An Empirical Model for Earthquake Probabilities in the San ... · WG90, WG99, and WG02, respectively. These probability estimates have provided policy and decision-makers with a rational

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Bulletin of the Seismological Society of America, Vol. 93, No. 1, pp. 1–13, February 2003

An Empirical Model for Earthquake Probabilities in the San Francisco Bay

Region, California, 2002–2031

by Paul A. Reasenberg, Thomas C. Hanks, and William H. Bakun

Abstract The moment magnitude M 7.8 earthquake in 1906 profoundly changedthe rate of seismic activity over much of northern California. The low rate of seismicactivity in the San Francisco Bay region (SFBR) since 1906, relative to that of thepreceding 55 yr, is often explained as a stress-shadow effect of the 1906 earthquake.However, existing elastic and visco-elastic models of stress change fail to fully ac-count for the duration of the lowered rate of earthquake activity. We use variationsin the rate of earthquakes as a basis for a simple empirical model for estimating theprobability of M �6.7 earthquakes in the SFBR. The model preserves the relativemagnitude distribution of sources predicted by the Working Group on CaliforniaEarthquake Probabilities’ (WGCEP, 1999; WGCEP, 2002) model of characterizedruptures on SFBR faults and is consistent with the occurrence of the four M �6.7earthquakes in the region since 1838. When the empirical model is extrapolated 30yr forward from 2002, it gives a probability of 0.42 for one or more M �6.7 in theSFBR. This result is lower than the probability of 0.5 estimated by WGCEP (1988),lower than the 30-yr Poisson probability of 0.60 obtained by WGCEP (1999) andWGCEP (2002), and lower than the 30-yr time-dependent probabilities of 0.67, 0.70,and 0.63 obtained by WGCEP (1990), WGCEP (1999), and WGCEP (2002), respec-tively, for the occurrence of one or more large earthquakes. This lower probabilityis consistent with the lack of adequate accounting for the 1906 stress-shadow in theseearlier reports. The empirical model represents one possible approach toward ac-counting for the stress-shadow effect of the 1906 earthquake. However, the discrep-ancy between our result and those obtained with other modeling methods underscoresthe fact that the physics controlling the timing of earthquakes is not well understood.Hence, we advise against using the empirical model alone (or any other single prob-ability model) for estimating the earthquake hazard and endorse the use of all credibleearthquake probability models for the region, including the empirical model, withappropriate weighting, as was done in WGCEP (2002).

Introduction

The U.S. Geological Survey has estimated the proba-bility of occurrence of large earthquakes in the San Fran-cisco Bay Region (SFBR) in a series of Working Group stud-ies (WGCEP, 1988; WGCEP, 1990; WGCEP, 1999; WGCEP,2002) (Fig. 1). Hereafter, we refer to these studies as WG88,WG90, WG99, and WG02, respectively. These probabilityestimates have provided policy and decision-makers with arational framework for assessing earthquake hazard and mit-igating earthquake risk in the region. At the heart of theprobability studies are statistical models of earthquake oc-currence on the region’s major faults that are based on geo-logic and geodetic data, the historical seismic record, andthe current understanding of the physical processes thatcause earthquakes.

The Working Group studies reveal an evolution in boththe methods and the quantity and quality of data used inearthquake probability modeling. On the data side, the evo-lution includes an increase in the number of fault segmentsin the SFBR that have been trenched to estimate long-termslip rates and the dates of large pre-1850 events (WG02);improved knowledge of deformation in the SFBR, in partderived from advances in Global Positioning System mea-surements (Prescott et al., 2001); increased information onhistorical seismicity (Bakun, 1999); an association of earth-quakes with faults in the SFBR (Wesson et al., 2002); and anew understanding of the 1836 event (Toppozada and Bor-chardt, 1998) and of the 1868 event (Yu and Segall, 1996).On the physics side, the evolution involves increasingly

2 P. A. Reasenberg, T. C. Hanks, and W. H. Bakun

Figure 1. Map showing the seven faults in the SanFrancisco Bay region (SFBR) characterized in WGCEP(1999) and WGCEP (2002) and considered in this ar-ticle. The SFBR is the area enclosed by dashed lines.

complex fault models that allow multi-segment ruptures andinclude the effect of aseismic slip (creep) and probabilitymodels that attempt to take account of the interactions be-tween faults. The present state of fault-interaction modelingin the SFBR—especially as it pertains to the 1906 earth-quake—is the primary concern of this article.

The occurrence of an earthquake redistributes stressthroughout the region, which in turn can affect the rates andprobabilities of subsequent earthquakes on nearby faults(Reasenberg and Simpson, 1992; Stein et al., 1997; Toda etal., 1998; Harris and Simpson, 1998). In the most generalterms, such stress effects increase with the size of the earth-quake and diminish with distance and time. Stress changesproduced by the largest earthquake in the SFBR’s historicalrecord—the 1906 San Francisco earthquake—are thought tohave caused the precipitous drop in earthquake activity inthe region (Sykes and Nishenko, 1984; Jaume and Sykes,1996) and may still be present today (Fig. 2). WG88 did notconsider the effect of stress changes produced by the 1906earthquake, but time-dependent models for earthquake prob-ability can take such effects into account in a variety of ways.WG90 estimated the change in loading stress on segmentsof the San Andreas fault caused by slip in the 1989 Loma

Prieta earthquake, but did not consider the changes causedby slip in 1906. WG99 and WG02 were the first studies toconsider and quantify the effects of the 1906 earthquake onearthquake probabilities in the SFBR. These studies usedelastic-halfspace dislocation calculations of the stresschanges and considered the potential effects of visco-elasticand rate-state-friction models as well. As expected, theseapproaches led to lower regional probabilities than were ob-tained when the effects of the 1906 earthquake were ignored.In this article, however, we argue that a probability model(such as those used by WG99 and WG02) that does notsufficiently explain the post-1906 regional quiescence mayalso overestimate the likelihood for earthquakes in the next30 yr.

We propose an alternative modeling approach for esti-mating the regional effects of the 1906 earthquake that relieson historical earthquake activity in the SFBR. Specifically,we consider the activity to result from the interactions of asystem of sub-parallel faults subjected to steady tectonicloading punctuated by stress changes (due to earthquakes)that occur over a range of length and time scales. Such asystem will tend to be inertial in its production of earth-quakes. That is, on a regional scale, the rate of earthquakeswill tend to change slowly when stressing rates are low (dur-ing periods of primarily tectonic loading) and increase ordecrease rapidly when the regional stressing rate is large andsudden, as happens in a large or great earthquake. We sug-gest that sudden and long-lived regional quiescence in theSFBR after the 1906 earthquake and the 10- to 15-yr periodsof quiescence after the 1838, 1868, and 1989 earthquakes(Sykes and Jaume, 1990; Toppozada et al., 2001) are ex-amples of such behavior. A corollary of this model is thatthe seismicity in the near future of such a system will (in theabsence of a major stress-changing event) be similar to thatof the recent past. In other words, we propose to use thecharacter of the recent earthquake activity as a proxy for thetime dependence of the physical processes governing faultinteractions. When we apply this approach to the SFBR,where earthquake activity has been low for nearly a century,we obtain a lower estimate of the regional probability forlarge earthquakes than obtained by WG02 and other Work-ing Groups. However, we advise against using the empiricalmodel alone (or any other single probability model) for es-timating the earthquake hazard in the region. Until the phys-ics involved in earthquake occurrence in the SFBR is betterunderstood, a prudent basis for estimating the earthquakehazard in the SFBR should combine the results of all credibleearthquake probability models for the region (including theempirical model), with appropriate weighting, as was donein WG02.

