Long-Term Earthquake Forecasts in the San Francisco Bay Area

Transcription

1 Long-Term Earthquake Forecasts in the San Francisco Bay Area Allan Goddard Lindh US Geological Survey Menlo Park, CA The remaining phenomenon of the same kind there will be no difficulty in reasoning out by the method of probabilities. A man may sometimes set aside meditations about eternal things, and for recreation turn to consider the truths of generation which are probable only; he will thus gain a pleasure not to be repented of, and secure for himself while he lives a wise and moderate pastime. Let us grant ourselves this indulgence, and go through the probabilities relating the same subjects which follow next in order. Plato, Timaeus Abstract n historic time the San Francisco Bay Area (SFBA) has been the site of four large earthquakes, including the M San Francisco earthquake, and most recently, the M Loma Prieta earthquake. Of the eight major fault segments considered here, two have not experienced large earthquakes in about 200 years, and the SF Peninsula segment of the 1906 rupture on the San Andreas appears from my calculations to be close to fully reloaded as well. have used simple geophysical and statistical models (elastic rebound model and Weibull distribution) to estimate the probability of large earthquakes (M6.7 or larger) in the SFBA in the coming decades. have used seismicity, geology, and geodesy to estimate segment boundaries, recurrence intervals, and the associated uncertainties. The results indicate that the SFBA has an approximately 82% chance of a large earthquake in the next 30 years, with four segments dominating the 30 yr probabilities; San Francisco Peninsula (33%), Southern Hayward (40%), Northern Hayward (28%) and Rodgers Cr (30%). Because of the proximity of these four segments to the urban portions of San Francisco and Oakland, the probability of these most vulnerable areas experiencing strong ground motion (an M6.7 within 25 km or less) one or more times within the next 30 years is about 70%. Because of the breadth and quality of our understanding of the earthquake machine in the SFBA, these probabilities depend in large part on the intrinsic variance in the earthquake recurrence process itself -- most conveniently expressed as the ratio of the standard deviation to the mean recurrence time, or intrinsic coefficient of variation (CV ). ve applied a new approach to estimating CV, using the time since the last characteristic event (the open-interval ) on well characterized segments. Combined with an estimate of the mean recurrence time on each segment, an estimate of the likelihood of each open interval can be computed, and a simple maximum likelihood procedure used to choose the best fitting value of CV. This approach has significant advantages over the more common approach of using a sequence of paleoseismic dates to estimate CV, since the completeness of paleoseismic records is always in question, and even one missed event seriously biases estimates of CV. The power in the technique lies not in establishing that a sequence of earthquake recurrence dates is very regular, but rather in establishing that there are no very short intervals large values of CV imply a large fraction of short recurrence intervals. Analysis of 10 major segments in California for which the requisite 1

2 data are available yields an estimate for CV of 0.2 or less; this value was used in the calculations outlined above. f we use this model to backcast seismicity rates for the last few centuries, we can fit reasonably well the large decrease in moderate seismicity following the 1906 earthquake, even though each segment is modeled independently. That is, no far-field stress perturbations from 1906 have been used in this work, nor do they appear to be needed to account for the gross variations in seismicity rate. This may call into question the wide-spread assumption that a stress shadow is self-evident following 1906, and suggests that more careful modeling is required to establish whether far-field stress changes are necessary to fit the observations. Prior statistical tests have shown that the observed decrease following 1906 is extremely unlikely, given a Poisson model for a null hypothesis. From the work presented here it appears one could not reject a simple timedependent model in the same manner, even though it includes no far-field stress effects. ntroduction Since the early 1980 s a variety of techniques have been applied to the question of how best to estimate the likelihood of large earthquakes along major plate boundaries. (The phrase large earthquakes is taken in this report to refer to events of M6.7 or larger. These are events, that like the LP earthquake will not only do severe damage in the immediate epicentral area, but will do significant damage to distances of km, and will thus be of regional significance in the SFBA.) For time periods of 100 years or greater, time spans appropriate for establishing construction standards for critical facilities, large engineered structures and the Uniform Building Code, there is general agreement that time stationary, or Poisson estimates are appropriate. That is, the rate of large earthquakes, in a given area, in the next 100 years is best estimated by using the long-term average rate of seismicity, based on historic seismicity, paleoseismic estimates of Holocene seismicity rates, and geodetic and geologic measures of the strain accumulation rate. For shorter time periods, however, the optimal strategy for computing the probabilities of large earthquakes is still a matter of discussion, with some workers modeling earthquake recurrence as a quasi-periodic process one that reflects the cycle of strain accumulation and release that drives the earthquake machine (Hagiwara, 1974; Rikitake, 1974; Lindh, 1983; Sykes and Nishenko, 1984). This is a critical question for insurance companies attempting to estimate their exposure, public officials and corporate executives attempting to allocate resources for upgrading vulnerable structures, and all those whose concerns are appropriately focused on a time-scale of decades rather than centuries. More than 100 years ago, G.K. Gilbert described a simple physical model for the occurrence of large earthquakes. Something of this sort happens with the mountain. The upthrust produces a local strain in the crust, involving a certain amount of compression and distortion, and this strain increases until it is sufficient to overcome the starting friction along the fractured surface. Suddenly, and almost instantaneously, there is an amount of motion sufficient to relieve the strain, and this is followed by a long period of quiet, during which the strain is gradually reimposed. The motion at the instant of yielding is so swift and so abruptly terminated as to constitute a shock, and this shock vibrates through the crust with diminishing force in all directions n fine, all the phenomena of an earthquake are produced. (Gilbert, 1884) 2

