Stress triggering and earthquake probability estimates

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109,, doi:10.1029/2003jb002437, 2004 Stress triggering and earthquake probability estimates Jeanne L. Hardebeck 1 Institute for Geophysics and Planetary Physics, Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California, USA Received 6 February 2003; revised 17 February 2004; accepted 25 February 2004; published 22 April 2004. [1] Stress triggering and fault interaction concepts are beginning to be incorporated into quantitative earthquake probability estimates. However, the current methods are limited in their range of compatible earthquake nucleation models. I introduce a new general method for translating stress changes into earthquake probability changes, which can potentially be used with any physical fault model. Given the large uncertainties in earthquake probability calculations, it is unclear whether the small probability changes resulting from stress transfer are significant and meaningful for the purposes of seismic hazard assessment. I present a case study of the effect of the 1992 M7.3 Landers, California, earthquake on the probability of a major earthquake on the San Andreas Fault in southern California. This is a large event in a well-studied region, which should maximize the probability change signal and minimize the uncertainty. Even for this scenario, the earthquake probability change for a 30 year period is not significant with respect to the initial uncertainty in the earthquake probability. This example, comparisons with other studies, and exploration of parameter space imply that the probability changes due to stress triggering are significant only for time intervals which are short compared to the repeat time of the target fault. Therefore stress change calculations will be useful in long-term seismic hazard assessment only for low slip-rate faults. Otherwise, stress triggering calculations are best utilized in the short-term immediately following a major earthquake. INDEX TERMS: 7209 Seismology: Earthquake dynamics and mechanics; 7223 Seismology: Seismic hazard assessment and prediction; 8164 Tectonophysics: Stresses crust and lithosphere; KEYWORDS: stress triggering, seismic hazard Citation: Hardebeck, J. L. (2004), Stress triggering and earthquake probability estimates, J. Geophys. Res., 109,, doi:10.1029/2003jb002437. 1. Introduction [2] The field of probabilistic seismic hazard assessment (PSHA) is moving away from time-independent models, in which the probability of a damaging earthquake on a given fault or in a given region is assumed stationary. Timedependent models of earthquake probability have been introduced, based on various geologic and geophysical data, such as the intervals between prior earthquakes [e.g., Working Group on California Earthquake Probabilities, 1995], fault slip rates [e.g., Working Group on California Earthquake Probabilities, 1995], observed seismicity patterns [e.g., Jackson and Kagan, 1999, 2000; Wiemer, 2000; Wiemer et al., 2002], and geodetic strain rates [e.g., Shen and Jackson, 2000; Anderson, 2002]. [3] Stress triggering and fault interaction concepts are also beginning to be incorporated into time-dependent earthquake probability estimates [Stein et al., 1997; Toda et al., 1998; Parsons et al., 2000; Toda and Stein, 2002; Working Group on California Earthquake Probabilities, 1 Now at U.S. Geological Survey, Menlo Park, California, USA. Copyright 2004 by the American Geophysical Union. 0148-0227/04/2003JB002437$09.00 1999]. A stress change due to an earthquake can move other faults toward or away from failure, changing the probability of potential earthquakes on these faults. Two methods for computing the probability change due to a stress change currently appear in the literature, but are limited in their usefulness because of implementation flaws and restrictions to specific classes of earthquake nucleation models. The first goal of this paper is to develop a new general recipe for incorporating the effect of a stress change into earthquake probability estimates. [4] The prior method introduced by Stein et al. [1997] and Toda et al. [1998], currently the most commonly used technique in the literature, is based on the earthquake nucleation model of Dieterich [1994]. The underlying physical model is a population of faults slipping (imperceptibly slowly) following rate- and state-dependent fault friction rules derived from laboratory experimentation, and failing when a critical slip rate is reached. A change in stress will change the slip rate of a fault and hence its time of failure. This model allows one to compute the change in seismicity rate due to a change in stress. Stein et al. [1997] and Toda et al. [1998] translate the change in seismicity rate to a change in the probability of an earthquake on a given fault. One 1of16

could slightly alter their technique for use with a different fault nucleation model, given an equation relating seismicity rate change to stress change. However, the implementation of the methodology is flawed, as discussed in Appendix A, so it would be preferable to design a new, more correct methodology. [5] In the (Brownian passage time) BPT earthquake recurrence model [Matthews et al., 2002], a fault state variable (which can represent stress or some other quantity) increases linearly through the seismic cycle, with the addition of some level of noise, and the fault fails when the state variable reaches a critical value. The effect of a stress change can be implemented by increasing or decreasing the state variable an appropriate amount. A major advantage of the BPT model is that the recurrence time distribution of a fault and the effect of a stress change are computed from a single model. The drawback of this method is the assumption that the state variable controlling earthquake nucleation increases linearly with time, and that the effect of a stress step on the state variable is also linear and does not depend on the time in the seismic cycle at which it occurs. In the rate- and state-dependent nucleation model of Dieterich [1994], however, the state variable would be the fault slip rate, which does not increase linearly through the seismic cycle. Additionally, the effect of a stress