International Geology Review Vol 44 2002 p 62ndash82Copyright copy 2002 by V H Winston amp Son Inc All rights reserved

0020-68140257562-21 $1000 62

Integrating Geologic and Geodetic Estimates of Slip Rateon the San Andreas Fault System

PAUL SEGALL

Department of Geophysics Stanford University Stanford California 94305

Abstract

Interseismic deformation measurements are generally interpreted in terms of steady slip on bur-ied elastic dislocations Although such models often yield slip rates that are in reasonable accordwith geologic observations they are inconsistent with our expectation of fault structure at depth andcannot explain transient postseismic deformation following large earthquakes An alternatetwo-dimensional model of repeating earthquakes that break an elastic plate of thickness H overly-ing a Maxwell viscoelastic half-space with relaxation time tR (Savage and Prescott 1978) involvesfive parameters H tR t T and where t is the time since the last quake T is the earthquake repeattime and is the slip rate Many parts of the San Andreas fault (SAF) system involve multipleparallel faults which further increases the number of parameters to be estimated All hope is notlost however if we make use of geologic constraints on slip rate as well as the measuredtime-dependent strain following the 1906 earthquake in addition to the present-day spatial distri-bution of deformation rate

GPS data from the Carrizo Plain segment of the SAF imply a considerably larger relaxation timethan inferred from the post-1906 strain rate transient recorded by repeated triangulation surveysThis may indicate that the simple Savage-Prescott model is deficient or that the crustal structureandor thermal profile differs significantly between northern and central California The data areconsistent with an average recurrence interval of ~280 years which agrees reasonably well withpaleoseismic estimates

To test the effect of regional variations in H and tR I analyze GPS-derived velocities from thenorthern San Francisco Bay area (Prescott et al 2001) and include the SAF the Hayward-RodgersCreek (HRC) and Concord-Green Valley faults (CGV) Optimization using constrained non-linearleast squares yields H ~18 km tR ~35 years TSAF = 280 years ~25 mmyr tHRC = 225 yearsTHRC = 429 years and ~7 mmyr The effect of the ConcordndashGreen Valley faults is modeledkinematically the optimal model involves 8 mmyr of slip beneath 2 km While neither the averagerecurrence interval on the SAF or HRC is well resolved the optimal inter-event time for the SAF isin reasonable agreement with paleoseismic results which indicate that the most recent event priorto 1906 occurred in the mid-1600s The geodetic data also favor the latest date allowed by historicaldata (~1775) for the most recent HRC earthquake a result not inconsistent with the paleoseismicfinding that the most recent event there occurred in the interval 1670ndash1776 The conclusion of thisstudy is that by combining the present-day deformation field post-1906 strain data and geologicbounds on slip rate and maximum earthquake slip it is possible to estimate parameters of consider-able geophysical interest including time since the most recent earthquake and average recurrenceintervals

Introduction

INTERSEISMIC DEFORMATION DATA collected alongthe San Andreas fault system reveals a broad zone ofstrain that extends roughly 100 km perpendicular tothe strike of the principal faults (eg Johnson et al1994) Approximately 35 mmyr of relative dis-placement is accommodated across this region (egPrescott et al 2001) Geodetic strain data such asthese are important components of seismic hazard

evaluations Simply stated the higher the strain ratethe bigger or more frequent the earthquakes mustbe to relieve that strain More quantitative informa-tion about earthquake occurrence can only be madethrough reference to a mechanical model of thedeformation field The simplest and most widelyused model is that of an infinitely long screwdislocation in an elastic half-space introduced bySavage and Burford in 1970 In this case the

smiddotsmiddot

smiddotSAFsmiddotHRC

ESTIMATES OF SLIP RATE 63

fault parallel velocity v(x) is a simple functionof slip rate and the fault locking depth d

Here x is distance nor-mal to the fault Given measured velocities v(xi) it isstraightforward to estimate that depth over whichfaults are locked and the average slip rate acrossthose faults (eg Williams et al 1994) Thisapproach has been generalized for finite length dis-locations (eg Matsursquoura et al 1986) and forfar-field motions consistent with rigid sphericalplates (eg Murray and Segall 2001) Slip ratesestimated from geodetic observations can be com-pared to those derived from geologic studies andalong with the average slip in large earthquakesform the basis of seismic hazard evaluations

The screw dislocation while a useful first-ordermodel of plate boundary deformation does not accu-rately represent fault structure at great depth withinthe earth Nor does it account for temporal varia-tions in strain rate that have been observed follow-ing large earthquakes More physically realisticmodels tend to fall into two categories The firstinvolves faulting in an elastic lithosphere overlyinga viscoelastic lower crust or upper mantle (Nur andMavko 1974 Savage and Prescott 1978 Lehner etal 1981 Cohen 1982 Rundle 1986 Lyzenga1991 Pollitz 1992) The viscoelastic medium mayextend to great depth or be limited to a channel offinite thickness Some studies have investigatednon-linear rheology (eg Reches et al 1994) Thesecond class of models explains time-dependentdeformation following earthquakes as arising fromvelocity strengthening slip on a downdip extensionof the coseismic fault surface (eg Tse and Rice1986) Most of these models can fit the present-daydistribution of strain rate across the San Andreasfault system with an appropriate choice of parame-ters A significant complexity in applying thesemodels however is the fact that in many places theSan Andreas fault system consists of multiple paral-lel faults In the San Francisco Bay region we havethe San Andreas HaywardndashRodgers Creek and theCalaverasndashConcordndashGreen Valley faults In south-ern California the subparallel Elsinore and SanJacinto faults are present in addition to the SanAndreas Because physically realistic models offaulting invariably involve a large number of param-eters and are slower to compute they are not oftenused in inversions of geodetic measurements Fur-thermore Savage (1990) has shown that it is diffi-cult to distinguish between these two classes ofmodels because the solution for the anti-plane vis-

coelastic problem has a corresponding elastic solu-tion involving time-dependent slip below thecoseismic rupture The latter two observations haveled to the feeling that more physically realistic mod-els are unwarranted

In this paper I argue that a key to moving for-ward with this problem is to include more informa-tion than simply the present-day deformation fieldAdditional information comes from the time depen-dence of deformation following large earthquakesand paleoseismic estimates of fault slip rates A pri-ori information about faultslip rate from geologicstudies can be used to complement information fromgeodetic studies In the present paper this is donein the context of rather simple models of the earth-quake cycle involving repeated earthquakes in anelastic layer overlying a Maxwell viscoelastichalf-space Although this model is itself highly ide-alized the results are sufficiently encouraging towarrant further investigations

The use of a priori information to regularize anill-posed inverse problem is well known (eg Jack-son 1979 Tarantola and Valette 1982 Menke1989) In this work I propose using Bayesrsquo rule tocombine geologic and geodetic information in orderto permit use of more sophisticated and hopefullyrealistic models than could be addressed with geo-detic data alone Previously Minster and Jordan(1987) used Bayesrsquo rule to combine geodetic andgeologic data to estimate the net motion across theSan Andreas fault system The general frameworkproposed here could be expanded to include infor-mation from other data sets as well but the focushere is on slip-rate data Fault slip rates are deter-mined from geologic observations such as measure-ments of offset stream channels and dates of thecorrelative sedimentary deposits From the point ofview of geodetic analysis this information places apriori constraints on the slip rate I denote the prob-ability density function (pdf) of the slip rate fromthe geologic observations as or perhaps moreappropriately as where g denotes geologicobservations such as measured offsets radiometricage determinationand so forth

On the other hand mechanical models of theearth allow us to predict geodetic observations giventhe slip rate on the fault and the other model param-eters such as the fault locking depth elastic-layerthickness viscosity of the lower crust and so forthDenoting the parameters of the mechanical modelother than slip rate as m then the model yields

smiddotv x( ) smiddot pcurren( ) x dcurren( )1ndashtan=

P smiddot( )P smiddot g( )

64 PAUL SEGALL

where d is a vector of geodetic dataBayesrsquo rule then states that

(1)

In words equation (1) states that the pdf of theslip-rate and fault-model parameters given geodeticobservations can be determined from the a prioridistributions of the slip-rate and the fault-modelparameters and the theoretical model linking theparameters to the geodetic observations The integralin the denominator ensures that the posterior proba-bility is properly normalized ie that the cumula-tive probability is unity Assuming that the priorprobability of the slip rate and the model parametersare independent then Thisis reasonable inasmuch as estimates of slip rate frompaleoseismic studies are independent of prior esti-mates of elastic layer thickness viscosity fault lock-

ing depth and so on While in general we have apriori constraints on these physical parameters forthe purposes of the present discussion I will oftenassume that the distribution is so broad as to benon-informative Given the independence of theprior distributions on s and m (1) reduces to

(2)

The posterior probability distribution is easilycomputed if the prior distribution is uniform oversome range That is we take the slip rate to beequally probable between specified upper and lowerbounds A simple illustration (Fig 1) can be infor-mative In Figure 1A the mean of the distributionfalls just outside the a priori bounds The posteriordistribution is thus highly concentrated near theupper bound from geologic data The geodetic and

P smiddot m( )

P smiddot m d( ) P smiddot m( )P d smiddot m( )

P smiddot( )P d smiddot m( ) smiddotd mds mograve

---------------------------------------------------------- =

P smiddot m( ) P smiddot( )P m( )=

P smiddot m d g( )

P smiddot g( )P m( )P d smiddot m( )

P smiddot( )P m( )P d smiddot m( ) smiddotd mds mograve

------------------------------------------------------------------------------

=

FIG 1 Example of application of Bayesrsquo theorem given a uniform prior distribution The prior distribution is uniformover the interval [01] and is shown in the dash-dot pattern The distribution from the data alone is dashed and theposterior distribution is solid In (D) the posterior distribution is essentially a delta function at 10

ESTIMATES OF SLIP RATE 65

geologic data together are much more informativethan either data set alone In Figure 1B the a prioribounds are so broad as to be uninformative the pos-terior distribution is equal to that obtained from thegeodetic data alone In Figure 1C the reverse istrue the geodetic data supply little information andthe posterior distribution reflects simply the a prioriinformation A fourth outcome also is possible asillustrated in Figure 1D where the two distributionsdo not overlap indicating that the two estimates areinconsistent The conclusion in this case is thateither the uncertainties in the geologic determina-tion of slip rate have been substantially mis-esti-mated or the physical model used to relate geodeticdata to slip rate is invalid The outcome in Figure1D could be the most useful in that assuming thegeologic bounds are valid it would invalidate anentire class of physical fault models

I start by considering the interseismic velocitydistribution across the Carrizo Plain segment of theSan Andreas fault (Fig 2) This area is chosenbecause it is the simplest part of the SAF systemuncomplicated by parallel faults The only otheractive structures at this latitude are the San Grego-riondashHosgri fault system which runs into an activedeformation zone in the Santa Maria Basin (Feigl etal 1990) and the fold-and-thrust belt northeast ofthe San Andreas at the eastern edge of the Coast

Ranges The latter structures involve motion largelynormal to the SAF and thus do not significantlyaffect the calculations here The San GregoriondashHos-gri deformation is accounted for crudely using asimple dislocation model

Assuming only a single San Andreas fault andmaking use of a two-dimensional dislocationmodel I find that the best-fitting locking depth is244 km (Fig 3) and the slip rate is 418 mmyr Incontrast Sieh and Jahns (1984) determined anaverage slip rate of 339 plusmn 29 mmyr for the past3700 yr and 358 + 54ndash41 mmyr for the past13250 yr based on offset fluvial deposits at Wal-lace Creek Allowing 3 mmyr of slip on the SanGregoriondashHosgri fault system which is modeled asa screw dislocation locked above 10 km the SanAndreas locking depth shallows slightly to 203km and the slip rate drops to 377 mmyr which isin reasonable accord with geologic estimates (Siehand Jahns 1984)

Of course the San Andreas fault is nottwo-dimensional Southeast of the Carrizo Plain theSan Andreas fault enters the Big Bend and theTransverse Ranges whereas the creeping zone islocated to the northwest The data analyzed here(Fig 2) were selected to be as far from possible fromthese locales yet some bias is unavoidable with atwo-dimensional model

FIG 2 Velocities determined by the Southern California Earthquake Center (SCEC) in the Carrizo Plain section ofthe San Andreas fault

66 PAUL SEGALL

It is important to determine the range of sliprates and locking depths that are consistent withthe GPS data A bootstrap resampling procedureshows that slip rates in the range of 34 to 41 mmyrare consistent with the geodetic observations (Fig4) The bootstrap was conducted in the followingmanner Let d be the vector of observations thepredicted data from the best-fitting model so that

is the vector of residuals The residualswere randomly resampled with replacement that issome elements may be chosen multiple times oth-ers not at all Denote the resampled residuals r Aresampled data vector is then usedto generate a model estimate and the process isrepeated many times The distribution of the boot-strapped model parameters is that shown in Figure4

It is well known that there is a tradeoff betweenthe locking depth and deep slip rate Increasing thelocking depth moves the dislocation farther from thesurface observation stations to achieve the samesurface strain requires a higher slip rate Applying apriori constraints on the slip rate from 31 mmyrand 37 mmyr based on the Sieh and Jahns (1984)data from Wallace Creek thus not only changes the

posterior distribution of slip rate but of the lockingdepth as well (Fig 5) The reason for this can beclearly seen from a scatter plot of the bootstrap res-amples as in Figure 6

Notice that the GPS data have significantly nar-rowed the distribution of admissible slip rates (Fig5) The posterior distribution is peaked at 35 to 37mmyr whereas the a priori distribution is uniformbetween 31 and 37 mmyr At the same time the apriori geologic information provided no constrainton the fault locking depth With geologic data alonewe cannot distinguish between a locked fault and afault that creeps steadily below some shallow lockedinterval The GPS data alone bound the lockingdepth to between 17 and 24 km Because of thestrong correlation between slip rate and lockingdepth the posterior distribution of the locking depthis narrowed to 17ndash20 km This simple exampleshows the power of using a priori geologic con-straints in allowing more information to be obtainedfrom geodetic data

Note that the bootstrap provides an estimate ofthe distribution of the model parameters whereasBayesrsquo rule requires P(d|m) as in equation (3)However for linear models with normally distrib-

d

r d dndash=

d d r+=

FIG 3 A Misfit as a function of depth for a simple screw dislocation model B Profile of fault parallel velocity acrossthe fault compared with prediction of a simple screw dislocation model C As in (B) but with 3 mmyr of slip beneaththe Hosgri fault

ESTIMATES OF SLIP RATE 67

uted observational errors the two approaches areequivalent Furthermore a uniform distribution withhard bounds may not be the best characterization ofthe prior probability distribution on slip rate Alter-

natively if the prior distribution and observationalerrors are normally distributed one can simplyapply the prior information as ldquopseudo datardquo (egJackson 1979)

FIG 4 Probability distributions for fault locking depth and slip rate from bootstrap results using GPS data I assume3 mmyr below a locking depth of 10 km on the Hosgri fault system

FIG 5 Posterior probability distribution from the bootstrap results using GPS data and assuming a uniform priordis-tribution on slip rate from 31 to 37 mmyr Compare with Figure 4 As before I assume 3 mmyr below a locking depthof 10 km on the Hosgri fault system

68 PAUL SEGALL

Viscoelastic Models

As noted in the Introduction the screw disloca-tion provides only a very limited description ofplate-boundary faulting A somewhat more realisticmodel involves an elastic layer of thickness H over-lying a Maxwell viscoelastic half-space (Fig 7) TheMaxwell material has relaxation time tR = 2hmicrowhere h is the viscosity and micro is the shear modulusAt time t = 0 slip Du occurs on the fault from thesurface to depth D pound H The velocity on the Earthrsquossurface as a function of position perpendicular to thefault x and time t is

(3)

where the spatial distribution is given by

(4)

The result follows from but is not stated in Nur andMavko (1974) The strain rate is simply obtained bydifferentiating the velocity with respect to x Noticethat the post-seismic velocity is a function of thecoseismic slip Du the depth of faulting D the elas-tic layer thickness H the material relaxation tR andthe time since the last large earthquake t Equation(3) does not include the effects of interseismic strainaccumulation Savage and Prescott (1978) extendedthe model by summing over an infinite sequence ofplate rupturing events (D = H) equally spaced intime with recurrence interval T The velocity field asa function of distance across strike (x) and time (t) isgiven by

(5)

where the temporal dependence depends only on thedimensionless ratios ttR and TtR

(6)

v x t( )

DuptR---------e

t tRcurrenndash t tRcurren( )n 1ndash

n 1ndash( )--------------------------Fn x( D H )

n 1=

yen

aring

=

Fn x D H ( )

D 2nH+x

---------------------egrave oslashaelig ouml1ndash

tan D 2nHndashx

--------------------egrave oslashaelig ouml1ndash

tan+

2xD

x2

2nH( )2D

2ndash+

------------------------------------------1ndash tan

=

=

v x t( ) smiddot

p--- Tn t tR T tRcurrencurren( )Fn x H H ( )

n 1=

yen

aring=

Tn t tR T tRcurrencurren( )

TtR-----

et tRcurrenndash

n 1ndash( )------------------ e

kT tRcurrenndash t kT+tR

--------------egrave oslashaelig ouml n 1ndash

k 0=

yen

aring

=

FIG 6 Tradeoff between slip rate and locking depth Each point is the result of a bootstrap resample of the SCECvelocity data The heavy points are those consistent with a uniform prior distribution on slip rate from 31 to 37 mmyrAs before I assume 3 mmyr below a locking depth of 10 km on the Hosgri fault system

ESTIMATES OF SLIP RATE 69

Because the far-field lithospheric velocity is drivensolely by the repeating earthquakes Representative examples are given in Figure 8 forvarious times through the earthquake cycle Noticethat when the asthenospheric relaxation time is ofthe order of the earthquake cycle time or longer TtR pound1 the surface velocity distribution is rathersteady with time (Fig 8A) On the other hand if therelaxation time is short compared to the recurrenceinterval TtR gt 1 there is a significant transientwith higher than average strain rates early in thecycle and lower than average strain rates late in thecycle

The Savage-Prescott model involves five param-eters lithospheric thickness H relaxation time tRtime since the last earthquake t recurrence time Tand long-term slip rate on the fault Because thetheory is linear it is straightforward to constructmodels with multiple parallel faults however in thiscase the number of parameters is even larger Forthree faults there are 11 parameters if one assumesthat the elastic thickness and relaxation time are thesame for all faults Note that the model is kinematicwith imposed slip events on the various faults at reg-ular intervals and thus does not account for thestress interactions between the faults

As a starting point I consider the Carrizo Plaindata with some of the parameters constrained byindependent observations The time since the lastgreat earthquake (1857) is t = 136 years To beginwith I assume a recurrence time of T = 160 yearsbased on the average interval found by Grant andSieh (1994) There are three remaining parametersto be estimated from the data tR H and Theoptimal estimates of the non-linear parameters tRand H are found by simple grid search (Fig 9) with

estimated by linear least squares The optimalmodel is found for H = 114 km and tR = 80 yearswith a slip rate of 402 mmyr There is a strongtradeoff between relaxation time and elastic-layerthickness such that models with elastic-layer thick-ness of up to 20 km and relaxation time of 180 yearsfit the data nearly equally well The reason for thetradeoff is as follows Larger values of H cause thedeformation to be spread out in space At the sametime the deformation becomes more diffuse withincreasing tT and TtR (Fig 8) and hence t tR = (tT) times (TtR ) Thus decreasing ttR causes the defor-mation to bemore localized which counters theeffect of increasing H Interestingly the slip ratedoes not vary with H and tR confirming that thisparameter is well constrained by the data (given thatall other parameters are fixed)

The inferred slip rate is however higher thanestimated from stream offsets at Wallace Creek of 33plusmn 3 mmyr (Sieh and Jahns 1984) Constraining theslip rate to 34 mmyr (Fig 10) does not fit the dataas well especially for the stations farthest west ofthe San Andreas The optimal model is H = 114 kmas before but the relaxation time increases to 180years

Relaxation times tR gt 50 years are long com-pared to what has been inferred for the post-1906strain rate transient north of the San Francisco Bayas measured by repeated triangulation surveys(Thatcher 1975a 1975b Kenner and Segall 2000)The relaxation time tR consistent with the post-1906transient can be inferred from the triangulation dataand the spatial derivative of equation (3) It wouldbe most consistent to employ the same Savage-Pres-cott model as used with the Carrizo Plain data how-ever this requires knowing the recurrence time oflarge earthquakes on the north coast segment of theSan Andreas fault An alternate approach employedhere is to remove the estimated secular deformationusing a model (see Kenner and Segall 2001) andthen fit the remaining strain rate data For 60meters of slip in the 1906 earthquake the optimal

smiddot Du Tcurren=

smiddot

smiddot

smiddot

FIG 7 Model of a faulted elastic plate overlying viscoelas-tic half-space The elastic layer has shear modulus micro the vis-coelast ic half-space shear modulus micro and viscosity hEarthquakes occur at a regular interval T such that thefar-field velocity is vyen2

70 PAUL SEGALL

model is H = 19 km with a relaxation time of 17years (Fig 11) substantially less than found fromthe Carrizo Plain interseismic data1

A tentative conclusion of this analysis is that theCarrizo Plain interseismic data require a relaxationtime in excess of 50 years whereas the post-1906triangulation data require relaxation times of 25years or less In order to test whether this discrep-ancy is truly required by the data I estimated theuncertainty in the two relaxation times using a boot-strap resampling procedure As discussed previ-ously the residuals to the best-fitting model wererandomly resampled with replacement and added

to the model predictions for each bootstrap resam-ple This approach seems best for small data setsand ensures that there were data at each observationposition (interseismic data) and time (postseismicdata)

For the post-1906 transient admissible relax-ation times range from 10 years to 30 years (Fig 12)On the other hand the present distribution of defor-mation on the Carrizo Plain demands relaxationtimes in excess of 35 years (Fig 13) Thus we cansafely conclude that there is no single relaxationtime that fits both data sets

There are a number of possible explanations forthis apparent discrepancy One possibility is thatthe model of a fully ruptured elastic plate overlyinga linear Maxwell half-space does not capture all ofthe relevant physics For example nonlinear rheol-ogy may be important The data are consistent withthe expectation that the effective viscosity is low

1 Note that the discrepancy is not due to the correction forinterseismic deformation in the post-1906 data Assuming arecurrence time of 250 years and a slip rate on the SAF of 25mmyr the optimal relaxation time is 228 years with an elas-tic-layer thickness of 167 km

FIG 8 Savage-Prescott model for (A) T tR = 1 and (B) T tR = 5 t is the time since the last earthquake T is theearthquake repeat time and tR is the Maxwell relaxation time In these calculations the earthquakes rupture the entireelastic layer

ESTIMATES OF SLIP RATE 71

early in the seismic cycle when the stresses arelarge and increases late in the cycle when thestresses have largely dissipated In addition recentreflection and refraction results in northern Califor-nia show the San Andreas fault cutting the lowercrust and possibly the Moho (eg Henstock et al1997 Parsons and Hart 1999) Aseismic slip onthe downdip extension of the seismogenic zone (asin Tse and Rice 1986) may take place in additionto a more deeply distributed relation Results fromKenner and Segall (2001) which include viscousrelaxation in a narrow fault zone and more deeplydistributed slip find the spatiotemporal pattern ofdeformation in the post-1906 triangulation data isbest fit with a short relaxation time in the fault zoneand a longer relaxation in the underlyinghalf-space Another possibility is that the thermo-mechanical structure of the lithosphere differsbetween the two areas It is also possible that along-strike variations in fault geometry (the Big Bend) or

slip behavior (the creeping zone) may bias the com-puted relaxation time although this seems lesslikely

Although the apparent relaxation times differbetween the Carrizo Plain and the northern SanFrancisco Bay region the elastic-layer thicknessestimates are consistent in the range of 13ndash20 kmCarrizo Plain slip-rate estimates cluster near theupper bound of 37 mmyr Interestingly the CarrizoPlain data yield recurrence times greater than 160years Whereas the distribution is reasonably broadboth the mean (288 years) and median (277) repeattimes fall between the 1988 working group estimateof 296 years (WGCEP 1988) and the 1995 workinggroup estimate of 206 years (WGCEP 1995) Theaverage coseismic slip for the predicted events isroughly 10 meters There are significant correlationsbetween parameters (between H and tR as dis-cussed previously as well as between tR and T)which means that any additional data that constrain

FIG 9 Comparison of Savage-Prescott model to the SCEC velocity data for the Carrizo Plain A Misfit as a functionof elastic-layer thickness and relaxation time B Best-fitting San Andreas slip rate corresponding to the elastic-layerthickness and relaxation time C Fit to the data Effect of the Hosgri fault is removed employing a dislocation model withslip rate of 3 mmyr and depth of 10 km

72 PAUL SEGALL

some of the parameters will help resolve others aswell

It should be emphasized that the paleoseismicobservations of Grant and Sieh (1994) show an irreg-ular pattern of earthquakes that cannot be explainedby the simple two-dimensional model used here tointerpret the geodetic observations Nevertheless itis interesting that the GPS data appear to have someresolving power for average recurrence interval

San Francisco Bay Data

In order to eliminate any geographic bias in theinterpretation associated with possibly differentcrustal structure and temperature distribution Inext consider the velocity profile across the SanAndreas fault north of the San Francisco Bay asdetermined by repeated GPS observations (Prescottet al 2001) as shown in Figure 14 The GPS sitesare sufficiently close to the post-1906 triangulationnetworks that it is reasonable to assume that the

crustal structure and thermal profile are the samefor both data sets In the following the GPS-derivedvelocities are rotated into fault-parallel and perpen-dicular components Only the fault-parallel motionsare modeled here although as pointed out by Pres-cott et al (2001) and Murray and Segall (2001)there is significant fault-normal motion in the east-ern coast ranges at the western edge of the CentralValley

The model is the same as discussed previouslywith three principal faults the San Andreas Hay-wardndashRodgers Creek and ConcordndashGreen Valleyfaults The parameters are the elastic plate thick-ness H Maxwell relaxation time tR the recurrencetime for great earthquakes on the SAF TSAF the sliprate on the SAF and the recurrence timeTHRC time since past event tHRC and slip rate on the Hayward-Rodgers Creek fault system Thetime since the last great earthquake on the northernSan Andreas fault tSAF is of course known Theeffect of the ConcordndashGreen Valley fault is modeled

smiddotSAFsmiddotHRC

FIG 10 Comparison of Savage-Prescott model to the SCEC velocity data for the Carrizo Plain with slip rate con-strained to 34 mmyr Effect of San GregoriondashHosgri fault has been removed with a dislocation model with slip rate of 3mmyr and depth of 10 km A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

ESTIMATES OF SLIP RATE 73

kinematically with a simple screw dislocationwhich adds two parameters the fault slip rate andlocking depth It is certainly inconsistent to ignoreviscoelastic effects for the ConcordndashGreen Valleyfault but I do so for three reasons (a) much less isknown about this fault (b) earthquakes there arepresumably least well modeled by infinitely longruptures and (c) it keeps the number of parametersreasonably tractable In essence the effect of theConcordndashGreen Valley is removed empirically and Imake no attempt to gain further insights into itsmechanical behavior Future work should addressand hopefully rectify this simplification

A priori bounds on the parameters are listed inTable 1 The lower bound on the lithospheric thick-ness comes from a conservative estimate of thedepth extent of slip in large earthquakes In fact itis more likely that slip in 1906 extended at least to15 km (Matthews and Segall 1993) as it certainlydid in the Loma Prieta earthquake (eg Beroza1991) The bounds on relaxation time come from thepreliminary analysis above (Figs 12 and 13) The

lower bound on TSAF comes from the elapsed timesince 1906 the upper bound from limits on theamount of coseismic slip (see below) The lowerbounds on tHRC and THRC are derived from the his-torical record there is no known earthquake on thenorthern HaywardndashRodgers Creek system since atleast 1775 (Toppozada and Borchardt 1998WGCEP 1999) The lower and upper bounds on sliprates were based loosely on the Northern CaliforniaWorking group (WGCEP 1999 Table 2) The Work-ing Group listed 2 s uncertainties of = 24 plusmn 3mmyr = 9 plusmn 2 mmyr and = 5 plusmn m 3mmyr

The estimation problem is to find the choice ofthe nine parameters in Table 1 that minimize themisfit to two geodetic data sets the present-day spa-tial distribution of deformation in the north SanFrancisco Bay region (Fig 14) and the time-depen-dent decay in strain rate following the 1906 earth-quake (Fig 11) For a non-linear problem such asthis one always needs to be concerned about localminima However preliminary testing with a simu-

smiddotSAFsmiddotHRC smiddotHRC

FIG 11 Post-1906 strain rate from triangulation data in the San Francisco Bay area Model fit to the model withpostseismic deformation only A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