3 This insight, that earthquakes involve a regular process of strain accumulation and release, was quantitatively confirmed for a large strike-slip earthquake by H.F. Reid following the great M San Francisco earthquake. n his report he noted, t seems probable that a very long period will elapse before another important earthquake occurs along that part of the San Andreas rift which broke in 1906; for we have seen that the strains causing the slip were probably accumulating for 100 years. (Reid, 1910) Gilbert and Reid s elastic rebound (ER) model, developed to explain the geodetic displacements observed after the 1906 earthquake, has come to form one of the backbones of modern earthquake science. Reid s projection of a 100-year latency period following 1906 was remarkably prescient, given that the 1989 M6.9 Loma Prieta earthquake occurred along the southern-most 40 km of the 1906 rupture (U.S.G.S., 1990; Sykes, 1996). The ER model was confirmed for large subduction-zone earthquakes in Japan {Shimazaki, 1980 #174} using the much longer historical record available in Japan; subsequent work on the historical catalog has confirmed and strengthened this result {Sykes, 2003 #532}. With the confirmation of continental drift as the driving force in the earthquake machine, the development of a global Plate Tectonic model, and the availability of a large amount of high quality geodetic data, Gilbert and Reid s original insights have been confirmed beyond any reasonable doubt when one is interpreting long-term seismic moment release rates, or the deformation following a large earthquake. However, when it comes to the question of whether a large earthquake actually will be followed by a longperiod of quiet, during which the strain is gradually reimposed, there is less than unanimous agreement, with some workers arguing that given the manifest uncertainties, a Poisson distribution still provides a credible estimate of the likelihood of future earthquakes (see for instance Jackson and Working Group on California Earthquake Probabilities, 1995; Schwartz and Working Group on California Earthquake Probabilities, 1999){Blanpied, 2003 #450}, and some have even argued that large earthquakes are clustered in time, with a power-law distribution (Kagan and Jackson, 1999). This is a difficult question to answer empirically, given that instrumental recordings of earthquakes only go back about a century, and that most large plateboundary earthquakes (LPBEQ) have recurrence intervals that range from years or more. t will take millennia to accumulate enough long recurrence histories of LPBEQ to directly resolve the question of which statistical distribution best models their recurrence. Fortunately, however, it is much easier to eliminate the Poisson distribution, because it predicts a large number of recurrences at short recurrence times, and the absence of short return times also places a crucial constraint on any quasi-periodic distribution used to implement the ER model. n this paper use the high quality data sets that have been assembled in the last few decades in the San Francisco Bay area to show that the history of large earthquakes in this area is sufficient to rule out a Poisson distribution with some confidence; the recurrence of large earthquakes is best fit by a quasi-periodic distribution with a CV of about 0.2. When used with a Weibull distribution to model the variance in the expected return time, this simple implementation of the ER model results in an estimate that the SFBA currently has about a 4% per year chance of a large earthquake, and an 81% chance of such an event in the next 30 years. Given the proximity of the most vulnerable portions of San Francisco and Oakland to the most dangerous segments of the San 3

4 Andreas and Hayward faults, this model suggests that those areas currently have a likelihood of strong ground motion of about 70% in the next 30 years. One might argue that what is undertaken here is too ambitious, that this field of study has become the provenance of large Working Groups, where the combined expertise of many people can be utilized (Dieterich et al., 1988; Dieterich et al., 1990; Jackson and Working Group on California Earthquake Probabilities, 1995; Schwartz and Working Group on California Earthquake Probabilities, 1999; Blanpied and Working Group on California Earthquake Probabilities, 2003) (These five reports will be referred to in this work as WG88, 90, 95, 99 & 03, respectively). wish to emphasize however that this work is a direct lineal descendent of probability estimates for California earthquakes completed between 1983 and 1988 (Lindh, 1983; Sykes and Nishenko, 1984; Lindh, 1988) and WG s 88 & 90, and can be considered a simple update of WG90, building very directly on the expertise and analysis of that report. t represents a very simple approach in which the number of free parameters are kept to a minimum, and the results are thereby falsifiable {Popper, 1959 #271}. t provides an alternative to the more complex stochastic approach that came into vogue recently with the WG95, 99 & 03 reports. Time will tell which approach proves more useful to society, and more accurately predicts the future occurrence of large earthquakes in California. The Elastic Rebound Model The Elastic Rebound model was originally formulated in terms of strain drop during a large earthquake, and the rate of strain accumulation between large events; the time to the next event was simply the ratio of the two (Reid, 1910). Unfortunately we only have geodetic strain observations for one SFBA earthquake in the 19 th century, and no geodetic data at all prior to We do have however, in some cases, geologic estimates of the surface offset in one or more past events, and of the average slip-rates. Thus we can reformulate the ER model in terms of slip in an earthquake ( D ), and slip-rate ( V V ), resulting in an estimate of the time between events, T ˆ D / V (Wallace, 1970); when the time of the last event is known, this makes possible an estimate of the time of the next event. n those cases where we have both geodetic and geologic observations, we can use an elastic dislocation model to combine the information into average estimates of D and/or V (Savage, 1980). This requires reliable estimates of the depth of locking on each fault segment and these are now available for the SFBA, based on a careful analysis of the maximum depth of microseismicity, combined with heat flow data to produce consistent estimates of the depth to the brittleductile transition (Williams, 2003). For the sake of clarity, and to underscore that the ER model as implemented here is sufficiently simple and clearly stated that it can be falsified, listed below are the premises on which this work is based. believe these premises cannot be refuted by current data or well-formed arguments, but if the seismicity of the SFBA in the coming decades does not occur as predicted here, the implication will be that one or more of these premises may be incorrect. Premise 1. The Gilbert-Reid elastic rebound model is approximately correct for LPBEQ s, when applied to a single fault segment. That is, T ˆ D / V where the D 4

