FIG. 5. Details of the models with successively smaller grid sizes at useful for such a prediction. However, if it occurs primarily 0.1 s before the earthquake. (A) The smallest grid size is 1080 m; h/h* through discrete earthquakes of magnitude 3 to 5, then it may = 2.5. (B)h = 360 m;h/h* = 0.83. (C)h = 120 m;h/h* = 0.28. Finally, be difficult to discern enough details of the pattern of accel- in C, the grid size is small enough that the cells begin to behave as a erating activity to allow a short-term prediction. group rather than individually. There is one tantalizing result from this model in terms of a path detected seismically and to increases in rate providing to intermediate- and short-term earthquake strain and strain prediction. In the 2.8 years after the foreshock (Fig. 4D), the that or might might not be detectable at the earth's surface. logarithm of the rate of moment increase on the eventual There might also be changes in other parameters such as hypocentral cell increases essentially linearly with logarithm of electric and magnetic fields (56). The question is whether or time to the mainshock (Fig. 7). Thus if there was some way to not these various changes could be used to predict the earth- follow this increased rate of moment release seismically (for quake. example, by monitoring microseismicity), then the data would Strain at earth's surface. Whether the pattern of strain rate "point" to the eventual time of the mainshock with increased on the Earth's surface can be used to detect that the main precision as the time of the mainshock approached. In prin- shock is coming is still debatable. Clearly the pattern itself cipal, such monitoring could also be done with a borehole develops soon enough prior to the mainshock in the 1-km grid strain meter in a deep borehole if one knew where to place it. size example shown in Fig. 4 to be useful if it could be observed, Such a linear increase on a log-log plot is exactly the form that although it only gives a few hours warning. However, the K. Aki (personal communication) suggested would make an magnitude of the increase in strain rate is not large. A few ideal earthquake prediction. The linear change of moment rate hours prior to the mainshock the maximum strain rates are on this log-log plot has a slope of approximately minus one in only 5 x 10-15 s-1. Given that one might want to detect this terms of logarithm of time to the earthquake. Thus, over the Downloaded by guest on October 1, 2021 Colloquium Paper: Tullis Proc. Natl. Acad. Sci. USA 93 (1996) 3809

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FIG. 6. Strain rate as a function of time prior to the earthquake with four grid sizes. These are calculated for the point on the earth's surface where the strain rate is the highest. In all cases the strain rate is smaller than can be detected in the time interval available, given the behavior of sensitive borehole instruments installed in the earth. essentially linear portion of this curve the moment rate and the quake the difference between the moment rate for a correct time to the earthquake are inversely proportional. guess and for a guess late by 1 day is only a factor of 2 in The way the linear increase in moment rate versus time to moment rate. Whether this could be distinguished given the the earthquake on a log-log plot (Fig. 7) might be used to discretization of seismic moment rate is questionable. One way predict the time of the earthquake is shown in Fig. 8. In to get some idea of what the moment rates are and how they practice one would not