74 PAUL SEGALL

lated annealing algorithm did not show evidence oflocal minima so the remaining analysis was doneusing a Sequential Quadratic Programming methodwith BFGS Hessian updates (eg Gill et al 1981)within Matlab The method allows upper and lowerbounds to be placed on all parameters as well asgeneralized inequality constraints The latter areneeded because tHRC is strictly bounded by tHRC poundTHRC In the context of the Savage and Prescottmodel earthquakes are periodic and it makes nosense for the elapsed time since the last event toexceed the average recurrence time In addition Ihave found it desirable to constrain the maximumcoseismic slip on both the San Andreas and Hay-ward faults This provides constraints of the form

and

Noting that the coseismic slip should be interpretedas an average over the fault in this two-dimensional

model I set meters and

metersIt should be noted that the GPS network does not

extend very far west of the San Andreas fault Forthis reason the data do not adequately constrain the

total motion across the plate boundary A reasonableapproach would be to include data from Pacificplate sites such as Hawaii I chose simply to con-strain the sum of the slip rates on all three faults tobe no greater than 40 mmyr (eg Argus and Gordon2001) This constraint significantly affects the solu-tion estimates without the cumulative slip con-straint tended to put the maximum slip rate allowedon all three faults exceeding the total displacementrate between the Sierra NevadandashGreat Valley blockand the Pacific plate

The optimal solution is found for H ~ 184 km tR~ 345 years TSAF = 280 years ~ 25 mmyrtHRC = 225 years THRC = 4286 years and ~ 7mmyr The ConcordndashGreen Valley fault slips at

~ 8 mmyr below 22 km A surprising resultis that the predicted recurrence time on the SanAndreas is reasonable in the sense of requiring 70-meter slip events every 280 years In comparisonSchwartz et al (1998) suggested that the penulti-mate event on the north coast segment of the SanAndreas occurred in the mid-1600s possibly in theinterval 1632ndash1659 corresponding to a recurrenceinterval of roughly 260 years Also surprising is thatthe time since the last earthquake on the Haywardndash

smiddotSAF TSAFtimes DuSAFmaxpound smiddotHRC THRCtimes DuHRC

maxpound

DuSAFmax

70= DuHRCmax

30=

smiddotSAFsmiddotHRC

smiddotHRC

FIG 12 Bootstrap results for post-1906 data in the San Francisco Bay area A Relaxation time B Elastic-layerthickness (1000 bootstrap resamples)

ESTIMATES OF SLIP RATE 75

Rodgers Creek fault is the minimum allowed 225years A comparison of the model fit to the data setsis shown in Figure 15 Generally speaking the dataare fit quite well with the possible exception of thestrain rate in the first post-1906 epoch The uncer-tainties in the post-1906 strain rates were actuallydecreased by a factor of two to ensure that these dataare fairly weighted compared to the GPS velocitiesDecreasing the uncertainties in the strain rates fur-ther to fit high strain rate in the immediatepost-1906 epoch causes a very significant increasein the misfit to the modern GPS data This is mostlikely a result of the large uncertainties in the trian-gulation data but may also indicate that the physi-cal model is inadequate

How is it that the geodetic data have any capabil-ity of resolving the recurrence interval on the SanAndreas fault Recall that there are three times inthe physical model the time since the last majorearthquake t the asthenospheric relaxation time tRand the recurrence interval T For the San Andreasfault we know tSAF ~93 years a priori and tR is rea-

sonably well resolved by the post-1906 strain ratetransient The estimation is thus free to vary T insuch a way as to optimize the fit to the velocity gra-dients near the San Andreas Constraints on themaximum slip in the earthquake help to furthertighten those constraints

In order to determine how well the data constrainthe model parameters I ran 500 bootstrap resam-ples and optimizations using the best-fitting modelas the starting point of the optimization As beforethe residuals were resampled and added to the pre-dicted data from the optimal model in order toensure that each observation point contributes to thesolution The results of the bootstrap are shown inFigure 16 The posterior distribution for recurrencetime on the San Andreas fault TSAF is quite broadwith a peak near the maximum allowed of (7000mm)(19 mmyr) ~370 years In contrast the poste-rior distribution for recurrence time on the Hay-wardndashRodgers Creek fault THRC is bimodal withprominent peaks near the minimum ~225 years andmaximum allowed value (3000 mm)(7 mmyr)

FIG 13 Bootstrap results for SCEC data along the Carrizo Plain segment of the San Andreas fault A Elastic-layerthickness B Relaxation time Note that 200 years was the upper bound considered C Recurrence time Recurrencetimes in excess of 400 years were not considered as these would lead to excessive coseismic slip D Slip rate

76 PAUL SEGALL

~ 428 years The distributions for tHRC the timesince the most recent HaywardndashRodgers Creekearthquake is strongly concentrated near the mini-

mum allowed range The distribution of slip rates onthe San Andreas and HaywardndashRodgers Creekfaults are bimodal with peaks at 19 mmyr and 25

FIG 14 GPS-derived velocities for the north San Francisco Bay region after Prescott et al (2001) AbbreviationsSAF = San Andreas fault HF = Hayward fault CF = Calaveras fault RGF = Rodgers Creek fault CGVF = ConcordndashGreen Valley fault

TABLE 1 A priori Bounds on Parameters

Parameter Symbol Min Max

Lithospheric thickness km H 10 30

Relaxation time yr tR 10 200

SAF recurrence interval yr TSAF 100 500

Time since last Hayward event yr tHRC 225 690

Hayward recurrence interval yr THRC 225 700

Green valley fault locking depth km d 05 15

SAF slip rate my 0019 0027

Hayward slip rate myr 0007 0013

Green valley slip rate myr 0002 0008

smiddotSAF

smiddotHRC

smiddotCGV

ESTIMATES OF SLIP RATE 77

mmyr (SAF) and 7 mmyr and 13 mmyr (HRC)These results still must be viewed with some cau-tion in that I have not demonstrated that the boot-strap optimizations were immune to local minima

The bootstrap calculations suggest that the con-straint on maximum coseismic slip does play astrong role in the results To explore this further Ivaried the maximum coseismic slip allowed on thetwo faults The results (Table 2) demonstrate that theSan Andreas recurrence interval is not fully con-trolled by the constraint on the maximum coseismicslip In contrast the recurrence interval on the Hay-ward fault is totally controlled by the maximum slipallowed (for a maximum slip of 10 meters the recur-rence interval reaches the upper bound of 700years) Notice also that the time since the last Hay-wardndashRodgers Creek event is always at the mini-mum value allowed (225 years) The fit to the data isslightly better for large infrequent earthquakes

Discussion

What has been learned from the modeling pre-sented here It seems clear that geodetic data

should not be analyzed in a vacuum The resolutionof the data alone is insufficient to address the prob-lems of greatest interest However when combinedwith prior information from geologic studies GPSdata appear to be able to resolve parameters of con-siderable interest such as the average recurrenceinterval between great earthquakes on the SanAndreas fault Using a simple model of a repeatingdislocation rupturing an elastic plate overlying aMaxwell visco-elastic medium I found an averagerecurrence interval of roughly 280 years on thenorth coast segment of the San Andreas faultAlthough the uncertainties in this estimate arelarge the result is in surprisingly good accordancewith recent paleoseismic results (Schwartz et al1998) that suggest an event similar to 1906 in themid 1600s

The results for the HaywardndashRodgers Creek faultare also not unreasonable According to the GPSdata the most recent event here occurred in ~1775regardless of the estimated recurrence time Thisresult is not inconsistent with the paleoseismic find-ing that the most recent event on the Rodgers Creekfault took place in the interval 1670ndash1776

TABLE 2 Influence of Maximum Coseismic Slip on Estimated Parameters1

Parameter

Lithospheric thickness H km 150 184 220

Relaxation time tR yr 357 345 340

SAF recurrence interval TSAF yr 220 280 337

Limiting interval yr 290 368 526

Time since last Hayward event tHRC yr 225 225 225

Hayward recurrence interval THRC yr 2857 4286 700

Limiting interval yr 285 428 714

Green Valley fault locking depth d km 11 22 47

SAF slip rate myr 0025 0025 00249

Hayward slip rate myr 0007 0007 00071

Green valley slip rate myr 0008 0008 0008

1Limiting interval is computed using the estimated slip rate for that fault not the minimum slip rate that would yield a longer upper bound

DuSAFmax

55=

DuHRCmax

20=

DuSAFmax

70=

DuHRCmax

30=

DuSAFmax

100=

DuHRCmax

50=

DuSAFmax

smiddotSAFcurren

DuHRCmax

smiddotHRCcurren

smiddotSAF

smiddotHRC

smiddotCGV

78 PAUL SEGALL

(Schwartz et al 1993) and on the northern Hay-ward fault in 1640ndash1776 (HFPEG 1999) The GPSdata alone yield limited information about the aver-age recurrence time on the HaywardndashRodgers Creekfault However adding the reasonable bound thatslip not exceed 2 meters on average yields a recur-rence interval of 285 years The average slip in the1868 Hayward fault earthquake was approximately2 meters (Yu and Segall 1996) In comparisontrenching at the El Cerrito site on the northern Hay-ward fault suggests a recurrence interval of possiblyless than 270 years but no more than 710 years(HFPEG 1999)

It should not need to be stated that earthquakesare neither periodic nor especially in the case of theHaywardndashRodgers Creek faults infinitely long Atwo-dimensional model cannot describe the situa-tion at a given trench site where ruptures mainly tothe north or south of the site cause some offset in the

trench Nevertheless the fact that the results fromsuch a simple model are not unreasonable is encour-aging One useful attribute of the models exploredhere are that they are testable in the sense of predict-ing future behavior Running the model forward intime to the next San Andreas earthquake yields thevelocity distributions shown in Figure 17 Of coursea north HaywardndashRodgers Creek earthquake mayoccur prior to the next San Andreas event

The Maxwell relaxation time is reasonablywell estimated by the data I find an optimal valueof about 35 years which corresponds to a viscos-ity of 16 times 1019 Pa-s It should be noted that theestimates of all of the model parameters is only asgood as the mechanical model used to interpretthe data If as seems quite likely there was after-slip on the San Andreas below the 1906 rupturezone the estimate of the viscosity is some combi-nation of the effects of afterslip and distributed

FIG 15 Fit of model to north San Francisco Bay data Maximum slip on Hayward fault is 30 m and on the SAF is70 m

ESTIMATES OF SLIP RATE 79

relaxation Li and Rice (1987) presented anapproximate method for including post-seismicfault slip at constant resistive stress with a relax-ing asthenosphere in the context of a repeatingperiodic earthquake sequence Combining suchan approach with appropriate constitutive laws forstable fault creep as in Tse and Rice (1986)would presumably provide a more physically real-istic description of fault behavior Clearly three-dimensional effects are also important especiallyfor earthquakes on the Rodgers Creek and Hay-ward faults Finally the effects of non-uniform

elastic properties are known to bias interpreta-tions based on simple homogeneous models Elas-tic moduli increase with depth which has theknown effect of biasing the depth distribution offault slip Similarly elastic properties may varywith lithology across the major fault zones whichthemselves may represent zones of more compli-ant rock Finite elements are certainly the mostgeneral way to compute forward models but arenot yet feasible for the optimization and bootstrapmethods as computed here which literally testedthousands of possible models

FIG 16 Five hundred bootstrap estimates of model parameters fit to the north San Francisco Bay data 90 confi-dence bounds are listed at the top of each panel The vertical line in the histograms for recurrence times TSAF and THRCshow the upper bounds corresponding to the minimum slip rate and maximum allowed coseismic slip respectively Thehorizontal bar in the panel for the San Andreas recurrence time shows the preferred estimate from Schwartz et al (1998)The horizontal bar in the panel for tH time since the last Hayward earthquake shows the range of dates compatible withpaleoseismic estimates of the penultimate HaywardndashRodgers Creek event

80 PAUL SEGALL

Acknowledgments

George Thompson set a standard that geophysi-cal models no matter how elegant must be consis-tent with geologic observations This work wasinspired by that example I would like to thank WillPrescott for discussions about the Bay Area defor-mation measurements and Mark Matthews for manyyears ago introducing me to Bayesrsquo theorem WayneThatcher Mark Murray Jeff McGuire and PeterCervelli provided valuable comments at early stagesin this study I thank Jim Savage Duncan Agnewand Shelley Kenner for their thoughtful reviewsGary Ernst and Simon Klemperer carefully editedthe manuscript

REFERENCES

Argus D F and Gordon R G 2001 Present tectonicmotion across the Coast Ranges and San Andreas faultsystem in central California Geological Society ofAmerica Bulletin v 113 p 1580ndash1592

Beroza G C 1991 Near-source modeling of the LomaPrieta earthquake Evidence for heterogeneous slipand implications for earthquake hazard Bulletin of theSeismological Society of America v 81 p 1603ndash1621

Cohen S C 1982 A multilayer model of time-dependentdeformation following an earthquake on a strike-slipfault Journal of Geophysical Research v 87 p 5409ndash5421

HFPEG (Hayward Fault Paleoearthquake Group) 1999Timing of paleoearthquakes on the northern HaywardfaultmdashPreliminary evidence in El Cerrito CaliforniaUS Geological Survey Open File Report 99-318

Henstock T J Levander A and Hole J A 1997Deformation in the lower crust of the San Andreas faultsystem in northern California Science v 278 p 650ndash653

Feigl K L King R W and Jordan T H 1990 Geo-detic measurement of tectonic deformation in theSanta Maria fold and thrust belt California Journal ofGeophysical Research v 95 p 2679ndash2699

Gill P E Murray W and Wright M H 1981 Practicaloptimization London UK Academic Press 401 p

Grant L B and Sieh K 1994 Paleoseismic evidence ofclustered earthquakes on the San Andreas fault in the

FIG 17 Simulation of Bay Area velocity field as a function of time Each line is separated by roughly 27 years

ESTIMATES OF SLIP RATE 63

fault parallel velocity v(x) is a simple functionof slip rate and the fault locking depth d

Here x is distance nor-mal to the fault Given measured velocities v(xi) it isstraightforward to estimate that depth over whichfaults are locked and the average slip rate acrossthose faults (eg Williams et al 1994) Thisapproach has been generalized for finite length dis-locations (eg Matsursquoura et al 1986) and forfar-field motions consistent with rigid sphericalplates (eg Murray and Segall 2001) Slip ratesestimated from geodetic observations can be com-pared to those derived from geologic studies andalong with the average slip in large earthquakesform the basis of seismic hazard evaluations

The screw dislocation while a useful first-ordermodel of plate boundary deformation does not accu-rately represent fault structure at great depth withinthe earth Nor does it account for temporal varia-tions in strain rate that have been observed follow-ing large earthquakes More physically realisticmodels tend to fall into two categories The firstinvolves faulting in an elastic lithosphere overlyinga viscoelastic lower crust or upper mantle (Nur andMavko 1974 Savage and Prescott 1978 Lehner etal 1981 Cohen 1982 Rundle 1986 Lyzenga1991 Pollitz 1992) The viscoelastic medium mayextend to great depth or be limited to a channel offinite thickness Some studies have investigatednon-linear rheology (eg Reches et al 1994) Thesecond class of models explains time-dependentdeformation following earthquakes as arising fromvelocity strengthening slip on a downdip extensionof the coseismic fault surface (eg Tse and Rice1986) Most of these models can fit the present-daydistribution of strain rate across the San Andreasfault system with an appropriate choice of parame-ters A significant complexity in applying thesemodels however is the fact that in many places theSan Andreas fault system consists of multiple paral-lel faults In the San Francisco Bay region we havethe San Andreas HaywardndashRodgers Creek and theCalaverasndashConcordndashGreen Valley faults In south-ern California the subparallel Elsinore and SanJacinto faults are present in addition to the SanAndreas Because physically realistic models offaulting invariably involve a large number of param-eters and are slower to compute they are not oftenused in inversions of geodetic measurements Fur-thermore Savage (1990) has shown that it is diffi-cult to distinguish between these two classes ofmodels because the solution for the anti-plane vis-

coelastic problem has a corresponding elastic solu-tion involving time-dependent slip below thecoseismic rupture The latter two observations haveled to the feeling that more physically realistic mod-els are unwarranted

In this paper I argue that a key to moving for-ward with this problem is to include more informa-tion than simply the present-day deformation fieldAdditional information comes from the time depen-dence of deformation following large earthquakesand paleoseismic estimates of fault slip rates A pri-ori information about faultslip rate from geologicstudies can be used to complement information fromgeodetic studies In the present paper this is donein the context of rather simple models of the earth-quake cycle involving repeated earthquakes in anelastic layer overlying a Maxwell viscoelastichalf-space Although this model is itself highly ide-alized the results are sufficiently encouraging towarrant further investigations

The use of a priori information to regularize anill-posed inverse problem is well known (eg Jack-son 1979 Tarantola and Valette 1982 Menke1989) In this work I propose using Bayesrsquo rule tocombine geologic and geodetic information in orderto permit use of more sophisticated and hopefullyrealistic models than could be addressed with geo-detic data alone Previously Minster and Jordan(1987) used Bayesrsquo rule to combine geodetic andgeologic data to estimate the net motion across theSan Andreas fault system The general frameworkproposed here could be expanded to include infor-mation from other data sets as well but the focushere is on slip-rate data Fault slip rates are deter-mined from geologic observations such as measure-ments of offset stream channels and dates of thecorrelative sedimentary deposits From the point ofview of geodetic analysis this information places apriori constraints on the slip rate I denote the prob-ability density function (pdf) of the slip rate fromthe geologic observations as or perhaps moreappropriately as where g denotes geologicobservations such as measured offsets radiometricage determinationand so forth

On the other hand mechanical models of theearth allow us to predict geodetic observations giventhe slip rate on the fault and the other model param-eters such as the fault locking depth elastic-layerthickness viscosity of the lower crust and so forthDenoting the parameters of the mechanical modelother than slip rate as m then the model yields

smiddotv x( ) smiddot pcurren( ) x dcurren( )1ndashtan=

P smiddot( )P smiddot g( )

64 PAUL SEGALL

where d is a vector of geodetic dataBayesrsquo rule then states that

(1)

In words equation (1) states that the pdf of theslip-rate and fault-model parameters given geodeticobservations can be determined from the a prioridistributions of the slip-rate and the fault-modelparameters and the theoretical model linking theparameters to the geodetic observations The integralin the denominator ensures that the posterior proba-bility is properly normalized ie that the cumula-tive probability is unity Assuming that the priorprobability of the slip rate and the model parametersare independent then Thisis reasonable inasmuch as estimates of slip rate frompaleoseismic studies are independent of prior esti-mates of elastic layer thickness viscosity fault lock-

ing depth and so on While in general we have apriori constraints on these physical parameters forthe purposes of the present discussion I will oftenassume that the distribution is so broad as to benon-informative Given the independence of theprior distributions on s and m (1) reduces to

(2)

The posterior probability distribution is easilycomputed if the prior distribution is uniform oversome range That is we take the slip rate to beequally probable between specified upper and lowerbounds A simple illustration (Fig 1) can be infor-mative In Figure 1A the mean of the distributionfalls just outside the a priori bounds The posteriordistribution is thus highly concentrated near theupper bound from geologic data The geodetic and

P smiddot m( )

P smiddot m d( ) P smiddot m( )P d smiddot m( )

P smiddot( )P d smiddot m( ) smiddotd mds mograve

---------------------------------------------------------- =

P smiddot m( ) P smiddot( )P m( )=

P smiddot m d g( )

P smiddot g( )P m( )P d smiddot m( )

P smiddot( )P m( )P d smiddot m( ) smiddotd mds mograve

------------------------------------------------------------------------------

=

FIG 1 Example of application of Bayesrsquo theorem given a uniform prior distribution The prior distribution is uniformover the interval [01] and is shown in the dash-dot pattern The distribution from the data alone is dashed and theposterior distribution is solid In (D) the posterior distribution is essentially a delta function at 10

ESTIMATES OF SLIP RATE 65

geologic data together are much more informativethan either data set alone In Figure 1B the a prioribounds are so broad as to be uninformative the pos-terior distribution is equal to that obtained from thegeodetic data alone In Figure 1C the reverse istrue the geodetic data supply little information andthe posterior distribution reflects simply the a prioriinformation A fourth outcome also is possible asillustrated in Figure 1D where the two distributionsdo not overlap indicating that the two estimates areinconsistent The conclusion in this case is thateither the uncertainties in the geologic determina-tion of slip rate have been substantially mis-esti-mated or the physical model used to relate geodeticdata to slip rate is invalid The outcome in Figure1D could be the most useful in that assuming thegeologic bounds are valid it would invalidate anentire class of physical fault models

I start by considering the interseismic velocitydistribution across the Carrizo Plain segment of theSan Andreas fault (Fig 2) This area is chosenbecause it is the simplest part of the SAF systemuncomplicated by parallel faults The only otheractive structures at this latitude are the San Grego-riondashHosgri fault system which runs into an activedeformation zone in the Santa Maria Basin (Feigl etal 1990) and the fold-and-thrust belt northeast ofthe San Andreas at the eastern edge of the Coast

Ranges The latter structures involve motion largelynormal to the SAF and thus do not significantlyaffect the calculations here The San GregoriondashHos-gri deformation is accounted for crudely using asimple dislocation model

Assuming only a single San Andreas fault andmaking use of a two-dimensional dislocationmodel I find that the best-fitting locking depth is244 km (Fig 3) and the slip rate is 418 mmyr Incontrast Sieh and Jahns (1984) determined anaverage slip rate of 339 plusmn 29 mmyr for the past3700 yr and 358 + 54ndash41 mmyr for the past13250 yr based on offset fluvial deposits at Wal-lace Creek Allowing 3 mmyr of slip on the SanGregoriondashHosgri fault system which is modeled asa screw dislocation locked above 10 km the SanAndreas locking depth shallows slightly to 203km and the slip rate drops to 377 mmyr which isin reasonable accord with geologic estimates (Siehand Jahns 1984)

Of course the San Andreas fault is nottwo-dimensional Southeast of the Carrizo Plain theSan Andreas fault enters the Big Bend and theTransverse Ranges whereas the creeping zone islocated to the northwest The data analyzed here(Fig 2) were selected to be as far from possible fromthese locales yet some bias is unavoidable with atwo-dimensional model

FIG 2 Velocities determined by the Southern California Earthquake Center (SCEC) in the Carrizo Plain section ofthe San Andreas fault

66 PAUL SEGALL

It is important to determine the range of sliprates and locking depths that are consistent withthe GPS data A bootstrap resampling procedureshows that slip rates in the range of 34 to 41 mmyrare consistent with the geodetic observations (Fig4) The bootstrap was conducted in the followingmanner Let d be the vector of observations thepredicted data from the best-fitting model so that

is the vector of residuals The residualswere randomly resampled with replacement that issome elements may be chosen multiple times oth-ers not at all Denote the resampled residuals r Aresampled data vector is then usedto generate a model estimate and the process isrepeated many times The distribution of the boot-strapped model parameters is that shown in Figure4

It is well known that there is a tradeoff betweenthe locking depth and deep slip rate Increasing thelocking depth moves the dislocation farther from thesurface observation stations to achieve the samesurface strain requires a higher slip rate Applying apriori constraints on the slip rate from 31 mmyrand 37 mmyr based on the Sieh and Jahns (1984)data from Wallace Creek thus not only changes the

posterior distribution of slip rate but of the lockingdepth as well (Fig 5) The reason for this can beclearly seen from a scatter plot of the bootstrap res-amples as in Figure 6

Notice that the GPS data have significantly nar-rowed the distribution of admissible slip rates (Fig5) The posterior distribution is peaked at 35 to 37mmyr whereas the a priori distribution is uniformbetween 31 and 37 mmyr At the same time the apriori geologic information provided no constrainton the fault locking depth With geologic data alonewe cannot distinguish between a locked fault and afault that creeps steadily below some shallow lockedinterval The GPS data alone bound the lockingdepth to between 17 and 24 km Because of thestrong correlation between slip rate and lockingdepth the posterior distribution of the locking depthis narrowed to 17ndash20 km This simple exampleshows the power of using a priori geologic con-straints in allowing more information to be obtainedfrom geodetic data

Note that the bootstrap provides an estimate ofthe distribution of the model parameters whereasBayesrsquo rule requires P(d|m) as in equation (3)However for linear models with normally distrib-

d

r d dndash=

d d r+=

FIG 3 A Misfit as a function of depth for a simple screw dislocation model B Profile of fault parallel velocity acrossthe fault compared with prediction of a simple screw dislocation model C As in (B) but with 3 mmyr of slip beneaththe Hosgri fault

ESTIMATES OF SLIP RATE 67

uted observational errors the two approaches areequivalent Furthermore a uniform distribution withhard bounds may not be the best characterization ofthe prior probability distribution on slip rate Alter-

natively if the prior distribution and observationalerrors are normally distributed one can simplyapply the prior information as ldquopseudo datardquo (egJackson 1979)

FIG 4 Probability distributions for fault locking depth and slip rate from bootstrap results using GPS data I assume3 mmyr below a locking depth of 10 km on the Hosgri fault system

FIG 5 Posterior probability distribution from the bootstrap results using GPS data and assuming a uniform priordis-tribution on slip rate from 31 to 37 mmyr Compare with Figure 4 As before I assume 3 mmyr below a locking depthof 10 km on the Hosgri fault system

68 PAUL SEGALL

Viscoelastic Models

As noted in the Introduction the screw disloca-tion provides only a very limited description ofplate-boundary faulting A somewhat more realisticmodel involves an elastic layer of thickness H over-lying a Maxwell viscoelastic half-space (Fig 7) TheMaxwell material has relaxation time tR = 2hmicrowhere h is the viscosity and micro is the shear modulusAt time t = 0 slip Du occurs on the fault from thesurface to depth D pound H The velocity on the Earthrsquossurface as a function of position perpendicular to thefault x and time t is

(3)

where the spatial distribution is given by

(4)

The result follows from but is not stated in Nur andMavko (1974) The strain rate is simply obtained bydifferentiating the velocity with respect to x Noticethat the post-seismic velocity is a function of thecoseismic slip Du the depth of faulting D the elas-tic layer thickness H the material relaxation tR andthe time since the last large earthquake t Equation(3) does not include the effects of interseismic strainaccumulation Savage and Prescott (1978) extendedthe model by summing over an infinite sequence ofplate rupturing events (D = H) equally spaced intime with recurrence interval T The velocity field asa function of distance across strike (x) and time (t) isgiven by

(5)

where the temporal dependence depends only on thedimensionless ratios ttR and TtR

(6)

v x t( )

DuptR---------e

t tRcurrenndash t tRcurren( )n 1ndash

n 1ndash( )--------------------------Fn x( D H )

n 1=

yen

aring

=

Fn x D H ( )

D 2nH+x

---------------------egrave oslashaelig ouml1ndash

tan D 2nHndashx

--------------------egrave oslashaelig ouml1ndash

tan+

2xD

x2

2nH( )2D

2ndash+

------------------------------------------1ndash tan

=

=

v x t( ) smiddot

p--- Tn t tR T tRcurrencurren( )Fn x H H ( )

n 1=

yen

aring=

Tn t tR T tRcurrencurren( )

TtR-----

et tRcurrenndash

n 1ndash( )------------------ e

kT tRcurrenndash t kT+tR

--------------egrave oslashaelig ouml n 1ndash

k 0=

yen

aring

=

FIG 6 Tradeoff between slip rate and locking depth Each point is the result of a bootstrap resample of the SCECvelocity data The heavy points are those consistent with a uniform prior distribution on slip rate from 31 to 37 mmyrAs before I assume 3 mmyr below a locking depth of 10 km on the Hosgri fault system

ESTIMATES OF SLIP RATE 69

Because the far-field lithospheric velocity is drivensolely by the repeating earthquakes Representative examples are given in Figure 8 forvarious times through the earthquake cycle Noticethat when the asthenospheric relaxation time is ofthe order of the earthquake cycle time or longer TtR pound1 the surface velocity distribution is rathersteady with time (Fig 8A) On the other hand if therelaxation time is short compared to the recurrenceinterval TtR gt 1 there is a significant transientwith higher than average strain rates early in thecycle and lower than average strain rates late in thecycle

The Savage-Prescott model involves five param-eters lithospheric thickness H relaxation time tRtime since the last earthquake t recurrence time Tand long-term slip rate on the fault Because thetheory is linear it is straightforward to constructmodels with multiple parallel faults however in thiscase the number of parameters is even larger Forthree faults there are 11 parameters if one assumesthat the elastic thickness and relaxation time are thesame for all faults Note that the model is kinematicwith imposed slip events on the various faults at reg-ular intervals and thus does not account for thestress interactions between the faults

As a starting point I consider the Carrizo Plaindata with some of the parameters constrained byindependent observations The time since the lastgreat earthquake (1857) is t = 136 years To beginwith I assume a recurrence time of T = 160 yearsbased on the average interval found by Grant andSieh (1994) There are three remaining parametersto be estimated from the data tR H and Theoptimal estimates of the non-linear parameters tRand H are found by simple grid search (Fig 9) with

estimated by linear least squares The optimalmodel is found for H = 114 km and tR = 80 yearswith a slip rate of 402 mmyr There is a strongtradeoff between relaxation time and elastic-layerthickness such that models with elastic-layer thick-ness of up to 20 km and relaxation time of 180 yearsfit the data nearly equally well The reason for thetradeoff is as follows Larger values of H cause thedeformation to be spread out in space At the sametime the deformation becomes more diffuse withincreasing tT and TtR (Fig 8) and hence t tR = (tT) times (TtR ) Thus decreasing ttR causes the defor-mation to bemore localized which counters theeffect of increasing H Interestingly the slip ratedoes not vary with H and tR confirming that thisparameter is well constrained by the data (given thatall other parameters are fixed)