5 displacement and velocity are for a single segment. (Some segments may require a correction for aseismic slip (Wallace, 1970), but fortunately this is a problem on only two of the SFBA segments considered here.) Premise 2. For most practical purposes, the recurrence times for a given segment have a simple quasi-periodic distribution in time; for simplicity use a Weibull distribution about T 1, the estimated time of the next event (Hagiwara, 1974; Brillinger, 1982; Cornell and Winterstein, 1988). Premise 3. The standard distribution of the distribution about T 1 can be approximated as 2 2 Tˆ P, where P is a measure of the parametric, or epistemic uncertainty, and is a measure of the intrinsic, or aleatory, uncertainty for a given segment. Prior studies of various recurrence data sets suggest that / ˆ T (called here CV ) is in the range (Nishenko and Buland, 1987; McCalpin and Slemmons, 1998; Ellsworth et al., 1999; Ogata, 1999). Premise 4. Quasi-stable segments are a feature of mature plate boundary faults, and can be identified in many cases on the basis of geologic, geodetic, seismological and geophysical criteria, even in those cases where there is no historic rupture. Premise 5. Characteristic segments. Slip events with about the same displacement occur repeatedly on approximately the same section of fault, but can sometimes combine with other segments to produce larger multi-segment ruptures. (Sieh (1996) called this the patch model ; it is a variant of his earlier uniform-slip model (Sieh, 1981).) Premise One is actually supported by a large quantity of seismological, geologic, and laboratory data, and some theoretical calculations; Premise Two is simply an assertion that for practical purposes the ER model can be implemented as a statistical distribution. Premises Three, Four and Five however are in a rather different category; the heterogeneity of the recurrence of LPBEQ s in space and time is not only an interesting scientific question, but the answers will largely determine whether practical long-term earthquake predictions will be feasible. Of course one can argue that if the heterogeneity in space and time is large enough, for practical purposes earthquake recurrence will effectively be Poissonian, and whether Premise One is technically correct will be irrelevant, for practical purposes. n this study will argue that the weight of the evidence very strongly supports modeling the recurrence of LPBEQ s in the SFBA as quasi-periodic events, with total uncertainties in estimates of the time of next events that average 30-40% or less of the mean recurrence time, thus making the calculation of useful time-dependent probability estimates possible. Segmentation This study begins by dividing each of the major active faults in the SFBA into discrete segments on the basis of estimates of the source and slip of past large earthquakes, patterns of microseismicity, and the presence of distinct geologic and/or geophysical anomalies along the major faults (Figure 1 and Table 1). The assumption is that each segment can be treated as a single unit, which will either fail in a single large earthquake, or in conjunction with adjacent segments to produce a larger event. This is 5

6 effectively an assumption that each segment is characterized by a single average slip-rate, and a single average displacement, each time it fails (Sieh, 1996). This is a discretized extension of the ER model to one-dimension, to accommodate the observation of variable slip along strike in some large strike-slip events (Sieh, 1978; Thatcher et al., 1997). n the case of multi-segment ruptures, the magnitude of the event will depend on the summed moments of the contributing segments; see the summed 1906 moment and magnitude at the bottom of Table 3. t should be noted that this report assumes that the slip on a segment is approximately the same, whether it fails as a single segment, or participates in a multi-segment rupture (Sieh, 1981). n addition to the evidence cited in Sieh (1981), evidence for this premise includes the strike-slip motion on the LP segment, which was approximately 2 m. in both the M earthquake (Sykes, 1996; Prentice and Ponti, 1997; Thatcher et al., 1997), and the M event (Lisowski et al., 1990). Figure 1 goes here Figure 1. Map of the SFBA showing faults, segments, seismicity and slip-rates. Segment ends are shown as crossing bars. Slip-rates are in mm/yr in parentheses following each segment name. The seismicity is shown as colored balloons, color coded by depth; yellow for shallowest events, through orange, red, purple, blue, to green for those below 15 km. Also shown are the locations of 5 geodetic monuments (black triangles) used in estimating 1906 slip on the SFP segment. The lines designating each segment are colorcoded to reflect the current 30-year probability from Table 7. Segment Table 1 -- Segmentation Char EQ M Length Width 1 South End North End (Km) (Km) Lat Lon Crit 2 Lat Lon Crit 2 Loma Prieta (LP) HSG C HSE G San Francisco Peninsula (SFP) HSE G SED San Andreas 1906 (7.8) ~ SED HE North (SAN) Hayward South (HS) SEG HSE G Hayward North (HN) 1868? ~ HSE G SEC Rodgers Cr. (RC) 1868? ~ SEC SE Calaveras North 1868? ~ SEG SE (CN) San Gregorio 1838? ~7 72 ~ SG S North (SGN) 1 Maximum Width of rupture from Williams (1999), based on heat flow and 95% cutoff on microseismicity depths. Values for SAN and SGN are guesses, based on nearby seismicity, and heat flow. 2 The criteria for determining segment boundaries used here are: H historic seismicity; S microseismicity patterns; E En-echelon offsets, or bends, in surface fault trace; fault intersections; G changes in basement geology; D variations in displacement in prior earthquakes, C changes in surface creep rate. 3 The 1906 San Francisco earthquake ruptured three segments, LP, SFP and SAN. We have no information on whether SAN has ever failed as a single event, and assume in this report that future events on the SAN segment will be of about M7.8, and will include LP and SFP as well. However since only the SFP segment 6