The WG99 and WG02 Models

WG99 and WG02 set out to estimate the probability ofone or more M �6.7 earthquakes in the SFBR in the 30-yrtime interval 2002–2031—not only for the region as a whole

An Empirical Model for Earthquake Probabilities in the San Francisco Bay Region, California, 2002–2031 3

Figure 2. (a) M � 5.5 earthquakes in SFBR from 1836 to 2000 (Bakun, 1999). (b)Various estimates of annual rate of earthquakes (M � 6.7) in SFBR from 1850 topresent: estimate based on a 15-yr sliding window over Bakun’s (1999) catalog (lightred line and smoothed heavy red line); long-term mean regional rate determined byWGCEP (2002) (dashed blue line); time-varying rate determined with WGCEP’s (2002)BPT model with stress interaction effects due to the 1906 and 1989 earthquakes (solidblue line); rate estimated with CNSS (2000) catalog (M � 3.0) 1942–1998, in sixintervals (green lines).

but also on a fault-by-fault (and fault segment-by-fault seg-ment) basis—by constructing a regional model composed ofseven characterized faults (each consisting of defined seg-ments that can rupture alone or in combination) and a back-ground that includes all other sources of earthquakes in theregion.

Long-term average rates of earthquakes of a variety ofsizes were built from segment-by-segment and fault-by-faultbehavior and summed across the SFBR to provide, in theend, regional rates of M �6.7 earthquakes and their sizedistribution. Because the WG99 and WG02 long-term mod-els are based mainly on geological data, the historical andinstrumental records of earthquakes in the SFBR can provideindependent checks. (Because the WG99 and WG02 long-term models differ in only minor ways, we will refer here-after only to the WG02 model.) Although this period of timeis just a fraction of one regional earthquake cycle, as definedby the mean time interval between repeating, 1906-sizeearthquakes, the model’s average rate of M 6.7 and largerevents and b-value of 0.9 are consistent with the historicaland instrumental records (Bakun, 1999).

Average rates of occurrence and their correspondingPoisson probabilities for M �6.7 earthquakes in the SFBR,however, are not the primary focus of the Working Groupstudies. Rather, the Working Groups have focused on theprobability of large earthquakes on individual faults during

specified contemporary intervals. From both theoretical con-siderations and empirical observations of the rate of seis-micity before and after the 1906 earthquake, it is generallyaccepted that the seismicity rate in the SFBR varies through-out a cycle of perhaps 250–500 yr (the average recurrencetime for a 1906-type rupture), perhaps spending little if anytime near its long-term average rate (Tocher, 1959; Ells-worth et al., 1981; Jaume and Sykes, 1996).

Superimposed on the 250–500 yr regional cycle areshorter periods of accelerated seismic activity observed be-fore the 1868 and 1989 earthquakes and quiescence afterthem (Sykes and Nishenko, 1984; Sykes and Jaume, 1990;Jaume and Sykes, 1996; Toppozada et al., 2001). The cyclicvariations in seismicity illustrate the need—and provide abasis—for time-dependent probability models.

The usual approach to modeling time-dependent earth-quake probability on a fault segment compares a distributionof recurrence times and the time since the last earthquake.For example, because it has been only 96 yr since the 1906earthquake, which repeats every 250–500 yr, the probabilityin a recurrence model of a repeat of the 1906 earthquakeduring the next 30 yr is judged to be 0.17, lower than thelong-term average probability of 0.20 (WG02).

WG02 used a Brownian Passage Time (BPT) model(Ellsworth et al., 1999; Matthews et al., 2002) to representearthquake recurrence on their characterized faults. The BPT


model is a statistical representation of the classic loading/release cycle for earthquakes (Lawson, 1908), which iswidely believed to underlie the brittle failure behavior of allfault systems. In the BPT model, fault loading is the sum ofa constant-rate loading process and random (Brownian) fluc-tuations. The distribution of recurrence times in the BPTmodel is similar to a lognormal distribution. Here, we com-pare the details of those models with our empirical model.

The 1906 Stress-Shadow and the WG02 Model

WG02 considered the effects of stress interactions—how the displacement in an earthquake on one fault segmentmay affect the likelihood of failure on adjoining or nearbyfault segments. Of particular concern is the length of thestress-shadow cast by the 1906 earthquake. The fulcrum ofthis concern is whether or not the SFBR has now fullyemerged from the 1906 stress-shadow, emerged on parts ofthe region (including the background) but not others, or re-mains fully under its influence. Several studies bear on thisquestion. Sykes and Nishenko (1984) argued, based on in-creased activity (1955–1982) east and southeast of San Fran-cisco, that this part of region was emerging from the stress-shadow. Bakun (1999) noted that the seismic moment ratein the decades before 1906 was 10 times that in the decadesafter 1906, but that the moment rate since 1977 has beencomparable to the pre-1906 rate.

The WG02 long-term regional model rate of 0.031 M�6.7 (95% confidence range, 0.024–0.041) yr�1 is consis-tent with the (admittedly brief) historical record. The occur-rence of four earthquakes since 1838 allows us to reject, with95% confidence, regional rates lower than 0.008 or greaterthan 0.047 (Fig. 2). However, the 84-yr-long (M �6.7) qui-escence from 1906 to 1989 is probably not consistent withthe recurrence probability model (including stress interac-tions) considered by WG99 and WG02. This model (BPT-step) includes Coulomb stress changes calculated for elastic-plate dislocation models of the 1906 and 1989 earthquakes.In the model, the regional drop in the 30-yr conditional prob-ability (from 0.70 to 0.44) in 1906 reflects both the 1906interaction effects (on other faults) and the “direct” relaxa-tion of the San Andreas fault due to the slip on it in 1906(Fig. 2b). By approximately 1990, the model’s regionalprobability has returned to its 1850–1900 average level. Theprobabilities calculated for the time since 1906 are too highto be consistent at the 90% confidence level with the 84-yr-long quiescence from 1906 to 1989.