know the actual time of the earthquake compare with what one can observe is to note that in Fig. 7 the to make a plot similar to that in Fig. 7. However, if the moment rate on the hypocentral cell reaches 35 mm/year logarithm of the moment rate data were plotted versus the corresponding to the average slip rate on the fault in the logarithm of various possible times until the earthquake, then only if the correct time is used will the plot remain linear. As 1017 shown in Fig. 8, the moment rate will either climb too rapidly 0 1016 as the time is approached if too long a time is used, or it will not climb rapidly enough if too short a time is used. The 1014 departure from linearity on the log-log plot becomes more E apparent as the time approaches that of the earthquake. As 1013 - 1012 shown in Fig. 8, it allows making increasingly more accurate U 1 30 MIDDLE time predictions as the earthquake approaches. In general, for 1011 CELLSMOUNTAIN the ideal model behavior, the departure from linearity result- ing from choosing an erroneous time becomes evident at a time z 109-"-,- lJ - r--^ L.....- ...." before the earthquake that is 1-3 times longer than the error LUw 8 35mm/yr for: 30 cells 1 cell in the time chosen. Thus, in Fig. 8, the departure from linearity 10.o7...... by making an error in the time of the earthquake of 1 day is O R SC E very evident 2 days before the earthquake but cannot be seen 106 R R AY N...... I...... 1 week before it. The same can be seen for longer and shorter times. TIME BEFORE EARTHQUAKE (LOG SCALE) Whether this behavior of the ideal model could be useful for earthquake prediction is not clear. The question is whether the FIG. 7. Linear increase on a log-log plot of the moment rate on the signal (the logarithm of the moment rate) can be detected with hypocentral cell for the model with 1-km cell sizes. This corresponds sufficient There are two The to a reciprocal relationship between the moment rate and the time accuracy. possible problems. until the earthquake. The moment rates on a variety of other cells are moment rate might be too low to observe even with down-hole also illustrated. They peak at various times corresponding to creep seismometers, and the discreteness of the microseismic events events that occur on only one cell. The moment rate for 30 1-km cells might preclude seeing any underlying pattern in the moment under Middle Mountain is also shown, as is the moment rate corre- rate. In the model, at a time 2 days before the actual earth- sponding to 1 and to 30 cells slipping at 35 mm/year. Downloaded by guest on October 1, 2021 3810 Colloquium Paper: Tullis Proc. Natl. Acad. Sci. USA 93 (1996) 1. Tse, S. T. & Rice, J. R. (1986) J. Geophys. Res. 91, 9452-9472. 1017I 2. Lorenzetti, E. A. & Tullis, T. E. (1989) J. Geophys. Res. 94, 12,343-12,361. Q IU0- Guesses \.. 3. Stuart, W. D. & Tullis, T. E. (1992) Eos Trans. AGU 73, Fall Meeting Abs. u late Supp., 406-407. 10158too 4. W. D. & T. E. J. Res. UC)O~~ 1;~~ 4 by time a Stuart, Tullis, (1995) Geophys. 100, 24,079-24,099. E 10 r indicated > 5. Rice, J. R. (1993) J. Geophys. Res. 98, 9885-9907. 1013 6. Beroza, G. C. & Jordan, T. H. (1990) J. Geophys. Res. 95, 2485-2510. L 7. Jordan, T. A., Ihme1, P. F. & Marone, C. J. (1993) Eos, Trans. AGU 74, Ziii Imin Spring Meeting Abs., Supp., 292. hr,u/1 s 8. Dieterich, J. H. (1994) J. Geophys. Res. 99, 2601-2618. uj Ol~~~l d a j r 9. Dieterich, J. H. (1992) Tectonophysics 211, 115-134. ^ 109 _ I 10. Marone, C., Vidale, J. E., Ellsworth, W. L. (1995) Geophys. Res. Lett., 22, Z IG.108. Es n of te te of tooearyltt 3095-3098. a a te e e 11. Vidale, J. E., Ellsworth, W., Cole, A. & Marone, C. (1994) Nature (London) se 1 m o in dicate. 