7 currently has a significant likelihood of failure, this assumption has little impact on probability estimates for the next 30 years. Some of geologic criteria used in this work to define segments were applied originally by Allen et al. (1965) and Allen (1968) to strike-slip faults. Mogi (1969), Sykes (1971), and Kelleher et al. (1973) extended the paradigm to great subduction zone events, and rwin and Barnes (1975) suggested a geologic mechanism for aseismic slip in the San Andreas system that helped clarify the role of the creeping section of the fault in central California. These segmentation criteria were augmented by seismicity and creep data for Parkfield (Lindh and Boore, 1981), and the Loma Prieta segment (Lindh et al., 1982), and appear to have successfully predicted the extent of rupture of the 1989 Loma Prieta earthquake (U.S.G.S., 1990; Sykes, 1996; Harris, 1998a). A similar application of seismic criteria to the central American trench by Harlow et al. (1981) correctly anticipated the extent of rupture and magnitude of an M7.5 subduction zone event off Nicaragua in 1992 (Harlow et al., 1992; Kerr, 1992). A summary of the successful use of seismicity to define seismic gaps is contained in Nishenko (1989); this multi-faceted approach to characterizing segments was also used by Sykes and Nishenko (1984), in assessing what they termed the tectonic style of a segment. While studies on the relation of segment boundaries to fault geometry (King and Nabelek, 1985; Sibson, 1985; Barka and Kadinsky-Cade, 1988), seismicity (Zoback et al., 1999) and crustal structure (Eberhardt-Phillips and Michael, 1998; Jachens and Zoback, 1999) continue to appear, and some theoretical buttressing is beginning to appear (Harris et al., 1991; Harris et al., 2002) it is clear that much more work, and a long testing period, will be required before these segmentation criteria can be applied with complete confidence. n particular it remains unclear whether segmentation is controlled by geometrical constraints in a relatively homogeneous crust, by variation in crustal and/or fault zone properties such as the coefficient of friction (Byerlee and Brace, 1968), porepressure (rwin and Barnes, 1975), the depth to the brittle-ductile transition (Lindh, 1991), or more likely, by an evolving combination of geometry and material properties, wherein geometrical complexities in a heterogeneous crust are perpetuated by finite displacements, that must be resolved in a complex network of interacting faults. Fundamentally however, the assumption used here is that rheology rules that is, the extent of rupture in large earthquakes is primarily controlled by the rocks juxtaposed across the fault, their physical condition, and secondarily, their geometry. have concluded, based on a careful review of the literature cited above, that this is the most defendable conclusion at this time. While there is not room here to discuss in detail the basis for most of the segment boundaries chosen, some are discussed in the literature cited above, and the most contentious boundary, that dividing the Hayward fault into two segments, is discussed in detail in Appendix B. n addition, a very terse summary is provided in Table 1 as to which specific criteria were considered most relevant in chosen each segment endpoint. t should be understood that the segment boundaries used here constitute a set of hypotheses about the extent of rupture of future earthquakes. This approach passed an important test with the Loma Prieta earthquake, but many more careful case studies will be required before the criteria used here can be regarded as anything like established knowledge. The process of verification could be expedited, of course, by their application and testing on other well-characterized plate boundaries. 7

8 t should also be noted that this approach differs from recent probability reports, including WG99 & 02, which have arrived at estimates of the slip in the last event (D) by a somewhat more complex process. They estimated the segment length and locking depth on each segment and then used an area-moment relation (Hanks and Bakun, 2002) to estimate D. This means that the uncertainties in segment length and in the area- Moment relation, translate into uncertainty in the average recurrence time. The simpler implementation of the ER model used here, and in WG88 & 90, results in recurrence estimates at a given point on a fault that are first-order independent of the details of segmentation. Thus the probabilities that are derived from this model, and any hazard calculations that might utilize them, are not contaminated by the uncertainties in the segment length, which is in many cases the least well-determined parameter. 8

9 Table 2 Symbols and definitions Segment Segment Name For each segment, the following parameters are (Seg.) defined. Char EQ Characteristic Earthquake, identified by year. n parentheses where no event has been observed in historic time, example given from nearby fault. M Nominal magnitude of Char EQ. ˆM 0 Estimated moment. ˆM 0 LWDR, where is the crustal rigidity, 11 taken here as 3 10 dyne-cm (Aki, 1967). ˆM The estimated moment magnitude. Mˆ 2/3 log Mˆ (Hanks and Kanamori, 1979) L Segment length. W Width of locked zone. R Seismic coupling factor; 1 for locked faults, 0 for creeping faults. As used in this study, a measure of the fraction of the segment that is fully locked (Bakun, 2003). P ntrinsic (or aleatory) uncertainty in the estimated recurrence time T ˆ, Tˆ * CV. The parametric (or epistemic) uncertainty in the estimated return time T,. 1 P 0 Tˆ CV The intrinsic (or aleatory) Coefficient of Variation on T ˆ. CV The total Coefficient of Variation on T 1, CV T / T 1 1. D D Slip in last event (mm). V V Slip-rate (mm/yr). Tˆ T ˆ Estimated recurrence interval. ˆ T D/ V, ˆ ˆ T ( T D / D V / V ) T 1 T 1 Time of the penultimate event. T Time of last event. 0 T 0 1 T 1 T Estimated time of the next event. T ˆ 1 T 0 T, T 1 0 Tˆ P T C Year for which probabilities are computed, 2002 in most cases here. T E Elapsed time since last event, TE TC T0. Segment Parameters For some of the segments the time of the last event ( T 0 ) is known from the historic record, for others it must be estimated from geologic trenching data; for these segments, this is a major contributor to the overall uncertainty (Table 3). The next step is to estimate the slip ( D D ) in the last earthquake on each segment. On the San Andreas this is relatively straightforward, because we have the geodetic data used to formulate the elastic rebound model (Reid, 1910). On most other 9