The inferred slip rate is however higher thanestimated from stream offsets at Wallace Creek of 33plusmn 3 mmyr (Sieh and Jahns 1984) Constraining theslip rate to 34 mmyr (Fig 10) does not fit the dataas well especially for the stations farthest west ofthe San Andreas The optimal model is H = 114 kmas before but the relaxation time increases to 180years

Relaxation times tR gt 50 years are long com-pared to what has been inferred for the post-1906strain rate transient north of the San Francisco Bayas measured by repeated triangulation surveys(Thatcher 1975a 1975b Kenner and Segall 2000)The relaxation time tR consistent with the post-1906transient can be inferred from the triangulation dataand the spatial derivative of equation (3) It wouldbe most consistent to employ the same Savage-Pres-cott model as used with the Carrizo Plain data how-ever this requires knowing the recurrence time oflarge earthquakes on the north coast segment of theSan Andreas fault An alternate approach employedhere is to remove the estimated secular deformationusing a model (see Kenner and Segall 2001) andthen fit the remaining strain rate data For 60meters of slip in the 1906 earthquake the optimal

smiddot Du Tcurren=

smiddot

smiddot

smiddot

FIG 7 Model of a faulted elastic plate overlying viscoelas-tic half-space The elastic layer has shear modulus micro the vis-coelast ic half-space shear modulus micro and viscosity hEarthquakes occur at a regular interval T such that thefar-field velocity is vyen2

70 PAUL SEGALL

model is H = 19 km with a relaxation time of 17years (Fig 11) substantially less than found fromthe Carrizo Plain interseismic data1

A tentative conclusion of this analysis is that theCarrizo Plain interseismic data require a relaxationtime in excess of 50 years whereas the post-1906triangulation data require relaxation times of 25years or less In order to test whether this discrep-ancy is truly required by the data I estimated theuncertainty in the two relaxation times using a boot-strap resampling procedure As discussed previ-ously the residuals to the best-fitting model wererandomly resampled with replacement and added

to the model predictions for each bootstrap resam-ple This approach seems best for small data setsand ensures that there were data at each observationposition (interseismic data) and time (postseismicdata)

For the post-1906 transient admissible relax-ation times range from 10 years to 30 years (Fig 12)On the other hand the present distribution of defor-mation on the Carrizo Plain demands relaxationtimes in excess of 35 years (Fig 13) Thus we cansafely conclude that there is no single relaxationtime that fits both data sets

There are a number of possible explanations forthis apparent discrepancy One possibility is thatthe model of a fully ruptured elastic plate overlyinga linear Maxwell half-space does not capture all ofthe relevant physics For example nonlinear rheol-ogy may be important The data are consistent withthe expectation that the effective viscosity is low

1 Note that the discrepancy is not due to the correction forinterseismic deformation in the post-1906 data Assuming arecurrence time of 250 years and a slip rate on the SAF of 25mmyr the optimal relaxation time is 228 years with an elas-tic-layer thickness of 167 km

FIG 8 Savage-Prescott model for (A) T tR = 1 and (B) T tR = 5 t is the time since the last earthquake T is theearthquake repeat time and tR is the Maxwell relaxation time In these calculations the earthquakes rupture the entireelastic layer

ESTIMATES OF SLIP RATE 71

early in the seismic cycle when the stresses arelarge and increases late in the cycle when thestresses have largely dissipated In addition recentreflection and refraction results in northern Califor-nia show the San Andreas fault cutting the lowercrust and possibly the Moho (eg Henstock et al1997 Parsons and Hart 1999) Aseismic slip onthe downdip extension of the seismogenic zone (asin Tse and Rice 1986) may take place in additionto a more deeply distributed relation Results fromKenner and Segall (2001) which include viscousrelaxation in a narrow fault zone and more deeplydistributed slip find the spatiotemporal pattern ofdeformation in the post-1906 triangulation data isbest fit with a short relaxation time in the fault zoneand a longer relaxation in the underlyinghalf-space Another possibility is that the thermo-mechanical structure of the lithosphere differsbetween the two areas It is also possible that along-strike variations in fault geometry (the Big Bend) or

slip behavior (the creeping zone) may bias the com-puted relaxation time although this seems lesslikely

Although the apparent relaxation times differbetween the Carrizo Plain and the northern SanFrancisco Bay region the elastic-layer thicknessestimates are consistent in the range of 13ndash20 kmCarrizo Plain slip-rate estimates cluster near theupper bound of 37 mmyr Interestingly the CarrizoPlain data yield recurrence times greater than 160years Whereas the distribution is reasonably broadboth the mean (288 years) and median (277) repeattimes fall between the 1988 working group estimateof 296 years (WGCEP 1988) and the 1995 workinggroup estimate of 206 years (WGCEP 1995) Theaverage coseismic slip for the predicted events isroughly 10 meters There are significant correlationsbetween parameters (between H and tR as dis-cussed previously as well as between tR and T)which means that any additional data that constrain

FIG 9 Comparison of Savage-Prescott model to the SCEC velocity data for the Carrizo Plain A Misfit as a functionof elastic-layer thickness and relaxation time B Best-fitting San Andreas slip rate corresponding to the elastic-layerthickness and relaxation time C Fit to the data Effect of the Hosgri fault is removed employing a dislocation model withslip rate of 3 mmyr and depth of 10 km

72 PAUL SEGALL

some of the parameters will help resolve others aswell

It should be emphasized that the paleoseismicobservations of Grant and Sieh (1994) show an irreg-ular pattern of earthquakes that cannot be explainedby the simple two-dimensional model used here tointerpret the geodetic observations Nevertheless itis interesting that the GPS data appear to have someresolving power for average recurrence interval

San Francisco Bay Data

In order to eliminate any geographic bias in theinterpretation associated with possibly differentcrustal structure and temperature distribution Inext consider the velocity profile across the SanAndreas fault north of the San Francisco Bay asdetermined by repeated GPS observations (Prescottet al 2001) as shown in Figure 14 The GPS sitesare sufficiently close to the post-1906 triangulationnetworks that it is reasonable to assume that the

crustal structure and thermal profile are the samefor both data sets In the following the GPS-derivedvelocities are rotated into fault-parallel and perpen-dicular components Only the fault-parallel motionsare modeled here although as pointed out by Pres-cott et al (2001) and Murray and Segall (2001)there is significant fault-normal motion in the east-ern coast ranges at the western edge of the CentralValley

The model is the same as discussed previouslywith three principal faults the San Andreas Hay-wardndashRodgers Creek and ConcordndashGreen Valleyfaults The parameters are the elastic plate thick-ness H Maxwell relaxation time tR the recurrencetime for great earthquakes on the SAF TSAF the sliprate on the SAF and the recurrence timeTHRC time since past event tHRC and slip rate on the Hayward-Rodgers Creek fault system Thetime since the last great earthquake on the northernSan Andreas fault tSAF is of course known Theeffect of the ConcordndashGreen Valley fault is modeled

smiddotSAFsmiddotHRC

FIG 10 Comparison of Savage-Prescott model to the SCEC velocity data for the Carrizo Plain with slip rate con-strained to 34 mmyr Effect of San GregoriondashHosgri fault has been removed with a dislocation model with slip rate of 3mmyr and depth of 10 km A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

ESTIMATES OF SLIP RATE 73

kinematically with a simple screw dislocationwhich adds two parameters the fault slip rate andlocking depth It is certainly inconsistent to ignoreviscoelastic effects for the ConcordndashGreen Valleyfault but I do so for three reasons (a) much less isknown about this fault (b) earthquakes there arepresumably least well modeled by infinitely longruptures and (c) it keeps the number of parametersreasonably tractable In essence the effect of theConcordndashGreen Valley is removed empirically and Imake no attempt to gain further insights into itsmechanical behavior Future work should addressand hopefully rectify this simplification

A priori bounds on the parameters are listed inTable 1 The lower bound on the lithospheric thick-ness comes from a conservative estimate of thedepth extent of slip in large earthquakes In fact itis more likely that slip in 1906 extended at least to15 km (Matthews and Segall 1993) as it certainlydid in the Loma Prieta earthquake (eg Beroza1991) The bounds on relaxation time come from thepreliminary analysis above (Figs 12 and 13) The

lower bound on TSAF comes from the elapsed timesince 1906 the upper bound from limits on theamount of coseismic slip (see below) The lowerbounds on tHRC and THRC are derived from the his-torical record there is no known earthquake on thenorthern HaywardndashRodgers Creek system since atleast 1775 (Toppozada and Borchardt 1998WGCEP 1999) The lower and upper bounds on sliprates were based loosely on the Northern CaliforniaWorking group (WGCEP 1999 Table 2) The Work-ing Group listed 2 s uncertainties of = 24 plusmn 3mmyr = 9 plusmn 2 mmyr and = 5 plusmn m 3mmyr

The estimation problem is to find the choice ofthe nine parameters in Table 1 that minimize themisfit to two geodetic data sets the present-day spa-tial distribution of deformation in the north SanFrancisco Bay region (Fig 14) and the time-depen-dent decay in strain rate following the 1906 earth-quake (Fig 11) For a non-linear problem such asthis one always needs to be concerned about localminima However preliminary testing with a simu-

smiddotSAFsmiddotHRC smiddotHRC

FIG 11 Post-1906 strain rate from triangulation data in the San Francisco Bay area Model fit to the model withpostseismic deformation only A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

74 PAUL SEGALL

lated annealing algorithm did not show evidence oflocal minima so the remaining analysis was doneusing a Sequential Quadratic Programming methodwith BFGS Hessian updates (eg Gill et al 1981)within Matlab The method allows upper and lowerbounds to be placed on all parameters as well asgeneralized inequality constraints The latter areneeded because tHRC is strictly bounded by tHRC poundTHRC In the context of the Savage and Prescottmodel earthquakes are periodic and it makes nosense for the elapsed time since the last event toexceed the average recurrence time In addition Ihave found it desirable to constrain the maximumcoseismic slip on both the San Andreas and Hay-ward faults This provides constraints of the form

and

Noting that the coseismic slip should be interpretedas an average over the fault in this two-dimensional

model I set meters and

metersIt should be noted that the GPS network does not

extend very far west of the San Andreas fault Forthis reason the data do not adequately constrain the

total motion across the plate boundary A reasonableapproach would be to include data from Pacificplate sites such as Hawaii I chose simply to con-strain the sum of the slip rates on all three faults tobe no greater than 40 mmyr (eg Argus and Gordon2001) This constraint significantly affects the solu-tion estimates without the cumulative slip con-straint tended to put the maximum slip rate allowedon all three faults exceeding the total displacementrate between the Sierra NevadandashGreat Valley blockand the Pacific plate

The optimal solution is found for H ~ 184 km tR~ 345 years TSAF = 280 years ~ 25 mmyrtHRC = 225 years THRC = 4286 years and ~ 7mmyr The ConcordndashGreen Valley fault slips at

~ 8 mmyr below 22 km A surprising resultis that the predicted recurrence time on the SanAndreas is reasonable in the sense of requiring 70-meter slip events every 280 years In comparisonSchwartz et al (1998) suggested that the penulti-mate event on the north coast segment of the SanAndreas occurred in the mid-1600s possibly in theinterval 1632ndash1659 corresponding to a recurrenceinterval of roughly 260 years Also surprising is thatthe time since the last earthquake on the Haywardndash

smiddotSAF TSAFtimes DuSAFmaxpound smiddotHRC THRCtimes DuHRC

maxpound

DuSAFmax

70= DuHRCmax

30=

smiddotSAFsmiddotHRC

smiddotHRC

FIG 12 Bootstrap results for post-1906 data in the San Francisco Bay area A Relaxation time B Elastic-layerthickness (1000 bootstrap resamples)

ESTIMATES OF SLIP RATE 75

Rodgers Creek fault is the minimum allowed 225years A comparison of the model fit to the data setsis shown in Figure 15 Generally speaking the dataare fit quite well with the possible exception of thestrain rate in the first post-1906 epoch The uncer-tainties in the post-1906 strain rates were actuallydecreased by a factor of two to ensure that these dataare fairly weighted compared to the GPS velocitiesDecreasing the uncertainties in the strain rates fur-ther to fit high strain rate in the immediatepost-1906 epoch causes a very significant increasein the misfit to the modern GPS data This is mostlikely a result of the large uncertainties in the trian-gulation data but may also indicate that the physi-cal model is inadequate

How is it that the geodetic data have any capabil-ity of resolving the recurrence interval on the SanAndreas fault Recall that there are three times inthe physical model the time since the last majorearthquake t the asthenospheric relaxation time tRand the recurrence interval T For the San Andreasfault we know tSAF ~93 years a priori and tR is rea-

sonably well resolved by the post-1906 strain ratetransient The estimation is thus free to vary T insuch a way as to optimize the fit to the velocity gra-dients near the San Andreas Constraints on themaximum slip in the earthquake help to furthertighten those constraints

In order to determine how well the data constrainthe model parameters I ran 500 bootstrap resam-ples and optimizations using the best-fitting modelas the starting point of the optimization As beforethe residuals were resampled and added to the pre-dicted data from the optimal model in order toensure that each observation point contributes to thesolution The results of the bootstrap are shown inFigure 16 The posterior distribution for recurrencetime on the San Andreas fault TSAF is quite broadwith a peak near the maximum allowed of (7000mm)(19 mmyr) ~370 years In contrast the poste-rior distribution for recurrence time on the Hay-wardndashRodgers Creek fault THRC is bimodal withprominent peaks near the minimum ~225 years andmaximum allowed value (3000 mm)(7 mmyr)

FIG 13 Bootstrap results for SCEC data along the Carrizo Plain segment of the San Andreas fault A Elastic-layerthickness B Relaxation time Note that 200 years was the upper bound considered C Recurrence time Recurrencetimes in excess of 400 years were not considered as these would lead to excessive coseismic slip D Slip rate

76 PAUL SEGALL

~ 428 years The distributions for tHRC the timesince the most recent HaywardndashRodgers Creekearthquake is strongly concentrated near the mini-

mum allowed range The distribution of slip rates onthe San Andreas and HaywardndashRodgers Creekfaults are bimodal with peaks at 19 mmyr and 25

FIG 14 GPS-derived velocities for the north San Francisco Bay region after Prescott et al (2001) AbbreviationsSAF = San Andreas fault HF = Hayward fault CF = Calaveras fault RGF = Rodgers Creek fault CGVF = ConcordndashGreen Valley fault

TABLE 1 A priori Bounds on Parameters

Parameter Symbol Min Max

Lithospheric thickness km H 10 30

Relaxation time yr tR 10 200

SAF recurrence interval yr TSAF 100 500

Time since last Hayward event yr tHRC 225 690

Hayward recurrence interval yr THRC 225 700

Green valley fault locking depth km d 05 15

SAF slip rate my 0019 0027

Hayward slip rate myr 0007 0013

Green valley slip rate myr 0002 0008

smiddotSAF

smiddotHRC

smiddotCGV

ESTIMATES OF SLIP RATE 77

mmyr (SAF) and 7 mmyr and 13 mmyr (HRC)These results still must be viewed with some cau-tion in that I have not demonstrated that the boot-strap optimizations were immune to local minima

The bootstrap calculations suggest that the con-straint on maximum coseismic slip does play astrong role in the results To explore this further Ivaried the maximum coseismic slip allowed on thetwo faults The results (Table 2) demonstrate that theSan Andreas recurrence interval is not fully con-trolled by the constraint on the maximum coseismicslip In contrast the recurrence interval on the Hay-ward fault is totally controlled by the maximum slipallowed (for a maximum slip of 10 meters the recur-rence interval reaches the upper bound of 700years) Notice also that the time since the last Hay-wardndashRodgers Creek event is always at the mini-mum value allowed (225 years) The fit to the data isslightly better for large infrequent earthquakes

Discussion

What has been learned from the modeling pre-sented here It seems clear that geodetic data

should not be analyzed in a vacuum The resolutionof the data alone is insufficient to address the prob-lems of greatest interest However when combinedwith prior information from geologic studies GPSdata appear to be able to resolve parameters of con-siderable interest such as the average recurrenceinterval between great earthquakes on the SanAndreas fault Using a simple model of a repeatingdislocation rupturing an elastic plate overlying aMaxwell visco-elastic medium I found an averagerecurrence interval of roughly 280 years on thenorth coast segment of the San Andreas faultAlthough the uncertainties in this estimate arelarge the result is in surprisingly good accordancewith recent paleoseismic results (Schwartz et al1998) that suggest an event similar to 1906 in themid 1600s

The results for the HaywardndashRodgers Creek faultare also not unreasonable According to the GPSdata the most recent event here occurred in ~1775regardless of the estimated recurrence time Thisresult is not inconsistent with the paleoseismic find-ing that the most recent event on the Rodgers Creekfault took place in the interval 1670ndash1776

TABLE 2 Influence of Maximum Coseismic Slip on Estimated Parameters1

Parameter

Lithospheric thickness H km 150 184 220

Relaxation time tR yr 357 345 340

SAF recurrence interval TSAF yr 220 280 337

Limiting interval yr 290 368 526

Time since last Hayward event tHRC yr 225 225 225

Hayward recurrence interval THRC yr 2857 4286 700

Limiting interval yr 285 428 714

Green Valley fault locking depth d km 11 22 47

SAF slip rate myr 0025 0025 00249

Hayward slip rate myr 0007 0007 00071

Green valley slip rate myr 0008 0008 0008

1Limiting interval is computed using the estimated slip rate for that fault not the minimum slip rate that would yield a longer upper bound

DuSAFmax

55=

DuHRCmax

20=

DuSAFmax

70=

DuHRCmax

30=

DuSAFmax

100=

DuHRCmax

50=

DuSAFmax

smiddotSAFcurren

DuHRCmax

smiddotHRCcurren

smiddotSAF

smiddotHRC

smiddotCGV

78 PAUL SEGALL

(Schwartz et al 1993) and on the northern Hay-ward fault in 1640ndash1776 (HFPEG 1999) The GPSdata alone yield limited information about the aver-age recurrence time on the HaywardndashRodgers Creekfault However adding the reasonable bound thatslip not exceed 2 meters on average yields a recur-rence interval of 285 years The average slip in the1868 Hayward fault earthquake was approximately2 meters (Yu and Segall 1996) In comparisontrenching at the El Cerrito site on the northern Hay-ward fault suggests a recurrence interval of possiblyless than 270 years but no more than 710 years(HFPEG 1999)

It should not need to be stated that earthquakesare neither periodic nor especially in the case of theHaywardndashRodgers Creek faults infinitely long Atwo-dimensional model cannot describe the situa-tion at a given trench site where ruptures mainly tothe north or south of the site cause some offset in the

trench Nevertheless the fact that the results fromsuch a simple model are not unreasonable is encour-aging One useful attribute of the models exploredhere are that they are testable in the sense of predict-ing future behavior Running the model forward intime to the next San Andreas earthquake yields thevelocity distributions shown in Figure 17 Of coursea north HaywardndashRodgers Creek earthquake mayoccur prior to the next San Andreas event

The Maxwell relaxation time is reasonablywell estimated by the data I find an optimal valueof about 35 years which corresponds to a viscos-ity of 16 times 1019 Pa-s It should be noted that theestimates of all of the model parameters is only asgood as the mechanical model used to interpretthe data If as seems quite likely there was after-slip on the San Andreas below the 1906 rupturezone the estimate of the viscosity is some combi-nation of the effects of afterslip and distributed

FIG 15 Fit of model to north San Francisco Bay data Maximum slip on Hayward fault is 30 m and on the SAF is70 m

ESTIMATES OF SLIP RATE 79

relaxation Li and Rice (1987) presented anapproximate method for including post-seismicfault slip at constant resistive stress with a relax-ing asthenosphere in the context of a repeatingperiodic earthquake sequence Combining suchan approach with appropriate constitutive laws forstable fault creep as in Tse and Rice (1986)would presumably provide a more physically real-istic description of fault behavior Clearly three-dimensional effects are also important especiallyfor earthquakes on the Rodgers Creek and Hay-ward faults Finally the effects of non-uniform

elastic properties are known to bias interpreta-tions based on simple homogeneous models Elas-tic moduli increase with depth which has theknown effect of biasing the depth distribution offault slip Similarly elastic properties may varywith lithology across the major fault zones whichthemselves may represent zones of more compli-ant rock Finite elements are certainly the mostgeneral way to compute forward models but arenot yet feasible for the optimization and bootstrapmethods as computed here which literally testedthousands of possible models

FIG 16 Five hundred bootstrap estimates of model parameters fit to the north San Francisco Bay data 90 confi-dence bounds are listed at the top of each panel The vertical line in the histograms for recurrence times TSAF and THRCshow the upper bounds corresponding to the minimum slip rate and maximum allowed coseismic slip respectively Thehorizontal bar in the panel for the San Andreas recurrence time shows the preferred estimate from Schwartz et al (1998)The horizontal bar in the panel for tH time since the last Hayward earthquake shows the range of dates compatible withpaleoseismic estimates of the penultimate HaywardndashRodgers Creek event

80 PAUL SEGALL

Acknowledgments

George Thompson set a standard that geophysi-cal models no matter how elegant must be consis-tent with geologic observations This work wasinspired by that example I would like to thank WillPrescott for discussions about the Bay Area defor-mation measurements and Mark Matthews for manyyears ago introducing me to Bayesrsquo theorem WayneThatcher Mark Murray Jeff McGuire and PeterCervelli provided valuable comments at early stagesin this study I thank Jim Savage Duncan Agnewand Shelley Kenner for their thoughtful reviewsGary Ernst and Simon Klemperer carefully editedthe manuscript

REFERENCES

Argus D F and Gordon R G 2001 Present tectonicmotion across the Coast Ranges and San Andreas faultsystem in central California Geological Society ofAmerica Bulletin v 113 p 1580ndash1592

Beroza G C 1991 Near-source modeling of the LomaPrieta earthquake Evidence for heterogeneous slipand implications for earthquake hazard Bulletin of theSeismological Society of America v 81 p 1603ndash1621

Cohen S C 1982 A multilayer model of time-dependentdeformation following an earthquake on a strike-slipfault Journal of Geophysical Research v 87 p 5409ndash5421

HFPEG (Hayward Fault Paleoearthquake Group) 1999Timing of paleoearthquakes on the northern HaywardfaultmdashPreliminary evidence in El Cerrito CaliforniaUS Geological Survey Open File Report 99-318

Henstock T J Levander A and Hole J A 1997Deformation in the lower crust of the San Andreas faultsystem in northern California Science v 278 p 650ndash653

Feigl K L King R W and Jordan T H 1990 Geo-detic measurement of tectonic deformation in theSanta Maria fold and thrust belt California Journal ofGeophysical Research v 95 p 2679ndash2699

Gill P E Murray W and Wright M H 1981 Practicaloptimization London UK Academic Press 401 p

Grant L B and Sieh K 1994 Paleoseismic evidence ofclustered earthquakes on the San Andreas fault in the

FIG 17 Simulation of Bay Area velocity field as a function of time Each line is separated by roughly 27 years

64 PAUL SEGALL

where d is a vector of geodetic dataBayesrsquo rule then states that

(1)

In words equation (1) states that the pdf of theslip-rate and fault-model parameters given geodeticobservations can be determined from the a prioridistributions of the slip-rate and the fault-modelparameters and the theoretical model linking theparameters to the geodetic observations The integralin the denominator ensures that the posterior proba-bility is properly normalized ie that the cumula-tive probability is unity Assuming that the priorprobability of the slip rate and the model parametersare independent then Thisis reasonable inasmuch as estimates of slip rate frompaleoseismic studies are independent of prior esti-mates of elastic layer thickness viscosity fault lock-

ing depth and so on While in general we have apriori constraints on these physical parameters forthe purposes of the present discussion I will oftenassume that the distribution is so broad as to benon-informative Given the independence of theprior distributions on s and m (1) reduces to

(2)

The posterior probability distribution is easilycomputed if the prior distribution is uniform oversome range That is we take the slip rate to beequally probable between specified upper and lowerbounds A simple illustration (Fig 1) can be infor-mative In Figure 1A the mean of the distributionfalls just outside the a priori bounds The posteriordistribution is thus highly concentrated near theupper bound from geologic data The geodetic and

P smiddot m( )

P smiddot m d( ) P smiddot m( )P d smiddot m( )

P smiddot( )P d smiddot m( ) smiddotd mds mograve

---------------------------------------------------------- =

P smiddot m( ) P smiddot( )P m( )=

P smiddot m d g( )

P smiddot g( )P m( )P d smiddot m( )

P smiddot( )P m( )P d smiddot m( ) smiddotd mds mograve

------------------------------------------------------------------------------

=

FIG 1 Example of application of Bayesrsquo theorem given a uniform prior distribution The prior distribution is uniformover the interval [01] and is shown in the dash-dot pattern The distribution from the data alone is dashed and theposterior distribution is solid In (D) the posterior distribution is essentially a delta function at 10

ESTIMATES OF SLIP RATE 65

geologic data together are much more informativethan either data set alone In Figure 1B the a prioribounds are so broad as to be uninformative the pos-terior distribution is equal to that obtained from thegeodetic data alone In Figure 1C the reverse istrue the geodetic data supply little information andthe posterior distribution reflects simply the a prioriinformation A fourth outcome also is possible asillustrated in Figure 1D where the two distributionsdo not overlap indicating that the two estimates areinconsistent The conclusion in this case is thateither the uncertainties in the geologic determina-tion of slip rate have been substantially mis-esti-mated or the physical model used to relate geodeticdata to slip rate is invalid The outcome in Figure1D could be the most useful in that assuming thegeologic bounds are valid it would invalidate anentire class of physical fault models

I start by considering the interseismic velocitydistribution across the Carrizo Plain segment of theSan Andreas fault (Fig 2) This area is chosenbecause it is the simplest part of the SAF systemuncomplicated by parallel faults The only otheractive structures at this latitude are the San Grego-riondashHosgri fault system which runs into an activedeformation zone in the Santa Maria Basin (Feigl etal 1990) and the fold-and-thrust belt northeast ofthe San Andreas at the eastern edge of the Coast

Ranges The latter structures involve motion largelynormal to the SAF and thus do not significantlyaffect the calculations here The San GregoriondashHos-gri deformation is accounted for crudely using asimple dislocation model

Assuming only a single San Andreas fault andmaking use of a two-dimensional dislocationmodel I find that the best-fitting locking depth is244 km (Fig 3) and the slip rate is 418 mmyr Incontrast Sieh and Jahns (1984) determined anaverage slip rate of 339 plusmn 29 mmyr for the past3700 yr and 358 + 54ndash41 mmyr for the past13250 yr based on offset fluvial deposits at Wal-lace Creek Allowing 3 mmyr of slip on the SanGregoriondashHosgri fault system which is modeled asa screw dislocation locked above 10 km the SanAndreas locking depth shallows slightly to 203km and the slip rate drops to 377 mmyr which isin reasonable accord with geologic estimates (Siehand Jahns 1984)

Of course the San Andreas fault is nottwo-dimensional Southeast of the Carrizo Plain theSan Andreas fault enters the Big Bend and theTransverse Ranges whereas the creeping zone islocated to the northwest The data analyzed here(Fig 2) were selected to be as far from possible fromthese locales yet some bias is unavoidable with atwo-dimensional model

FIG 2 Velocities determined by the Southern California Earthquake Center (SCEC) in the Carrizo Plain section ofthe San Andreas fault

66 PAUL SEGALL

It is important to determine the range of sliprates and locking depths that are consistent withthe GPS data A bootstrap resampling procedureshows that slip rates in the range of 34 to 41 mmyrare consistent with the geodetic observations (Fig4) The bootstrap was conducted in the followingmanner Let d be the vector of observations thepredicted data from the best-fitting model so that

is the vector of residuals The residualswere randomly resampled with replacement that issome elements may be chosen multiple times oth-ers not at all Denote the resampled residuals r Aresampled data vector is then usedto generate a model estimate and the process isrepeated many times The distribution of the boot-strapped model parameters is that shown in Figure4

It is well known that there is a tradeoff betweenthe locking depth and deep slip rate Increasing thelocking depth moves the dislocation farther from thesurface observation stations to achieve the samesurface strain requires a higher slip rate Applying apriori constraints on the slip rate from 31 mmyrand 37 mmyr based on the Sieh and Jahns (1984)data from Wallace Creek thus not only changes the

posterior distribution of slip rate but of the lockingdepth as well (Fig 5) The reason for this can beclearly seen from a scatter plot of the bootstrap res-amples as in Figure 6

Notice that the GPS data have significantly nar-rowed the distribution of admissible slip rates (Fig5) The posterior distribution is peaked at 35 to 37mmyr whereas the a priori distribution is uniformbetween 31 and 37 mmyr At the same time the apriori geologic information provided no constrainton the fault locking depth With geologic data alonewe cannot distinguish between a locked fault and afault that creeps steadily below some shallow lockedinterval The GPS data alone bound the lockingdepth to between 17 and 24 km Because of thestrong correlation between slip rate and lockingdepth the posterior distribution of the locking depthis narrowed to 17ndash20 km This simple exampleshows the power of using a priori geologic con-straints in allowing more information to be obtainedfrom geodetic data