368, 624-646. creeping108w section at 2.5 months beforethe35m rainshfor1 Whether 12. Dieterich, J. H. (1972) J. Geophys. Res. 77, 3690-3697. 13. Beeler, N. M., Tullis, T. E. & Weeks, J. D. (1994) Geophys. Res. Lett. 21, 107 1987-1990. 14. Rice, J. R. & Ruina, A. L. (1983) Trans ASME, J. Appl. Mech. 50, 343-349. 100tYRiMdetec1R 1R 1 AY -R1 1 1 IN pSEp 15. Tullis, T. E. (1988) Pure Appl. Geophys. 126, 555-588. GUESSED TIME BEFORE EARTHQUAKE (log scale) 16. Dieterich, J. H. (1978) Pure Appl. Geophys. 116, 790-806. 17. Dieterich, J. H. (1979) J. Geophys. Res. 84, 2161-2168. FIG. 8. Estimation of the time of the earthquake by plotting the 18. Dieterich, J. H. & Kilgore, B. D. (1994) Pure Appl. Geophys. 143, 283-302. moment rate versus a variety of possible times to the rand 19. Gu, J. C., Rice, J. R., Ruina, A. L. & Tse, S. (1984) J. Mech. Phys. Solids 32, which time the most linear 167-196. seeing gave plot. 20. Rice, J. R. & Gu, J. C. (1983) Pure Appl. Geophys. 121, 187-219. 21. Blanpied, M. L. & Tullis, T. E. (1986) Pure Appl. Geophys. 124, 43-73. creeping section at 2.5 months before the mainshock. Whether 22. Gu, Y. & Wong, T.-f. (1991) J. Geophys. Res. 96, 21,677-21,691. thioisis a detectable rate for a -km square patch on the fault 23. Gu, Y. & Wong, T.-f. (1992) Rock Mechanics Proceedings ofthe 33rd U.S. with a sampling time of a few weeks depends on the sensitivity Symposium, (A. A. Balkema, Rotterdam), pp. 151-158. of the seismic system. It mightbe detectable by using the 24. Gu, Y. & Wong, T.-f. (1994) AGU Geophys. Monogr. Ser. 83, 15-35. Middle Mountain at Park- 25. Weeks, J. D. & Tullis, T. E. (1985) J. Geophys. Res. 90, 7821-7826. down-hole seismic array around 26. Tullis, T. E. & Weeks, J. D. (1986) Pure Appl. Geophys. 124, 10-42. field. By a day before the earthquake, the moment rate on the 27. Blanpied, M. L., Lockner, D. A. & Byerlee, J. D. (1995) J. Geophys. Res. hypcentralc ell increases by a factore of 60 over the rate 100, 13,045-13,064. corresponding to 35 mm/year, but the available sampling time 28. Beeler, N. M., Tullis, T. E., Blanpied, M. L. & Weeks, J. D. (1996) J. to detect this is much smaller. If this increase in moment rate Geophys. Res. 101, in press. 29. Blanpied, M. L., Lockner, D. A. & Byerlee, J. D. (1991) Geophys. Res. Lett. occurs only via a few discrete earthquakes, then the increasing 18, 609-612. moment rate may not be clearly seen as part of a gradually 30. Reinen, L. A., Weeks, J. D. & Tullis, T. E. (1991) Geophys. Res. Lett. 18, accelerating signal. These earthquakes would be foreshocks, 1921-1924. not be clear for them to be 31. Reinen, L. A., Tullis, T. E. & Weeks, J. D. (1992) Geophys. Res. Lett. 19, but the pattern might enough 1535-1538. recognized as such. 32. Reinen, L. A., Weeks, J. D. & Tullis, T. E. (1994) Pure Appl. Geophys. 143, Thus the problems of predicting the model earthquake seem 318-358. similar to the problems of predicting real earthquakes, even if 33. Reinen, L. A. (1993) Ph.D. thesis (Brown Univ., Providence, RI). preceded by foreshocks. Namely, how does one discriminate a 34. Dieterich, J. H. (1981) AGU Geophys. Monogr. Ser. 24, 103-120. of moment rate the eventual 35. Ruina, A. L. (1983) J. Geophys. Res. 88, 10359-10370. pattern increasing preceding 36. Linker, M. F. & Dieterich, J. H. (1992) J. Geophys. Res. 97, 4923-4940. mainshock when that increasing moment rate is strongly 37. Chester, F. M. (1994) J. Geophys. Res. 99, 7247-7261. discretized. Some kind of temporaral averaging of the moment 38. Marone, C., Raleigh, C. B. & Scholz, C. H. (1990) J. Geophys. Res. 95, rate may be a solution, but the accelerating pattern could be 7007-7025. lost in the averaging. 