10 segments we have paleoseismic estimates of the slip in the last event (Budding et al., 1991; Simpson et al., 1998; Hall et al., 1999; Lienkaemper et al., 2002), or in some cases an average slip per event spanning several events (Niemi and Hall, 1992; Kelson et al., 1996). The values used here are listed in Table 3. Thanks to the high quality geodetic and geologic data sets that have been assembled in the SFBA in the last few decades (Lisowski et al., 1991; Kelson et al., 1992; Lienkaemper and Working Group on Northern California Earthquake Potential, 1996; Argus and Gordon, 2001), the slip-rate ( V ) on each segment is the best-resolved piece of the puzzle (Table 3). (The parametric uncertainty P is determined by summing the variances for the time of the last event ( T 0 ) and the mean recurrence interval ( T ˆ ), , where ˆ ( / / ), see Table 2.) P 0 Tˆ ˆ T T D D V V The slip-rate estimates used here are based on balancing long- and short-term geologic rates with geodetic estimates, and include the requirement of continuity of sliprate along strike; that is variations in slip-rate only occur at fault intersections, and summed slip-rates match across intersections. The slip-rates have the additional constraint that along any traverse normal to the plate boundary, the rates must sum to the overall plate motion rate, which is well known from a variety of techniques to be mm/yr, with uncertainty estimates of 1-2 mm/yr. (Kelson et al., 1992; Atwater and Stock, 1998; Freymueller et al., 1999; Argus and Gordon, 2001; Prescott et al., 2001). This results in cumulative probability estimates for the region that are more robust than those for individual fault estimates, since if the slip-rate, and resulting probabilities are underestimated on one fault segment, they will be overestimated on a parallel fault strand. Seg. R ˆM 0 Mˆ 0 V Table 3 Segment Parameters SGN E Sum E Numbers in bold are input parameters. 1 The computed moment for the 1906 earthquake is the sum of the moments computed for the LP, SFP and NSA segments. T T0 D D V V dyne-cm cm cm LP E SFP E NSA E HS E HN E RC E CN E cm/ yr cm/ yr P Tˆ 10

11 The slip-rates used here agree in most cases with those of Lienkaemper et al. (1996) with the following exceptions: 1. have used 20 mm/yr for the LP and SFP segments. WG99 & 02 placed too much reliance on the single paleoseismic result of Hall et al. (1999), while ignoring the compelling geologic (Fumal et al., 1999) and geodetic evidence (Argus and Gordon, 2001; Prescott et al., 2001), and the theoretical modeling of Geist and Andrews (2000). 2. have used 4 mm/yr for the SGN segment (Simpson et al., 1998). The 7 mm/yr used by WG99 & 02 was obtained by differencing the rates for the SAN and SFP segments; differencing two uncertain numbers is not a credible procedure in the face of the direct geologic and geodetic evidence for a slip-rate on the SGN segment (Argus and Gordon, 2001). Moreover, as argued above, the value used by WG99&02 for the SFP segment is too low. 3. have used 10 mm/yr for the Hayward and RC faults, based on the stream channel offset of Williams (1999). 4. have used 5 mm/yr for the CN segment, based on the careful paleoseismic work of Kelson et al. (1996). 5. The slip-rates in Table 3 sum to 39 mm/yr on profiles normal to the San Andreas, leaving 1-2 mm/yr for the Mt. Lewis-Greenville fault system (Figure 1), which is shown on the State of California fault maps as a through-going fault (Bortugno et al., 1992), and appears to offset a number of geologic and topographic features by 3-5 km. The seismic evidence leaves little doubt that it is absorbing some small fraction of the plate boundary motion (Oppenheimer, 1992; Walter et al., 1998). t is not included as a segment in this report because there is no evidence it can or has produced earthquakes larger than M6, and the slip-rate, based on the rates-of-change seen geodetically on lowangle fault crossing lines, is apparently very low (Savage et al., 1998). t should be noted that the overall picture of slip rates presented here, 24 mm/yr west of SF Bay, and mm/yr to the east, receives strong support from recent work on Tertiary offsets and rates. Graymer et al. (2002) show that for the last 6 Myrs, total offset rates in the East Bay have averaged 17 mm/yr, and on the Hayward system 10 mm/yr. They show that in detail the locus of surface slip has moved between the various fault strands, just as Graymer (2002) has shown for the Hayward-Mission-Chabot system, but that average overall rates remain relatively constant. The match to current slip-rates in the East Bay could not be better, and this indirectly confirms the San Andreas and San Gregorio rates, given the strength of the estimates on the total plate-boundary rates (Argus and Gordon, 2001), and their constancy in time (Atwater and Stock, 1998). The uncertainties cited here assume a Gaussian distribution, and when they are combined with the recurrence model, variances are summed, as if all the distributions involved are Gaussian. This is not strictly true, of course, because a Weibull distribution is used to model earthquake recurrence. This has no practical consequences, however, because for a CV in the vicinity of 0.3, the Weibull and Gaussian distributions are very similar (Rothschild and Logothetis, 1986). The uncertainties used here are in all cases an estimate of the one standard deviation uncertainty ( ), although it should be understood that these are, in many cases, approximations at best. Where the data come 11