A reliance on elastic-plate calculations for estimatingthe Coulomb stress changes in the BPT-step model may havelead to the underestimation of the 1906 stress-shadow. Re-alizing this, WG02 considered applying the potential effectsof visco-elastic and rate-state-friction calculations in mod-eling the 1906 stress change (Kenner and Segall, 1999; Par-sons, 2002). However, these models were not used and theyalso fell short of explaining the duration of the post-1906quiescence (WG02). Hence, in the absence of an adequate

physical model for representing the stress effects of the 1906earthquake, we seek a surrogate, empirical model.

Empirical Model

In this section we describe an approach to modeling the1906 stress effects that uses the time variation of the his-torical regional seismicity rate as a proxy for the time-dependent physical interactions between faults. We explorethis approach by examining in detail its application to theSFBR, in conjunction with the long-term (WG02) faultmodel.

We noted in the previous section that the time depen-dence in the BPT model derives from the assumed quasi-regular, quasi-identical ruptures of a suite of specified faultsegments and combinations of segments. These renewalmodels require (at least bounding) knowledge of the timesof last ruptures, an estimate of the mean recurrence times ofsegment ruptures, and an estimate of the intrinsic aperiodic-ity of the recurrence process. Simpler probability modelsmake fewer assumptions and require fewer data to constrainthem, but they provide correspondingly fewer details aboutthe time-dependent probability on individual fault segments.The simplest probability model is the Poisson model. Withno assumptions made about the timing of each segment rup-ture other than its mean long-term rate, the Poisson proba-bility is time independent.

Slightly more complex than the Poisson is the time-varying (or nonstationary) Poisson model (Ross, 1983, Mat-thews and Reasenberg, 1988), which we use here. Like thePoisson model, the time-varying Poisson model allows us toavoid explicitly specifying the timing of the recurrence ofindividual fault segments or ruptures, yet still allows the ratein the region to vary with time. We specify the model bydefining a function, f(t), with mean value 1.0, that modulates(multiplies) the long-term regional mean rate. This modelassumes that there is some spatial coherence (fault to fault)in the earthquake activity, but says nothing about whetherthis coherence is due to stress interactions, time-varyingdriving forces, or anything else. Our model describes onlythe time dependence; the relative magnitude distributions ofthe rupture sources, which are defined in the long-term faultmodel (WG99, WG02), are time-independent.

We define f(t) as the ratio of the time-varying regionalrate at time t to the WG99 long-term regional rate (M �6.7)of 0.031 yr�1. For probability estimates for the coming 30yrs, we extrapolate f(t) to the period 2002–2031 using a va-riety of models. We use the mean of these extrapolated val-ues of f(t) as a multiplicative factor, which is applied to thelong-term rate for the region, the individual, characterizedfault systems, and the background. Thus, for each fault, thebackground, and the region as a whole, WG02’s relativecontributions of the characterized ruptures and the smaller,uncharacterized earthquakes are preserved; the rate densityat all magnitudes varies with f(t), as shown in Figure 3.

In calculating f(t) for the historical period, we estimate


Region (b = 0.9)

Fault A

Fault B

Log

Num

ber

Magnitude

f(t)

f(t)

f(t)

Background

f(t)

Figure 3. Cartoon illustrating the modulation ofthe long-term average regional rates in the SFBR, us-ing the time-varying rate function, f(t). The curvesrepresent the long-term mean rate of earthquakes inthe region, on two hypothetical faults in the region,and in the background. The up–down arrows repre-sent the hypothesized change with time, f(t), that eachfault, the background, and the region experience. Allfour of these curves move up and down together, withf(t). Here, Fault B, like the Calaveras fault, is highlyproductive of smaller earthquakes (with relativelyhigh b-value), and has two characterized rupturesdominating its larger magnitude spectrum. Fault A,like the San Andreas fault, has fewer small events, alower b-value, and one, larger-magnitude character-ized rupture. The regional curve is the sum of allfaults and the background.

Figure 4. Annual rate of M � 3.0 earthquakes inCNSS catalog for SFBR, as represented by the meanrate in non-overlapping 1-yr intervals. Large spike inthe rate estimated with the raw catalog (solid line) in1989–1990 reflects aftershocks of Loma Prieta earth-quake. Smaller spikes in 1986, 1984, and 1979–1980correspond to Mt. Lewis (M 5.6), Morgan Hill (M6.2) and Coyote Lake (M 5.7) and Livermore (M 5.8)aftershock sequences, respectively. Declustering thecatalog (broken line) effectively removes these after-shock sequences. Even with aftershock sequences re-moved, an up-trend in regional rate is apparent start-ing in the 1970s. Rate (declustered) after 1989 isbelow the mean declustered rate (horizontal line),suggesting a post-Loma Prieta quiescence (Toppo-zada et al., 2001).

the changes in regional rate of M �6.7 over time using theoccurrence of smaller earthquakes and a Gutenberg–Richter(G–R) scaling relation with b � 0.9 (Bakun, 1999). Our useof a G–R model is limited to this step alone, where we cal-culate the ratio of the regional M �6.7 rate, as a function oftime, to its long-term mean. Specifically, we are not assum-ing that the rate of larger earthquakes on a fault is definedby the regional G–R relation, or by a fault-specific G–R re-lation. Rather, we assume that the M �6.7 rate on a fault isequal to the modeled long-term rate on that fault (which ingeneral may have a complex magnitude distribution, as dic-tated by its particular rupture model) multiplied by f(t).

Estimation of f(t) for the Nineteenthand Twentieth Centuries

We use two earthquake catalogs to estimate f(t) for thehistorical period. The catalog of M �5.5 earthquakes inSFBR since 1850 (Bakun, 1999) provides uniform coveragein the region complete to M 5.5 over an interval that spansthe 1906 earthquake (Fig. 2a). This catalog is useful for char-acterizing the seismic quiescence following the 1906 earth-quake, or rate increase preceding it, or both. We also use theCouncil of the National Seismic System (CNSS, 2000) cat-

alog of M �3.0 earthquakes in the SFBR since 1942 (Fig.4), the year when the University of California began listingmagnitudes (but not moment magnitude) for all locatedevents in the region. While this catalog cannot capture thefull effect of a 1906 stress-shadow, it serves as a check onthe contemporary rate of earthquake activity estimated withthe Bakun (1999) catalog and provides an estimate of themost recent rates.

We estimate the time-varying regional rate of the M�5.5 earthquakes in Bakun’s (1999) catalog using a slidingaveraging window. A small window width h nearly repro-duces the raw data, while a large window width provides asmoothed estimate of the rate. Matthews and Reasenberg(1988) explored the effects of under- and over-smoothingseismicity data in the search of its underlying rate. Whilethey chose an automatic method for selecting h, known asleast-squares cross validation, they noted that “there is noway to escape completely the subjectivity involved in selec-tion of the bandwidth parameter.”