39. Tullis, T. E., Beeler, N. M. & Weeks, J. 0. (1993) Eos Trans. AGU, 74, Conclusions. The behavior of the model Parkfield earth- Spring Meeting Abs. Supp., 296. 40. Okubo, P. G. & Dieterich, J. H. (1984) J. Geophys. Res. 89, 5815-5827. quakes discussed here is based on laboratory determinations of 41. Okubo, P. G. & Dieterich, J. H. (1986) AGU Geophys. Monogr. Ser. 37, rock friction constitutive laws. The model earthquakes show 25-35. premonitory slip prior to the eventual mainshock, but the 42. Ohnaka, M. (1986) AGU Geophys. Monogr. Ser. 37, 13-24. and extent it is small. Like 43. Spray, J. G. (1987) J. Struct. Geol. 9, 49-60. spatial temporal of relatively 44. Spray, J. G. (1988) Contrib. Mineral. Petrol. 99, 464-475. prediction of real earthquakes, the prediction of these model 45. Shimamoto, T. (1994) J. Tectonic Res. Group Japan 39, 103-114. earthquakes is not an easy task. It is encouraging for the 46. Brune, J., Brown, S. & Johnson, P. (1992) Tectonophysics 218, 59-67. usefulness ofthe models that they are similar in manyways to 47. Heaton, T. H. (1990) Phys. Earth Planet. Inter. 64, 1-20. the behavior of real faults. However, even if in one 48. Beeler, N. M. & Tullis, T. E. (1996) Bull. Seismol. Soc. Am., in press. principle 49. Perrin, G., Rice, J. R. & Zheng, G. (1994) J. Mech. Phys. Solids, 43, could predict the model earthquakes, this may be impossible 1461-1495. in a realistic field setting. This is due to one's inability to place 50. Brune, J. N., Henyey, T. L. & Roy, R. F. (1969) J. Geophys. Res. 74, instruments close enough to the accelerating part of the fault 3821-3827. to detect the motions and to recognize an accelerating slip 51. Lachenbruch, A. H. & Sass, J. H. (1973) Proceedings of the Conference on when it occurs via discrete events. Tectonic Problems of the San Andreas Fault System, eds. Kovach, R. L. & pattern Nur, A. (Stanford Univ. Press, Palo Alto, CA), pp. 192-205. The mainshock grows out of accelerating local creep and 52. Lachenbruch, A. H. & Sass, J. H. (1980) J. Geophys. Res. 85, 6185-6222. failure as the stress reaches sufficiently high levels,just as in the 53. Hickman, S. H. (1991) Rev. Geophys. Suppl. 759-775. laboratory experiments. The modeling suggests that monitor- 54. Zoback, M. D., Zoback, M. L., Mount, V. S., Suppe, J., Eaton, J. P., Healy, ing microseismicity may be the most sensitive way to detect the J. H., Oppenheimer, D., Reasenberg, P., Jones, L., Raleigh, C. B., Wong, failure. L. G., Scotti, O. & Wentworth, C. (1987) Science 238, 1105-1111. growing 55. Rice, J. R. (1992) Fault Mechanics and Transport Properties of Rocks, Academic Press, London, 476-503. Much of the modeling on which this work is based was done in 56. Stuart, W. D., Banks, P. O., Sasai, Y. & Liu, S.-W. (1995) J. Geophys. Res. collaboration with William Stuart and I am grateful for his generosity 100, 101-110. in sharing his computer programs and his insight. My work was 57. Agnew, D. C. (1986) Rev. Geophys. 24, 579-624. supported by U.S. Geological Survey Grants 1434-93-G-2278 and 58. Johnston, M. J. S., Borcherdt, R. d. & Linde, A. T. (1986) J. Geophys. Res. 1434-94-G-2422 and by National Science Foundation Grants EAR- 91, 11,497-11,502. 88-16791, 9206649, 9220005, and 9317038. 59. Wyatt, F. K. (1988) J. Geophys. Res. 93, 7923-7242. Downloaded by guest on October 1, 2021