change on the slip rate is nonlinear and is dependent on the time in the seismic cycle at which it occurs. The BPT method could potentially be altered to incorporate the rate- and statedependent nucleation model, but it was not designed to easily accommodate such models. In this paper, I develop a more general method that can potentially be used with any quantitative model of fault failure. [6] There are many uncertainties inherent in the computation of stress changes, for example, in the earthquake slip model, in the assumed earth structure, and in the orientation and probable slip direction of the target fault. Additional uncertainty arises from estimating the earthquake probability prior to the stress change, and from modeling the physics of earthquake initiation. These potentially large uncertainties raise the question of whether the small probability changes resulting from stress transfer are significant and meaningful for the purposes of seismic hazard assessment. Consequently, the second goal of this paper is to test the significance of computed probability changes with respect to the uncertainties. I will quantify this uncertainty by propagating the error in the input parameters through the probability change calculations. [7] I present an example study where the effects of uncertainty should be minimized: the effect of the 1992 M7.3 Landers, California, earthquake on the San Andreas Fault in southern California. The event is large and close to the fault, so the stress change signal is large. The Landers slip distribution is relatively well constrained [e.g., Wald and Heaton, 1994], minimizing the uncertainty in the stress change calculation. Additionally, the earthquake history and slip rate of the San Andreas Fault have been extensively studied (see the references compiled by the Working Group on California Earthquake Probabilities [1995]), minimizing the uncertainty in the pre-landers earthquake probability estimates. Even for this best-case example, I find that the earthquake probability change is not significant with respect to the initial uncertainty in the earthquake probability. 2. Method [8] I introduce a new method for incorporating stress changes into the estimation of earthquake probability. The idea is to map the probability density function (pdf) for the time of failure of a fault to a new pdf reflecting the effects of a change in stress. This new technique has an advantage over prior methods in that it can be used with any physical model of earthquake nucleation, and can therefore continue to be used as our knowledge of fault mechanics improves. It can also be used to directly compare the implications of different physical models. [9] The method takes three inputs: (1) the initial time-offailure pdf for an earthquake of defined magnitude on the fault of interest, (2) the computed stress change on the fault, and (3) a rule relating the post-stress-change time to failure of a fault to the pre-stress-change time to failure for any given stress change. This rule is derived from some quantitative model of earthquake nucleation, and must preserve the order of failure (i.e., if one fault is initially closer to failure than another, it will still be closer to failure if both faults experience the same stress change.) The method is modular in that it can be used for any pdf, any stress change, and any quantitative nucleation model. [10] The time-of-failure pdf can be thought of in two ways. It is usually understood to describe the probable time of failure of a single, real fault for which the exact failure time is not known. Alternatively, consider a collection of hypothetical faults, each with an exactly known time of failure. One of the hypothetical faults corresponds to the real fault, but we do not know which one, and each has an equal probability. The pdf for the time of failure of the real fault can be thought of as the density of the times of failure for an appropriate collection of hypothetical faults (Figure 1). [11] The effect of the stress change on the pdf can also be considered in terms of this collection of hypothetical faults. The stress change is applied to each of the hypothetical faults, and the given model of earthquake nucleation is used to map the pre-stress-change time of failure to the poststress-change time of failure. The stress change moves each hypothetical source toward or away from failure, shifting and perhaps deforming the distribution of their times of failure. The new distribution of times of failure is equivalent to the new time-of-failure pdf, including the stress change, for the real fault (Figure 1). It is impractical to generate a very large set of hypothetical faults and track each one individually, so the method is implemented equivalently but somewhat differently. [12] The implementation of the method is based on tracking a subset of the hypothetical faults, and filling in the probability between them. The hypothetical sources are initially sorted by their time to failure, and they remain in the same order of failure as they experience the same stress change (for example, Figure 1 of Dieterich [1994] illustrates this for the rate-and-state friction model.) Because the ordering must be preserved, the same number of hypothetical sources always fail between two given reference sources. Therefore the probability of the real fault 2of16