Figure 5 shows the Bakun (1999) catalog smoothed withsliding windows of width h � 5, 15, and 50 yr. While theseestimates of f(t) vary in their details, they all show a regionalrate of M �5.5 earthquakes in the past 50 yr below its av-erage for most of the past 150 yr. Of course, with the 5-yearwindow, the average rate is now zero, since the last M �5.5event was in 1989. The 15- and 50-yr windows give esti-mates of the annual rate of M �6.7 earthquakes in the in-tervals 1985–2000 and 1950–2000 of 0.011 and 0.013, re-spectively, or 35% and 42% of the WG02 geologicallydetermined mean regional rate of 0.031. The similarity of


Mag

nitu

de

1850 1900 1950 2000

5.0

6.0

1838

1868

1906

1989

Rat

e

1850 1900 1950 2000

0.0

0.06

5 -year sliding window

Rat

e

1850 1900 1950 2000

0.0

0.06


Rat

e

1850 1900 1950 2000

0.0

0.06


7.0

8.0

Figure 5. Effect of the choice of averaging win-dow width in estimating f(t). (a) Earthquakes M �5.5 in catalog of Bakun (1999). (b) Average rate ofM � 6.7 earthquakes estimated from the M � 5.5earthquakes in Bakun (1999) using an h � 5 yr slid-ing averaging window and extrapolated to M � 6.7through a Gutenberg–Richter relation with b � 0.9.(c) Same, except averaging window width is h � 15years. (d) Same, except averaging window width is h� 50 years. The windows are lagging, meaning thateach point represents the rate averaged over the hyears before it.

these contemporary (15- and 50-yr) average rate estimates,while perhaps fortuitous, suggests to us that the estimatesare not very sensitive to the choice of smoothing windowwidth. We use the 15-yr average in the following analysisand in Figures 2b and 6. The annual rate of M �6.7 earth-quakes in the SFBR from 1850–2000 (estimated from theM �5.5 population with b � 0.9) is 0.030 yr�1 or 97% ofthe geologically based estimate. Considering that the obser-vation period is less than one cycle for the region, this agree-ment may be fortuitous. Our estimates of f(t) based onM �3.0 and M �5.5 data are in good agreement (Fig. 6),suggesting that 0.9 is the appropriate b-value.

Extrapolating f(t) into the Twenty-first Century

To extrapolate f(t) to the period 2002–2031, we em-ployed a variety of models selected to span the range ofplausible possibilities. Since the 30-yr period of interest isshort compared with both the length of the earthquake cycleas defined by 1906-type earthquakes and the duration of theputative stress-shadow, the extrapolated rate is not particu-larly sensitive to the choice of extrapolation method used.The first six models are based on earthquake rates in theSFBR since 1906 (Table 1, Fig. 6). In model A, the extrap-olated rate in 2002–2031 is based on the mean rate of earth-quakes (M �3.0) in the CNSS (2000) catalog between 1942and 1998, the longest period for which that catalog is com-plete at this magnitude level. Model B uses the correspond-

ing mean rate between 1984 and 1998, the contemporaryperiod with the highest rate. Models C and D are based onthe mean rates of M �5.5 events in the periods 1906–2000and 1979–2000, respectively. Again, these are, respectively,the longest and highest-rate contemporary periods since1906. Models E and F are based on least-squares fits to theaverage rates in the CNSS catalog (M �3.0) in approxi-mately 10-yr-long bins in the intervals 1942–2000 and1970–2000, respectively.

The concept of a regional seismic cycle for the SFBR,as suggested by Mogi (1981) and Ellsworth et al. (1981)underlies models G through J. The hypothesized cycle iscontrolled by the occurrence of 1906-size ruptures andmarked by a period of quiescence after, and increased activitybefore, the cycle-controlling earthquake. Based on the re-gional seismicity, regional cycles for the SFBR have been con-sidered by Tocher (1959), Bufe and Varnes (1993), Jaumeand Sykes (1996). Such models seem credible in view of agrowing body of observations suggesting the stress change–induced suppression of earthquake activity elsewhere in Cali-fornia. With the use of elastic models of stress change, Harrisand Simpson (1996) suggested that the great 1857 Fort Tejonearthquake may have suppressed or delayed earthquake ac-tivity by at least 50 yr on faults in southern California thatwere relaxed by that earthquake. Simpson and Reasenberg(1994) estimated that the relaxation of stress on some faultsin the SFBR persisted for nearly 50 yr after the 1906 earth-quake. Reasenberg and Simpson (1997) observed a decreasein earthquake activity on the Hayward fault consistent withmodeled stress changes produced by the Loma Prieta earth-quake. Bakun (2000) showed that the rate of M �7.0 earth-quakes along the north coast of California, near the north-west end of the 1906 rupture, decreased after 1906.

Inspection of the earthquake activity in the SFBR (Fig.2a) suggests a post-1906 quiescence, or a pre-1906 buildup,or both. We assume that the drop in regional activity after1906 was an effect of the 1906 earthquake (presumably astress-shadow effect) and allow that this effect may still bepresent to some extent in the region. We assume that theregional cycle repeats and the region will eventually fullyemerge from the stress-shadow and return to pre-1906 levelsof activity before the next 1906-type earthquake. Againstthis backdrop of a region-wide seismicity cycle associatedwith 1906-size events, the occurrence of M �7 earthquakeswithin the region adds spatial and temporal heterogeneity toearthquake activity in the region that is not specifically ac-counted for in our model.

Following Bakun (1999), we represent the seismic cyclein the SFBR for a 1906-type earthquake by hypothesizing arepeat of the pre-1906 earthquake activity during the 55 yrbefore the next such event (Fig. 6). For the occurrence timeof this future event, we use 250 or 450 yr after 1906, esti-mates consistent with Bakun’s (1999) and WG02’s modeledmean recurrence time, respectively. Bakun (1999) used thistechnique to estimate a range for the seismic moment deficitto be released in future earthquakes before the next 1906-


Figure 6. Comparison of estimates of regional rate of M � 6.7 earthquakes (left axis) andcorresponding 30-yr probabilities (right axis). Black lines show empirical models A to J over thenext 400 yr, based on extrapolations of historical data (models A to F) and on four possiblerealizations of a regional seismic cycle model that are consistent with observed earthquake activityin the SFBR (models G to J). The hypothetical cycle has mean recurrence times of 250 and 450yr. The 1906 earthquake and its postulated successor are indicated with thin (solid and dashed)black lines representing the regional rate (M � 6.7) for their (assumed generic) aftershock se-quences (Reasenberg and Jones, 1989, 1994). For these cycles, the average rate observed in 1850–1905 is assumed to repeat in the 55-yr intervals ending in 2156 and 2351, respectively (solid redhorizontal lines). Smoothed estimated rate inferred from M � 5.5 earthquakes using a 15-yr slidingwindow is shown with solid red line. Estimates of regional rate based on M � 3.0 earthquakesare shown by green lines. The 30-yr conditional probability estimates made with WGCEP (2002)BPT model with stress effects associated with the 1906 and 1989 earthquakes are shown with blueline. The WGCEP (2002) Poisson (long-term) probability is dashed blue line.