Note that the bootstrap provides an estimate ofthe distribution of the model parameters whereasBayesrsquo rule requires P(d|m) as in equation (3)However for linear models with normally distrib-

d

r d dndash=

d d r+=

FIG 3 A Misfit as a function of depth for a simple screw dislocation model B Profile of fault parallel velocity acrossthe fault compared with prediction of a simple screw dislocation model C As in (B) but with 3 mmyr of slip beneaththe Hosgri fault

ESTIMATES OF SLIP RATE 67

uted observational errors the two approaches areequivalent Furthermore a uniform distribution withhard bounds may not be the best characterization ofthe prior probability distribution on slip rate Alter-

natively if the prior distribution and observationalerrors are normally distributed one can simplyapply the prior information as ldquopseudo datardquo (egJackson 1979)

FIG 4 Probability distributions for fault locking depth and slip rate from bootstrap results using GPS data I assume3 mmyr below a locking depth of 10 km on the Hosgri fault system

FIG 5 Posterior probability distribution from the bootstrap results using GPS data and assuming a uniform priordis-tribution on slip rate from 31 to 37 mmyr Compare with Figure 4 As before I assume 3 mmyr below a locking depthof 10 km on the Hosgri fault system

68 PAUL SEGALL

Viscoelastic Models

As noted in the Introduction the screw disloca-tion provides only a very limited description ofplate-boundary faulting A somewhat more realisticmodel involves an elastic layer of thickness H over-lying a Maxwell viscoelastic half-space (Fig 7) TheMaxwell material has relaxation time tR = 2hmicrowhere h is the viscosity and micro is the shear modulusAt time t = 0 slip Du occurs on the fault from thesurface to depth D pound H The velocity on the Earthrsquossurface as a function of position perpendicular to thefault x and time t is

(3)

where the spatial distribution is given by

(4)

The result follows from but is not stated in Nur andMavko (1974) The strain rate is simply obtained bydifferentiating the velocity with respect to x Noticethat the post-seismic velocity is a function of thecoseismic slip Du the depth of faulting D the elas-tic layer thickness H the material relaxation tR andthe time since the last large earthquake t Equation(3) does not include the effects of interseismic strainaccumulation Savage and Prescott (1978) extendedthe model by summing over an infinite sequence ofplate rupturing events (D = H) equally spaced intime with recurrence interval T The velocity field asa function of distance across strike (x) and time (t) isgiven by

(5)

where the temporal dependence depends only on thedimensionless ratios ttR and TtR

(6)

v x t( )

DuptR---------e

t tRcurrenndash t tRcurren( )n 1ndash

n 1ndash( )--------------------------Fn x( D H )

n 1=

yen

aring

=

Fn x D H ( )

D 2nH+x

---------------------egrave oslashaelig ouml1ndash

tan D 2nHndashx

--------------------egrave oslashaelig ouml1ndash

tan+

2xD

x2

2nH( )2D

2ndash+

------------------------------------------1ndash tan

=

=

v x t( ) smiddot

p--- Tn t tR T tRcurrencurren( )Fn x H H ( )

n 1=

yen

aring=

Tn t tR T tRcurrencurren( )

TtR-----

et tRcurrenndash

n 1ndash( )------------------ e

kT tRcurrenndash t kT+tR

--------------egrave oslashaelig ouml n 1ndash

k 0=

yen

aring

=

FIG 6 Tradeoff between slip rate and locking depth Each point is the result of a bootstrap resample of the SCECvelocity data The heavy points are those consistent with a uniform prior distribution on slip rate from 31 to 37 mmyrAs before I assume 3 mmyr below a locking depth of 10 km on the Hosgri fault system

ESTIMATES OF SLIP RATE 69

Because the far-field lithospheric velocity is drivensolely by the repeating earthquakes Representative examples are given in Figure 8 forvarious times through the earthquake cycle Noticethat when the asthenospheric relaxation time is ofthe order of the earthquake cycle time or longer TtR pound1 the surface velocity distribution is rathersteady with time (Fig 8A) On the other hand if therelaxation time is short compared to the recurrenceinterval TtR gt 1 there is a significant transientwith higher than average strain rates early in thecycle and lower than average strain rates late in thecycle

The Savage-Prescott model involves five param-eters lithospheric thickness H relaxation time tRtime since the last earthquake t recurrence time Tand long-term slip rate on the fault Because thetheory is linear it is straightforward to constructmodels with multiple parallel faults however in thiscase the number of parameters is even larger Forthree faults there are 11 parameters if one assumesthat the elastic thickness and relaxation time are thesame for all faults Note that the model is kinematicwith imposed slip events on the various faults at reg-ular intervals and thus does not account for thestress interactions between the faults

As a starting point I consider the Carrizo Plaindata with some of the parameters constrained byindependent observations The time since the lastgreat earthquake (1857) is t = 136 years To beginwith I assume a recurrence time of T = 160 yearsbased on the average interval found by Grant andSieh (1994) There are three remaining parametersto be estimated from the data tR H and Theoptimal estimates of the non-linear parameters tRand H are found by simple grid search (Fig 9) with

estimated by linear least squares The optimalmodel is found for H = 114 km and tR = 80 yearswith a slip rate of 402 mmyr There is a strongtradeoff between relaxation time and elastic-layerthickness such that models with elastic-layer thick-ness of up to 20 km and relaxation time of 180 yearsfit the data nearly equally well The reason for thetradeoff is as follows Larger values of H cause thedeformation to be spread out in space At the sametime the deformation becomes more diffuse withincreasing tT and TtR (Fig 8) and hence t tR = (tT) times (TtR ) Thus decreasing ttR causes the defor-mation to bemore localized which counters theeffect of increasing H Interestingly the slip ratedoes not vary with H and tR confirming that thisparameter is well constrained by the data (given thatall other parameters are fixed)

The inferred slip rate is however higher thanestimated from stream offsets at Wallace Creek of 33plusmn 3 mmyr (Sieh and Jahns 1984) Constraining theslip rate to 34 mmyr (Fig 10) does not fit the dataas well especially for the stations farthest west ofthe San Andreas The optimal model is H = 114 kmas before but the relaxation time increases to 180years

Relaxation times tR gt 50 years are long com-pared to what has been inferred for the post-1906strain rate transient north of the San Francisco Bayas measured by repeated triangulation surveys(Thatcher 1975a 1975b Kenner and Segall 2000)The relaxation time tR consistent with the post-1906transient can be inferred from the triangulation dataand the spatial derivative of equation (3) It wouldbe most consistent to employ the same Savage-Pres-cott model as used with the Carrizo Plain data how-ever this requires knowing the recurrence time oflarge earthquakes on the north coast segment of theSan Andreas fault An alternate approach employedhere is to remove the estimated secular deformationusing a model (see Kenner and Segall 2001) andthen fit the remaining strain rate data For 60meters of slip in the 1906 earthquake the optimal

smiddot Du Tcurren=

smiddot

smiddot

smiddot

FIG 7 Model of a faulted elastic plate overlying viscoelas-tic half-space The elastic layer has shear modulus micro the vis-coelast ic half-space shear modulus micro and viscosity hEarthquakes occur at a regular interval T such that thefar-field velocity is vyen2

70 PAUL SEGALL

model is H = 19 km with a relaxation time of 17years (Fig 11) substantially less than found fromthe Carrizo Plain interseismic data1

A tentative conclusion of this analysis is that theCarrizo Plain interseismic data require a relaxationtime in excess of 50 years whereas the post-1906triangulation data require relaxation times of 25years or less In order to test whether this discrep-ancy is truly required by the data I estimated theuncertainty in the two relaxation times using a boot-strap resampling procedure As discussed previ-ously the residuals to the best-fitting model wererandomly resampled with replacement and added

to the model predictions for each bootstrap resam-ple This approach seems best for small data setsand ensures that there were data at each observationposition (interseismic data) and time (postseismicdata)

For the post-1906 transient admissible relax-ation times range from 10 years to 30 years (Fig 12)On the other hand the present distribution of defor-mation on the Carrizo Plain demands relaxationtimes in excess of 35 years (Fig 13) Thus we cansafely conclude that there is no single relaxationtime that fits both data sets

There are a number of possible explanations forthis apparent discrepancy One possibility is thatthe model of a fully ruptured elastic plate overlyinga linear Maxwell half-space does not capture all ofthe relevant physics For example nonlinear rheol-ogy may be important The data are consistent withthe expectation that the effective viscosity is low

1 Note that the discrepancy is not due to the correction forinterseismic deformation in the post-1906 data Assuming arecurrence time of 250 years and a slip rate on the SAF of 25mmyr the optimal relaxation time is 228 years with an elas-tic-layer thickness of 167 km

FIG 8 Savage-Prescott model for (A) T tR = 1 and (B) T tR = 5 t is the time since the last earthquake T is theearthquake repeat time and tR is the Maxwell relaxation time In these calculations the earthquakes rupture the entireelastic layer

ESTIMATES OF SLIP RATE 71

early in the seismic cycle when the stresses arelarge and increases late in the cycle when thestresses have largely dissipated In addition recentreflection and refraction results in northern Califor-nia show the San Andreas fault cutting the lowercrust and possibly the Moho (eg Henstock et al1997 Parsons and Hart 1999) Aseismic slip onthe downdip extension of the seismogenic zone (asin Tse and Rice 1986) may take place in additionto a more deeply distributed relation Results fromKenner and Segall (2001) which include viscousrelaxation in a narrow fault zone and more deeplydistributed slip find the spatiotemporal pattern ofdeformation in the post-1906 triangulation data isbest fit with a short relaxation time in the fault zoneand a longer relaxation in the underlyinghalf-space Another possibility is that the thermo-mechanical structure of the lithosphere differsbetween the two areas It is also possible that along-strike variations in fault geometry (the Big Bend) or

slip behavior (the creeping zone) may bias the com-puted relaxation time although this seems lesslikely

Although the apparent relaxation times differbetween the Carrizo Plain and the northern SanFrancisco Bay region the elastic-layer thicknessestimates are consistent in the range of 13ndash20 kmCarrizo Plain slip-rate estimates cluster near theupper bound of 37 mmyr Interestingly the CarrizoPlain data yield recurrence times greater than 160years Whereas the distribution is reasonably broadboth the mean (288 years) and median (277) repeattimes fall between the 1988 working group estimateof 296 years (WGCEP 1988) and the 1995 workinggroup estimate of 206 years (WGCEP 1995) Theaverage coseismic slip for the predicted events isroughly 10 meters There are significant correlationsbetween parameters (between H and tR as dis-cussed previously as well as between tR and T)which means that any additional data that constrain

FIG 9 Comparison of Savage-Prescott model to the SCEC velocity data for the Carrizo Plain A Misfit as a functionof elastic-layer thickness and relaxation time B Best-fitting San Andreas slip rate corresponding to the elastic-layerthickness and relaxation time C Fit to the data Effect of the Hosgri fault is removed employing a dislocation model withslip rate of 3 mmyr and depth of 10 km

72 PAUL SEGALL

some of the parameters will help resolve others aswell

It should be emphasized that the paleoseismicobservations of Grant and Sieh (1994) show an irreg-ular pattern of earthquakes that cannot be explainedby the simple two-dimensional model used here tointerpret the geodetic observations Nevertheless itis interesting that the GPS data appear to have someresolving power for average recurrence interval

San Francisco Bay Data

In order to eliminate any geographic bias in theinterpretation associated with possibly differentcrustal structure and temperature distribution Inext consider the velocity profile across the SanAndreas fault north of the San Francisco Bay asdetermined by repeated GPS observations (Prescottet al 2001) as shown in Figure 14 The GPS sitesare sufficiently close to the post-1906 triangulationnetworks that it is reasonable to assume that the

crustal structure and thermal profile are the samefor both data sets In the following the GPS-derivedvelocities are rotated into fault-parallel and perpen-dicular components Only the fault-parallel motionsare modeled here although as pointed out by Pres-cott et al (2001) and Murray and Segall (2001)there is significant fault-normal motion in the east-ern coast ranges at the western edge of the CentralValley

The model is the same as discussed previouslywith three principal faults the San Andreas Hay-wardndashRodgers Creek and ConcordndashGreen Valleyfaults The parameters are the elastic plate thick-ness H Maxwell relaxation time tR the recurrencetime for great earthquakes on the SAF TSAF the sliprate on the SAF and the recurrence timeTHRC time since past event tHRC and slip rate on the Hayward-Rodgers Creek fault system Thetime since the last great earthquake on the northernSan Andreas fault tSAF is of course known Theeffect of the ConcordndashGreen Valley fault is modeled

smiddotSAFsmiddotHRC

FIG 10 Comparison of Savage-Prescott model to the SCEC velocity data for the Carrizo Plain with slip rate con-strained to 34 mmyr Effect of San GregoriondashHosgri fault has been removed with a dislocation model with slip rate of 3mmyr and depth of 10 km A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

ESTIMATES OF SLIP RATE 73

kinematically with a simple screw dislocationwhich adds two parameters the fault slip rate andlocking depth It is certainly inconsistent to ignoreviscoelastic effects for the ConcordndashGreen Valleyfault but I do so for three reasons (a) much less isknown about this fault (b) earthquakes there arepresumably least well modeled by infinitely longruptures and (c) it keeps the number of parametersreasonably tractable In essence the effect of theConcordndashGreen Valley is removed empirically and Imake no attempt to gain further insights into itsmechanical behavior Future work should addressand hopefully rectify this simplification

A priori bounds on the parameters are listed inTable 1 The lower bound on the lithospheric thick-ness comes from a conservative estimate of thedepth extent of slip in large earthquakes In fact itis more likely that slip in 1906 extended at least to15 km (Matthews and Segall 1993) as it certainlydid in the Loma Prieta earthquake (eg Beroza1991) The bounds on relaxation time come from thepreliminary analysis above (Figs 12 and 13) The

lower bound on TSAF comes from the elapsed timesince 1906 the upper bound from limits on theamount of coseismic slip (see below) The lowerbounds on tHRC and THRC are derived from the his-torical record there is no known earthquake on thenorthern HaywardndashRodgers Creek system since atleast 1775 (Toppozada and Borchardt 1998WGCEP 1999) The lower and upper bounds on sliprates were based loosely on the Northern CaliforniaWorking group (WGCEP 1999 Table 2) The Work-ing Group listed 2 s uncertainties of = 24 plusmn 3mmyr = 9 plusmn 2 mmyr and = 5 plusmn m 3mmyr

The estimation problem is to find the choice ofthe nine parameters in Table 1 that minimize themisfit to two geodetic data sets the present-day spa-tial distribution of deformation in the north SanFrancisco Bay region (Fig 14) and the time-depen-dent decay in strain rate following the 1906 earth-quake (Fig 11) For a non-linear problem such asthis one always needs to be concerned about localminima However preliminary testing with a simu-

smiddotSAFsmiddotHRC smiddotHRC

FIG 11 Post-1906 strain rate from triangulation data in the San Francisco Bay area Model fit to the model withpostseismic deformation only A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

74 PAUL SEGALL

lated annealing algorithm did not show evidence oflocal minima so the remaining analysis was doneusing a Sequential Quadratic Programming methodwith BFGS Hessian updates (eg Gill et al 1981)within Matlab The method allows upper and lowerbounds to be placed on all parameters as well asgeneralized inequality constraints The latter areneeded because tHRC is strictly bounded by tHRC poundTHRC In the context of the Savage and Prescottmodel earthquakes are periodic and it makes nosense for the elapsed time since the last event toexceed the average recurrence time In addition Ihave found it desirable to constrain the maximumcoseismic slip on both the San Andreas and Hay-ward faults This provides constraints of the form

and

Noting that the coseismic slip should be interpretedas an average over the fault in this two-dimensional

model I set meters and

metersIt should be noted that the GPS network does not

extend very far west of the San Andreas fault Forthis reason the data do not adequately constrain the

total motion across the plate boundary A reasonableapproach would be to include data from Pacificplate sites such as Hawaii I chose simply to con-strain the sum of the slip rates on all three faults tobe no greater than 40 mmyr (eg Argus and Gordon2001) This constraint significantly affects the solu-tion estimates without the cumulative slip con-straint tended to put the maximum slip rate allowedon all three faults exceeding the total displacementrate between the Sierra NevadandashGreat Valley blockand the Pacific plate

The optimal solution is found for H ~ 184 km tR~ 345 years TSAF = 280 years ~ 25 mmyrtHRC = 225 years THRC = 4286 years and ~ 7mmyr The ConcordndashGreen Valley fault slips at

~ 8 mmyr below 22 km A surprising resultis that the predicted recurrence time on the SanAndreas is reasonable in the sense of requiring 70-meter slip events every 280 years In comparisonSchwartz et al (1998) suggested that the penulti-mate event on the north coast segment of the SanAndreas occurred in the mid-1600s possibly in theinterval 1632ndash1659 corresponding to a recurrenceinterval of roughly 260 years Also surprising is thatthe time since the last earthquake on the Haywardndash

smiddotSAF TSAFtimes DuSAFmaxpound smiddotHRC THRCtimes DuHRC

maxpound

DuSAFmax

70= DuHRCmax

30=

smiddotSAFsmiddotHRC

smiddotHRC

FIG 12 Bootstrap results for post-1906 data in the San Francisco Bay area A Relaxation time B Elastic-layerthickness (1000 bootstrap resamples)

ESTIMATES OF SLIP RATE 75

Rodgers Creek fault is the minimum allowed 225years A comparison of the model fit to the data setsis shown in Figure 15 Generally speaking the dataare fit quite well with the possible exception of thestrain rate in the first post-1906 epoch The uncer-tainties in the post-1906 strain rates were actuallydecreased by a factor of two to ensure that these dataare fairly weighted compared to the GPS velocitiesDecreasing the uncertainties in the strain rates fur-ther to fit high strain rate in the immediatepost-1906 epoch causes a very significant increasein the misfit to the modern GPS data This is mostlikely a result of the large uncertainties in the trian-gulation data but may also indicate that the physi-cal model is inadequate

How is it that the geodetic data have any capabil-ity of resolving the recurrence interval on the SanAndreas fault Recall that there are three times inthe physical model the time since the last majorearthquake t the asthenospheric relaxation time tRand the recurrence interval T For the San Andreasfault we know tSAF ~93 years a priori and tR is rea-

sonably well resolved by the post-1906 strain ratetransient The estimation is thus free to vary T insuch a way as to optimize the fit to the velocity gra-dients near the San Andreas Constraints on themaximum slip in the earthquake help to furthertighten those constraints

In order to determine how well the data constrainthe model parameters I ran 500 bootstrap resam-ples and optimizations using the best-fitting modelas the starting point of the optimization As beforethe residuals were resampled and added to the pre-dicted data from the optimal model in order toensure that each observation point contributes to thesolution The results of the bootstrap are shown inFigure 16 The posterior distribution for recurrencetime on the San Andreas fault TSAF is quite broadwith a peak near the maximum allowed of (7000mm)(19 mmyr) ~370 years In contrast the poste-rior distribution for recurrence time on the Hay-wardndashRodgers Creek fault THRC is bimodal withprominent peaks near the minimum ~225 years andmaximum allowed value (3000 mm)(7 mmyr)

FIG 13 Bootstrap results for SCEC data along the Carrizo Plain segment of the San Andreas fault A Elastic-layerthickness B Relaxation time Note that 200 years was the upper bound considered C Recurrence time Recurrencetimes in excess of 400 years were not considered as these would lead to excessive coseismic slip D Slip rate

76 PAUL SEGALL

~ 428 years The distributions for tHRC the timesince the most recent HaywardndashRodgers Creekearthquake is strongly concentrated near the mini-

mum allowed range The distribution of slip rates onthe San Andreas and HaywardndashRodgers Creekfaults are bimodal with peaks at 19 mmyr and 25

FIG 14 GPS-derived velocities for the north San Francisco Bay region after Prescott et al (2001) AbbreviationsSAF = San Andreas fault HF = Hayward fault CF = Calaveras fault RGF = Rodgers Creek fault CGVF = ConcordndashGreen Valley fault

TABLE 1 A priori Bounds on Parameters

Parameter Symbol Min Max

Lithospheric thickness km H 10 30

Relaxation time yr tR 10 200

SAF recurrence interval yr TSAF 100 500

Time since last Hayward event yr tHRC 225 690

Hayward recurrence interval yr THRC 225 700

Green valley fault locking depth km d 05 15

SAF slip rate my 0019 0027

Hayward slip rate myr 0007 0013

Green valley slip rate myr 0002 0008

smiddotSAF

smiddotHRC

smiddotCGV

ESTIMATES OF SLIP RATE 77

mmyr (SAF) and 7 mmyr and 13 mmyr (HRC)These results still must be viewed with some cau-tion in that I have not demonstrated that the boot-strap optimizations were immune to local minima

The bootstrap calculations suggest that the con-straint on maximum coseismic slip does play astrong role in the results To explore this further Ivaried the maximum coseismic slip allowed on thetwo faults The results (Table 2) demonstrate that theSan Andreas recurrence interval is not fully con-trolled by the constraint on the maximum coseismicslip In contrast the recurrence interval on the Hay-ward fault is totally controlled by the maximum slipallowed (for a maximum slip of 10 meters the recur-rence interval reaches the upper bound of 700years) Notice also that the time since the last Hay-wardndashRodgers Creek event is always at the mini-mum value allowed (225 years) The fit to the data isslightly better for large infrequent earthquakes

Discussion

What has been learned from the modeling pre-sented here It seems clear that geodetic data

should not be analyzed in a vacuum The resolutionof the data alone is insufficient to address the prob-lems of greatest interest However when combinedwith prior information from geologic studies GPSdata appear to be able to resolve parameters of con-siderable interest such as the average recurrenceinterval between great earthquakes on the SanAndreas fault Using a simple model of a repeatingdislocation rupturing an elastic plate overlying aMaxwell visco-elastic medium I found an averagerecurrence interval of roughly 280 years on thenorth coast segment of the San Andreas faultAlthough the uncertainties in this estimate arelarge the result is in surprisingly good accordancewith recent paleoseismic results (Schwartz et al1998) that suggest an event similar to 1906 in themid 1600s

The results for the HaywardndashRodgers Creek faultare also not unreasonable According to the GPSdata the most recent event here occurred in ~1775regardless of the estimated recurrence time Thisresult is not inconsistent with the paleoseismic find-ing that the most recent event on the Rodgers Creekfault took place in the interval 1670ndash1776

TABLE 2 Influence of Maximum Coseismic Slip on Estimated Parameters1

Parameter

Lithospheric thickness H km 150 184 220

Relaxation time tR yr 357 345 340

SAF recurrence interval TSAF yr 220 280 337

Limiting interval yr 290 368 526

Time since last Hayward event tHRC yr 225 225 225

Hayward recurrence interval THRC yr 2857 4286 700

Limiting interval yr 285 428 714

Green Valley fault locking depth d km 11 22 47

SAF slip rate myr 0025 0025 00249

Hayward slip rate myr 0007 0007 00071

Green valley slip rate myr 0008 0008 0008

1Limiting interval is computed using the estimated slip rate for that fault not the minimum slip rate that would yield a longer upper bound

DuSAFmax

55=

DuHRCmax

20=

DuSAFmax

70=

DuHRCmax

30=

DuSAFmax

100=

DuHRCmax

50=

DuSAFmax

smiddotSAFcurren

DuHRCmax

smiddotHRCcurren

smiddotSAF

smiddotHRC

smiddotCGV

78 PAUL SEGALL

(Schwartz et al 1993) and on the northern Hay-ward fault in 1640ndash1776 (HFPEG 1999) The GPSdata alone yield limited information about the aver-age recurrence time on the HaywardndashRodgers Creekfault However adding the reasonable bound thatslip not exceed 2 meters on average yields a recur-rence interval of 285 years The average slip in the1868 Hayward fault earthquake was approximately2 meters (Yu and Segall 1996) In comparisontrenching at the El Cerrito site on the northern Hay-ward fault suggests a recurrence interval of possiblyless than 270 years but no more than 710 years(HFPEG 1999)

It should not need to be stated that earthquakesare neither periodic nor especially in the case of theHaywardndashRodgers Creek faults infinitely long Atwo-dimensional model cannot describe the situa-tion at a given trench site where ruptures mainly tothe north or south of the site cause some offset in the

trench Nevertheless the fact that the results fromsuch a simple model are not unreasonable is encour-aging One useful attribute of the models exploredhere are that they are testable in the sense of predict-ing future behavior Running the model forward intime to the next San Andreas earthquake yields thevelocity distributions shown in Figure 17 Of coursea north HaywardndashRodgers Creek earthquake mayoccur prior to the next San Andreas event

The Maxwell relaxation time is reasonablywell estimated by the data I find an optimal valueof about 35 years which corresponds to a viscos-ity of 16 times 1019 Pa-s It should be noted that theestimates of all of the model parameters is only asgood as the mechanical model used to interpretthe data If as seems quite likely there was after-slip on the San Andreas below the 1906 rupturezone the estimate of the viscosity is some combi-nation of the effects of afterslip and distributed

FIG 15 Fit of model to north San Francisco Bay data Maximum slip on Hayward fault is 30 m and on the SAF is70 m

ESTIMATES OF SLIP RATE 79

relaxation Li and Rice (1987) presented anapproximate method for including post-seismicfault slip at constant resistive stress with a relax-ing asthenosphere in the context of a repeatingperiodic earthquake sequence Combining suchan approach with appropriate constitutive laws forstable fault creep as in Tse and Rice (1986)would presumably provide a more physically real-istic description of fault behavior Clearly three-dimensional effects are also important especiallyfor earthquakes on the Rodgers Creek and Hay-ward faults Finally the effects of non-uniform

elastic properties are known to bias interpreta-tions based on simple homogeneous models Elas-tic moduli increase with depth which has theknown effect of biasing the depth distribution offault slip Similarly elastic properties may varywith lithology across the major fault zones whichthemselves may represent zones of more compli-ant rock Finite elements are certainly the mostgeneral way to compute forward models but arenot yet feasible for the optimization and bootstrapmethods as computed here which literally testedthousands of possible models

FIG 16 Five hundred bootstrap estimates of model parameters fit to the north San Francisco Bay data 90 confi-dence bounds are listed at the top of each panel The vertical line in the histograms for recurrence times TSAF and THRCshow the upper bounds corresponding to the minimum slip rate and maximum allowed coseismic slip respectively Thehorizontal bar in the panel for the San Andreas recurrence time shows the preferred estimate from Schwartz et al (1998)The horizontal bar in the panel for tH time since the last Hayward earthquake shows the range of dates compatible withpaleoseismic estimates of the penultimate HaywardndashRodgers Creek event

80 PAUL SEGALL

Acknowledgments

George Thompson set a standard that geophysi-cal models no matter how elegant must be consis-tent with geologic observations This work wasinspired by that example I would like to thank WillPrescott for discussions about the Bay Area defor-mation measurements and Mark Matthews for manyyears ago introducing me to Bayesrsquo theorem WayneThatcher Mark Murray Jeff McGuire and PeterCervelli provided valuable comments at early stagesin this study I thank Jim Savage Duncan Agnewand Shelley Kenner for their thoughtful reviewsGary Ernst and Simon Klemperer carefully editedthe manuscript

REFERENCES

Argus D F and Gordon R G 2001 Present tectonicmotion across the Coast Ranges and San Andreas faultsystem in central California Geological Society ofAmerica Bulletin v 113 p 1580ndash1592

Beroza G C 1991 Near-source modeling of the LomaPrieta earthquake Evidence for heterogeneous slipand implications for earthquake hazard Bulletin of theSeismological Society of America v 81 p 1603ndash1621

Cohen S C 1982 A multilayer model of time-dependentdeformation following an earthquake on a strike-slipfault Journal of Geophysical Research v 87 p 5409ndash5421

HFPEG (Hayward Fault Paleoearthquake Group) 1999Timing of paleoearthquakes on the northern HaywardfaultmdashPreliminary evidence in El Cerrito CaliforniaUS Geological Survey Open File Report 99-318

Henstock T J Levander A and Hole J A 1997Deformation in the lower crust of the San Andreas faultsystem in northern California Science v 278 p 650ndash653

Feigl K L King R W and Jordan T H 1990 Geo-detic measurement of tectonic deformation in theSanta Maria fold and thrust belt California Journal ofGeophysical Research v 95 p 2679ndash2699

Gill P E Murray W and Wright M H 1981 Practicaloptimization London UK Academic Press 401 p

Grant L B and Sieh K 1994 Paleoseismic evidence ofclustered earthquakes on the San Andreas fault in the

FIG 17 Simulation of Bay Area velocity field as a function of time Each line is separated by roughly 27 years

ESTIMATES OF SLIP RATE 65

geologic data together are much more informativethan either data set alone In Figure 1B the a prioribounds are so broad as to be uninformative the pos-terior distribution is equal to that obtained from thegeodetic data alone In Figure 1C the reverse istrue the geodetic data supply little information andthe posterior distribution reflects simply the a prioriinformation A fourth outcome also is possible asillustrated in Figure 1D where the two distributionsdo not overlap indicating that the two estimates areinconsistent The conclusion in this case is thateither the uncertainties in the geologic determina-tion of slip rate have been substantially mis-esti-mated or the physical model used to relate geodeticdata to slip rate is invalid The outcome in Figure1D could be the most useful in that assuming thegeologic bounds are valid it would invalidate anentire class of physical fault models