Figure 1. An example of the idea behind the new method for incorporating stress changes into earthquake probability estimates. The time-of-failure probability density function (pdf) for an earthquake on a given fault is assumed to be a log-normal distribution with T rep = 1000 years and s = 500 years (see Appendix B). The pdf can be thought of in terms of a collection of hypothetical faults, each with an exactly known time of failure. The numbered boxes represent these hypothetical faults. Each has an equal probability of being the real fault, so the density of their times to failure is equivalent to the pdf. The top panel shows a pdf (black curve) without the effects of a stress change. A stress change of Dt = 1 bar is applied at time t = 500 years. The stress change moves each of the hypothetical sources toward failure by a different amount, shifting and deforming the distribution of their times of failure, as shown in the bottom panel. The time shift for each hypothetical source is computed assuming the rate and state model of Dieterich [1994], with As = 1 bar and stressing rate _t = 0.01 bar/yr. The new distribution of times of failure is equivalent to the new pdf (black curve) for the real fault, including the stress change. The pdf shown in black is computed using the new method described in this paper. failing at a time between them is the same after the stress change as before. The new pdf can be constructed by keeping the area under the curve constant between each pair of reference sources. [13] Consider a stress step at time 0 (the time axis can always be shifted to be 0 at the time of the stress change). Let P be the probability that an earthquake would have occurred between time T0 old and T1 old, given the original pdf. This means that immediately before the stress step, there was a probability P that the fault would fail between two hypothetical reference sources with times to failure T0 old and T1 old. After the stress step, there should still be a P probability that the fault would fail between these two reference sources, which have new times to failure T0 new and T1 new. Therefore P will also be the probability that the event would occur between time T0 new and T1 new for the new pdf that includes the stress change. The new pdf, f new (t), with the stress change, can therefore be constructed from the original pdf, f old (t), without the stress change, by ensuring that Z T1old Z T1new P ¼ f old ðtþdt ¼ f new ðtþdt ð1þ T0 old T0 new for any pair of reference sources. [14] The implementation of the method is illustrated in Figure 2. Prior to the stress change, the two pdfs are the same, so f new (t) = f old (t) for t < 0. I then step through selected reference sources with increasing time to failure. The construction is simplest if the reference sources are evenly spaced for the new pdf, so I choose appropriate T0 new and T1 new accordingly, and then use the rule relating pre- and post-stress-change time to failure to find T0 old and T1 old. The function f old (t) is assumed to be known exactly, and P can be found by integrating f old (t) between T0 old and T1 old. Then the new pdf can be constructed by defining f new ðt0 new t < T1 new Þ ¼ P=ðT1 new T0 new Þ: ð2þ With closely spaced reference sources, f new (t) will approximate the continuous pdf. [15] For seismic hazard assessment, we are often interested in the conditional probability, the probability that the event will occur between time t i and time t j given that it did not occur prior to time t i. This is found from P cond ¼ R tj t i where f(t) =f old (t) orf(t) =f new (t). f ðtþdt R 1 t i f ðtþdt ; ð3þ 3of16

Figure 2. The implementation of the new method for incorporating stress changes into earthquake probability estimates. The probability of the true fault failing at a time between two given reference hypothetical sources should be the same before and after the stress change. (For instance, in Figure 1, there are always 19 hypothetical sources between sources 70 and 90, so the equivalent probability is also constant.) The pdf with the stress change can therefore be constructed from the pdf without the stress change as shown. The post-stress-change time of failure of each reference fault is found, and the probability P1 that the earthquake occurs before reference fault 1 is kept constant, as is the probability P2 that the earthquake occurs between reference faults 1 and 2, etc. With closely spaced reference faults, the constructed pdf will approximate a continuous pdf. The pdfs are the same as in Figure 1, with reference sources chosen every 20 sources (sources 70, 90, 110, etc., from Figure 1.) [16] This method can be implemented for any deterministic physical model for which the effect of a stress change on the time to failure of a fault can be quantified. Therefore the method could be used with coseismic static stress changes for anything from a simple critical failure stress model, to more complex fault constitutive models [e.g., Dieterich, 1994; Ruina, 1983; Perrin et al., 1995], to models in which pore fluid effects are also considered [e.g., Fitzenz and Miller, 2001]. Post-seismic stress changes, or long-term changes in stressing rate, could be implemented directly or through repeated application of appropriate stress changes for successive time steps. Coseismic dynamic stress changes also could be implemented, for physical models involving a change in fault state and hence time to failure, if these physical changes can be quantified and related to failure time. Stochastic earthquake nucleation models, such as the BPT model, cannot be directly implemented because of the deterministic nature of the methodology. [17] In this paper, I implement the method for coseismic static stress changes using the rate- and state-dependent earthquake nucleation model of Dieterich [1994]. The equations relating T old and T new are derived in Appendix B. Also shown in Appendix B is the integration of f old (t) fora log-normal pdf. The log-normal distribution was chosen for use in this paper because of its prevalence in the literature. It has the disadvantage that conditional probability decreases to zero at large times. However, I will not need to compute conditional probabilities at large times in the following examples. 3. Effect of the Landers Earthquake on the San Andreas Fault [18] I use a real example to test the significance of the computed earthquake probability change following from a stress change, with respect to the uncertainty in both the earthquake probability and the probability change. I choose a large earthquake in a well-studied region in order to maximize the probability change signal and minimize the uncertainty. Therefore this should be a best-case test of whether the computed earthquake probability changes are significant, and will indicate whether it is meaningful, at this time, to include stress change calculations in earthquake probability estimates. [19] The 1992 M7.3 Landers earthquake occurred in southern California just to the northeast of the San Andreas Fault (SAF), the major plate-boundary fault in the region (Figure 3). This event altered the surrounding stress field [e.g., Hauksson, 1994; Hardebeck and Hauksson, 2001] and triggered many earthquakes in both the near field [e.g., King et al., 1994; Hardebeck et al., 1998; Kilb et al., 2000] and the far field [e.g., Hill et al., 1993]. Because of the proximity of this earthquake to the SAF, there was much interest in the static stress change on the SAF due to Landers. Several studies [Harris and Simpson, 1992; Jaumé 4of16