Table 1Extrapolation of Regional Rate

Model Data Time Period Method

Mean rate(M � 6.7) in2002–2031*

Mean off(t) in

2002–2031†Mean of f(t)

over one cycle

A M � 3.0 1942–1998 mean 0.014 0.47 —B M � 3.0 1984–1998 mean 0.016 0.53 —C M � 5.5 1906–2000 mean 0.011 0.37 —D M � 5.5 1979–2000 mean 0.020 0.67 —E M � 3.0 1942–1998 linear trend 0.016 0.53 —F M � 3.0 1970–1998 linear trend 0.020 0.67 —G M � 5.5 1970–2156 seismic cycle—linear rise 0.027 0.90 1.00H M � 5.5 1970–2351 seismic cycle—linear rise 0.020 0.67 1.15I M � 5.5 1970–2156 seismic cycle—exponential rise 0.019 0.63 0.96J M � 5.5 1970–2351 seismic cycle—exponential rise 0.014 0.47 0.74

*Rates are extrapolated to M � 6.7 by assuming a Gutenberg–Richter relation with b � 0.9.†Based on a long-term mean annual rate (M � 6.7) of 0.031.

size event. Here, we use the technique to define additionalmodels for estimating f(t) in the next 30 yr. We begin withthe contemporary regional rate and hypothesize that the fu-ture rate will vary smoothly, ultimately increasing to the pre-1906 rate time-shifted 250 or 450 yr forward. In essence, we

are proposing possible scenarios for the shape of f(t) inMogi’s (1981) stages I and II.

For models G and H we assume a linear rate increasepassing through the contemporary rate (1970–2000 mean)and the assumed pre-2156 and pre-2356 average rates, re-


spectively (Fig. 6). Models I and J assume an exponentialrate increase, with a characteristic time constant arbitrarilychosen to be 50 yr, connecting the contemporary rate to thepre-2156 and pre-2356 rate. For each of these models, weuse the extrapolated or modeled mean rate between 2002and 2031 to estimate earthquake probability in that time in-terval. The means of f(t) over the interval 2002–2031 forthese scenarios are summarized in Table 1. The extrapolatedregional annual rates (M �6.7) in our 10 models range from0.011 to 0.027 and average 0.018, about midway betweenthe current annual rate of 0.012 and the WG02 modeledlong-term mean rate of 0.031. The 30-yr probability corre-sponding to an annual rate of 0.018 is 0.42 and the meanvalue of f(t) in this time period is � 0.58.f(t)

Models G–J each describe a complete seismic cycle, forwhich the mean seismic moment rate can be calculated. Inconstructing these models, we arbitrarily chose the mean re-currence time and shape of the f(t) over the cycle. The shapeof f(t) may be further constrained by requiring that its meanover one cycle equal 1.0. Such a constraint was not appliedin these hypothetical models; the means of models G–J rangefrom 0.74 to 1.15 (Table 1).

Deaggregating Probability onto SFBR Faults

If we wanted to estimate only the regional rate of earth-quakes in the next 30 yr, we would stop here and concludethat the regional rate will be slightly higher than the rate ofthe past 30 yr. However, we can extend the regional empir-ical model to one that deaggregates the rates (and 30-yr prob-abilities) onto the faults in the SFBR by assuming that seis-micity in the region is correlated. A regional correlation inactivity might be produced by regional stress changes pro-duced by the 1906 earthquake or by other changes in re-gional stress not associated with a SFBR earthquake. Forestimating the conditional probabilities in 2002–2031, weassume that a major factor causing the twentieth centuryquiescence is the stress change produced by the 1906 earth-quake and that the effect of this stress change is similar onall of the characterized faults in the region, including the SanAndreas fault, and on faults of arbitrary orientation in thebackground. We assume, however, that the physics under-lying the post-1906 quiescence on the San Andreas fault isnot based on a stress-shadow effect; rather, that it is basedon the release of strain by slip in the 1906 earthquake, inaccordance with the model of Reid (1910). We ignore ratechanges associated with shocks within subregions similar insize to those associated with the 1868 or 1989 earthquakes,except insofar as they contribute to the regional activity. Amore detailed model that incorporates the effects on activityassociated with smaller earthquakes in the region (or otherpossible causes) is beyond the scope of this study.

The simplest de-aggregation model assumes that f(t) de-scribes the time variations in rate on each fault and in thebackground, as illustrated in Figure 3. That is, each fault andthe background experience the same proportionate rate fluc-tuations about its mean as the region does about its mean.

To test this assumption, we compare the rate (based on theinferred a-value) on each fault and in the background in twotime periods (1850–1905 and 1951–1998) with the corre-sponding long-term rate estimated in WG02. Although thistest provides little temporal resolution, it is well suited forour purpose because it is sensitive to changes spanning 1906.

Wesson et al. (2002) estimated the probability that theinstrumental M �3.0 earthquakes, 1951–1998, and the his-torical M �5.5 earthquakes, 1850–2001, were associatedwith each of the seven characterized faults and the back-ground, as defined in WG02. We use these associations andremove aftershocks using a “declustering” algorithm (Rea-senberg, 1985). We decluster the catalog in order to reducethe considerable effect the aftershocks of the 1979 CoyoteLake, 1984 Morgan Hill, and 1989 Loma Prieta earth-quakes on the distributions. We construct G–R magnitude–frequency distributions of independent events for each faultand for the background. First, we obtain for each fault andfor the background the least-squares best fit to a G–R distri-bution for the instrumental M �3.0 earthquakes, 1951–1998(Table 2). (The magnitude range for the fit varies, dependingon the data available on each fault and in the background.)Next, we obtain the a-value for each fault and the back-ground using the least-squares best fit for the M �5.5 1850–1905 catalog. We assume that b on each fault does not varywith time and has the value estimated for the 1951–1998catalog. With b thus fixed, we treat a as the free parameterand compare the modeled rates on each fault before and after1906.

For the San Andreas, Hayward–Rodgers Creek, andCalaveras faults and the background, the G–R models thusdetermined are detailed in Table 2 and shown as solid anddotted black lines in Figure 7, along with the cumulativemagnitude distributions on which the models are based.(These distributions for the faults and the background in-corporate the Wesson et al. [2002] associations.) Suppres-sion of activity after 1906 relative to the period before 1906is apparent in the vertical offsets in the G–R models. Theassumption of G–R behavior for M �6 appears to be sup-ported by these data where sufficient earthquakes have oc-curred (the San Andreas and Calaveras faults, the back-ground, and the region). Wesson et al. (2002) associatedfewer earthquakes with the Hayward–Rodgers Creek faultfor M �4.3 than expected for the G–R model. However, thenumber of M �4.3 events expected since 1950 is only 3.4,a rather small number with concomitant high uncertainty.Activity in 1951–1998 on the Calaveras fault exceeds theG–R model for M �5.8, reflecting the moderate earthquakesthat have occurred there since 1979 (e.g., the 1979 CoyoteLake and 1984 Morgan Hill earthquakes).