I start by considering the interseismic velocitydistribution across the Carrizo Plain segment of theSan Andreas fault (Fig 2) This area is chosenbecause it is the simplest part of the SAF systemuncomplicated by parallel faults The only otheractive structures at this latitude are the San Grego-riondashHosgri fault system which runs into an activedeformation zone in the Santa Maria Basin (Feigl etal 1990) and the fold-and-thrust belt northeast ofthe San Andreas at the eastern edge of the Coast

Ranges The latter structures involve motion largelynormal to the SAF and thus do not significantlyaffect the calculations here The San GregoriondashHos-gri deformation is accounted for crudely using asimple dislocation model

Assuming only a single San Andreas fault andmaking use of a two-dimensional dislocationmodel I find that the best-fitting locking depth is244 km (Fig 3) and the slip rate is 418 mmyr Incontrast Sieh and Jahns (1984) determined anaverage slip rate of 339 plusmn 29 mmyr for the past3700 yr and 358 + 54ndash41 mmyr for the past13250 yr based on offset fluvial deposits at Wal-lace Creek Allowing 3 mmyr of slip on the SanGregoriondashHosgri fault system which is modeled asa screw dislocation locked above 10 km the SanAndreas locking depth shallows slightly to 203km and the slip rate drops to 377 mmyr which isin reasonable accord with geologic estimates (Siehand Jahns 1984)

Of course the San Andreas fault is nottwo-dimensional Southeast of the Carrizo Plain theSan Andreas fault enters the Big Bend and theTransverse Ranges whereas the creeping zone islocated to the northwest The data analyzed here(Fig 2) were selected to be as far from possible fromthese locales yet some bias is unavoidable with atwo-dimensional model

FIG 2 Velocities determined by the Southern California Earthquake Center (SCEC) in the Carrizo Plain section ofthe San Andreas fault

66 PAUL SEGALL

It is important to determine the range of sliprates and locking depths that are consistent withthe GPS data A bootstrap resampling procedureshows that slip rates in the range of 34 to 41 mmyrare consistent with the geodetic observations (Fig4) The bootstrap was conducted in the followingmanner Let d be the vector of observations thepredicted data from the best-fitting model so that

is the vector of residuals The residualswere randomly resampled with replacement that issome elements may be chosen multiple times oth-ers not at all Denote the resampled residuals r Aresampled data vector is then usedto generate a model estimate and the process isrepeated many times The distribution of the boot-strapped model parameters is that shown in Figure4

It is well known that there is a tradeoff betweenthe locking depth and deep slip rate Increasing thelocking depth moves the dislocation farther from thesurface observation stations to achieve the samesurface strain requires a higher slip rate Applying apriori constraints on the slip rate from 31 mmyrand 37 mmyr based on the Sieh and Jahns (1984)data from Wallace Creek thus not only changes the

posterior distribution of slip rate but of the lockingdepth as well (Fig 5) The reason for this can beclearly seen from a scatter plot of the bootstrap res-amples as in Figure 6

Notice that the GPS data have significantly nar-rowed the distribution of admissible slip rates (Fig5) The posterior distribution is peaked at 35 to 37mmyr whereas the a priori distribution is uniformbetween 31 and 37 mmyr At the same time the apriori geologic information provided no constrainton the fault locking depth With geologic data alonewe cannot distinguish between a locked fault and afault that creeps steadily below some shallow lockedinterval The GPS data alone bound the lockingdepth to between 17 and 24 km Because of thestrong correlation between slip rate and lockingdepth the posterior distribution of the locking depthis narrowed to 17ndash20 km This simple exampleshows the power of using a priori geologic con-straints in allowing more information to be obtainedfrom geodetic data

Note that the bootstrap provides an estimate ofthe distribution of the model parameters whereasBayesrsquo rule requires P(d|m) as in equation (3)However for linear models with normally distrib-

d

r d dndash=

d d r+=

FIG 3 A Misfit as a function of depth for a simple screw dislocation model B Profile of fault parallel velocity acrossthe fault compared with prediction of a simple screw dislocation model C As in (B) but with 3 mmyr of slip beneaththe Hosgri fault

ESTIMATES OF SLIP RATE 67

uted observational errors the two approaches areequivalent Furthermore a uniform distribution withhard bounds may not be the best characterization ofthe prior probability distribution on slip rate Alter-

natively if the prior distribution and observationalerrors are normally distributed one can simplyapply the prior information as ldquopseudo datardquo (egJackson 1979)

FIG 4 Probability distributions for fault locking depth and slip rate from bootstrap results using GPS data I assume3 mmyr below a locking depth of 10 km on the Hosgri fault system

FIG 5 Posterior probability distribution from the bootstrap results using GPS data and assuming a uniform priordis-tribution on slip rate from 31 to 37 mmyr Compare with Figure 4 As before I assume 3 mmyr below a locking depthof 10 km on the Hosgri fault system

68 PAUL SEGALL

Viscoelastic Models

As noted in the Introduction the screw disloca-tion provides only a very limited description ofplate-boundary faulting A somewhat more realisticmodel involves an elastic layer of thickness H over-lying a Maxwell viscoelastic half-space (Fig 7) TheMaxwell material has relaxation time tR = 2hmicrowhere h is the viscosity and micro is the shear modulusAt time t = 0 slip Du occurs on the fault from thesurface to depth D pound H The velocity on the Earthrsquossurface as a function of position perpendicular to thefault x and time t is

(3)

where the spatial distribution is given by

(4)

The result follows from but is not stated in Nur andMavko (1974) The strain rate is simply obtained bydifferentiating the velocity with respect to x Noticethat the post-seismic velocity is a function of thecoseismic slip Du the depth of faulting D the elas-tic layer thickness H the material relaxation tR andthe time since the last large earthquake t Equation(3) does not include the effects of interseismic strainaccumulation Savage and Prescott (1978) extendedthe model by summing over an infinite sequence ofplate rupturing events (D = H) equally spaced intime with recurrence interval T The velocity field asa function of distance across strike (x) and time (t) isgiven by

(5)

where the temporal dependence depends only on thedimensionless ratios ttR and TtR

(6)

v x t( )

DuptR---------e

t tRcurrenndash t tRcurren( )n 1ndash

n 1ndash( )--------------------------Fn x( D H )

n 1=

yen

aring

=

Fn x D H ( )

D 2nH+x

---------------------egrave oslashaelig ouml1ndash

tan D 2nHndashx

--------------------egrave oslashaelig ouml1ndash

tan+

2xD

x2

2nH( )2D

2ndash+

------------------------------------------1ndash tan

=

=

v x t( ) smiddot

p--- Tn t tR T tRcurrencurren( )Fn x H H ( )

n 1=

yen

aring=

Tn t tR T tRcurrencurren( )

TtR-----

et tRcurrenndash

n 1ndash( )------------------ e

kT tRcurrenndash t kT+tR

--------------egrave oslashaelig ouml n 1ndash

k 0=

yen

aring

=

FIG 6 Tradeoff between slip rate and locking depth Each point is the result of a bootstrap resample of the SCECvelocity data The heavy points are those consistent with a uniform prior distribution on slip rate from 31 to 37 mmyrAs before I assume 3 mmyr below a locking depth of 10 km on the Hosgri fault system

ESTIMATES OF SLIP RATE 69

Because the far-field lithospheric velocity is drivensolely by the repeating earthquakes Representative examples are given in Figure 8 forvarious times through the earthquake cycle Noticethat when the asthenospheric relaxation time is ofthe order of the earthquake cycle time or longer TtR pound1 the surface velocity distribution is rathersteady with time (Fig 8A) On the other hand if therelaxation time is short compared to the recurrenceinterval TtR gt 1 there is a significant transientwith higher than average strain rates early in thecycle and lower than average strain rates late in thecycle

The Savage-Prescott model involves five param-eters lithospheric thickness H relaxation time tRtime since the last earthquake t recurrence time Tand long-term slip rate on the fault Because thetheory is linear it is straightforward to constructmodels with multiple parallel faults however in thiscase the number of parameters is even larger Forthree faults there are 11 parameters if one assumesthat the elastic thickness and relaxation time are thesame for all faults Note that the model is kinematicwith imposed slip events on the various faults at reg-ular intervals and thus does not account for thestress interactions between the faults

As a starting point I consider the Carrizo Plaindata with some of the parameters constrained byindependent observations The time since the lastgreat earthquake (1857) is t = 136 years To beginwith I assume a recurrence time of T = 160 yearsbased on the average interval found by Grant andSieh (1994) There are three remaining parametersto be estimated from the data tR H and Theoptimal estimates of the non-linear parameters tRand H are found by simple grid search (Fig 9) with

estimated by linear least squares The optimalmodel is found for H = 114 km and tR = 80 yearswith a slip rate of 402 mmyr There is a strongtradeoff between relaxation time and elastic-layerthickness such that models with elastic-layer thick-ness of up to 20 km and relaxation time of 180 yearsfit the data nearly equally well The reason for thetradeoff is as follows Larger values of H cause thedeformation to be spread out in space At the sametime the deformation becomes more diffuse withincreasing tT and TtR (Fig 8) and hence t tR = (tT) times (TtR ) Thus decreasing ttR causes the defor-mation to bemore localized which counters theeffect of increasing H Interestingly the slip ratedoes not vary with H and tR confirming that thisparameter is well constrained by the data (given thatall other parameters are fixed)

The inferred slip rate is however higher thanestimated from stream offsets at Wallace Creek of 33plusmn 3 mmyr (Sieh and Jahns 1984) Constraining theslip rate to 34 mmyr (Fig 10) does not fit the dataas well especially for the stations farthest west ofthe San Andreas The optimal model is H = 114 kmas before but the relaxation time increases to 180years

Relaxation times tR gt 50 years are long com-pared to what has been inferred for the post-1906strain rate transient north of the San Francisco Bayas measured by repeated triangulation surveys(Thatcher 1975a 1975b Kenner and Segall 2000)The relaxation time tR consistent with the post-1906transient can be inferred from the triangulation dataand the spatial derivative of equation (3) It wouldbe most consistent to employ the same Savage-Pres-cott model as used with the Carrizo Plain data how-ever this requires knowing the recurrence time oflarge earthquakes on the north coast segment of theSan Andreas fault An alternate approach employedhere is to remove the estimated secular deformationusing a model (see Kenner and Segall 2001) andthen fit the remaining strain rate data For 60meters of slip in the 1906 earthquake the optimal

smiddot Du Tcurren=

smiddot

smiddot

smiddot

FIG 7 Model of a faulted elastic plate overlying viscoelas-tic half-space The elastic layer has shear modulus micro the vis-coelast ic half-space shear modulus micro and viscosity hEarthquakes occur at a regular interval T such that thefar-field velocity is vyen2

70 PAUL SEGALL

model is H = 19 km with a relaxation time of 17years (Fig 11) substantially less than found fromthe Carrizo Plain interseismic data1

A tentative conclusion of this analysis is that theCarrizo Plain interseismic data require a relaxationtime in excess of 50 years whereas the post-1906triangulation data require relaxation times of 25years or less In order to test whether this discrep-ancy is truly required by the data I estimated theuncertainty in the two relaxation times using a boot-strap resampling procedure As discussed previ-ously the residuals to the best-fitting model wererandomly resampled with replacement and added

to the model predictions for each bootstrap resam-ple This approach seems best for small data setsand ensures that there were data at each observationposition (interseismic data) and time (postseismicdata)

For the post-1906 transient admissible relax-ation times range from 10 years to 30 years (Fig 12)On the other hand the present distribution of defor-mation on the Carrizo Plain demands relaxationtimes in excess of 35 years (Fig 13) Thus we cansafely conclude that there is no single relaxationtime that fits both data sets

There are a number of possible explanations forthis apparent discrepancy One possibility is thatthe model of a fully ruptured elastic plate overlyinga linear Maxwell half-space does not capture all ofthe relevant physics For example nonlinear rheol-ogy may be important The data are consistent withthe expectation that the effective viscosity is low

1 Note that the discrepancy is not due to the correction forinterseismic deformation in the post-1906 data Assuming arecurrence time of 250 years and a slip rate on the SAF of 25mmyr the optimal relaxation time is 228 years with an elas-tic-layer thickness of 167 km

FIG 8 Savage-Prescott model for (A) T tR = 1 and (B) T tR = 5 t is the time since the last earthquake T is theearthquake repeat time and tR is the Maxwell relaxation time In these calculations the earthquakes rupture the entireelastic layer

ESTIMATES OF SLIP RATE 71

early in the seismic cycle when the stresses arelarge and increases late in the cycle when thestresses have largely dissipated In addition recentreflection and refraction results in northern Califor-nia show the San Andreas fault cutting the lowercrust and possibly the Moho (eg Henstock et al1997 Parsons and Hart 1999) Aseismic slip onthe downdip extension of the seismogenic zone (asin Tse and Rice 1986) may take place in additionto a more deeply distributed relation Results fromKenner and Segall (2001) which include viscousrelaxation in a narrow fault zone and more deeplydistributed slip find the spatiotemporal pattern ofdeformation in the post-1906 triangulation data isbest fit with a short relaxation time in the fault zoneand a longer relaxation in the underlyinghalf-space Another possibility is that the thermo-mechanical structure of the lithosphere differsbetween the two areas It is also possible that along-strike variations in fault geometry (the Big Bend) or

slip behavior (the creeping zone) may bias the com-puted relaxation time although this seems lesslikely

Although the apparent relaxation times differbetween the Carrizo Plain and the northern SanFrancisco Bay region the elastic-layer thicknessestimates are consistent in the range of 13ndash20 kmCarrizo Plain slip-rate estimates cluster near theupper bound of 37 mmyr Interestingly the CarrizoPlain data yield recurrence times greater than 160years Whereas the distribution is reasonably broadboth the mean (288 years) and median (277) repeattimes fall between the 1988 working group estimateof 296 years (WGCEP 1988) and the 1995 workinggroup estimate of 206 years (WGCEP 1995) Theaverage coseismic slip for the predicted events isroughly 10 meters There are significant correlationsbetween parameters (between H and tR as dis-cussed previously as well as between tR and T)which means that any additional data that constrain

FIG 9 Comparison of Savage-Prescott model to the SCEC velocity data for the Carrizo Plain A Misfit as a functionof elastic-layer thickness and relaxation time B Best-fitting San Andreas slip rate corresponding to the elastic-layerthickness and relaxation time C Fit to the data Effect of the Hosgri fault is removed employing a dislocation model withslip rate of 3 mmyr and depth of 10 km

72 PAUL SEGALL

some of the parameters will help resolve others aswell

It should be emphasized that the paleoseismicobservations of Grant and Sieh (1994) show an irreg-ular pattern of earthquakes that cannot be explainedby the simple two-dimensional model used here tointerpret the geodetic observations Nevertheless itis interesting that the GPS data appear to have someresolving power for average recurrence interval

San Francisco Bay Data

In order to eliminate any geographic bias in theinterpretation associated with possibly differentcrustal structure and temperature distribution Inext consider the velocity profile across the SanAndreas fault north of the San Francisco Bay asdetermined by repeated GPS observations (Prescottet al 2001) as shown in Figure 14 The GPS sitesare sufficiently close to the post-1906 triangulationnetworks that it is reasonable to assume that the

crustal structure and thermal profile are the samefor both data sets In the following the GPS-derivedvelocities are rotated into fault-parallel and perpen-dicular components Only the fault-parallel motionsare modeled here although as pointed out by Pres-cott et al (2001) and Murray and Segall (2001)there is significant fault-normal motion in the east-ern coast ranges at the western edge of the CentralValley

The model is the same as discussed previouslywith three principal faults the San Andreas Hay-wardndashRodgers Creek and ConcordndashGreen Valleyfaults The parameters are the elastic plate thick-ness H Maxwell relaxation time tR the recurrencetime for great earthquakes on the SAF TSAF the sliprate on the SAF and the recurrence timeTHRC time since past event tHRC and slip rate on the Hayward-Rodgers Creek fault system Thetime since the last great earthquake on the northernSan Andreas fault tSAF is of course known Theeffect of the ConcordndashGreen Valley fault is modeled

smiddotSAFsmiddotHRC

FIG 10 Comparison of Savage-Prescott model to the SCEC velocity data for the Carrizo Plain with slip rate con-strained to 34 mmyr Effect of San GregoriondashHosgri fault has been removed with a dislocation model with slip rate of 3mmyr and depth of 10 km A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

ESTIMATES OF SLIP RATE 73

kinematically with a simple screw dislocationwhich adds two parameters the fault slip rate andlocking depth It is certainly inconsistent to ignoreviscoelastic effects for the ConcordndashGreen Valleyfault but I do so for three reasons (a) much less isknown about this fault (b) earthquakes there arepresumably least well modeled by infinitely longruptures and (c) it keeps the number of parametersreasonably tractable In essence the effect of theConcordndashGreen Valley is removed empirically and Imake no attempt to gain further insights into itsmechanical behavior Future work should addressand hopefully rectify this simplification

A priori bounds on the parameters are listed inTable 1 The lower bound on the lithospheric thick-ness comes from a conservative estimate of thedepth extent of slip in large earthquakes In fact itis more likely that slip in 1906 extended at least to15 km (Matthews and Segall 1993) as it certainlydid in the Loma Prieta earthquake (eg Beroza1991) The bounds on relaxation time come from thepreliminary analysis above (Figs 12 and 13) The

lower bound on TSAF comes from the elapsed timesince 1906 the upper bound from limits on theamount of coseismic slip (see below) The lowerbounds on tHRC and THRC are derived from the his-torical record there is no known earthquake on thenorthern HaywardndashRodgers Creek system since atleast 1775 (Toppozada and Borchardt 1998WGCEP 1999) The lower and upper bounds on sliprates were based loosely on the Northern CaliforniaWorking group (WGCEP 1999 Table 2) The Work-ing Group listed 2 s uncertainties of = 24 plusmn 3mmyr = 9 plusmn 2 mmyr and = 5 plusmn m 3mmyr

The estimation problem is to find the choice ofthe nine parameters in Table 1 that minimize themisfit to two geodetic data sets the present-day spa-tial distribution of deformation in the north SanFrancisco Bay region (Fig 14) and the time-depen-dent decay in strain rate following the 1906 earth-quake (Fig 11) For a non-linear problem such asthis one always needs to be concerned about localminima However preliminary testing with a simu-

smiddotSAFsmiddotHRC smiddotHRC

FIG 11 Post-1906 strain rate from triangulation data in the San Francisco Bay area Model fit to the model withpostseismic deformation only A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

74 PAUL SEGALL

lated annealing algorithm did not show evidence oflocal minima so the remaining analysis was doneusing a Sequential Quadratic Programming methodwith BFGS Hessian updates (eg Gill et al 1981)within Matlab The method allows upper and lowerbounds to be placed on all parameters as well asgeneralized inequality constraints The latter areneeded because tHRC is strictly bounded by tHRC poundTHRC In the context of the Savage and Prescottmodel earthquakes are periodic and it makes nosense for the elapsed time since the last event toexceed the average recurrence time In addition Ihave found it desirable to constrain the maximumcoseismic slip on both the San Andreas and Hay-ward faults This provides constraints of the form

and

Noting that the coseismic slip should be interpretedas an average over the fault in this two-dimensional

model I set meters and

metersIt should be noted that the GPS network does not

extend very far west of the San Andreas fault Forthis reason the data do not adequately constrain the

total motion across the plate boundary A reasonableapproach would be to include data from Pacificplate sites such as Hawaii I chose simply to con-strain the sum of the slip rates on all three faults tobe no greater than 40 mmyr (eg Argus and Gordon2001) This constraint significantly affects the solu-tion estimates without the cumulative slip con-straint tended to put the maximum slip rate allowedon all three faults exceeding the total displacementrate between the Sierra NevadandashGreat Valley blockand the Pacific plate

The optimal solution is found for H ~ 184 km tR~ 345 years TSAF = 280 years ~ 25 mmyrtHRC = 225 years THRC = 4286 years and ~ 7mmyr The ConcordndashGreen Valley fault slips at

~ 8 mmyr below 22 km A surprising resultis that the predicted recurrence time on the SanAndreas is reasonable in the sense of requiring 70-meter slip events every 280 years In comparisonSchwartz et al (1998) suggested that the penulti-mate event on the north coast segment of the SanAndreas occurred in the mid-1600s possibly in theinterval 1632ndash1659 corresponding to a recurrenceinterval of roughly 260 years Also surprising is thatthe time since the last earthquake on the Haywardndash

smiddotSAF TSAFtimes DuSAFmaxpound smiddotHRC THRCtimes DuHRC

maxpound

DuSAFmax

70= DuHRCmax

30=

smiddotSAFsmiddotHRC

smiddotHRC

FIG 12 Bootstrap results for post-1906 data in the San Francisco Bay area A Relaxation time B Elastic-layerthickness (1000 bootstrap resamples)

ESTIMATES OF SLIP RATE 75

Rodgers Creek fault is the minimum allowed 225years A comparison of the model fit to the data setsis shown in Figure 15 Generally speaking the dataare fit quite well with the possible exception of thestrain rate in the first post-1906 epoch The uncer-tainties in the post-1906 strain rates were actuallydecreased by a factor of two to ensure that these dataare fairly weighted compared to the GPS velocitiesDecreasing the uncertainties in the strain rates fur-ther to fit high strain rate in the immediatepost-1906 epoch causes a very significant increasein the misfit to the modern GPS data This is mostlikely a result of the large uncertainties in the trian-gulation data but may also indicate that the physi-cal model is inadequate

How is it that the geodetic data have any capabil-ity of resolving the recurrence interval on the SanAndreas fault Recall that there are three times inthe physical model the time since the last majorearthquake t the asthenospheric relaxation time tRand the recurrence interval T For the San Andreasfault we know tSAF ~93 years a priori and tR is rea-

sonably well resolved by the post-1906 strain ratetransient The estimation is thus free to vary T insuch a way as to optimize the fit to the velocity gra-dients near the San Andreas Constraints on themaximum slip in the earthquake help to furthertighten those constraints

In order to determine how well the data constrainthe model parameters I ran 500 bootstrap resam-ples and optimizations using the best-fitting modelas the starting point of the optimization As beforethe residuals were resampled and added to the pre-dicted data from the optimal model in order toensure that each observation point contributes to thesolution The results of the bootstrap are shown inFigure 16 The posterior distribution for recurrencetime on the San Andreas fault TSAF is quite broadwith a peak near the maximum allowed of (7000mm)(19 mmyr) ~370 years In contrast the poste-rior distribution for recurrence time on the Hay-wardndashRodgers Creek fault THRC is bimodal withprominent peaks near the minimum ~225 years andmaximum allowed value (3000 mm)(7 mmyr)

FIG 13 Bootstrap results for SCEC data along the Carrizo Plain segment of the San Andreas fault A Elastic-layerthickness B Relaxation time Note that 200 years was the upper bound considered C Recurrence time Recurrencetimes in excess of 400 years were not considered as these would lead to excessive coseismic slip D Slip rate

76 PAUL SEGALL

~ 428 years The distributions for tHRC the timesince the most recent HaywardndashRodgers Creekearthquake is strongly concentrated near the mini-

mum allowed range The distribution of slip rates onthe San Andreas and HaywardndashRodgers Creekfaults are bimodal with peaks at 19 mmyr and 25

FIG 14 GPS-derived velocities for the north San Francisco Bay region after Prescott et al (2001) AbbreviationsSAF = San Andreas fault HF = Hayward fault CF = Calaveras fault RGF = Rodgers Creek fault CGVF = ConcordndashGreen Valley fault

TABLE 1 A priori Bounds on Parameters

Parameter Symbol Min Max

Lithospheric thickness km H 10 30

Relaxation time yr tR 10 200

SAF recurrence interval yr TSAF 100 500

Time since last Hayward event yr tHRC 225 690

Hayward recurrence interval yr THRC 225 700

Green valley fault locking depth km d 05 15

SAF slip rate my 0019 0027

Hayward slip rate myr 0007 0013

Green valley slip rate myr 0002 0008

smiddotSAF

smiddotHRC

smiddotCGV

ESTIMATES OF SLIP RATE 77

mmyr (SAF) and 7 mmyr and 13 mmyr (HRC)These results still must be viewed with some cau-tion in that I have not demonstrated that the boot-strap optimizations were immune to local minima

The bootstrap calculations suggest that the con-straint on maximum coseismic slip does play astrong role in the results To explore this further Ivaried the maximum coseismic slip allowed on thetwo faults The results (Table 2) demonstrate that theSan Andreas recurrence interval is not fully con-trolled by the constraint on the maximum coseismicslip In contrast the recurrence interval on the Hay-ward fault is totally controlled by the maximum slipallowed (for a maximum slip of 10 meters the recur-rence interval reaches the upper bound of 700years) Notice also that the time since the last Hay-wardndashRodgers Creek event is always at the mini-mum value allowed (225 years) The fit to the data isslightly better for large infrequent earthquakes

Discussion

What has been learned from the modeling pre-sented here It seems clear that geodetic data

should not be analyzed in a vacuum The resolutionof the data alone is insufficient to address the prob-lems of greatest interest However when combinedwith prior information from geologic studies GPSdata appear to be able to resolve parameters of con-siderable interest such as the average recurrenceinterval between great earthquakes on the SanAndreas fault Using a simple model of a repeatingdislocation rupturing an elastic plate overlying aMaxwell visco-elastic medium I found an averagerecurrence interval of roughly 280 years on thenorth coast segment of the San Andreas faultAlthough the uncertainties in this estimate arelarge the result is in surprisingly good accordancewith recent paleoseismic results (Schwartz et al1998) that suggest an event similar to 1906 in themid 1600s

The results for the HaywardndashRodgers Creek faultare also not unreasonable According to the GPSdata the most recent event here occurred in ~1775regardless of the estimated recurrence time Thisresult is not inconsistent with the paleoseismic find-ing that the most recent event on the Rodgers Creekfault took place in the interval 1670ndash1776

TABLE 2 Influence of Maximum Coseismic Slip on Estimated Parameters1

Parameter

Lithospheric thickness H km 150 184 220

Relaxation time tR yr 357 345 340

SAF recurrence interval TSAF yr 220 280 337

Limiting interval yr 290 368 526

Time since last Hayward event tHRC yr 225 225 225

Hayward recurrence interval THRC yr 2857 4286 700

Limiting interval yr 285 428 714

Green Valley fault locking depth d km 11 22 47

SAF slip rate myr 0025 0025 00249

Hayward slip rate myr 0007 0007 00071

Green valley slip rate myr 0008 0008 0008

1Limiting interval is computed using the estimated slip rate for that fault not the minimum slip rate that would yield a longer upper bound

DuSAFmax

55=

DuHRCmax

20=

DuSAFmax

70=

DuHRCmax

30=

DuSAFmax

100=

DuHRCmax

50=

DuSAFmax

smiddotSAFcurren

DuHRCmax

smiddotHRCcurren

smiddotSAF

smiddotHRC

smiddotCGV

78 PAUL SEGALL

(Schwartz et al 1993) and on the northern Hay-ward fault in 1640ndash1776 (HFPEG 1999) The GPSdata alone yield limited information about the aver-age recurrence time on the HaywardndashRodgers Creekfault However adding the reasonable bound thatslip not exceed 2 meters on average yields a recur-rence interval of 285 years The average slip in the1868 Hayward fault earthquake was approximately2 meters (Yu and Segall 1996) In comparisontrenching at the El Cerrito site on the northern Hay-ward fault suggests a recurrence interval of possiblyless than 270 years but no more than 710 years(HFPEG 1999)

It should not need to be stated that earthquakesare neither periodic nor especially in the case of theHaywardndashRodgers Creek faults infinitely long Atwo-dimensional model cannot describe the situa-tion at a given trench site where ruptures mainly tothe north or south of the site cause some offset in the

trench Nevertheless the fact that the results fromsuch a simple model are not unreasonable is encour-aging One useful attribute of the models exploredhere are that they are testable in the sense of predict-ing future behavior Running the model forward intime to the next San Andreas earthquake yields thevelocity distributions shown in Figure 17 Of coursea north HaywardndashRodgers Creek earthquake mayoccur prior to the next San Andreas event

The Maxwell relaxation time is reasonablywell estimated by the data I find an optimal valueof about 35 years which corresponds to a viscos-ity of 16 times 1019 Pa-s It should be noted that theestimates of all of the model parameters is only asgood as the mechanical model used to interpretthe data If as seems quite likely there was after-slip on the San Andreas below the 1906 rupturezone the estimate of the viscosity is some combi-nation of the effects of afterslip and distributed

FIG 15 Fit of model to north San Francisco Bay data Maximum slip on Hayward fault is 30 m and on the SAF is70 m

ESTIMATES OF SLIP RATE 79

relaxation Li and Rice (1987) presented anapproximate method for including post-seismicfault slip at constant resistive stress with a relax-ing asthenosphere in the context of a repeatingperiodic earthquake sequence Combining suchan approach with appropriate constitutive laws forstable fault creep as in Tse and Rice (1986)would presumably provide a more physically real-istic description of fault behavior Clearly three-dimensional effects are also important especiallyfor earthquakes on the Rodgers Creek and Hay-ward faults Finally the effects of non-uniform

elastic properties are known to bias interpreta-tions based on simple homogeneous models Elas-tic moduli increase with depth which has theknown effect of biasing the depth distribution offault slip Similarly elastic properties may varywith lithology across the major fault zones whichthemselves may represent zones of more compli-ant rock Finite elements are certainly the mostgeneral way to compute forward models but arenot yet feasible for the optimization and bootstrapmethods as computed here which literally testedthousands of possible models