Figure 3. Map of the region of this study, showing the M7.3 Landers mainshock surface rupture and the trace of the San Andreas Fault. Approximate boundaries for the San Bernardino segment of the San Andreas are indicated. and Sykes, 1992; Stein et al., 1992] computed relatively large stress changes on the San Bernardino segment of the SAF, which suggests it moved significantly toward failure. [20] Here I compute the change in the conditional probability of a major earthquake on the San Bernardino segment of the SAF resulting from the Landers-induced stress change. I use the method introduced above, and propagate the uncertainties in the input parameters to estimate the uncertainty of the computed probability change, using a Monte Carlo method. I then compare the probability change to the uncertainty in order to assess its significance. 3.1. Input Parameters [21] The input parameters, the stress change and the initial earthquake time-of-failure pdf, are taken from the literature. The uncertainty of each parameter is also taken from the literature, or estimated from the variation in the published values. The uncertainty of the time-of-failure pdf is represented by generating 6000 pdf s that span a range of possibilities. The rate- and state-dependent friction model of Dieterich [1994] is assumed to be the correct fault failure model, and the parameter As and its uncertainty are estimated from values in the literature. 3.1.1. Stress Change [22] I use the results of three studies [Harris and Simpson, 1992; Jaumé and Sykes, 1992; Stein et al., 1992] that compute the coseismic static stress change due to Landers as a function of distance along the SAF. The static stress change varies significantly along strike. I assume that the relevant value for stress triggering is the maximum Coulomb stress change, as this stress change may trigger rupture initiation. The Coulomb stress change, DCS, is defined as DCS ¼ Dt þ m 0 Ds; ð4þ where Dt and Ds are the change in shear and normal stress, respectively, and m 0 is the effective coefficient of friction. I use m 0 = 0.4, as this value is common in the literature [King et al., 1994]. [23] The maximum DCS for the San Bernardino segment of the SAF from the calculations of Harris and Simpson [1992], Jaumé and Sykes [1992], and Stein et al. [1992] is 5.9 bar, 7.5 bar, and 5.6 bar, respectively. From this, I infer that the maximum DCS on the SAF is 6.3 ± 1.0 bar (±1s). The use of the maximum Coulomb stress change implies the assumption that the next event will nucleate at the location of the greatest stress change, implying that the entire fault is equally prepared for failure. I make this assumption because we do not have any other information indicating where the next event on the SAF is likely to nucleate. [24] I have chosen not to include post-seismic stress changes in these example calculations. Post-seismic stress changes may occur due to visco-elastic relaxation, after-slip and/or the re-equilibration of pore fluid pressure. These additional stress changes may increase the over all stress change, and lead to an increase in probability change. However, the post-seismic stress change appears to be small relative to the coseismic stress change for this example. Pollitz and Sacks [2002] compute the stress change due to the visco-elastic response following Landers for a 15-year period. The total post-seismic stress change on the San Bernardino segment of the SAF is slightly greater than 1 bar, significantly less than the coseismic change. 3.1.2. Earthquake Probability [25] The second input to the calculation is the time-offailure pdf for the San Bernardino segment of the SAF, without the effect of the Landers-induced stress change. This is somewhat problematic because there is currently debate as to how slip in the region is partitioned between the SAF, the San Jacinto Fault (SJF) and the Eastern California Shear Zone. Some work implies a slip rate on the San Bernardino segment of 22 25 mm/yr [e.g., Weldon and Sieh, 1985; Bennett et al., 1996]. Other studies, however, find a high slip rate for the SJF [e.g., Savage and Prescott, 1976; King and Savage, 1983; Johnson et al., 1994; Kendrick et al., 2002], which implies a low slip rate, 15 mm/yr, on the San Bernardino segment of the SAF. I will present results for both the high-slip-rate and low-sliprate models. [26] I generate a suite of pdf s following the Working Group on California Earthquake Probabilities [1995]. They compute pdf s in three different ways, one based on the times of previous earthquakes obtained from paleoseismology, and two based on the observed slip rate of the fault and the estimated slip per event. I implement two of these methods, their dates paleoseismology method, and their renewal slip rate method. For each method, I generate 3000 pdf s, using both the high-slip-rate and the low-sliprate models. [27] The dates method is based on paleoseismic dates for previous large earthquakes on the segment. For the highslip-rate model, I use the set of earthquakes observed at Wrightwood and Pitman Canyon [Fumal et al., 1993; Seitz et al., 1997], which are given dates of 1200, 1450, 1590, 1690 and 1812 AD. For the low-slip-rate model, I use the two events observed at Plunge Creek [McGill et al., 2002], dated 1450 and 1630 AD. A log-normal distribution is fit to 5of16