We quantify the comparison in activity in the pre-1906and post-1951 periods by comparing, for each period, themodeled rate of M �6.7 earthquakes on individual faultsand in the background to the corresponding long-term rateestimated by WG02.

To do this, we extend the definition of f(t) to character-


Table 2Gutenberg–Richter Models

1951–1998 Models 1850–1905 Models

Fault*

Long-termRate†

MagnitudeRange

for Model a b Rate‡ Rate Ratio§

MagnitudeRange

for Model a Rate|

UpperRangeof Rate

LowerRangeof Rate Rate Ratio§

SAF 0.0073 3.0–4.8 2.73 � 0.08 0.82 � 0.02 0.0017 0.234 (0.145–0.380) 5.5–6.1 3.72 � 0.11 0.0168 0.0291 0.0097 2.286 (1.318–3.966)H/RC 0.0086 3.0–4.2 2.79 � 0.08 0.92 � 0.02 0.0004 0.049 (0.029–0.083) 5.5–6.5 4.15 � 0.12 0.0096 0.0181 0.0051 1.114 (0.592–2.096)CAL 0.0033 3.0–5.0 3.02 � 0.08 0.81 � 0.02 0.0039 1.203 (0.736–1.966) 5.5–6.1 3.72 � 0.10 0.0197 0.0339 0.0114 6.045 (3.514–10.40)C/GV 0.0011 3.0–3.9 2.09 � 0.19 0.90 � 0.05 0.0001 0.106 (0.030–0.383) 5.5–6.3 3.57 � 0.32 0.0035 0.0171 0.0007 3.234 (0.658–15.89)SGR 0.0035 3.2–4.1 0.86 � 0.11 0.51 � 0.03 0.0028 0.799 (0.382–1.672) 5.5–5.9 1.42 � 0.19 0.0100 0.0248 0.0040 2.883 (1.162–7.150)GVL 0.0010 3.3–4.1 2.26 � 0.18 0.86 � 0.05 0.0003 0.302 (0.092–0.988) 5.5–6.1 2.86 � 0.23 0.0013 0.0046 0.0003 1.201 (0.326–4.423)MTD 0.0009 3.5–4.0 3.65 � 0.38 0.89 � 0.11 0.0049 5.496 (0.430–70.28) 5.5–6.0 2.92 � 0.53 0.0009 0.0162 0.0001 1.022 (0.057–18.29)Bkgd 0.0051 3.0–4.8 3.67 � 0.03 0.89 � 0.01 0.0051 0.998 (0.807–1.236) 5.5–6.4 4.31 � 0.04 0.0221 0.0281 0.0174 4.336 (3.418–5.499)Region 0.0308 3.0–5.0 3.82 � 0.04 0.86 � 0.01 0.0118 0.383 (0.301–0.486) 5.5–6.8 4.58 � 0.06 0.0665 0.0890 0.0497 2.158 (1.612–2.888)

*SAF, San Andreas fault; H/RC, Hayward/Rodgers Creek fault; CAL, Calaveras fault; C/GV, Concord/Green Valley fault; SGR, San Gregorio fault;GVL, Greenville fault; MTD, Mount Diablo fault.

†Annual rate of M � 6.7 earthquakes in WG02 long-term model inferred from G–R relation.‡Annual rate of M � 6.7 earthquakes in 1951–1998 inferred from G–R relation.§Ratio of model rate to long-term rate of M � 6.7 earthquakes (range).|Annual rate of M � 6.7 earthquakes in 1850–1905.

ize the rates of earthquakes on individual characterized faultsand the background, relative to their respective long-termmeans. That is, for each fault, the background, and the region(denoted by k) we define and as thek kf(t) f(t)pre-1906 post-1951

mean rate of M �6.7 earthquakes in 1850–1905 and 1951–1998, respectively, divided by the corresponding long-termrate for that fault, the background, or the region estimatedby WG02 (see Table 2, Fig. 8). To compare models basedon catalogs spanning different magnitude ranges, we assumethat a single G–R relation for each fault accurately describesthe seismicity over the magnitude range 3.0 � M � �6.The 1951–1998 data are consistent with this assumption onthe San Andreas and Calaveras faults, in the background,and the region (Fig. 7).

Earthquake rates observed before 1906 are almost ev-erywhere above the respective long-term mean levels (Fig.8). In the region, the mean rate in the pre-1906 period is 2.2times the long-term mean ( � 2.2). The values ofRegionf(t)pre-1906

for the individual faults and the region range fromkf(t)pre-1906

1.1 (Hayward) to 6.0 (Calaveras). In the background,� 4.3. Overall, there is considerable uniformityBackgroundf(t)pre-1906

in among the faults (Fig. 8). This uniformity maykf(t)pre-1906

be, in part, an artifact of the relatively large location errorsassociated with these events and the resulting blurring of theassociation matrix found by Wesson et al. (2002).

In contrast, are more varied and generallykf(t)post-1951

below the respective long-term average levels (i.e., �1), es-pecially on the San Andreas and Hayward–Rodgers Creekand Concord–Green Valley faults. A notable exception isthe Calaveras fault, where the rate in 1951–1998 is slightlyabove its long-term rate ( � 1.2). The rate in theCalaverasf(t)post-1951

background in 1951–1998 is virtually the same as the long-term rate ( � 1.0). Wesson et al. (2002) noteBackgroundf(t)post-1951

that the rates of M �5.5 earthquakes in the background inboth 1951–1998 and 1971–1998 equal the long-term rate,

while the first half of the twentieth century saw a quiescencein the background and suggest that in the background, whichincludes thrust sources as well as strike-slip faults (both setswith diverse orientations), the post-1906 quiescence mayhave ended in the 1950s.

While the levels of activity relative to the respectivelong-term levels on the three most active faults (San An-dreas, Hayward–Rodgers Creek, and Calaveras) all de-creased in 1906, their ranking in activity level (relative totheir respective long-term rates) remained the same. That is,both before 1906 and after 1951, the relative rate on theCalaveras was the highest, and the relative rate on the Hay-ward fault was the lowest of these three faults.