FIG 16 Five hundred bootstrap estimates of model parameters fit to the north San Francisco Bay data 90 confi-dence bounds are listed at the top of each panel The vertical line in the histograms for recurrence times TSAF and THRCshow the upper bounds corresponding to the minimum slip rate and maximum allowed coseismic slip respectively Thehorizontal bar in the panel for the San Andreas recurrence time shows the preferred estimate from Schwartz et al (1998)The horizontal bar in the panel for tH time since the last Hayward earthquake shows the range of dates compatible withpaleoseismic estimates of the penultimate HaywardndashRodgers Creek event

80 PAUL SEGALL

Acknowledgments

George Thompson set a standard that geophysi-cal models no matter how elegant must be consis-tent with geologic observations This work wasinspired by that example I would like to thank WillPrescott for discussions about the Bay Area defor-mation measurements and Mark Matthews for manyyears ago introducing me to Bayesrsquo theorem WayneThatcher Mark Murray Jeff McGuire and PeterCervelli provided valuable comments at early stagesin this study I thank Jim Savage Duncan Agnewand Shelley Kenner for their thoughtful reviewsGary Ernst and Simon Klemperer carefully editedthe manuscript

REFERENCES

Argus D F and Gordon R G 2001 Present tectonicmotion across the Coast Ranges and San Andreas faultsystem in central California Geological Society ofAmerica Bulletin v 113 p 1580ndash1592

Beroza G C 1991 Near-source modeling of the LomaPrieta earthquake Evidence for heterogeneous slipand implications for earthquake hazard Bulletin of theSeismological Society of America v 81 p 1603ndash1621

Cohen S C 1982 A multilayer model of time-dependentdeformation following an earthquake on a strike-slipfault Journal of Geophysical Research v 87 p 5409ndash5421

HFPEG (Hayward Fault Paleoearthquake Group) 1999Timing of paleoearthquakes on the northern HaywardfaultmdashPreliminary evidence in El Cerrito CaliforniaUS Geological Survey Open File Report 99-318

Henstock T J Levander A and Hole J A 1997Deformation in the lower crust of the San Andreas faultsystem in northern California Science v 278 p 650ndash653

Feigl K L King R W and Jordan T H 1990 Geo-detic measurement of tectonic deformation in theSanta Maria fold and thrust belt California Journal ofGeophysical Research v 95 p 2679ndash2699

Gill P E Murray W and Wright M H 1981 Practicaloptimization London UK Academic Press 401 p

Grant L B and Sieh K 1994 Paleoseismic evidence ofclustered earthquakes on the San Andreas fault in the

FIG 17 Simulation of Bay Area velocity field as a function of time Each line is separated by roughly 27 years

66 PAUL SEGALL

It is important to determine the range of sliprates and locking depths that are consistent withthe GPS data A bootstrap resampling procedureshows that slip rates in the range of 34 to 41 mmyrare consistent with the geodetic observations (Fig4) The bootstrap was conducted in the followingmanner Let d be the vector of observations thepredicted data from the best-fitting model so that

is the vector of residuals The residualswere randomly resampled with replacement that issome elements may be chosen multiple times oth-ers not at all Denote the resampled residuals r Aresampled data vector is then usedto generate a model estimate and the process isrepeated many times The distribution of the boot-strapped model parameters is that shown in Figure4

It is well known that there is a tradeoff betweenthe locking depth and deep slip rate Increasing thelocking depth moves the dislocation farther from thesurface observation stations to achieve the samesurface strain requires a higher slip rate Applying apriori constraints on the slip rate from 31 mmyrand 37 mmyr based on the Sieh and Jahns (1984)data from Wallace Creek thus not only changes the

posterior distribution of slip rate but of the lockingdepth as well (Fig 5) The reason for this can beclearly seen from a scatter plot of the bootstrap res-amples as in Figure 6

Notice that the GPS data have significantly nar-rowed the distribution of admissible slip rates (Fig5) The posterior distribution is peaked at 35 to 37mmyr whereas the a priori distribution is uniformbetween 31 and 37 mmyr At the same time the apriori geologic information provided no constrainton the fault locking depth With geologic data alonewe cannot distinguish between a locked fault and afault that creeps steadily below some shallow lockedinterval The GPS data alone bound the lockingdepth to between 17 and 24 km Because of thestrong correlation between slip rate and lockingdepth the posterior distribution of the locking depthis narrowed to 17ndash20 km This simple exampleshows the power of using a priori geologic con-straints in allowing more information to be obtainedfrom geodetic data

Note that the bootstrap provides an estimate ofthe distribution of the model parameters whereasBayesrsquo rule requires P(d|m) as in equation (3)However for linear models with normally distrib-

d

r d dndash=

d d r+=

FIG 3 A Misfit as a function of depth for a simple screw dislocation model B Profile of fault parallel velocity acrossthe fault compared with prediction of a simple screw dislocation model C As in (B) but with 3 mmyr of slip beneaththe Hosgri fault

ESTIMATES OF SLIP RATE 67

uted observational errors the two approaches areequivalent Furthermore a uniform distribution withhard bounds may not be the best characterization ofthe prior probability distribution on slip rate Alter-

natively if the prior distribution and observationalerrors are normally distributed one can simplyapply the prior information as ldquopseudo datardquo (egJackson 1979)

FIG 4 Probability distributions for fault locking depth and slip rate from bootstrap results using GPS data I assume3 mmyr below a locking depth of 10 km on the Hosgri fault system

FIG 5 Posterior probability distribution from the bootstrap results using GPS data and assuming a uniform priordis-tribution on slip rate from 31 to 37 mmyr Compare with Figure 4 As before I assume 3 mmyr below a locking depthof 10 km on the Hosgri fault system

68 PAUL SEGALL

Viscoelastic Models

As noted in the Introduction the screw disloca-tion provides only a very limited description ofplate-boundary faulting A somewhat more realisticmodel involves an elastic layer of thickness H over-lying a Maxwell viscoelastic half-space (Fig 7) TheMaxwell material has relaxation time tR = 2hmicrowhere h is the viscosity and micro is the shear modulusAt time t = 0 slip Du occurs on the fault from thesurface to depth D pound H The velocity on the Earthrsquossurface as a function of position perpendicular to thefault x and time t is

(3)

where the spatial distribution is given by

(4)

The result follows from but is not stated in Nur andMavko (1974) The strain rate is simply obtained bydifferentiating the velocity with respect to x Noticethat the post-seismic velocity is a function of thecoseismic slip Du the depth of faulting D the elas-tic layer thickness H the material relaxation tR andthe time since the last large earthquake t Equation(3) does not include the effects of interseismic strainaccumulation Savage and Prescott (1978) extendedthe model by summing over an infinite sequence ofplate rupturing events (D = H) equally spaced intime with recurrence interval T The velocity field asa function of distance across strike (x) and time (t) isgiven by

(5)

where the temporal dependence depends only on thedimensionless ratios ttR and TtR

(6)

v x t( )

DuptR---------e

t tRcurrenndash t tRcurren( )n 1ndash

n 1ndash( )--------------------------Fn x( D H )

n 1=

yen

aring

=

Fn x D H ( )

D 2nH+x

---------------------egrave oslashaelig ouml1ndash

tan D 2nHndashx

--------------------egrave oslashaelig ouml1ndash

tan+

2xD

x2

2nH( )2D

2ndash+

------------------------------------------1ndash tan

=

=

v x t( ) smiddot

p--- Tn t tR T tRcurrencurren( )Fn x H H ( )

n 1=

yen

aring=

Tn t tR T tRcurrencurren( )

TtR-----

et tRcurrenndash

n 1ndash( )------------------ e

kT tRcurrenndash t kT+tR

--------------egrave oslashaelig ouml n 1ndash

k 0=

yen

aring

=

FIG 6 Tradeoff between slip rate and locking depth Each point is the result of a bootstrap resample of the SCECvelocity data The heavy points are those consistent with a uniform prior distribution on slip rate from 31 to 37 mmyrAs before I assume 3 mmyr below a locking depth of 10 km on the Hosgri fault system

ESTIMATES OF SLIP RATE 69

Because the far-field lithospheric velocity is drivensolely by the repeating earthquakes Representative examples are given in Figure 8 forvarious times through the earthquake cycle Noticethat when the asthenospheric relaxation time is ofthe order of the earthquake cycle time or longer TtR pound1 the surface velocity distribution is rathersteady with time (Fig 8A) On the other hand if therelaxation time is short compared to the recurrenceinterval TtR gt 1 there is a significant transientwith higher than average strain rates early in thecycle and lower than average strain rates late in thecycle

The Savage-Prescott model involves five param-eters lithospheric thickness H relaxation time tRtime since the last earthquake t recurrence time Tand long-term slip rate on the fault Because thetheory is linear it is straightforward to constructmodels with multiple parallel faults however in thiscase the number of parameters is even larger Forthree faults there are 11 parameters if one assumesthat the elastic thickness and relaxation time are thesame for all faults Note that the model is kinematicwith imposed slip events on the various faults at reg-ular intervals and thus does not account for thestress interactions between the faults

As a starting point I consider the Carrizo Plaindata with some of the parameters constrained byindependent observations The time since the lastgreat earthquake (1857) is t = 136 years To beginwith I assume a recurrence time of T = 160 yearsbased on the average interval found by Grant andSieh (1994) There are three remaining parametersto be estimated from the data tR H and Theoptimal estimates of the non-linear parameters tRand H are found by simple grid search (Fig 9) with

estimated by linear least squares The optimalmodel is found for H = 114 km and tR = 80 yearswith a slip rate of 402 mmyr There is a strongtradeoff between relaxation time and elastic-layerthickness such that models with elastic-layer thick-ness of up to 20 km and relaxation time of 180 yearsfit the data nearly equally well The reason for thetradeoff is as follows Larger values of H cause thedeformation to be spread out in space At the sametime the deformation becomes more diffuse withincreasing tT and TtR (Fig 8) and hence t tR = (tT) times (TtR ) Thus decreasing ttR causes the defor-mation to bemore localized which counters theeffect of increasing H Interestingly the slip ratedoes not vary with H and tR confirming that thisparameter is well constrained by the data (given thatall other parameters are fixed)

The inferred slip rate is however higher thanestimated from stream offsets at Wallace Creek of 33plusmn 3 mmyr (Sieh and Jahns 1984) Constraining theslip rate to 34 mmyr (Fig 10) does not fit the dataas well especially for the stations farthest west ofthe San Andreas The optimal model is H = 114 kmas before but the relaxation time increases to 180years

Relaxation times tR gt 50 years are long com-pared to what has been inferred for the post-1906strain rate transient north of the San Francisco Bayas measured by repeated triangulation surveys(Thatcher 1975a 1975b Kenner and Segall 2000)The relaxation time tR consistent with the post-1906transient can be inferred from the triangulation dataand the spatial derivative of equation (3) It wouldbe most consistent to employ the same Savage-Pres-cott model as used with the Carrizo Plain data how-ever this requires knowing the recurrence time oflarge earthquakes on the north coast segment of theSan Andreas fault An alternate approach employedhere is to remove the estimated secular deformationusing a model (see Kenner and Segall 2001) andthen fit the remaining strain rate data For 60meters of slip in the 1906 earthquake the optimal

smiddot Du Tcurren=

smiddot

smiddot

smiddot

FIG 7 Model of a faulted elastic plate overlying viscoelas-tic half-space The elastic layer has shear modulus micro the vis-coelast ic half-space shear modulus micro and viscosity hEarthquakes occur at a regular interval T such that thefar-field velocity is vyen2

70 PAUL SEGALL

model is H = 19 km with a relaxation time of 17years (Fig 11) substantially less than found fromthe Carrizo Plain interseismic data1

A tentative conclusion of this analysis is that theCarrizo Plain interseismic data require a relaxationtime in excess of 50 years whereas the post-1906triangulation data require relaxation times of 25years or less In order to test whether this discrep-ancy is truly required by the data I estimated theuncertainty in the two relaxation times using a boot-strap resampling procedure As discussed previ-ously the residuals to the best-fitting model wererandomly resampled with replacement and added

to the model predictions for each bootstrap resam-ple This approach seems best for small data setsand ensures that there were data at each observationposition (interseismic data) and time (postseismicdata)

For the post-1906 transient admissible relax-ation times range from 10 years to 30 years (Fig 12)On the other hand the present distribution of defor-mation on the Carrizo Plain demands relaxationtimes in excess of 35 years (Fig 13) Thus we cansafely conclude that there is no single relaxationtime that fits both data sets

There are a number of possible explanations forthis apparent discrepancy One possibility is thatthe model of a fully ruptured elastic plate overlyinga linear Maxwell half-space does not capture all ofthe relevant physics For example nonlinear rheol-ogy may be important The data are consistent withthe expectation that the effective viscosity is low

1 Note that the discrepancy is not due to the correction forinterseismic deformation in the post-1906 data Assuming arecurrence time of 250 years and a slip rate on the SAF of 25mmyr the optimal relaxation time is 228 years with an elas-tic-layer thickness of 167 km

FIG 8 Savage-Prescott model for (A) T tR = 1 and (B) T tR = 5 t is the time since the last earthquake T is theearthquake repeat time and tR is the Maxwell relaxation time In these calculations the earthquakes rupture the entireelastic layer

ESTIMATES OF SLIP RATE 71

early in the seismic cycle when the stresses arelarge and increases late in the cycle when thestresses have largely dissipated In addition recentreflection and refraction results in northern Califor-nia show the San Andreas fault cutting the lowercrust and possibly the Moho (eg Henstock et al1997 Parsons and Hart 1999) Aseismic slip onthe downdip extension of the seismogenic zone (asin Tse and Rice 1986) may take place in additionto a more deeply distributed relation Results fromKenner and Segall (2001) which include viscousrelaxation in a narrow fault zone and more deeplydistributed slip find the spatiotemporal pattern ofdeformation in the post-1906 triangulation data isbest fit with a short relaxation time in the fault zoneand a longer relaxation in the underlyinghalf-space Another possibility is that the thermo-mechanical structure of the lithosphere differsbetween the two areas It is also possible that along-strike variations in fault geometry (the Big Bend) or

slip behavior (the creeping zone) may bias the com-puted relaxation time although this seems lesslikely

Although the apparent relaxation times differbetween the Carrizo Plain and the northern SanFrancisco Bay region the elastic-layer thicknessestimates are consistent in the range of 13ndash20 kmCarrizo Plain slip-rate estimates cluster near theupper bound of 37 mmyr Interestingly the CarrizoPlain data yield recurrence times greater than 160years Whereas the distribution is reasonably broadboth the mean (288 years) and median (277) repeattimes fall between the 1988 working group estimateof 296 years (WGCEP 1988) and the 1995 workinggroup estimate of 206 years (WGCEP 1995) Theaverage coseismic slip for the predicted events isroughly 10 meters There are significant correlationsbetween parameters (between H and tR as dis-cussed previously as well as between tR and T)which means that any additional data that constrain

FIG 9 Comparison of Savage-Prescott model to the SCEC velocity data for the Carrizo Plain A Misfit as a functionof elastic-layer thickness and relaxation time B Best-fitting San Andreas slip rate corresponding to the elastic-layerthickness and relaxation time C Fit to the data Effect of the Hosgri fault is removed employing a dislocation model withslip rate of 3 mmyr and depth of 10 km

72 PAUL SEGALL

some of the parameters will help resolve others aswell

It should be emphasized that the paleoseismicobservations of Grant and Sieh (1994) show an irreg-ular pattern of earthquakes that cannot be explainedby the simple two-dimensional model used here tointerpret the geodetic observations Nevertheless itis interesting that the GPS data appear to have someresolving power for average recurrence interval

San Francisco Bay Data

In order to eliminate any geographic bias in theinterpretation associated with possibly differentcrustal structure and temperature distribution Inext consider the velocity profile across the SanAndreas fault north of the San Francisco Bay asdetermined by repeated GPS observations (Prescottet al 2001) as shown in Figure 14 The GPS sitesare sufficiently close to the post-1906 triangulationnetworks that it is reasonable to assume that the

crustal structure and thermal profile are the samefor both data sets In the following the GPS-derivedvelocities are rotated into fault-parallel and perpen-dicular components Only the fault-parallel motionsare modeled here although as pointed out by Pres-cott et al (2001) and Murray and Segall (2001)there is significant fault-normal motion in the east-ern coast ranges at the western edge of the CentralValley

The model is the same as discussed previouslywith three principal faults the San Andreas Hay-wardndashRodgers Creek and ConcordndashGreen Valleyfaults The parameters are the elastic plate thick-ness H Maxwell relaxation time tR the recurrencetime for great earthquakes on the SAF TSAF the sliprate on the SAF and the recurrence timeTHRC time since past event tHRC and slip rate on the Hayward-Rodgers Creek fault system Thetime since the last great earthquake on the northernSan Andreas fault tSAF is of course known Theeffect of the ConcordndashGreen Valley fault is modeled

smiddotSAFsmiddotHRC

FIG 10 Comparison of Savage-Prescott model to the SCEC velocity data for the Carrizo Plain with slip rate con-strained to 34 mmyr Effect of San GregoriondashHosgri fault has been removed with a dislocation model with slip rate of 3mmyr and depth of 10 km A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

ESTIMATES OF SLIP RATE 73

kinematically with a simple screw dislocationwhich adds two parameters the fault slip rate andlocking depth It is certainly inconsistent to ignoreviscoelastic effects for the ConcordndashGreen Valleyfault but I do so for three reasons (a) much less isknown about this fault (b) earthquakes there arepresumably least well modeled by infinitely longruptures and (c) it keeps the number of parametersreasonably tractable In essence the effect of theConcordndashGreen Valley is removed empirically and Imake no attempt to gain further insights into itsmechanical behavior Future work should addressand hopefully rectify this simplification

A priori bounds on the parameters are listed inTable 1 The lower bound on the lithospheric thick-ness comes from a conservative estimate of thedepth extent of slip in large earthquakes In fact itis more likely that slip in 1906 extended at least to15 km (Matthews and Segall 1993) as it certainlydid in the Loma Prieta earthquake (eg Beroza1991) The bounds on relaxation time come from thepreliminary analysis above (Figs 12 and 13) The

lower bound on TSAF comes from the elapsed timesince 1906 the upper bound from limits on theamount of coseismic slip (see below) The lowerbounds on tHRC and THRC are derived from the his-torical record there is no known earthquake on thenorthern HaywardndashRodgers Creek system since atleast 1775 (Toppozada and Borchardt 1998WGCEP 1999) The lower and upper bounds on sliprates were based loosely on the Northern CaliforniaWorking group (WGCEP 1999 Table 2) The Work-ing Group listed 2 s uncertainties of = 24 plusmn 3mmyr = 9 plusmn 2 mmyr and = 5 plusmn m 3mmyr

The estimation problem is to find the choice ofthe nine parameters in Table 1 that minimize themisfit to two geodetic data sets the present-day spa-tial distribution of deformation in the north SanFrancisco Bay region (Fig 14) and the time-depen-dent decay in strain rate following the 1906 earth-quake (Fig 11) For a non-linear problem such asthis one always needs to be concerned about localminima However preliminary testing with a simu-

smiddotSAFsmiddotHRC smiddotHRC

FIG 11 Post-1906 strain rate from triangulation data in the San Francisco Bay area Model fit to the model withpostseismic deformation only A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

74 PAUL SEGALL

lated annealing algorithm did not show evidence oflocal minima so the remaining analysis was doneusing a Sequential Quadratic Programming methodwith BFGS Hessian updates (eg Gill et al 1981)within Matlab The method allows upper and lowerbounds to be placed on all parameters as well asgeneralized inequality constraints The latter areneeded because tHRC is strictly bounded by tHRC poundTHRC In the context of the Savage and Prescottmodel earthquakes are periodic and it makes nosense for the elapsed time since the last event toexceed the average recurrence time In addition Ihave found it desirable to constrain the maximumcoseismic slip on both the San Andreas and Hay-ward faults This provides constraints of the form

and

Noting that the coseismic slip should be interpretedas an average over the fault in this two-dimensional

model I set meters and

metersIt should be noted that the GPS network does not

extend very far west of the San Andreas fault Forthis reason the data do not adequately constrain the

total motion across the plate boundary A reasonableapproach would be to include data from Pacificplate sites such as Hawaii I chose simply to con-strain the sum of the slip rates on all three faults tobe no greater than 40 mmyr (eg Argus and Gordon2001) This constraint significantly affects the solu-tion estimates without the cumulative slip con-straint tended to put the maximum slip rate allowedon all three faults exceeding the total displacementrate between the Sierra NevadandashGreat Valley blockand the Pacific plate

The optimal solution is found for H ~ 184 km tR~ 345 years TSAF = 280 years ~ 25 mmyrtHRC = 225 years THRC = 4286 years and ~ 7mmyr The ConcordndashGreen Valley fault slips at

~ 8 mmyr below 22 km A surprising resultis that the predicted recurrence time on the SanAndreas is reasonable in the sense of requiring 70-meter slip events every 280 years In comparisonSchwartz et al (1998) suggested that the penulti-mate event on the north coast segment of the SanAndreas occurred in the mid-1600s possibly in theinterval 1632ndash1659 corresponding to a recurrenceinterval of roughly 260 years Also surprising is thatthe time since the last earthquake on the Haywardndash

smiddotSAF TSAFtimes DuSAFmaxpound smiddotHRC THRCtimes DuHRC

maxpound

DuSAFmax

70= DuHRCmax

30=

smiddotSAFsmiddotHRC

smiddotHRC

FIG 12 Bootstrap results for post-1906 data in the San Francisco Bay area A Relaxation time B Elastic-layerthickness (1000 bootstrap resamples)

ESTIMATES OF SLIP RATE 75

Rodgers Creek fault is the minimum allowed 225years A comparison of the model fit to the data setsis shown in Figure 15 Generally speaking the dataare fit quite well with the possible exception of thestrain rate in the first post-1906 epoch The uncer-tainties in the post-1906 strain rates were actuallydecreased by a factor of two to ensure that these dataare fairly weighted compared to the GPS velocitiesDecreasing the uncertainties in the strain rates fur-ther to fit high strain rate in the immediatepost-1906 epoch causes a very significant increasein the misfit to the modern GPS data This is mostlikely a result of the large uncertainties in the trian-gulation data but may also indicate that the physi-cal model is inadequate

How is it that the geodetic data have any capabil-ity of resolving the recurrence interval on the SanAndreas fault Recall that there are three times inthe physical model the time since the last majorearthquake t the asthenospheric relaxation time tRand the recurrence interval T For the San Andreasfault we know tSAF ~93 years a priori and tR is rea-

sonably well resolved by the post-1906 strain ratetransient The estimation is thus free to vary T insuch a way as to optimize the fit to the velocity gra-dients near the San Andreas Constraints on themaximum slip in the earthquake help to furthertighten those constraints

In order to determine how well the data constrainthe model parameters I ran 500 bootstrap resam-ples and optimizations using the best-fitting modelas the starting point of the optimization As beforethe residuals were resampled and added to the pre-dicted data from the optimal model in order toensure that each observation point contributes to thesolution The results of the bootstrap are shown inFigure 16 The posterior distribution for recurrencetime on the San Andreas fault TSAF is quite broadwith a peak near the maximum allowed of (7000mm)(19 mmyr) ~370 years In contrast the poste-rior distribution for recurrence time on the Hay-wardndashRodgers Creek fault THRC is bimodal withprominent peaks near the minimum ~225 years andmaximum allowed value (3000 mm)(7 mmyr)

FIG 13 Bootstrap results for SCEC data along the Carrizo Plain segment of the San Andreas fault A Elastic-layerthickness B Relaxation time Note that 200 years was the upper bound considered C Recurrence time Recurrencetimes in excess of 400 years were not considered as these would lead to excessive coseismic slip D Slip rate

76 PAUL SEGALL

~ 428 years The distributions for tHRC the timesince the most recent HaywardndashRodgers Creekearthquake is strongly concentrated near the mini-

mum allowed range The distribution of slip rates onthe San Andreas and HaywardndashRodgers Creekfaults are bimodal with peaks at 19 mmyr and 25

FIG 14 GPS-derived velocities for the north San Francisco Bay region after Prescott et al (2001) AbbreviationsSAF = San Andreas fault HF = Hayward fault CF = Calaveras fault RGF = Rodgers Creek fault CGVF = ConcordndashGreen Valley fault

TABLE 1 A priori Bounds on Parameters

Parameter Symbol Min Max

Lithospheric thickness km H 10 30

Relaxation time yr tR 10 200

SAF recurrence interval yr TSAF 100 500

Time since last Hayward event yr tHRC 225 690

Hayward recurrence interval yr THRC 225 700

Green valley fault locking depth km d 05 15

SAF slip rate my 0019 0027

Hayward slip rate myr 0007 0013

Green valley slip rate myr 0002 0008

smiddotSAF

smiddotHRC

smiddotCGV

ESTIMATES OF SLIP RATE 77

mmyr (SAF) and 7 mmyr and 13 mmyr (HRC)These results still must be viewed with some cau-tion in that I have not demonstrated that the boot-strap optimizations were immune to local minima

The bootstrap calculations suggest that the con-straint on maximum coseismic slip does play astrong role in the results To explore this further Ivaried the maximum coseismic slip allowed on thetwo faults The results (Table 2) demonstrate that theSan Andreas recurrence interval is not fully con-trolled by the constraint on the maximum coseismicslip In contrast the recurrence interval on the Hay-ward fault is totally controlled by the maximum slipallowed (for a maximum slip of 10 meters the recur-rence interval reaches the upper bound of 700years) Notice also that the time since the last Hay-wardndashRodgers Creek event is always at the mini-mum value allowed (225 years) The fit to the data isslightly better for large infrequent earthquakes

Discussion

What has been learned from the modeling pre-sented here It seems clear that geodetic data

should not be analyzed in a vacuum The resolutionof the data alone is insufficient to address the prob-lems of greatest interest However when combinedwith prior information from geologic studies GPSdata appear to be able to resolve parameters of con-siderable interest such as the average recurrenceinterval between great earthquakes on the SanAndreas fault Using a simple model of a repeatingdislocation rupturing an elastic plate overlying aMaxwell visco-elastic medium I found an averagerecurrence interval of roughly 280 years on thenorth coast segment of the San Andreas faultAlthough the uncertainties in this estimate arelarge the result is in surprisingly good accordancewith recent paleoseismic results (Schwartz et al1998) that suggest an event similar to 1906 in themid 1600s

The results for the HaywardndashRodgers Creek faultare also not unreasonable According to the GPSdata the most recent event here occurred in ~1775regardless of the estimated recurrence time Thisresult is not inconsistent with the paleoseismic find-ing that the most recent event on the Rodgers Creekfault took place in the interval 1670ndash1776

TABLE 2 Influence of Maximum Coseismic Slip on Estimated Parameters1

Parameter

Lithospheric thickness H km 150 184 220

Relaxation time tR yr 357 345 340

SAF recurrence interval TSAF yr 220 280 337

Limiting interval yr 290 368 526

Time since last Hayward event tHRC yr 225 225 225

Hayward recurrence interval THRC yr 2857 4286 700

Limiting interval yr 285 428 714

Green Valley fault locking depth d km 11 22 47

SAF slip rate myr 0025 0025 00249

Hayward slip rate myr 0007 0007 00071

Green valley slip rate myr 0008 0008 0008

1Limiting interval is computed using the estimated slip rate for that fault not the minimum slip rate that would yield a longer upper bound

DuSAFmax

55=

DuHRCmax

20=

DuSAFmax

70=

DuHRCmax

30=

DuSAFmax

100=

DuHRCmax

50=

DuSAFmax

smiddotSAFcurren

DuHRCmax

smiddotHRCcurren

smiddotSAF

smiddotHRC

smiddotCGV

78 PAUL SEGALL

(Schwartz et al 1993) and on the northern Hay-ward fault in 1640ndash1776 (HFPEG 1999) The GPSdata alone yield limited information about the aver-age recurrence time on the HaywardndashRodgers Creekfault However adding the reasonable bound thatslip not exceed 2 meters on average yields a recur-rence interval of 285 years The average slip in the1868 Hayward fault earthquake was approximately2 meters (Yu and Segall 1996) In comparisontrenching at the El Cerrito site on the northern Hay-ward fault suggests a recurrence interval of possiblyless than 270 years but no more than 710 years(HFPEG 1999)

It should not need to be stated that earthquakesare neither periodic nor especially in the case of theHaywardndashRodgers Creek faults infinitely long Atwo-dimensional model cannot describe the situa-tion at a given trench site where ruptures mainly tothe north or south of the site cause some offset in the

trench Nevertheless the fact that the results fromsuch a simple model are not unreasonable is encour-aging One useful attribute of the models exploredhere are that they are testable in the sense of predict-ing future behavior Running the model forward intime to the next San Andreas earthquake yields thevelocity distributions shown in Figure 17 Of coursea north HaywardndashRodgers Creek earthquake mayoccur prior to the next San Andreas event