the observed interevent times, including the interval from last event to the present. The remaining pdf s are generated by randomly choosing a new set of interevent times from this initial probability distribution and fitting a new lognormal distribution to them. [28] The renewal model is based on the observed slip rate and event offset of the fault segment. For the high-sliprate model, I use the v = 24.5 ± 3.5 mm/yr (±1s) estimated by Weldon and Sieh [1985]. For the low-slip-rate model, I take the minimum value of the 14 25 mm/yr range of Harden and Matti [1989] and assume a similar uncertainty, so I use v = 14 ± 3.5 mm/yr (±1s). From these values, the mean recurrence time is found, T r = u/v, where u is the assumed slip per event, chosen from the normal distribution u = 3.5 ± 0.5 m (±1s). The coefficient of variation for the log-normal distribution, which controls the width of the distribution (see Appendix B), is randomly selected from the distribution b = 0.5 ± 0.1 (±1s). 3.1.3. Rate and State Parameters [29] Two parameters must be chosen in order to use the rate- and state-dependent friction model of Dieterich [1994]. The first is As, where A is a friction parameter and s is the effective normal stress. The parameter As was estimated by Dieterich [1994] from the durations of aftershock sequences to be 0.1 of the mainshock stress drop, or on the order of 1 bar. Toda et al. [1998] estimate from seismicity rate changes that As is 0.35 ± 0.15 bar. I therefore choose As from either the distribution 1.0 ± 0.25 bar (±1s) or 0.35 ± 0.15 bar (±1s). [30] The second parameter is _t, the background stressing rate. The stressing rate follows directly from the chosen fault slip and recurrence time for the renewal model, given the assumed fault dimensions of the segment [Working Group on California Earthquake Probabilities, 1995]. For each pdf that is generated, the corresponding _t is used. For the dates method, the slip per event is chosen from the normal distribution u = 3.5 ± 0.5 m (±1s), and the stressing rate for each pdf is found from the mean recurrence interval and the estimated event stress drop. This is of course a simple approximation of the relationship between slip, slip rate, stress drop, stressing rate and recurrence time, which assumes that these parameters are constant in space and time, which may not be the case. 3.2. Conditional Probability Change [31] I compute the conditional probability change and its uncertainty using a Monte Carlo method. I randomly choose the time-of-failure pdf for the San Bernardino segment of the SAF from the suite of pdf s described above. Independently, I choose the stress change, DCS, and the rate-andstate friction parameter, As, from the distributions given above. [32] I find the conditional probability for a given time interval both with and without the Landers-induced stress change for each of the 6000 Monte Carlo iterations, using the new methodology introduced in this paper. Each pair of conditional probability values is used to compute the conditional probability change. From the results of all of the trials, the expected value of the conditional probability with and without the stress change, the expected value of the probability change, and the uncertainty of each of these values is determined. [33] I compute the conditional probability for time intervals starting both at the time of the Landers mainshock, June 28, 1992; and at the start time of the Working Group on California Earthquake Probabilities [1995] study, January 1, 1994. The lengths of the time intervals range from several days to 30 years. This allows me to investigate any time-dependence in the significance of the probability calculations. 3.3. Results [34] The conditional probability change for an earthquake on the San Bernardino segment of the SAF for various time periods is summarized in Figure 4. Each panel shows the conditional probability for a set of different-lengthed time intervals, all beginning at the same time. The horizontal axis shows the end time of the time interval, so the length of the time period increases to the right. The top panel shows the probability change for time intervals beginning the day of the Landers mainshock, representing calculations one might have made in 1992 right after the mainshock, for the next 1 year, 5 years, 10 years, etc. The bottom panel shows the results for time intervals beginning January 1, 1994, representing calculations one might have made in 1994, for instance had stress interaction been included in the Working Group on California Earthquake Probabilities [1995] study. The second set of calculations includes the conditional information that the SAF did not rupture between 1992 and 1994. [35] For time periods beginning at the time of the Landers event, for all intervals longer than 10 days, the conditional probability change is remarkably stable at 0.1, for both the high- and low-slip-rate models. The uncertainty estimates from the Monte Carlo trials span a relatively narrow range of values, indicating that the conditional probability change is distinct from zero and significant with respect to its uncertainty. For time periods beginning in 1994, however, the conditional probability change is always <0.01, and the uncertainty estimates imply that it is indistinguishable from zero and hence not significant. The results are sensitive to the start time of the interval because the difference between the pdf s with and without the stress change are greatest immediately following the Landers earthquake. In other words, the highest probability of an event occurring on the SAF was between 1992 and 1994, the calculations starting in 1992 include this time, while the calculations starting in 1994 do not. [36] Next I compare the conditional probability including the stress change to the conditional probability not including the stress change, in order to test whether the conditional probability change is significant with respect to the uncertainty in the earthquake probability (Figure 5). Results are shown for time periods beginning at the time of the Landers earthquake. Although the probability increase is uniformly 0.1, this change is significant for short time periods, but not for longer time periods. For short time intervals, the conditional probability of an earthquake in that interval is very small, so an increase in probability of 0.1 is a substantial change. Hence, at short times, the confidence regions do not overlap. However, for longer time intervals, the conditional probability and its uncertainty are larger than 0.1, so the confidence regions for the probability with and without the stress change overlap considerably. The cross- 6of16