12 from paleoseismic estimates, which are often given as a bounding range of dates or offset distances, the bounds have been treated as 2 estimates; neglecting the tails in this manner is effectively equivalent to allowing some small possibility that there is additional variance in the problem, such as, for instance, an incomplete or misinterpreted stratigraphic section. Given these parameter estimates, it is now possible to estimate the expected return time ( T 1 1 ) of the next Characteristic event on each segment (Table 4), since the essence of the ER model as implemented here is that T 1 is simply T 0 D / V. For use in later calculations, Table 4 also includes estimates of times of some prior events (T -1, T -2 ), and their associated uncertainties. Table 4 Event times: observed (bold), estimated from geologic data, and predicted (italic) Seg. T -2 2 T -1 1 T 0 0 T +1 1 LP SFP SAN HS HN RC CN SGN Dates in Bold are historical, while those in normal type are estimated from paleoseismic data, except for the T -2 date for LP, which is a guess, based on historical reports of strong shaking at San Juan Bautista in the days following 11 October, 1800 (Townley and Allen, 1939; Toppozada et al., 1981). The predicted dates and uncertainties (in italic) are from Table 7; the predicted dates are 0 event, the uncertainties include T Tˆ T ˆ,, and in the case of segments without an historical and T 0, the uncertainty in T 0. Statistical Model To translate these return times into estimates of the probability of future earthquakes a statistical distribution is required. The Poisson model is the simplest it requires only a single number, the annual rate of earthquakes on a given segment ( / Tˆ 1 ) and has long been used in seismic hazard work. However in the SFBA, where we have some detailed understanding of the earthquake machine, the Poisson model is not appropriate. The clearest evidence of this is the length of the open intervals on most of the segments considered here. The Poisson distribution predicts that almost 40% of the segments should have a recurrence within the first half of the mean recurrence time (Lindh, 1988); not only would such early recurrences violate the letter and the spirit of the ER model, but they have not occurred in the SFBA in the last 200 years, as is clear by simply examining the historic record. The HN and RC segments have not had an event 12

13 greater than M6.7 in 200 years or more (Bakun, 1999) and their recurrence intervals are estimated to be in the year range; clearly they have not had any events in the first half of their recurrence cycle. Similarly, the HS segment had an M6.8 event 134 years ago; the estimate used here for the recurrence interval is 150 years. f we consider the portion of the San Andreas Fault that slipped in 1906, only the LP sub-segment has had a subsequent event. Estimates of Tˆ on that segment prior to 1989 were between 94 ( 39 ) (Lindh, 1988), and 136 ( 58 ) years (Dieterich et al., 1988), and it failed after 83 years. No other earthquakes larger than M6 have occurred along the 1906 rupture since 1906; Tˆ SFP has been variously estimated to lie between 115 years and 188 years, and estimates for Tˆ SAN range from 200 to 256 years. Thus there is no evidence in the historic record for any short recurrence times, let alone 40% in less than half the estimated mean recurrence time by the Poisson distribution. Admittedly the number of cases is small, but these arguments will be quantified below, where Poisson recurrence is tested against a quasi-periodic model. Statistical Distribution have chosen to use the Weibull distribution to model the variation in earthquake recurrence times, for it s simplicity and ease of use, and because the details of the distribution chosen, whether it be Gaussian, Weibull, Gamma, Brownian Passage-Time or log-normal, makes little difference (Utsu, 1984; Ogata, 1999). Moreover, the Weibull hazard function has the essential property that it is monotonic positive with increasing time, unlike the log-normal distribution which has a decreasing hazard function at times beyond the mean recurrence time (see for instance Figure 3 of Ellsworth et al. (1999)). We have abundant geodetic data showing that strain accumulation occurs at a constant rate between large earthquakes (Savage et al., 1998), and cannot conceive of a reason a statistical distribution which does not at least match this fact. For a Weibull recurrence model, the problem is easily formulated in terms of familiar functions. (This presentation borrows heavily from that of Dieterich et al (1990), whose notation have used wherever possible. The only significant difference is that they used a lognormal PDF, where have used a Weibull.) The fraction of all earthquake recurrence times in an interval ( t, t T ), where T is the time interval for which the probability is being calculated can be simply expressed as: P( TE t TE T ) F( TE T;, ) F( TE ;, ), (1) where T E is the time elapsed since the last event and Ft ( ;, ) is the cumulative Weibull probability; ( t / ) F( t;, ) 1 e, (2) and 1/CV, and T ˆ / ( CV 1) (Cornell and Winterstein, 1988). Cornell and Winterstein s approximations are intuitively very nice, since the first parameter is just the inverse of the CV, and the second is approximately 10% more than the mean recurrence time for values of CV between 0.2 and 0.7. (Unfortunately there is no standard set of symbols for the Weibull distribution; for convenience have used those from Microsoft EXCEL. Note that to avoid overflow there is no Gamma function in EXCEL, so one must use the expression EXP(GAMMALN(CV+1) to compute.) 13