The Maxwell relaxation time is reasonablywell estimated by the data I find an optimal valueof about 35 years which corresponds to a viscos-ity of 16 times 1019 Pa-s It should be noted that theestimates of all of the model parameters is only asgood as the mechanical model used to interpretthe data If as seems quite likely there was after-slip on the San Andreas below the 1906 rupturezone the estimate of the viscosity is some combi-nation of the effects of afterslip and distributed

FIG 15 Fit of model to north San Francisco Bay data Maximum slip on Hayward fault is 30 m and on the SAF is70 m

ESTIMATES OF SLIP RATE 79

relaxation Li and Rice (1987) presented anapproximate method for including post-seismicfault slip at constant resistive stress with a relax-ing asthenosphere in the context of a repeatingperiodic earthquake sequence Combining suchan approach with appropriate constitutive laws forstable fault creep as in Tse and Rice (1986)would presumably provide a more physically real-istic description of fault behavior Clearly three-dimensional effects are also important especiallyfor earthquakes on the Rodgers Creek and Hay-ward faults Finally the effects of non-uniform

elastic properties are known to bias interpreta-tions based on simple homogeneous models Elas-tic moduli increase with depth which has theknown effect of biasing the depth distribution offault slip Similarly elastic properties may varywith lithology across the major fault zones whichthemselves may represent zones of more compli-ant rock Finite elements are certainly the mostgeneral way to compute forward models but arenot yet feasible for the optimization and bootstrapmethods as computed here which literally testedthousands of possible models

FIG 16 Five hundred bootstrap estimates of model parameters fit to the north San Francisco Bay data 90 confi-dence bounds are listed at the top of each panel The vertical line in the histograms for recurrence times TSAF and THRCshow the upper bounds corresponding to the minimum slip rate and maximum allowed coseismic slip respectively Thehorizontal bar in the panel for the San Andreas recurrence time shows the preferred estimate from Schwartz et al (1998)The horizontal bar in the panel for tH time since the last Hayward earthquake shows the range of dates compatible withpaleoseismic estimates of the penultimate HaywardndashRodgers Creek event

80 PAUL SEGALL

Acknowledgments

George Thompson set a standard that geophysi-cal models no matter how elegant must be consis-tent with geologic observations This work wasinspired by that example I would like to thank WillPrescott for discussions about the Bay Area defor-mation measurements and Mark Matthews for manyyears ago introducing me to Bayesrsquo theorem WayneThatcher Mark Murray Jeff McGuire and PeterCervelli provided valuable comments at early stagesin this study I thank Jim Savage Duncan Agnewand Shelley Kenner for their thoughtful reviewsGary Ernst and Simon Klemperer carefully editedthe manuscript

REFERENCES

Argus D F and Gordon R G 2001 Present tectonicmotion across the Coast Ranges and San Andreas faultsystem in central California Geological Society ofAmerica Bulletin v 113 p 1580ndash1592

Beroza G C 1991 Near-source modeling of the LomaPrieta earthquake Evidence for heterogeneous slipand implications for earthquake hazard Bulletin of theSeismological Society of America v 81 p 1603ndash1621

Cohen S C 1982 A multilayer model of time-dependentdeformation following an earthquake on a strike-slipfault Journal of Geophysical Research v 87 p 5409ndash5421

HFPEG (Hayward Fault Paleoearthquake Group) 1999Timing of paleoearthquakes on the northern HaywardfaultmdashPreliminary evidence in El Cerrito CaliforniaUS Geological Survey Open File Report 99-318

Henstock T J Levander A and Hole J A 1997Deformation in the lower crust of the San Andreas faultsystem in northern California Science v 278 p 650ndash653

Feigl K L King R W and Jordan T H 1990 Geo-detic measurement of tectonic deformation in theSanta Maria fold and thrust belt California Journal ofGeophysical Research v 95 p 2679ndash2699

Gill P E Murray W and Wright M H 1981 Practicaloptimization London UK Academic Press 401 p

Grant L B and Sieh K 1994 Paleoseismic evidence ofclustered earthquakes on the San Andreas fault in the

FIG 17 Simulation of Bay Area velocity field as a function of time Each line is separated by roughly 27 years

ESTIMATES OF SLIP RATE 67

uted observational errors the two approaches areequivalent Furthermore a uniform distribution withhard bounds may not be the best characterization ofthe prior probability distribution on slip rate Alter-

natively if the prior distribution and observationalerrors are normally distributed one can simplyapply the prior information as ldquopseudo datardquo (egJackson 1979)

FIG 4 Probability distributions for fault locking depth and slip rate from bootstrap results using GPS data I assume3 mmyr below a locking depth of 10 km on the Hosgri fault system

FIG 5 Posterior probability distribution from the bootstrap results using GPS data and assuming a uniform priordis-tribution on slip rate from 31 to 37 mmyr Compare with Figure 4 As before I assume 3 mmyr below a locking depthof 10 km on the Hosgri fault system

68 PAUL SEGALL

Viscoelastic Models

As noted in the Introduction the screw disloca-tion provides only a very limited description ofplate-boundary faulting A somewhat more realisticmodel involves an elastic layer of thickness H over-lying a Maxwell viscoelastic half-space (Fig 7) TheMaxwell material has relaxation time tR = 2hmicrowhere h is the viscosity and micro is the shear modulusAt time t = 0 slip Du occurs on the fault from thesurface to depth D pound H The velocity on the Earthrsquossurface as a function of position perpendicular to thefault x and time t is

(3)

where the spatial distribution is given by

(4)

The result follows from but is not stated in Nur andMavko (1974) The strain rate is simply obtained bydifferentiating the velocity with respect to x Noticethat the post-seismic velocity is a function of thecoseismic slip Du the depth of faulting D the elas-tic layer thickness H the material relaxation tR andthe time since the last large earthquake t Equation(3) does not include the effects of interseismic strainaccumulation Savage and Prescott (1978) extendedthe model by summing over an infinite sequence ofplate rupturing events (D = H) equally spaced intime with recurrence interval T The velocity field asa function of distance across strike (x) and time (t) isgiven by

(5)

where the temporal dependence depends only on thedimensionless ratios ttR and TtR

(6)

v x t( )

DuptR---------e

t tRcurrenndash t tRcurren( )n 1ndash

n 1ndash( )--------------------------Fn x( D H )

n 1=

yen

aring

=

Fn x D H ( )

D 2nH+x

---------------------egrave oslashaelig ouml1ndash

tan D 2nHndashx

--------------------egrave oslashaelig ouml1ndash

tan+

2xD

x2

2nH( )2D

2ndash+

------------------------------------------1ndash tan

=

=

v x t( ) smiddot

p--- Tn t tR T tRcurrencurren( )Fn x H H ( )

n 1=

yen

aring=

Tn t tR T tRcurrencurren( )

TtR-----

et tRcurrenndash

n 1ndash( )------------------ e

kT tRcurrenndash t kT+tR

--------------egrave oslashaelig ouml n 1ndash

k 0=

yen

aring

=

FIG 6 Tradeoff between slip rate and locking depth Each point is the result of a bootstrap resample of the SCECvelocity data The heavy points are those consistent with a uniform prior distribution on slip rate from 31 to 37 mmyrAs before I assume 3 mmyr below a locking depth of 10 km on the Hosgri fault system

ESTIMATES OF SLIP RATE 69

Because the far-field lithospheric velocity is drivensolely by the repeating earthquakes Representative examples are given in Figure 8 forvarious times through the earthquake cycle Noticethat when the asthenospheric relaxation time is ofthe order of the earthquake cycle time or longer TtR pound1 the surface velocity distribution is rathersteady with time (Fig 8A) On the other hand if therelaxation time is short compared to the recurrenceinterval TtR gt 1 there is a significant transientwith higher than average strain rates early in thecycle and lower than average strain rates late in thecycle

The Savage-Prescott model involves five param-eters lithospheric thickness H relaxation time tRtime since the last earthquake t recurrence time Tand long-term slip rate on the fault Because thetheory is linear it is straightforward to constructmodels with multiple parallel faults however in thiscase the number of parameters is even larger Forthree faults there are 11 parameters if one assumesthat the elastic thickness and relaxation time are thesame for all faults Note that the model is kinematicwith imposed slip events on the various faults at reg-ular intervals and thus does not account for thestress interactions between the faults

As a starting point I consider the Carrizo Plaindata with some of the parameters constrained byindependent observations The time since the lastgreat earthquake (1857) is t = 136 years To beginwith I assume a recurrence time of T = 160 yearsbased on the average interval found by Grant andSieh (1994) There are three remaining parametersto be estimated from the data tR H and Theoptimal estimates of the non-linear parameters tRand H are found by simple grid search (Fig 9) with

estimated by linear least squares The optimalmodel is found for H = 114 km and tR = 80 yearswith a slip rate of 402 mmyr There is a strongtradeoff between relaxation time and elastic-layerthickness such that models with elastic-layer thick-ness of up to 20 km and relaxation time of 180 yearsfit the data nearly equally well The reason for thetradeoff is as follows Larger values of H cause thedeformation to be spread out in space At the sametime the deformation becomes more diffuse withincreasing tT and TtR (Fig 8) and hence t tR = (tT) times (TtR ) Thus decreasing ttR causes the defor-mation to bemore localized which counters theeffect of increasing H Interestingly the slip ratedoes not vary with H and tR confirming that thisparameter is well constrained by the data (given thatall other parameters are fixed)

The inferred slip rate is however higher thanestimated from stream offsets at Wallace Creek of 33plusmn 3 mmyr (Sieh and Jahns 1984) Constraining theslip rate to 34 mmyr (Fig 10) does not fit the dataas well especially for the stations farthest west ofthe San Andreas The optimal model is H = 114 kmas before but the relaxation time increases to 180years

Relaxation times tR gt 50 years are long com-pared to what has been inferred for the post-1906strain rate transient north of the San Francisco Bayas measured by repeated triangulation surveys(Thatcher 1975a 1975b Kenner and Segall 2000)The relaxation time tR consistent with the post-1906transient can be inferred from the triangulation dataand the spatial derivative of equation (3) It wouldbe most consistent to employ the same Savage-Pres-cott model as used with the Carrizo Plain data how-ever this requires knowing the recurrence time oflarge earthquakes on the north coast segment of theSan Andreas fault An alternate approach employedhere is to remove the estimated secular deformationusing a model (see Kenner and Segall 2001) andthen fit the remaining strain rate data For 60meters of slip in the 1906 earthquake the optimal

smiddot Du Tcurren=

smiddot

smiddot

smiddot

FIG 7 Model of a faulted elastic plate overlying viscoelas-tic half-space The elastic layer has shear modulus micro the vis-coelast ic half-space shear modulus micro and viscosity hEarthquakes occur at a regular interval T such that thefar-field velocity is vyen2

70 PAUL SEGALL

model is H = 19 km with a relaxation time of 17years (Fig 11) substantially less than found fromthe Carrizo Plain interseismic data1

A tentative conclusion of this analysis is that theCarrizo Plain interseismic data require a relaxationtime in excess of 50 years whereas the post-1906triangulation data require relaxation times of 25years or less In order to test whether this discrep-ancy is truly required by the data I estimated theuncertainty in the two relaxation times using a boot-strap resampling procedure As discussed previ-ously the residuals to the best-fitting model wererandomly resampled with replacement and added

to the model predictions for each bootstrap resam-ple This approach seems best for small data setsand ensures that there were data at each observationposition (interseismic data) and time (postseismicdata)

For the post-1906 transient admissible relax-ation times range from 10 years to 30 years (Fig 12)On the other hand the present distribution of defor-mation on the Carrizo Plain demands relaxationtimes in excess of 35 years (Fig 13) Thus we cansafely conclude that there is no single relaxationtime that fits both data sets

There are a number of possible explanations forthis apparent discrepancy One possibility is thatthe model of a fully ruptured elastic plate overlyinga linear Maxwell half-space does not capture all ofthe relevant physics For example nonlinear rheol-ogy may be important The data are consistent withthe expectation that the effective viscosity is low

1 Note that the discrepancy is not due to the correction forinterseismic deformation in the post-1906 data Assuming arecurrence time of 250 years and a slip rate on the SAF of 25mmyr the optimal relaxation time is 228 years with an elas-tic-layer thickness of 167 km

FIG 8 Savage-Prescott model for (A) T tR = 1 and (B) T tR = 5 t is the time since the last earthquake T is theearthquake repeat time and tR is the Maxwell relaxation time In these calculations the earthquakes rupture the entireelastic layer

ESTIMATES OF SLIP RATE 71

early in the seismic cycle when the stresses arelarge and increases late in the cycle when thestresses have largely dissipated In addition recentreflection and refraction results in northern Califor-nia show the San Andreas fault cutting the lowercrust and possibly the Moho (eg Henstock et al1997 Parsons and Hart 1999) Aseismic slip onthe downdip extension of the seismogenic zone (asin Tse and Rice 1986) may take place in additionto a more deeply distributed relation Results fromKenner and Segall (2001) which include viscousrelaxation in a narrow fault zone and more deeplydistributed slip find the spatiotemporal pattern ofdeformation in the post-1906 triangulation data isbest fit with a short relaxation time in the fault zoneand a longer relaxation in the underlyinghalf-space Another possibility is that the thermo-mechanical structure of the lithosphere differsbetween the two areas It is also possible that along-strike variations in fault geometry (the Big Bend) or

slip behavior (the creeping zone) may bias the com-puted relaxation time although this seems lesslikely

Although the apparent relaxation times differbetween the Carrizo Plain and the northern SanFrancisco Bay region the elastic-layer thicknessestimates are consistent in the range of 13ndash20 kmCarrizo Plain slip-rate estimates cluster near theupper bound of 37 mmyr Interestingly the CarrizoPlain data yield recurrence times greater than 160years Whereas the distribution is reasonably broadboth the mean (288 years) and median (277) repeattimes fall between the 1988 working group estimateof 296 years (WGCEP 1988) and the 1995 workinggroup estimate of 206 years (WGCEP 1995) Theaverage coseismic slip for the predicted events isroughly 10 meters There are significant correlationsbetween parameters (between H and tR as dis-cussed previously as well as between tR and T)which means that any additional data that constrain

FIG 9 Comparison of Savage-Prescott model to the SCEC velocity data for the Carrizo Plain A Misfit as a functionof elastic-layer thickness and relaxation time B Best-fitting San Andreas slip rate corresponding to the elastic-layerthickness and relaxation time C Fit to the data Effect of the Hosgri fault is removed employing a dislocation model withslip rate of 3 mmyr and depth of 10 km

72 PAUL SEGALL

some of the parameters will help resolve others aswell

It should be emphasized that the paleoseismicobservations of Grant and Sieh (1994) show an irreg-ular pattern of earthquakes that cannot be explainedby the simple two-dimensional model used here tointerpret the geodetic observations Nevertheless itis interesting that the GPS data appear to have someresolving power for average recurrence interval

San Francisco Bay Data

In order to eliminate any geographic bias in theinterpretation associated with possibly differentcrustal structure and temperature distribution Inext consider the velocity profile across the SanAndreas fault north of the San Francisco Bay asdetermined by repeated GPS observations (Prescottet al 2001) as shown in Figure 14 The GPS sitesare sufficiently close to the post-1906 triangulationnetworks that it is reasonable to assume that the

crustal structure and thermal profile are the samefor both data sets In the following the GPS-derivedvelocities are rotated into fault-parallel and perpen-dicular components Only the fault-parallel motionsare modeled here although as pointed out by Pres-cott et al (2001) and Murray and Segall (2001)there is significant fault-normal motion in the east-ern coast ranges at the western edge of the CentralValley

The model is the same as discussed previouslywith three principal faults the San Andreas Hay-wardndashRodgers Creek and ConcordndashGreen Valleyfaults The parameters are the elastic plate thick-ness H Maxwell relaxation time tR the recurrencetime for great earthquakes on the SAF TSAF the sliprate on the SAF and the recurrence timeTHRC time since past event tHRC and slip rate on the Hayward-Rodgers Creek fault system Thetime since the last great earthquake on the northernSan Andreas fault tSAF is of course known Theeffect of the ConcordndashGreen Valley fault is modeled

smiddotSAFsmiddotHRC

FIG 10 Comparison of Savage-Prescott model to the SCEC velocity data for the Carrizo Plain with slip rate con-strained to 34 mmyr Effect of San GregoriondashHosgri fault has been removed with a dislocation model with slip rate of 3mmyr and depth of 10 km A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

ESTIMATES OF SLIP RATE 73

kinematically with a simple screw dislocationwhich adds two parameters the fault slip rate andlocking depth It is certainly inconsistent to ignoreviscoelastic effects for the ConcordndashGreen Valleyfault but I do so for three reasons (a) much less isknown about this fault (b) earthquakes there arepresumably least well modeled by infinitely longruptures and (c) it keeps the number of parametersreasonably tractable In essence the effect of theConcordndashGreen Valley is removed empirically and Imake no attempt to gain further insights into itsmechanical behavior Future work should addressand hopefully rectify this simplification

A priori bounds on the parameters are listed inTable 1 The lower bound on the lithospheric thick-ness comes from a conservative estimate of thedepth extent of slip in large earthquakes In fact itis more likely that slip in 1906 extended at least to15 km (Matthews and Segall 1993) as it certainlydid in the Loma Prieta earthquake (eg Beroza1991) The bounds on relaxation time come from thepreliminary analysis above (Figs 12 and 13) The

lower bound on TSAF comes from the elapsed timesince 1906 the upper bound from limits on theamount of coseismic slip (see below) The lowerbounds on tHRC and THRC are derived from the his-torical record there is no known earthquake on thenorthern HaywardndashRodgers Creek system since atleast 1775 (Toppozada and Borchardt 1998WGCEP 1999) The lower and upper bounds on sliprates were based loosely on the Northern CaliforniaWorking group (WGCEP 1999 Table 2) The Work-ing Group listed 2 s uncertainties of = 24 plusmn 3mmyr = 9 plusmn 2 mmyr and = 5 plusmn m 3mmyr

The estimation problem is to find the choice ofthe nine parameters in Table 1 that minimize themisfit to two geodetic data sets the present-day spa-tial distribution of deformation in the north SanFrancisco Bay region (Fig 14) and the time-depen-dent decay in strain rate following the 1906 earth-quake (Fig 11) For a non-linear problem such asthis one always needs to be concerned about localminima However preliminary testing with a simu-

smiddotSAFsmiddotHRC smiddotHRC

FIG 11 Post-1906 strain rate from triangulation data in the San Francisco Bay area Model fit to the model withpostseismic deformation only A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

74 PAUL SEGALL

lated annealing algorithm did not show evidence oflocal minima so the remaining analysis was doneusing a Sequential Quadratic Programming methodwith BFGS Hessian updates (eg Gill et al 1981)within Matlab The method allows upper and lowerbounds to be placed on all parameters as well asgeneralized inequality constraints The latter areneeded because tHRC is strictly bounded by tHRC poundTHRC In the context of the Savage and Prescottmodel earthquakes are periodic and it makes nosense for the elapsed time since the last event toexceed the average recurrence time In addition Ihave found it desirable to constrain the maximumcoseismic slip on both the San Andreas and Hay-ward faults This provides constraints of the form

and

Noting that the coseismic slip should be interpretedas an average over the fault in this two-dimensional

model I set meters and

metersIt should be noted that the GPS network does not

extend very far west of the San Andreas fault Forthis reason the data do not adequately constrain the

total motion across the plate boundary A reasonableapproach would be to include data from Pacificplate sites such as Hawaii I chose simply to con-strain the sum of the slip rates on all three faults tobe no greater than 40 mmyr (eg Argus and Gordon2001) This constraint significantly affects the solu-tion estimates without the cumulative slip con-straint tended to put the maximum slip rate allowedon all three faults exceeding the total displacementrate between the Sierra NevadandashGreat Valley blockand the Pacific plate

The optimal solution is found for H ~ 184 km tR~ 345 years TSAF = 280 years ~ 25 mmyrtHRC = 225 years THRC = 4286 years and ~ 7mmyr The ConcordndashGreen Valley fault slips at

~ 8 mmyr below 22 km A surprising resultis that the predicted recurrence time on the SanAndreas is reasonable in the sense of requiring 70-meter slip events every 280 years In comparisonSchwartz et al (1998) suggested that the penulti-mate event on the north coast segment of the SanAndreas occurred in the mid-1600s possibly in theinterval 1632ndash1659 corresponding to a recurrenceinterval of roughly 260 years Also surprising is thatthe time since the last earthquake on the Haywardndash

smiddotSAF TSAFtimes DuSAFmaxpound smiddotHRC THRCtimes DuHRC

maxpound

DuSAFmax

70= DuHRCmax

30=

smiddotSAFsmiddotHRC

smiddotHRC

FIG 12 Bootstrap results for post-1906 data in the San Francisco Bay area A Relaxation time B Elastic-layerthickness (1000 bootstrap resamples)

ESTIMATES OF SLIP RATE 75

Rodgers Creek fault is the minimum allowed 225years A comparison of the model fit to the data setsis shown in Figure 15 Generally speaking the dataare fit quite well with the possible exception of thestrain rate in the first post-1906 epoch The uncer-tainties in the post-1906 strain rates were actuallydecreased by a factor of two to ensure that these dataare fairly weighted compared to the GPS velocitiesDecreasing the uncertainties in the strain rates fur-ther to fit high strain rate in the immediatepost-1906 epoch causes a very significant increasein the misfit to the modern GPS data This is mostlikely a result of the large uncertainties in the trian-gulation data but may also indicate that the physi-cal model is inadequate

How is it that the geodetic data have any capabil-ity of resolving the recurrence interval on the SanAndreas fault Recall that there are three times inthe physical model the time since the last majorearthquake t the asthenospheric relaxation time tRand the recurrence interval T For the San Andreasfault we know tSAF ~93 years a priori and tR is rea-

sonably well resolved by the post-1906 strain ratetransient The estimation is thus free to vary T insuch a way as to optimize the fit to the velocity gra-dients near the San Andreas Constraints on themaximum slip in the earthquake help to furthertighten those constraints

In order to determine how well the data constrainthe model parameters I ran 500 bootstrap resam-ples and optimizations using the best-fitting modelas the starting point of the optimization As beforethe residuals were resampled and added to the pre-dicted data from the optimal model in order toensure that each observation point contributes to thesolution The results of the bootstrap are shown inFigure 16 The posterior distribution for recurrencetime on the San Andreas fault TSAF is quite broadwith a peak near the maximum allowed of (7000mm)(19 mmyr) ~370 years In contrast the poste-rior distribution for recurrence time on the Hay-wardndashRodgers Creek fault THRC is bimodal withprominent peaks near the minimum ~225 years andmaximum allowed value (3000 mm)(7 mmyr)

FIG 13 Bootstrap results for SCEC data along the Carrizo Plain segment of the San Andreas fault A Elastic-layerthickness B Relaxation time Note that 200 years was the upper bound considered C Recurrence time Recurrencetimes in excess of 400 years were not considered as these would lead to excessive coseismic slip D Slip rate

76 PAUL SEGALL

~ 428 years The distributions for tHRC the timesince the most recent HaywardndashRodgers Creekearthquake is strongly concentrated near the mini-

mum allowed range The distribution of slip rates onthe San Andreas and HaywardndashRodgers Creekfaults are bimodal with peaks at 19 mmyr and 25

FIG 14 GPS-derived velocities for the north San Francisco Bay region after Prescott et al (2001) AbbreviationsSAF = San Andreas fault HF = Hayward fault CF = Calaveras fault RGF = Rodgers Creek fault CGVF = ConcordndashGreen Valley fault

TABLE 1 A priori Bounds on Parameters

Parameter Symbol Min Max

Lithospheric thickness km H 10 30

Relaxation time yr tR 10 200

SAF recurrence interval yr TSAF 100 500

Time since last Hayward event yr tHRC 225 690

Hayward recurrence interval yr THRC 225 700

Green valley fault locking depth km d 05 15

SAF slip rate my 0019 0027

Hayward slip rate myr 0007 0013

Green valley slip rate myr 0002 0008

smiddotSAF

smiddotHRC

smiddotCGV

ESTIMATES OF SLIP RATE 77

mmyr (SAF) and 7 mmyr and 13 mmyr (HRC)These results still must be viewed with some cau-tion in that I have not demonstrated that the boot-strap optimizations were immune to local minima

The bootstrap calculations suggest that the con-straint on maximum coseismic slip does play astrong role in the results To explore this further Ivaried the maximum coseismic slip allowed on thetwo faults The results (Table 2) demonstrate that theSan Andreas recurrence interval is not fully con-trolled by the constraint on the maximum coseismicslip In contrast the recurrence interval on the Hay-ward fault is totally controlled by the maximum slipallowed (for a maximum slip of 10 meters the recur-rence interval reaches the upper bound of 700years) Notice also that the time since the last Hay-wardndashRodgers Creek event is always at the mini-mum value allowed (225 years) The fit to the data isslightly better for large infrequent earthquakes

Discussion

What has been learned from the modeling pre-sented here It seems clear that geodetic data

should not be analyzed in a vacuum The resolutionof the data alone is insufficient to address the prob-lems of greatest interest However when combinedwith prior information from geologic studies GPSdata appear to be able to resolve parameters of con-siderable interest such as the average recurrenceinterval between great earthquakes on the SanAndreas fault Using a simple model of a repeatingdislocation rupturing an elastic plate overlying aMaxwell visco-elastic medium I found an averagerecurrence interval of roughly 280 years on thenorth coast segment of the San Andreas faultAlthough the uncertainties in this estimate arelarge the result is in surprisingly good accordancewith recent paleoseismic results (Schwartz et al1998) that suggest an event similar to 1906 in themid 1600s

The results for the HaywardndashRodgers Creek faultare also not unreasonable According to the GPSdata the most recent event here occurred in ~1775regardless of the estimated recurrence time Thisresult is not inconsistent with the paleoseismic find-ing that the most recent event on the Rodgers Creekfault took place in the interval 1670ndash1776

TABLE 2 Influence of Maximum Coseismic Slip on Estimated Parameters1

Parameter

Lithospheric thickness H km 150 184 220

Relaxation time tR yr 357 345 340

SAF recurrence interval TSAF yr 220 280 337

Limiting interval yr 290 368 526

Time since last Hayward event tHRC yr 225 225 225

Hayward recurrence interval THRC yr 2857 4286 700

Limiting interval yr 285 428 714

Green Valley fault locking depth d km 11 22 47

SAF slip rate myr 0025 0025 00249

Hayward slip rate myr 0007 0007 00071

Green valley slip rate myr 0008 0008 0008

1Limiting interval is computed using the estimated slip rate for that fault not the minimum slip rate that would yield a longer upper bound

DuSAFmax

55=

DuHRCmax

20=

DuSAFmax

70=

DuHRCmax

30=

DuSAFmax

100=

DuHRCmax

50=

DuSAFmax

smiddotSAFcurren

DuHRCmax

smiddotHRCcurren

smiddotSAF

smiddotHRC

smiddotCGV

78 PAUL SEGALL

(Schwartz et al 1993) and on the northern Hay-ward fault in 1640ndash1776 (HFPEG 1999) The GPSdata alone yield limited information about the aver-age recurrence time on the HaywardndashRodgers Creekfault However adding the reasonable bound thatslip not exceed 2 meters on average yields a recur-rence interval of 285 years The average slip in the1868 Hayward fault earthquake was approximately2 meters (Yu and Segall 1996) In comparisontrenching at the El Cerrito site on the northern Hay-ward fault suggests a recurrence interval of possiblyless than 270 years but no more than 710 years(HFPEG 1999)

It should not need to be stated that earthquakesare neither periodic nor especially in the case of theHaywardndashRodgers Creek faults infinitely long Atwo-dimensional model cannot describe the situa-tion at a given trench site where ruptures mainly tothe north or south of the site cause some offset in the

trench Nevertheless the fact that the results fromsuch a simple model are not unreasonable is encour-aging One useful attribute of the models exploredhere are that they are testable in the sense of predict-ing future behavior Running the model forward intime to the next San Andreas earthquake yields thevelocity distributions shown in Figure 17 Of coursea north HaywardndashRodgers Creek earthquake mayoccur prior to the next San Andreas event

The Maxwell relaxation time is reasonablywell estimated by the data I find an optimal valueof about 35 years which corresponds to a viscos-ity of 16 times 1019 Pa-s It should be noted that theestimates of all of the model parameters is only asgood as the mechanical model used to interpretthe data If as seems quite likely there was after-slip on the San Andreas below the 1906 rupturezone the estimate of the viscosity is some combi-nation of the effects of afterslip and distributed

FIG 15 Fit of model to north San Francisco Bay data Maximum slip on Hayward fault is 30 m and on the SAF is70 m

ESTIMATES OF SLIP RATE 79

relaxation Li and Rice (1987) presented anapproximate method for including post-seismicfault slip at constant resistive stress with a relax-ing asthenosphere in the context of a repeatingperiodic earthquake sequence Combining suchan approach with appropriate constitutive laws forstable fault creep as in Tse and Rice (1986)would presumably provide a more physically real-istic description of fault behavior Clearly three-dimensional effects are also important especiallyfor earthquakes on the Rodgers Creek and Hay-ward faults Finally the effects of non-uniform

elastic properties are known to bias interpreta-tions based on simple homogeneous models Elas-tic moduli increase with depth which has theknown effect of biasing the depth distribution offault slip Similarly elastic properties may varywith lithology across the major fault zones whichthemselves may represent zones of more compli-ant rock Finite elements are certainly the mostgeneral way to compute forward models but arenot yet feasible for the optimization and bootstrapmethods as computed here which literally testedthousands of possible models