Figure 4. Conditional probability change for an earthquake on the San Bernardino segment of the SAF, due to the Landers earthquake. (top) The conditional probability change for time intervals beginning the day of the Landers mainshock, June 28, 1992. (bottom) The conditional probability change for time intervals beginning January 1, 1994. The horizontal axis shows the end time of the time interval, so the length of the time period for the conditional probability increases to the right. The median result for 6000 Monte Carlo trials and the middle 80% are shown for both the high-slip-rate and low-slip-rate models of the San Bernardino segment. (This figure does not resemble the 1/t curves in plots such as Figure 4B of Parsons et al. [2000], because it is not plotting the same thing. This plot uses different-length time periods with the same starting time, while those use equal-length time period with different starting times.) over point is around 5 15 years, depending on how much of an overlap is considered significant. Although the conditional probability is somewhat less for the low-slip-rate model, the significance of the probability change through time is similar. 4. Discussion [37] Whether or not a given conditional probability change is meaningful for seismic hazard assessment is somewhat subjective. I will assume that a probability change less than the computed 1s uncertainty in the probability is not a significant change. Adjusting an estimated earthquake probability within its 1s confidence range should not, for instance, lead to any significant (>1s) changes in calculated ground motion probabilities or major shifts in public policy related to the earthquake potential. [38] For the example presented above, examining the effect of the Landers earthquake on the SAF, the changes in earthquake conditional probability are only meaningful for short (<10 years) time intervals starting relatively soon (<0.5 year) after the mainshock. This implies that stress change calculations are useful only for short-term probability estimates immediately following a major earthquake, not for long-term probability estimates using past earthquakes. For example, for a calculation similar to the Working Group on California Earthquake Probabilities [1995] estimate of the earthquake potential on the San Bernardino segment of the SAF during 1994 2024, the effect of Landers raises the probability from 0.346 ± 0.177 to 0.353 ± 0.180, which is clearly not a significant increase. [39] Three other studies [Toda et al., 1998; Parsons et al., 2000; Toda and Stein, 2002] report the confidence ranges of their computed conditional probability changes, also using a Monte Carlo method to compute uncertainty. Two of these studies obtain only probability changes that I would not consider significant because the change is within the 1s uncertainty of the probability. Parsons et al. [2000] computed the conditional probability change for three faults near Istanbul, Turkey, for 10- and 30-year periods, none of which are significant at the 1s level. However, when the probability of an earthquake on any of the faults during a 10-year period is computed, the probability increase due to stress transfer is just barely significant. Toda and Stein [2002] compute the probability change for an earthquake on the Parkfield segment of the SAF due to two nearby 7of16

Figure 5. Conditional probability for an earthquake on the San Bernardino segment of the SAF, with and without the effect of the Landers earthquake, for time intervals beginning the day of the Landers mainshock, June 28, 1992. (top) High-slip-rate models for the San Bernardino segment. (bottom) Lowslip-rate model for the San Bernardino segment. The horizontal axis shows the end time of the time interval, so the length of the time period for the conditional probability increases to the right. The median result for 6000 Monte Carlo trials and the middle 80% are shown. (This figure does not resemble the 1/t curves in plots such as Figure 4B of Parsons et al. [2000], because it is not plotting the same thing. This plot uses different-length time periods with the same starting time, while those use equal-length time period with different starting times.) M 6.0 earthquakes. The 10-year conditional probability decrease is approximately half of the 1s uncertainty. [40] The only work to present conditional probability changes that are significant at the 1s level is Toda et al. [1998], which studied the effect of the 1995 M6.9 Kobe, Japan, earthquake on nearby major faults. (I consider their reported stress changes significant only for fault segments where the uncertainty ranges for their Weibull probability with and without the effects of Kobe are both given in their Table 3 and are nonoverlapping.) They found significant decreases in probability for 10- and 30-year periods for the western Arima-Takatsuki Tectonic Line (ATTL), the Naruto segment of the Median Tectonic Line (MTL), the Uemachi Fault and the Ikoma Fault. [41] There are three factors which may contribute to the significance of the probability changes found by Toda et al. [1998]. First, Toda et al. [1998] use a method that often overestimates the probability change (see Appendix A), which may make the probability changes appear more significant. Second, tests show that a stress decrease tends to lead to a larger and more significant probability change than a similar-sized increase (Figure 6). Third, the conditional earthquake probabilities for the faults studied by Toda et al. [1998] are very low compared to the probabilities for the SAF, because their estimated repeat times are much longer. A probability change added to a small probability is more significant than the same probability change added to a larger probability. This last effect is probably the most important, as Toda and Stein [2002] use the same method as Toda et al. [1998] and study a situation with a similar-sized stress decrease, but find a probability decrease that is not significant. [42] I further explore the significance of the conditional probability changes by computing probabilities over a range of parameter values (Figure 6). Each panel shows results for a stress change occurring at a different point in the seismic cycle. The percent change in conditional probability is shown as a function of the stress change, normalized by the seismic cycle stress, and the time interval, normalized by the length of the seismic cycle. The normalization means that Figure 6 applies to faults with any stressing rate. The shading indicates areas where the probability changes are not significant, assuming input parameter uncertainty on the order of that in the San Bernardino example. [43] As expected, larger stress changes produce larger and more significant probability changes. Probability changes tend to be larger and more significant for stress changes occurring earlier in the seismic cycle. This is because the 8of16