Although we have assumed the effect of the 1906 earth-quake to be the same on all faults, more likely it was het-erogeneous. For example, the particularly low rate on theHayward–Rodgers Creek fault may be due to its proximityto the slip in the 1906 event. Perhaps the particularly highcontemporary rate of activity on the Calaveras fault signalsthat this fault is in an area that has emerged from the 1906stress-shadow, as suggested by Sykes and Jaume (1990) andin accord with plate-tectonic loading models proposed bySimpson and Reasenberg (1994). Alternatively, this patterncould possibly be explained by supposing that WG02 un-derestimated the long-term rate for the Calaveras fault andoverestimated the long-term rate for the Hayward–RodgersCreek fault. And, of course, this pattern could simply meanthat departures in rate on these faults from their long-termlevels have persisted throughout the nineteenth and twenti-eth centuries.

We thus find some, if not conclusive, support in thesecomparisons for our conjecture that the regional seismic qui-escence since 1906 occurred uniformly throughout the re-gion. A common decrease in rate on virtually every fault inthe twentieth century, and a common increase during the


10-4

10-3

10-2

10-1

100

101 a) San Andreas b) Hayward -Rodgers Creek c) Calaveras

d) Background e) Region

3 4 5 6 7 8 3 4 5 6 7 8 3 4 5 6 7 8

3 4 5 6 7 8 3 4 5 6 7 8

MagnitudeMagnitude

10-4

10-3

10-2

10-1

100

101

10-4

10-3

10-2

10-1

100

101

10-4

10-3

10-2

10-1

100

101

10-4

10-3

10-2

10-1

100

101

Ann

ual R

ate

Ann

ual R

ate

b = 0.82 b = 0.92 b = 0.81

b = 0.89b = 0.89

(T = 378 yrs)1906

1868

1989

Figure 7. Long-term cumulative magnitude-frequency distributions observed in the SFBR andmodeled by WGCEP (2002). Solid red curves are mean long-term rates of earthquakes in the WGCEP(2002) model. Dotted red lines indicate the model’s 95% confidence range. Symbols are historicalseismicity determined with Wesson et al.’s (2002) fault associations. Rates in various catalogs andtime and magnitude ranges are shown: M � 5.5 for 1850–2001 (open squares); M � 3, declustered,for 1951–1998 (solid circles); post-1906 M � 5.5 (solid squares); pre-1906 M � 5.5 (open dia-monds). The solid black line is the Gutenberg–Richter (G–R) relation fit to the 1951–1998 rates,and the dotted black line is the G–R fit to the pre-1906 rates with b fixed. The G–R relations providegood fits to each of these data sets, but the a-values vary systematically: pre-1906 activity hasconsistently higher a-value, post-1906 consistently lower a-value, reflecting the well-known post-1906 seismic quiescence in the SFBR. The 1906 earthquake is plotted with a recurrence time of378 yr, with M 7.8. M � 6.7 earthquakes labeled with their year of occurrence. (a) San Andreasfault. A post-1906 G–R distribution is continuous from M 3.0 to M �6.0. (b) Hayward-RodgersCreek fault system. Paucity of M � 4 earthquakes on these faults since 1906 precludes the abilityto fit a G–R model above M �4. (c) Calaveras fault. Post-1906 M � 5 earthquakes are well-fitwith a G–R model, however the rate of M � 5 earthquakes rises above this trend, reflecting recentactivity there since 1979. (d) Background seismicity. Again, a G–R distribution for the post-1906earthquakes is continuous from M 3 to M �6. (e) Regional seismicity. Post-1906 seismicity iswell-fit with G–R model from M 3.0 to M 6.2. WGCEP (2002) long-term model is consistent with1850–2001 M � 5.5 observed rates, which lie between pre-1906 and post-1906 rates.

second half of the nineteenth century (both relative to therespective estimated long-term mean rates) suggest a com-parable response of all the characterized faults and back-ground to the 1906 earthquake. Departures from uniformresponse among the faults and background in the 1951–1998period may be due to a variety of effects, including aleatory(sampling) error, actual variations in unrelated to thef(t)

1906 earthquake on individual faults and, possibly, errors inthe geologically-based long-term rate estimates in WG02.

Discussion

Time-dependent models for earthquake probability inthe San Francisco Bay region typically employ three ele-


0.01

0.1

1

10

100

Region Bkgd SAF H/RC CAL C/GV SGR

Rat

io o

f Rat

e to

Lon

g-te

rm M

ean

Rat

e 1850-19051951-1998

Figure 8. The mean rates of M � 6.7 earthquakesin 1850–1905 (red) and 1951–2000 (blue) on faults,the background and in the region are shown as a ratiorelative to their respective long-term rates determinedby WGCEP (2002). That is, the rates have been “fault-normalized,” rather than “region-normalized.” Tickmarks indicate ratio to the long-term rate, error barrepresents the �1r range. In the San Francisco Bayregion, the rate was approximately 2.2 times the long-term rate before 1906 and approximately 0.4 timesthe long-term rate after 1951. The rate (relative to thelong-term) decreased on each fault individually andin the background.

ments: one for modeling the long-term behavior of the faultsin the region; one for modeling the quasi-periodic occur-rence of characterized earthquakes on isolated, tectonicallyloaded fault segments; and one for modeling departures fromsuch behavior that result from the interactions among themajor faults caused by large earthquakes. Our model com-bines the second and third elements. We use the time-independent long-term earthquake model for the SFBR de-termined by WG02 to describe the long-term rate andmagnitude distribution of earthquakes in the region. To thiswe apply a time-varying factor that is based on regional seis-micity. The time-varying part of the model is a proxy forboth the periodic behavior of the faults and their interactions,as these are expressed through the historical seismicity. Thisapproach is motivated by a sharp contrast in the regional rateof M �5.5 earthquakes before and after 1906 and a gradualincrease in the rate of small and moderate earthquakes since1942, relative to long-term mean rates. These observationssuggest that, in the absence of a large earthquake in the re-gion, the rate of earthquakes in the region vary only slowlyand that a sudden increase in the rate (and probability) oflarge earthquakes in the near future is unlikely.

The most important historical interaction effect in theSFBR is widely believed to be the stress changes associatedwith the 1906 earthquake. Elastic-plate models for estimat-ing these stress changes, and simple models for the evolutionof the loading of these faults subsequent to such static stresschanges, are utilized by WG02 for estimating the effect ofthe 1906 earthquake on the earthquake probability. These

models suggest that the 30-yr probability for M �6.7 earth-quakes in the region is approximately equal to that whichprevailed during the second half of the nineteenth century.This would imply that the earthquake activity in the SFBRwill significantly increase in the next 30 yr, relative to thelast 50 yr. However, the continuing seismic quiescence inthe SFBR after 1906, relative to the long-term rates and theobserved nineteenth century activity, suggests that thesemodels may underestimate the geographical extent and du-ration of the stress relaxation effects of the 1906 earthquake.More extensive effects of the 1906 earthquake can be seenin more complex models of stress transfer based on visco-elastic response of the crust (e.g., Kenner and Segall, 1999;Parsons, 2002), however these models also fall short of fullyexplaining the twentieth century activity lull. Informal ex-periments with rate-state-modeled effects can cause a “seis-micity shadow” to persist even after the causative stress-shadow is ended, but a reliable model of this sort has notbeen developed (R. W. Simpson, personal comm., 2002).Our empirical model, which is a proxy for physical inter-action models using viscoelastic, rate-state-friction or otherrheologies, suggests that regional activity will not increasemuch in the next 30 yr.