14 The probabilities reported here employ the additional knowledge that the earthquake has not occurred prior to time T E. The probability of an event in time conditional on the earthquake not having occurred prior to T E, is: P( TE T TE T ) P( TE T TE T T TE ) 1 P(0 T T ) E ( TE / ) (( TE T )/ ) F( TE T ) F( TE) e e. ( T / ) FT ( ) E e E COEFFCENT OF VARATON Since we have used geologic and geodetic data to estimate the recurrence T, parameters, and the parametric uncertainty for each segment, the only remaining question is what value to use for CV, the intrinsic variability in the recurrence process. ( will make the usual assumption that a single CV can be used for all the segments. This may not be true -- faults with larger displacements might have more regular recurrence, for instance but at least all the faults considered here are strike-slip, and all have displacements of tens of km. or more.) Table 5 Likelihood Test on the ntrinsic Coefficient of Variation ( Seg T 0 Tˆ P CV CV ) T E LP SFP SAN HS HN RC CN SGN LP SFP HS SAN Likelihood (7 segments) 8.64E E E E E E-02 Likelihood (8 segments) 4.24E E E E E E-03 Likelihood (9 segments) 1.31E E E E E E-04 Likelihood (10 segments) 2.00E E E E E E-05 14

15 Likelihood (11 segments) Likelihood (12 segments) 4.28E E E E E E E E E E E E-07 A maximum-likelihood procedure was used to find an optimal value of CV, with the input data being simply the estimates of T E and T ˆ (and their uncertainties) from Table 5, supplemented by four closed recurrence intervals from the historic and geologic record. (Since the Weibull distribution becomes an exponential distribution when CV equals 1.0, this provides a convenient means of testing whether the events are Poisson distributed at the same time.) The methodology used here is similar to earlier attempts to estimate average recurrence times and CV from historic or paleoseismic data (see for instance Davis et. al (1989) and Ogata (1999)), using both open and closed recurrence intervals. Here, however, have used prior estimates of ˆ T and and geodetic data, and only solved for a single parameter -- a more stable estimate. P from historic, geologic CV -- resulting hopefully in The problem separates very nicely into two parts. The Likelihood of each open interval (the first eight rows in Table 5) is simply the probability of no event occurring on that segment before time T E, or ( TE / ) P(0 T T E ) = F( TE ;, ) 1 e (4) and the Likelihood of this set of eight open intervals is simply, 8 L18 P(0 T TE ). (5) i 1 While these probabilities are functions of and, T E, and are functions of T ˆ and CV. Since prior estimates are used of T E, T ˆ and their uncertainties (Tables 3 & 4), CV is left as the only adjustable parameter in the Likelihood procedure. The resulting Likelihood values are shown in the row labeled 8 segments in Table 5; they rise monotonically from a minimum at CV =1, to a gentle plateau for values of 0.2 and less (Figure 2). Also shown is a curve for 7 segments, which omits the value for the HN segment, since as will be discussed below, the recurrence parameters for that segment are rather more uncertain than the others. Also listed in Table 5 are Likelihood s for four closed intervals for which we have reasonable data; the details concerning these segments are included in a later section. These intervals are: to 1989 on the Loma Prieta segment, called here LP, to 1906 on the San Francisco Peninsula segment (SFP ), ( 35 ) to 1868 on the Hayward South segment (HS ), and ( 25 ) to 1906 on the San Andreas North segment (SAN ). For these closed intervals the probability is calculated for a time T around the observed T E, P( T T t T T ) F( T T;, ) F( T T;, ). (6) E E E E 15

16 Likelihood (For the numbers shown in Table 5 a T of 15 yrs. was used, but the same answer is obtained for any small value.) This results in a Likelihood that includes the closed intervals: n L1 n L1 8 * P( TE T T TE T ), (7) i 9 where the value of n depends on how many of the closed intervals are included. The Likelihood values resulting from these calculations are listed in the bottom six rows of Table 5 for values of CV ranging from 0.0 to 1.0; they are plotted in Figure 2, with each curve normalized by the Likelihood at CV =0.2. Likelihood values have been computed for six different scenarios one for the eight open intervals, four more by adding sequentially each of the closed intervals, and one by omitting the HN segment to emphasize that the answer does not depend on data selection. n addition have tested the possibility that slip rates have been incorrectly partitioned between the various fault strands by juggling slip rates, subject to the constraint they sum to 39 mm/yr across the plate boundary; this has a negligible effect on the most likely values of CV. have also tested the huge earthquake hypothesis, which asserts that rare very large events absorb a disproportionate share of the moment rate (Jackson, 1996) by halving the slip rates everywhere; this also had no significant effect on the distribution of Likelihood s. t is clear that smaller values of CV are strongly favored; for continuity with earlier work have used a CV of 0.2 (Buland and Nishenko, 1987; Lindh 1988; Dieterich et al., 1988 & 1990), which as will show below, also provides a better overall fit to the historic seismicity of the SFBA than a Poisson model, or any model with a large CV. 3.5 Likelihood of Observed ntervals Normalized to CV = ntervals 8 ntervals 9 ntervals 10 ntervals 11 ntervals 12 ntervals CV Figure 2. Likelihood plotted against CV, the data are from the six bottom columns of Table 5. The 8 intervals curve is the Likelihood of the observed open-intervals on the eight segments studied (the T E in Table 5). That is, what is plotted is the Likelihood that these eight segments have each not had a characteristic event in time T E for each segment The four lower curves 16