Figure 6. Contours indicate percent probability change as a function of the stress change Dt (normalized by the seismic cycle stress Ds) and the length of the time interval Dt (normalized by the event repeat time on the target fault Trep) which is assumed to start at the time of the stress change. Because of the normalization, these Figures are appropriate for faults with any stressing rate (any combination of Ds and Trep). Each panel shows results for a stress change occurring at a different time during the seismic cycle. The shaded region indicates roughly where the conditional probability change will not be statistically significant at the 1s level, assuming that the probability uncertainty is similar to that in Figure 5. I assume that the original probability distribution is log-normal with a coefficient of variation b = 0.5, and that As is 0.1 of the seismic cycle stress (following Dieterich [1994]). The symbols show results using data from this study (square), Toda et al. [1998] (triangles), Parsons et al. [2000] (circles), and Toda and Stein [2002] (diamond), given in Table 1. For each datum, I choose the panel which closest represents the time in the seismic cycle at which the stress change occurred, and plot the normalized stress change versus normalized time interval. The percent probability change can be read from the contours. If the datum falls within the shaded region, the probability change is probably not significant. 9of16

Table 1. Faults for Which Stress-Interaction Earthquake Probabilities Have Been Computed Dt, a Ds, b t s, c Dt, d e T rep, Probability Change Fault bar bar years years years Estimated f Published g This Study San Bernardino SAF h 6 100 180 30 146 20% 21% Parsons et al. [2000] Yalova 1 19 105 30 190 30% 50% Prince s Island 0.8 21 233 30 210 20% 35% Marmara 0.3 54 490 30 540 10% 18% Toda and Stein [2002] Parkfield SAF 0.3 2.2 17 10 22 25% 22% Toda et al. [1998] East ATTL i 2.82 7 400 30 1000 >500% 425% West ATTL 1.92 6 1000 30 1000 95% 99% Tokushima MTL j 0.09 10 400 30 250 0% 43% Naruto MTL 0.99 11 1000 30 380 50% 87% Wakayama MTL 0.068 8.5 1000 30 500 0% 11% Yamasaki 0.09 5 1130 30 1000 20% 15% Branch 0.36 5 1130 30 1000 45% 50% Hanaore 0.01 3 1100 30 1000 0% 2% Uemachi 0.21 4 1000 30 1000 35% 34% Ikoma 0.14 4 1000 30 1000 25% 24% a Static Coulomb stress change from the cited papers. b The seismic cycle stress, found from the repeat time and the estimated stressing rate given by Parsons et al. [2000], Toda and Stein [2002], or Toda et al. [1998, Table 2]. For the San Bernardino segment, I assume an earthquake stress drop of 100 bar. c The elapsed time since the last earthquake, from the cited paper. d The time interval for the conditional probability calculations. e The assumed repeat time of the fault, from the cited paper. f The estimated percent probability change, read from Figure 6. g The published percent probability change. For Parsons et al. [2000], the percent change between the 30-year interaction and background probability in Table 1. For Toda and Stein [2002], from Figure 12. For Toda et al. [1998], the percent change between the 30-year Weibull probability with and without Kobe from their Table 3. (For example, the probability for the Ikoma Fault goes from 0.0245 to 0.0185, or a change of 0.006 out of 0.0245, a 24% change.) h SAF, San Andreas Fault. i ATTL, Arima-Takatsuki Tectonic Line. j MTL, Median Tectonic Line. model of Dieterich [1994] implies a larger change in time to failure the earlier the stress step is applied. The probability change is also larger and more significant for shorter time intervals, because, for the fault model of Dieterich [1994], earthquake rate (and probability) change is greatest immediately following a stress change. For example, for a stress increase of 0.05 the seismic cycle stress (or 1.5 bar assuming a 30 bar earthquake stress drop) occurring at a time 75% of the way through the seismic cycle, the probability change is significant for time intervals less than 0.12 of the seismic cycle (15 years for a high-slip-rate fault like the SAF, >150 years for a low-slip-rate fault like those that ruptured in the Landers earthquake). For smaller stress changes of <0.03 of the seismic cycle stress (or <0.9 bar), however, the probability change is never significant. [44] The conditional probability change and its significance can be roughly determined using Figure 6, without having to implement the technique introduced above, as long as the repeat time and the time since the last event can be estimated. Data from the example in this study, as well as from Toda et al. [1998], Parsons et al. [2000], and Toda and Stein [2002], are shown in Figure 6, and the relevant parameters listed in Table 1. Only five faults (all from the Kobe earthquake study of Toda et al. [1998]) exhibit significant probability change. Two of these faults experienced relatively large stress changes of 0.3 0.4 of the assumed seismic cycle stress (the East and West sections of the ATTL), while the other three are relatively low slip rate faults with lower stress changes, decreases of <0.1 of the assumed seismic cycle stress (the Naruto segment of the MTL, a branch of the Yamasaki Fault, and the Uemachi Fault). [45] Taken together, these results imply that conditional probability changes computed from stress changes are usually meaningful only for time intervals which are short compared to the repeat time of the target fault. Therefore stress change calculations may be useful in long-term (30 year) seismic hazard assessment for low slip-rate faults. For high slip-rate faults, which, because of their more frequent earthquakes, are of more societal importance and tend to dominate PSHA calculations, stress changes are useful mainly for short-term probability estimates immediately following a major earthquake. This implies a limited usefulness for stress change modeling in PSHA. Stress triggering concepts would be of most practical value in hazard assessment as part of a short-term, near-real-time response to major earthquakes. The forecasts from stress change calculations should be tested against, and perhaps integrated with, short-term empirical earthquake probability estimates based on seismicity patterns [e.g., Wiemer, 2000; Wiemer et al., 2002] or other data. [46] Long-term (100 year) decreases in regional seismic activity, or stress shadows, have been observed following very large earthquakes, such as the 1857 M8 1 4 Fort Tejon 10 of 16