We estimated earthquake probabilities for individualfaults within the region, based on the assumption that thechange in rate for the region is appropriate for every fault inthe region. That is, we assumed that the 1906 stress-shadowaffected all faults equally. We explored the validity of thisassumption and found that the rate changes on all of theindividual principal faults in the SFBR are positively corre-lated with the regional rate change. The correlation was notuniform, however, perhaps because of aleatory error or errorin the estimated long-term mean rates in some cases. Wecannot rule out the possibility that the effect of the stress-shadow is not uniform among the faults, or that some faults(particularly the Calaveras fault) may have emerged fromthe stress-shadow before others, as has been suggested byprevious studies (Sykes and Nishenko, 1984; Simpson andReasenberg, 1994) as well as by our observations (Fig. 8).

Toppozada et al. (2001) observed that the occurrence ofM �5.5 shocks was higher in the SFBR in the decades beforethe 1838, 1868, and 1989 earthquakes and in the decadesbefore 1906. These observations can be explained by assum-ing a cycle of accelerated moderate-size seismicity before,and suppressed activity after, M �7 and larger earthquakes,whose duration and spatial extent scale with main shockmagnitude (Sykes and Jaume, 1990; Jaume and Sykes,1996). Our empirical model, which is based on regional ac-tivity, is unlikely to be able to identify such smaller-scalepatterns. However, we think it likely that the rate (and thusprobability) of earthquakes in the SFBR at this time is af-fected by both a regional stress-shadow effect associatedwith the 1906 earthquake and more localized, shorter-livedphenomena associated with the Loma Prieta earthquake and,possibly, other similar-size earthquakes yet to occur. To the


degree that the 1906 effect prevails, our model is expectedto be accurate.

In view of these sources of uncertainty about the inter-pretation of our de-aggregation calculations, we are moreconfident of our regional probability model and less confi-dant of our de-aggregated probabilities for the individualfaults. This is not a novel situation and appears to be anintrinsic aspect of regional and fault-specific earthquakeprobability estimates. In fact, every California WorkingGroup has emphasized the robustness of their earthquakeprobability estimates aggregated over the region, while ac-knowledging the larger uncertainties in their probabilities forindividual faults. Our estimate of 0.42 for the probability ofone or more M �6.7 earthquakes in the SFBR in 2002–2031is lower than the regional probability of 0.5 estimated byWG88, lower than the Poisson probability of 0.6 obtainedby WG99, and lower than the P � 0.67, P � 0.70 andP � 0.63 30-yr probabilities estimated by WG90, WG99,and WG02, respectively.

Quite simply, these contrasts are largely the result ofdifferences in (or lack of differences, in the Poisson case)methods for estimating the effect of the 1906 earthquake inthe SFBR. The WG02 BPT model for the probability in2002–2031 (Fig. 6), which relies on elastic models of stresstransfer, predicts activity well above the long-term regionalmean rate, approximately at the level observed in the secondhalf of the nineteenth century, and suggests that the regionreturned to above-average rates in the mid-twentieth century.In contrast, an empirical model suggests that most of theregion has yet to return to its long-term mean level ofactivity.

Throughout this article, we have emphasized that time-dependent modeling of earthquake probability is still verymuch in its infancy, and many important factors that mightinfluence the timing of major earthquakes are at best poorlyunderstood. We have presented an empirical model for theSan Francisco Bay region in order to capture a factor (thequieting effect of the 1906 earthquake on the region) thatmay not be fully accounted for in other models. The empir-ical model presents no new physics; rather, it serves as aproxy for the yet-to-be-understood physics involved in theinteraction of faults. We believe that an appropriate appli-cation of this model is one made in conjunction with othermodels of earthquake occurrence in the region that use avariety of methods for estimating both earthquake recurrenceand interaction effects. We advise against using the empir-ical model (or any other single probability model) alone forestimating the earthquake hazard in the region. Until thephysics involved in earthquake occurrence in the SFBR ismuch better understood, a prudent basis for estimating theearthquake hazard in the SFBR should combine the resultsof all credible earthquake probability models for the region,as was done in WG02.

Conclusions

• Earthquake activity throughout the SFBR has been lowersince the occurrence of the 1906 M 7.8 earthquake than it

had been in the half-century before it. The decrease inactivity is observed on each of the seven characterizedfaults in the region and in the background.

• This suppression of activity can be explained as a stress-shadow effect associated with the 1906 earthquake. How-ever, currently available analytical models of fault inter-action in the SFBR may underestimate this effect.

• We have presented an empirical model based on the rela-tive level and trend of regional seismicity as a proxy forthe time dependence of the physical processes governingfault interactions. This model may provide a more accurateestimate of the probability of large earthquakes in theSFBR in the next 30 yr than current analytical models do,at least until the physics underlying fault interactions isbetter understood.

• Our empirical model suggests that the regional rate ofM �6.7 earthquakes will be slightly higher in the next 30yr than it was in the past 30 yr, as the regional stress-shadow effect continues to erode. With this model, theprobability of one or more M �6.7 earthquakes occurringin the SFBR in the next 30 yr is lower than estimates ob-tained by recent Working Groups.

• The 1906 stress-shadow effect may not be homogeneousthroughout the region. Furthermore, cycles in moderate-size earthquake activity associated with M �7 earthquakesin the SFBR, like the 1989 M 6.9 Loma Prieta earthquake,may currently affect the earthquake rate on individualfaults or in limited areas. Because neither of these effectsis specifically accounted for in the empirical model, we areless certain of the fault-by-fault probabilities obtained withthe model than we are with the regional probability.

Acknowledgments

We are grateful to Joe Andrews, Joan Gomberg, Bob Simpson, LynnSykes, and an anonymous reviewer for constructive reviews of the manu-script. Our work benefited greatly from discussions with Mike Blanpied,Bill Ellsworth, Al Lindh, Andy Michael, Bob Simpson, Rob Wesson, andMary Lou Zoback.

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U.S. Geological Survey345 Middlefield RoadMenlo Park, California [email protected]: 650-329-5002

(P.A.R.)

U.S. Geological Survey345 Middlefield RoadMenlo Park CA 94025

(T.C.H.)

U.S. Geological Survey345 Middlefield RoadMenlo Park, California 94025

(W.H.B.)

Manuscript received 11 January 2002.

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