17 include the additional Likelihood s, added one at a time, that the closed recurrence intervals -- such as the interval from 1906 to have occurred. The 7 intervals segment is the 8 intervals data, with the probabilities for the HN segment removed. Since a Weibull distribution becomes an exponential distribution as CV 1, the values on the far right side represent the Likelihood of a Poisson model reproducing the observed pattern of open- and closed intervals. This is hardly a surprising result if we examine the published literature on this question. n a careful study of recent paleoseismic data from five onshore faults in Japan, Ogata (1999) obtained best-fitting values for CV of 0.2 or less for all five of the faults studied. (Note that Ogata does not provide estimates of CV, or the variance, in his tables. However, the variance can be computed from his parameter estimates, using the formulas in Rothschild and Logothetis(1986), and the CV computed by taking the ratio of the square-root of the variance to the Mean Return Times (MRT), which Ogata does provide. Note also that there is a typo in Table 6 of Ogata. The formula for the RMT for the 1 Weibull distribution should read RMT (1/ ) / 1/ assignments.), for his parameter McCalpin and Slemmons (1998) compiled a detailed summary of 41 paleoseismic chronologies, for each of which they computed estimates of T ˆ and T ˆ. They recognize the difficulty in estimating from short runs of data, and provide a summary figure in ˆ T which they show that as the number of events in each data set increases, the ratio / Tˆ approaches 0.3. Since this estimate includes dating and stratigraphic uncertainties, as well as, it is clearly consistent with values of CV of 0.3 or less. Ellsworth et al. (1999) reanalyzed, and briefly summarize some of this same data set; they also added a large number of seismically determined recurrence sequences for small events from Central California (Ellsworth, 1995; Nadeau and Johnson, 1998). While they were critical of the analysis of McCalpin and Slemmons (1998), they obtained essentially the same result; 70% of their estimates of ( their asymmetry, roughly comparable to / Tˆ ) lie in a symmetrical peak centered between 0.0 and Tˆ While they chose to average all their estimates of -- obtaining a value of 0.44 ( 0.22) this is clearly inappropriate, since their data are not normally distributed, but have a very long right hand tail (their Figure 6c). n addition, many of the large values of they report should have been treated as outliers. For instance, they obtained large values of (0.68 to 0.96) for three paleoseismic data sets from the San Andreas in southern California, but it is extremely unlikely that each of these data sets represent repeats of individual characteristic earthquakes, given the tectonic and seismological complexity of the region surrounding the intersection of the intersection of the Banning and Mission Strands of the San Andreas, the San Jacinto and Cucamunga faults. Sieh (1996) presented a patch model interpretation of some of that same data, in which he argues that the complexity at Pallett Creek is due to its position near the junction of very large earthquake ruptures; large events to the north like the great 1857 Fort Tejon earthquake, and comparable events to the south which rupture the San Andreas from their to the Salton Sea, like the presumed 1680 event. Fumal et al. (2002) have recently presented an update of this model, and argue on the basis of very Tˆ 17

18 good data at Wrightwood and Thousand Palms Oasis that the last five events on the southernmost 200 km of the San Andreas have occurred at intervals of 215 ( 25) years, implying a very small CV. (They do note, however, the problem this short recurrence interval causes when compared to the approximately 326 yrs that have passed since the last large event along this portion of the San Andreas. Lindh (1988) tried to resolve this by averaging the apparent recurrence interval with the longer open interval since the last event; the problem cries out for the approach of Ogata (1999) in which a Maximum Likelihood procedure is applied to the entire data set, providing a single estimate of the recurrence time and CV.) Thus Fumal et al. (2002) effectively show that the apparent complexity of the paleoseismic record at Wrightwood is very likely the result of being in the overlap zone between great adjacent ruptures, just as Sieh (1996) argued for the nearby Pallett Creek site. Similarly Ellsworth et al. obtained an of 0.77, corresponding to very irregular recurrence, for M9 events on the Cascadia subduction zone. However, the data set they analyzed was derived from onshore paleoseismic studies (Atwater and Hemphill-Haley, 1997), and it appears from offshore turbidite records that the onshore record is somewhat misleading (Adams, 1990). Having collected addition data offshore of Washington and Oregon, Goldfinger and Nelson (1999) have concluded that the sequence was remarkably regular; 12 turbidite events have occurred during 7200 years or on average every 600 years. Existing AMS ages show 700 yr between the first and second MA [Mazama Ash] events; the same periodicity between most events in all cores also is suggested by the consistent thickness of hemipelagic sediment representing about 600 yr between turbidite beds. With regard to the question of regularity of recurrence of great subduction earthquakes, it should also be noted that Ellsworth et al. obtained an of 0.23 for the sequence of nine paleoearthquakes described by Plafker et al. (1992) and Plafker and Rubin (1994) for prior occurrences of the great 1964 Alaskan earthquake. Thus disagree strongly with the conclusion of WG95, Ellsworth et al. (1999), and WG99&03 that =0.5 is a reasonable estimate of the aperiodicity for large strikeslip earthquakes, and believe that WG99&02 compounded the error by using a weighted average of quasi-periodic and Poisson models, effectively using an even larger CV as will show below this choice accounts for most of the anomalously low probabilities they obtained for faults in the SFBA. With regard to the question of whether Poisson recurrence remains a credible model for earthquake recurrence, this question can be addressed directly by the analysis presented in Figure 2 and Table 5, since a Weibull distribution goes to an exponential distribution as CV approaches 1. f we take CV =0.2 as our preferred model, and wish to compare it s performance to a Poisson model, we need only compare the Likelihood values in the CV =0.2 and 1.0 columns in Table 5; these are tabulated in Table 6. No. of segments Table 6 -- Likelihood Ratios Likelihood CV =0.2 CV =1.0 Likelihood Ratio AC E E