and 1906 M8 1 4 San Francisco earthquakes on the SAF [Harris and Simpson, 1998]. In these cases, the computed earthquake probability change for the entire region may be significant for a much longer time than for the single-fault calculations discussed here. Even if the probability decrease for a single fault weren t significant, the integrated probability change over a large population of such faults might be. Additionally, the significance of the probability change would be larger than for the Landers example because of the much larger stress changes produced by these events. There also may be long-lasting or permanent effects of large stress changes that are not predicted by the rate and state model, which allows a permanent seismicity rate change only for a permanent change in stressing rate. Alternatively, the stress shadows may be the result of long-term postseismic stress changes not included in a simple stress-step model. 5. Conclusions [47] I have introduced a new technique for incorporating stress changes into time-dependent estimates of earthquake probability. It is a general method, in that it can be used with any quantitative model of earthquake nucleation. This is an improvement over prior methods, which are either flawed or based on specific physical models, as the new technique can continue to be used as the field of fault mechanics advances. [48] I test whether the computed earthquake probability change following a stress change is significant with respect to the uncertainty. I do this by propagating the uncertainty in the input parameters using a Monte Carlo technique. As an example study, I compute the probability change for a major earthquake on the San Bernardino segment of the SAF due to the stress change induced by the 1992 M7.3 Landers earthquake. This scenario should have the largest signal and lowest uncertainty in southern California. Even for this case, the probability change is significant only for time intervals less than 10 years starting within 0.5 year after the Landers earthquake. [49] This example, comparisons with other studies, and exploration of parameter space imply that the probability changes are generally significant only for time intervals which are short compared to the repeat time of the fault. Therefore stress change calculations will be useful in longterm seismic hazard assessment only for low slip-rate faults. Otherwise, stress changes are best utilized in the short-term immediately following a major earthquake. Appendix A: Previous Approach [50] The technique for incorporating stress changes into the estimation of earthquake probability that is predominant in the literature [Stein et al., 1997; Toda et al., 1998] is implemented incorrectly. The resulting errors in the computed conditional probability can be seen by comparing the results with synthetic data sets (some examples appear in Figure A1). The error is dependent on the input parameters, but for most parameter values, it is an overestimate. A1. Method [51] The method is based on the model of Dieterich [1994], which quantitatively relates changes in stress to changes in seismicity rate, using an empirical fault friction law derived from laboratory experiments. The methodology summarized here is explained in detail BY Stein et al. [1997] and Toda et al. [1998]. It assumes that a probability density function (pdf) for the time of failure of a fault, without the effects of the stress change, is given, and that the stress change is a step change in shear stress. The earthquake probability including the stress change is then computed in two steps. A1.1. Step 1 [52] In the first step, the initial pdf is shifted forward in time by Dt, the time that it would take to accumulate the given stress change at the background stressing rate. The change in time to failure is Dt = Dt/_t, where Dt is the shear stress change and _t is the stressing rate. A negative Dt results in a shift back in time. [53] In preparation for the second step, the conditional probability, P cond, for the time period of interest (t i to t j )is computed using the shifted pdf. This probability can also be thought of as an earthquake rate, r, for a stationary Poisson model: r ¼ 1 lnð1 P cond Þ: ða1þ t j t i A1.2. Step 2 [54] The second step consists of perturbing this earthquake rate, r, based on the relationship between stress and seismicity rate found by Dieterich [1994]. For a stress step at time t i, the expected number of events occurring between time t i and time t j is N ¼ r ðt j t i Þ þ t a ln 1 þ ½expð Dt=AsÞ 1Šexp½ t j t i =ta Š ða2þ expð Dt=AsÞ where t a = As/_t, A is a rate-and-state friction parameter (generally in the range of 0.005 to 0.02), s is the effective normal stress, and r is the background earthquake rate, found here from equation (A1). (If the time of interest does not start at the time of the stress change, two equations of the same form as equation (A2) can be subtracted to find the expected number of events for the time period [see Toda et al., 1998].) The net conditional probability of an event occurring between time t i and t j can be found from N assuming a stationary Poisson model: P net ¼ 1 expð NÞ: ða3þ A2. Problems [55] There are three problems with the implementation of the method. A2.1. Inconsistent Physical Model [56] The most fundamental problem is the lack of a consistent underlying physical model. Steps 1 and 2 implicitly assume different physical models for fault failure which are mutually inconsistent. [57] Step 1 is based on the model that a fault fails when it reaches a critical stress threshold. In this model, a 11 of 16