FIG 16 Five hundred bootstrap estimates of model parameters fit to the north San Francisco Bay data 90 confi-dence bounds are listed at the top of each panel The vertical line in the histograms for recurrence times TSAF and THRCshow the upper bounds corresponding to the minimum slip rate and maximum allowed coseismic slip respectively Thehorizontal bar in the panel for the San Andreas recurrence time shows the preferred estimate from Schwartz et al (1998)The horizontal bar in the panel for tH time since the last Hayward earthquake shows the range of dates compatible withpaleoseismic estimates of the penultimate HaywardndashRodgers Creek event

80 PAUL SEGALL

Acknowledgments

George Thompson set a standard that geophysi-cal models no matter how elegant must be consis-tent with geologic observations This work wasinspired by that example I would like to thank WillPrescott for discussions about the Bay Area defor-mation measurements and Mark Matthews for manyyears ago introducing me to Bayesrsquo theorem WayneThatcher Mark Murray Jeff McGuire and PeterCervelli provided valuable comments at early stagesin this study I thank Jim Savage Duncan Agnewand Shelley Kenner for their thoughtful reviewsGary Ernst and Simon Klemperer carefully editedthe manuscript

REFERENCES

Argus D F and Gordon R G 2001 Present tectonicmotion across the Coast Ranges and San Andreas faultsystem in central California Geological Society ofAmerica Bulletin v 113 p 1580ndash1592

Beroza G C 1991 Near-source modeling of the LomaPrieta earthquake Evidence for heterogeneous slipand implications for earthquake hazard Bulletin of theSeismological Society of America v 81 p 1603ndash1621

Cohen S C 1982 A multilayer model of time-dependentdeformation following an earthquake on a strike-slipfault Journal of Geophysical Research v 87 p 5409ndash5421

HFPEG (Hayward Fault Paleoearthquake Group) 1999Timing of paleoearthquakes on the northern HaywardfaultmdashPreliminary evidence in El Cerrito CaliforniaUS Geological Survey Open File Report 99-318

Henstock T J Levander A and Hole J A 1997Deformation in the lower crust of the San Andreas faultsystem in northern California Science v 278 p 650ndash653

Feigl K L King R W and Jordan T H 1990 Geo-detic measurement of tectonic deformation in theSanta Maria fold and thrust belt California Journal ofGeophysical Research v 95 p 2679ndash2699

Gill P E Murray W and Wright M H 1981 Practicaloptimization London UK Academic Press 401 p

Grant L B and Sieh K 1994 Paleoseismic evidence ofclustered earthquakes on the San Andreas fault in the

FIG 17 Simulation of Bay Area velocity field as a function of time Each line is separated by roughly 27 years

68 PAUL SEGALL

Viscoelastic Models

As noted in the Introduction the screw disloca-tion provides only a very limited description ofplate-boundary faulting A somewhat more realisticmodel involves an elastic layer of thickness H over-lying a Maxwell viscoelastic half-space (Fig 7) TheMaxwell material has relaxation time tR = 2hmicrowhere h is the viscosity and micro is the shear modulusAt time t = 0 slip Du occurs on the fault from thesurface to depth D pound H The velocity on the Earthrsquossurface as a function of position perpendicular to thefault x and time t is

(3)

where the spatial distribution is given by

(4)

The result follows from but is not stated in Nur andMavko (1974) The strain rate is simply obtained bydifferentiating the velocity with respect to x Noticethat the post-seismic velocity is a function of thecoseismic slip Du the depth of faulting D the elas-tic layer thickness H the material relaxation tR andthe time since the last large earthquake t Equation(3) does not include the effects of interseismic strainaccumulation Savage and Prescott (1978) extendedthe model by summing over an infinite sequence ofplate rupturing events (D = H) equally spaced intime with recurrence interval T The velocity field asa function of distance across strike (x) and time (t) isgiven by

(5)

where the temporal dependence depends only on thedimensionless ratios ttR and TtR

(6)

v x t( )

DuptR---------e

t tRcurrenndash t tRcurren( )n 1ndash

n 1ndash( )--------------------------Fn x( D H )

n 1=

yen

aring

=

Fn x D H ( )

D 2nH+x

---------------------egrave oslashaelig ouml1ndash

tan D 2nHndashx

--------------------egrave oslashaelig ouml1ndash

tan+

2xD

x2

2nH( )2D

2ndash+

------------------------------------------1ndash tan

=

=

v x t( ) smiddot

p--- Tn t tR T tRcurrencurren( )Fn x H H ( )

n 1=

yen

aring=

Tn t tR T tRcurrencurren( )

TtR-----

et tRcurrenndash

n 1ndash( )------------------ e

kT tRcurrenndash t kT+tR

--------------egrave oslashaelig ouml n 1ndash

k 0=

yen

aring

=

FIG 6 Tradeoff between slip rate and locking depth Each point is the result of a bootstrap resample of the SCECvelocity data The heavy points are those consistent with a uniform prior distribution on slip rate from 31 to 37 mmyrAs before I assume 3 mmyr below a locking depth of 10 km on the Hosgri fault system

ESTIMATES OF SLIP RATE 69

Because the far-field lithospheric velocity is drivensolely by the repeating earthquakes Representative examples are given in Figure 8 forvarious times through the earthquake cycle Noticethat when the asthenospheric relaxation time is ofthe order of the earthquake cycle time or longer TtR pound1 the surface velocity distribution is rathersteady with time (Fig 8A) On the other hand if therelaxation time is short compared to the recurrenceinterval TtR gt 1 there is a significant transientwith higher than average strain rates early in thecycle and lower than average strain rates late in thecycle

The Savage-Prescott model involves five param-eters lithospheric thickness H relaxation time tRtime since the last earthquake t recurrence time Tand long-term slip rate on the fault Because thetheory is linear it is straightforward to constructmodels with multiple parallel faults however in thiscase the number of parameters is even larger Forthree faults there are 11 parameters if one assumesthat the elastic thickness and relaxation time are thesame for all faults Note that the model is kinematicwith imposed slip events on the various faults at reg-ular intervals and thus does not account for thestress interactions between the faults

As a starting point I consider the Carrizo Plaindata with some of the parameters constrained byindependent observations The time since the lastgreat earthquake (1857) is t = 136 years To beginwith I assume a recurrence time of T = 160 yearsbased on the average interval found by Grant andSieh (1994) There are three remaining parametersto be estimated from the data tR H and Theoptimal estimates of the non-linear parameters tRand H are found by simple grid search (Fig 9) with

estimated by linear least squares The optimalmodel is found for H = 114 km and tR = 80 yearswith a slip rate of 402 mmyr There is a strongtradeoff between relaxation time and elastic-layerthickness such that models with elastic-layer thick-ness of up to 20 km and relaxation time of 180 yearsfit the data nearly equally well The reason for thetradeoff is as follows Larger values of H cause thedeformation to be spread out in space At the sametime the deformation becomes more diffuse withincreasing tT and TtR (Fig 8) and hence t tR = (tT) times (TtR ) Thus decreasing ttR causes the defor-mation to bemore localized which counters theeffect of increasing H Interestingly the slip ratedoes not vary with H and tR confirming that thisparameter is well constrained by the data (given thatall other parameters are fixed)

The inferred slip rate is however higher thanestimated from stream offsets at Wallace Creek of 33plusmn 3 mmyr (Sieh and Jahns 1984) Constraining theslip rate to 34 mmyr (Fig 10) does not fit the dataas well especially for the stations farthest west ofthe San Andreas The optimal model is H = 114 kmas before but the relaxation time increases to 180years

Relaxation times tR gt 50 years are long com-pared to what has been inferred for the post-1906strain rate transient north of the San Francisco Bayas measured by repeated triangulation surveys(Thatcher 1975a 1975b Kenner and Segall 2000)The relaxation time tR consistent with the post-1906transient can be inferred from the triangulation dataand the spatial derivative of equation (3) It wouldbe most consistent to employ the same Savage-Pres-cott model as used with the Carrizo Plain data how-ever this requires knowing the recurrence time oflarge earthquakes on the north coast segment of theSan Andreas fault An alternate approach employedhere is to remove the estimated secular deformationusing a model (see Kenner and Segall 2001) andthen fit the remaining strain rate data For 60meters of slip in the 1906 earthquake the optimal

smiddot Du Tcurren=

smiddot

smiddot

smiddot

FIG 7 Model of a faulted elastic plate overlying viscoelas-tic half-space The elastic layer has shear modulus micro the vis-coelast ic half-space shear modulus micro and viscosity hEarthquakes occur at a regular interval T such that thefar-field velocity is vyen2

70 PAUL SEGALL

model is H = 19 km with a relaxation time of 17years (Fig 11) substantially less than found fromthe Carrizo Plain interseismic data1

A tentative conclusion of this analysis is that theCarrizo Plain interseismic data require a relaxationtime in excess of 50 years whereas the post-1906triangulation data require relaxation times of 25years or less In order to test whether this discrep-ancy is truly required by the data I estimated theuncertainty in the two relaxation times using a boot-strap resampling procedure As discussed previ-ously the residuals to the best-fitting model wererandomly resampled with replacement and added

to the model predictions for each bootstrap resam-ple This approach seems best for small data setsand ensures that there were data at each observationposition (interseismic data) and time (postseismicdata)

For the post-1906 transient admissible relax-ation times range from 10 years to 30 years (Fig 12)On the other hand the present distribution of defor-mation on the Carrizo Plain demands relaxationtimes in excess of 35 years (Fig 13) Thus we cansafely conclude that there is no single relaxationtime that fits both data sets

There are a number of possible explanations forthis apparent discrepancy One possibility is thatthe model of a fully ruptured elastic plate overlyinga linear Maxwell half-space does not capture all ofthe relevant physics For example nonlinear rheol-ogy may be important The data are consistent withthe expectation that the effective viscosity is low

1 Note that the discrepancy is not due to the correction forinterseismic deformation in the post-1906 data Assuming arecurrence time of 250 years and a slip rate on the SAF of 25mmyr the optimal relaxation time is 228 years with an elas-tic-layer thickness of 167 km

FIG 8 Savage-Prescott model for (A) T tR = 1 and (B) T tR = 5 t is the time since the last earthquake T is theearthquake repeat time and tR is the Maxwell relaxation time In these calculations the earthquakes rupture the entireelastic layer

ESTIMATES OF SLIP RATE 71

early in the seismic cycle when the stresses arelarge and increases late in the cycle when thestresses have largely dissipated In addition recentreflection and refraction results in northern Califor-nia show the San Andreas fault cutting the lowercrust and possibly the Moho (eg Henstock et al1997 Parsons and Hart 1999) Aseismic slip onthe downdip extension of the seismogenic zone (asin Tse and Rice 1986) may take place in additionto a more deeply distributed relation Results fromKenner and Segall (2001) which include viscousrelaxation in a narrow fault zone and more deeplydistributed slip find the spatiotemporal pattern ofdeformation in the post-1906 triangulation data isbest fit with a short relaxation time in the fault zoneand a longer relaxation in the underlyinghalf-space Another possibility is that the thermo-mechanical structure of the lithosphere differsbetween the two areas It is also possible that along-strike variations in fault geometry (the Big Bend) or

slip behavior (the creeping zone) may bias the com-puted relaxation time although this seems lesslikely

Although the apparent relaxation times differbetween the Carrizo Plain and the northern SanFrancisco Bay region the elastic-layer thicknessestimates are consistent in the range of 13ndash20 kmCarrizo Plain slip-rate estimates cluster near theupper bound of 37 mmyr Interestingly the CarrizoPlain data yield recurrence times greater than 160years Whereas the distribution is reasonably broadboth the mean (288 years) and median (277) repeattimes fall between the 1988 working group estimateof 296 years (WGCEP 1988) and the 1995 workinggroup estimate of 206 years (WGCEP 1995) Theaverage coseismic slip for the predicted events isroughly 10 meters There are significant correlationsbetween parameters (between H and tR as dis-cussed previously as well as between tR and T)which means that any additional data that constrain

FIG 9 Comparison of Savage-Prescott model to the SCEC velocity data for the Carrizo Plain A Misfit as a functionof elastic-layer thickness and relaxation time B Best-fitting San Andreas slip rate corresponding to the elastic-layerthickness and relaxation time C Fit to the data Effect of the Hosgri fault is removed employing a dislocation model withslip rate of 3 mmyr and depth of 10 km

72 PAUL SEGALL

some of the parameters will help resolve others aswell

It should be emphasized that the paleoseismicobservations of Grant and Sieh (1994) show an irreg-ular pattern of earthquakes that cannot be explainedby the simple two-dimensional model used here tointerpret the geodetic observations Nevertheless itis interesting that the GPS data appear to have someresolving power for average recurrence interval

San Francisco Bay Data

In order to eliminate any geographic bias in theinterpretation associated with possibly differentcrustal structure and temperature distribution Inext consider the velocity profile across the SanAndreas fault north of the San Francisco Bay asdetermined by repeated GPS observations (Prescottet al 2001) as shown in Figure 14 The GPS sitesare sufficiently close to the post-1906 triangulationnetworks that it is reasonable to assume that the

crustal structure and thermal profile are the samefor both data sets In the following the GPS-derivedvelocities are rotated into fault-parallel and perpen-dicular components Only the fault-parallel motionsare modeled here although as pointed out by Pres-cott et al (2001) and Murray and Segall (2001)there is significant fault-normal motion in the east-ern coast ranges at the western edge of the CentralValley

The model is the same as discussed previouslywith three principal faults the San Andreas Hay-wardndashRodgers Creek and ConcordndashGreen Valleyfaults The parameters are the elastic plate thick-ness H Maxwell relaxation time tR the recurrencetime for great earthquakes on the SAF TSAF the sliprate on the SAF and the recurrence timeTHRC time since past event tHRC and slip rate on the Hayward-Rodgers Creek fault system Thetime since the last great earthquake on the northernSan Andreas fault tSAF is of course known Theeffect of the ConcordndashGreen Valley fault is modeled

smiddotSAFsmiddotHRC

FIG 10 Comparison of Savage-Prescott model to the SCEC velocity data for the Carrizo Plain with slip rate con-strained to 34 mmyr Effect of San GregoriondashHosgri fault has been removed with a dislocation model with slip rate of 3mmyr and depth of 10 km A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

ESTIMATES OF SLIP RATE 73

kinematically with a simple screw dislocationwhich adds two parameters the fault slip rate andlocking depth It is certainly inconsistent to ignoreviscoelastic effects for the ConcordndashGreen Valleyfault but I do so for three reasons (a) much less isknown about this fault (b) earthquakes there arepresumably least well modeled by infinitely longruptures and (c) it keeps the number of parametersreasonably tractable In essence the effect of theConcordndashGreen Valley is removed empirically and Imake no attempt to gain further insights into itsmechanical behavior Future work should addressand hopefully rectify this simplification

A priori bounds on the parameters are listed inTable 1 The lower bound on the lithospheric thick-ness comes from a conservative estimate of thedepth extent of slip in large earthquakes In fact itis more likely that slip in 1906 extended at least to15 km (Matthews and Segall 1993) as it certainlydid in the Loma Prieta earthquake (eg Beroza1991) The bounds on relaxation time come from thepreliminary analysis above (Figs 12 and 13) The

lower bound on TSAF comes from the elapsed timesince 1906 the upper bound from limits on theamount of coseismic slip (see below) The lowerbounds on tHRC and THRC are derived from the his-torical record there is no known earthquake on thenorthern HaywardndashRodgers Creek system since atleast 1775 (Toppozada and Borchardt 1998WGCEP 1999) The lower and upper bounds on sliprates were based loosely on the Northern CaliforniaWorking group (WGCEP 1999 Table 2) The Work-ing Group listed 2 s uncertainties of = 24 plusmn 3mmyr = 9 plusmn 2 mmyr and = 5 plusmn m 3mmyr

The estimation problem is to find the choice ofthe nine parameters in Table 1 that minimize themisfit to two geodetic data sets the present-day spa-tial distribution of deformation in the north SanFrancisco Bay region (Fig 14) and the time-depen-dent decay in strain rate following the 1906 earth-quake (Fig 11) For a non-linear problem such asthis one always needs to be concerned about localminima However preliminary testing with a simu-

smiddotSAFsmiddotHRC smiddotHRC

FIG 11 Post-1906 strain rate from triangulation data in the San Francisco Bay area Model fit to the model withpostseismic deformation only A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

74 PAUL SEGALL

lated annealing algorithm did not show evidence oflocal minima so the remaining analysis was doneusing a Sequential Quadratic Programming methodwith BFGS Hessian updates (eg Gill et al 1981)within Matlab The method allows upper and lowerbounds to be placed on all parameters as well asgeneralized inequality constraints The latter areneeded because tHRC is strictly bounded by tHRC poundTHRC In the context of the Savage and Prescottmodel earthquakes are periodic and it makes nosense for the elapsed time since the last event toexceed the average recurrence time In addition Ihave found it desirable to constrain the maximumcoseismic slip on both the San Andreas and Hay-ward faults This provides constraints of the form

and

Noting that the coseismic slip should be interpretedas an average over the fault in this two-dimensional

model I set meters and

metersIt should be noted that the GPS network does not

extend very far west of the San Andreas fault Forthis reason the data do not adequately constrain the

total motion across the plate boundary A reasonableapproach would be to include data from Pacificplate sites such as Hawaii I chose simply to con-strain the sum of the slip rates on all three faults tobe no greater than 40 mmyr (eg Argus and Gordon2001) This constraint significantly affects the solu-tion estimates without the cumulative slip con-straint tended to put the maximum slip rate allowedon all three faults exceeding the total displacementrate between the Sierra NevadandashGreat Valley blockand the Pacific plate

The optimal solution is found for H ~ 184 km tR~ 345 years TSAF = 280 years ~ 25 mmyrtHRC = 225 years THRC = 4286 years and ~ 7mmyr The ConcordndashGreen Valley fault slips at

~ 8 mmyr below 22 km A surprising resultis that the predicted recurrence time on the SanAndreas is reasonable in the sense of requiring 70-meter slip events every 280 years In comparisonSchwartz et al (1998) suggested that the penulti-mate event on the north coast segment of the SanAndreas occurred in the mid-1600s possibly in theinterval 1632ndash1659 corresponding to a recurrenceinterval of roughly 260 years Also surprising is thatthe time since the last earthquake on the Haywardndash

smiddotSAF TSAFtimes DuSAFmaxpound smiddotHRC THRCtimes DuHRC

maxpound

DuSAFmax

70= DuHRCmax

30=

smiddotSAFsmiddotHRC

smiddotHRC

FIG 12 Bootstrap results for post-1906 data in the San Francisco Bay area A Relaxation time B Elastic-layerthickness (1000 bootstrap resamples)

ESTIMATES OF SLIP RATE 75

Rodgers Creek fault is the minimum allowed 225years A comparison of the model fit to the data setsis shown in Figure 15 Generally speaking the dataare fit quite well with the possible exception of thestrain rate in the first post-1906 epoch The uncer-tainties in the post-1906 strain rates were actuallydecreased by a factor of two to ensure that these dataare fairly weighted compared to the GPS velocitiesDecreasing the uncertainties in the strain rates fur-ther to fit high strain rate in the immediatepost-1906 epoch causes a very significant increasein the misfit to the modern GPS data This is mostlikely a result of the large uncertainties in the trian-gulation data but may also indicate that the physi-cal model is inadequate

How is it that the geodetic data have any capabil-ity of resolving the recurrence interval on the SanAndreas fault Recall that there are three times inthe physical model the time since the last majorearthquake t the asthenospheric relaxation time tRand the recurrence interval T For the San Andreasfault we know tSAF ~93 years a priori and tR is rea-

sonably well resolved by the post-1906 strain ratetransient The estimation is thus free to vary T insuch a way as to optimize the fit to the velocity gra-dients near the San Andreas Constraints on themaximum slip in the earthquake help to furthertighten those constraints

In order to determine how well the data constrainthe model parameters I ran 500 bootstrap resam-ples and optimizations using the best-fitting modelas the starting point of the optimization As beforethe residuals were resampled and added to the pre-dicted data from the optimal model in order toensure that each observation point contributes to thesolution The results of the bootstrap are shown inFigure 16 The posterior distribution for recurrencetime on the San Andreas fault TSAF is quite broadwith a peak near the maximum allowed of (7000mm)(19 mmyr) ~370 years In contrast the poste-rior distribution for recurrence time on the Hay-wardndashRodgers Creek fault THRC is bimodal withprominent peaks near the minimum ~225 years andmaximum allowed value (3000 mm)(7 mmyr)

FIG 13 Bootstrap results for SCEC data along the Carrizo Plain segment of the San Andreas fault A Elastic-layerthickness B Relaxation time Note that 200 years was the upper bound considered C Recurrence time Recurrencetimes in excess of 400 years were not considered as these would lead to excessive coseismic slip D Slip rate

76 PAUL SEGALL

~ 428 years The distributions for tHRC the timesince the most recent HaywardndashRodgers Creekearthquake is strongly concentrated near the mini-

mum allowed range The distribution of slip rates onthe San Andreas and HaywardndashRodgers Creekfaults are bimodal with peaks at 19 mmyr and 25

FIG 14 GPS-derived velocities for the north San Francisco Bay region after Prescott et al (2001) AbbreviationsSAF = San Andreas fault HF = Hayward fault CF = Calaveras fault RGF = Rodgers Creek fault CGVF = ConcordndashGreen Valley fault

TABLE 1 A priori Bounds on Parameters

Parameter Symbol Min Max

Lithospheric thickness km H 10 30

Relaxation time yr tR 10 200

SAF recurrence interval yr TSAF 100 500

Time since last Hayward event yr tHRC 225 690

Hayward recurrence interval yr THRC 225 700

Green valley fault locking depth km d 05 15

SAF slip rate my 0019 0027

Hayward slip rate myr 0007 0013

Green valley slip rate myr 0002 0008

smiddotSAF

smiddotHRC

smiddotCGV

ESTIMATES OF SLIP RATE 77

mmyr (SAF) and 7 mmyr and 13 mmyr (HRC)These results still must be viewed with some cau-tion in that I have not demonstrated that the boot-strap optimizations were immune to local minima

The bootstrap calculations suggest that the con-straint on maximum coseismic slip does play astrong role in the results To explore this further Ivaried the maximum coseismic slip allowed on thetwo faults The results (Table 2) demonstrate that theSan Andreas recurrence interval is not fully con-trolled by the constraint on the maximum coseismicslip In contrast the recurrence interval on the Hay-ward fault is totally controlled by the maximum slipallowed (for a maximum slip of 10 meters the recur-rence interval reaches the upper bound of 700years) Notice also that the time since the last Hay-wardndashRodgers Creek event is always at the mini-mum value allowed (225 years) The fit to the data isslightly better for large infrequent earthquakes

Discussion

What has been learned from the modeling pre-sented here It seems clear that geodetic data

should not be analyzed in a vacuum The resolutionof the data alone is insufficient to address the prob-lems of greatest interest However when combinedwith prior information from geologic studies GPSdata appear to be able to resolve parameters of con-siderable interest such as the average recurrenceinterval between great earthquakes on the SanAndreas fault Using a simple model of a repeatingdislocation rupturing an elastic plate overlying aMaxwell visco-elastic medium I found an averagerecurrence interval of roughly 280 years on thenorth coast segment of the San Andreas faultAlthough the uncertainties in this estimate arelarge the result is in surprisingly good accordancewith recent paleoseismic results (Schwartz et al1998) that suggest an event similar to 1906 in themid 1600s

The results for the HaywardndashRodgers Creek faultare also not unreasonable According to the GPSdata the most recent event here occurred in ~1775regardless of the estimated recurrence time Thisresult is not inconsistent with the paleoseismic find-ing that the most recent event on the Rodgers Creekfault took place in the interval 1670ndash1776

TABLE 2 Influence of Maximum Coseismic Slip on Estimated Parameters1

Parameter

Lithospheric thickness H km 150 184 220

Relaxation time tR yr 357 345 340

SAF recurrence interval TSAF yr 220 280 337

Limiting interval yr 290 368 526

Time since last Hayward event tHRC yr 225 225 225

Hayward recurrence interval THRC yr 2857 4286 700

Limiting interval yr 285 428 714

Green Valley fault locking depth d km 11 22 47

SAF slip rate myr 0025 0025 00249

Hayward slip rate myr 0007 0007 00071

Green valley slip rate myr 0008 0008 0008

1Limiting interval is computed using the estimated slip rate for that fault not the minimum slip rate that would yield a longer upper bound

DuSAFmax

55=

DuHRCmax

20=

DuSAFmax

70=

DuHRCmax

30=

DuSAFmax

100=

DuHRCmax

50=

DuSAFmax

smiddotSAFcurren

DuHRCmax

smiddotHRCcurren

smiddotSAF

smiddotHRC

smiddotCGV

78 PAUL SEGALL

(Schwartz et al 1993) and on the northern Hay-ward fault in 1640ndash1776 (HFPEG 1999) The GPSdata alone yield limited information about the aver-age recurrence time on the HaywardndashRodgers Creekfault However adding the reasonable bound thatslip not exceed 2 meters on average yields a recur-rence interval of 285 years The average slip in the1868 Hayward fault earthquake was approximately2 meters (Yu and Segall 1996) In comparisontrenching at the El Cerrito site on the northern Hay-ward fault suggests a recurrence interval of possiblyless than 270 years but no more than 710 years(HFPEG 1999)

It should not need to be stated that earthquakesare neither periodic nor especially in the case of theHaywardndashRodgers Creek faults infinitely long Atwo-dimensional model cannot describe the situa-tion at a given trench site where ruptures mainly tothe north or south of the site cause some offset in the

trench Nevertheless the fact that the results fromsuch a simple model are not unreasonable is encour-aging One useful attribute of the models exploredhere are that they are testable in the sense of predict-ing future behavior Running the model forward intime to the next San Andreas earthquake yields thevelocity distributions shown in Figure 17 Of coursea north HaywardndashRodgers Creek earthquake mayoccur prior to the next San Andreas event

The Maxwell relaxation time is reasonablywell estimated by the data I find an optimal valueof about 35 years which corresponds to a viscos-ity of 16 times 1019 Pa-s It should be noted that theestimates of all of the model parameters is only asgood as the mechanical model used to interpretthe data If as seems quite likely there was after-slip on the San Andreas below the 1906 rupturezone the estimate of the viscosity is some combi-nation of the effects of afterslip and distributed

FIG 15 Fit of model to north San Francisco Bay data Maximum slip on Hayward fault is 30 m and on the SAF is70 m

ESTIMATES OF SLIP RATE 79

relaxation Li and Rice (1987) presented anapproximate method for including post-seismicfault slip at constant resistive stress with a relax-ing asthenosphere in the context of a repeatingperiodic earthquake sequence Combining suchan approach with appropriate constitutive laws forstable fault creep as in Tse and Rice (1986)would presumably provide a more physically real-istic description of fault behavior Clearly three-dimensional effects are also important especiallyfor earthquakes on the Rodgers Creek and Hay-ward faults Finally the effects of non-uniform

elastic properties are known to bias interpreta-tions based on simple homogeneous models Elas-tic moduli increase with depth which has theknown effect of biasing the depth distribution offault slip Similarly elastic properties may varywith lithology across the major fault zones whichthemselves may represent zones of more compli-ant rock Finite elements are certainly the mostgeneral way to compute forward models but arenot yet feasible for the optimization and bootstrapmethods as computed here which literally testedthousands of possible models

FIG 16 Five hundred bootstrap estimates of model parameters fit to the north San Francisco Bay data 90 confi-dence bounds are listed at the top of each panel The vertical line in the histograms for recurrence times TSAF and THRCshow the upper bounds corresponding to the minimum slip rate and maximum allowed coseismic slip respectively Thehorizontal bar in the panel for the San Andreas recurrence time shows the preferred estimate from Schwartz et al (1998)The horizontal bar in the panel for tH time since the last Hayward earthquake shows the range of dates compatible withpaleoseismic estimates of the penultimate HaywardndashRodgers Creek event

80 PAUL SEGALL

Acknowledgments

George Thompson set a standard that geophysi-cal models no matter how elegant must be consis-tent with geologic observations This work wasinspired by that example I would like to thank WillPrescott for discussions about the Bay Area defor-mation measurements and Mark Matthews for manyyears ago introducing me to Bayesrsquo theorem WayneThatcher Mark Murray Jeff McGuire and PeterCervelli provided valuable comments at early stagesin this study I thank Jim Savage Duncan Agnewand Shelley Kenner for their thoughtful reviewsGary Ernst and Simon Klemperer carefully editedthe manuscript

REFERENCES

Argus D F and Gordon R G 2001 Present tectonicmotion across the Coast Ranges and San Andreas faultsystem in central California Geological Society ofAmerica Bulletin v 113 p 1580ndash1592

Beroza G C 1991 Near-source modeling of the LomaPrieta earthquake Evidence for heterogeneous slipand implications for earthquake hazard Bulletin of theSeismological Society of America v 81 p 1603ndash1621

Cohen S C 1982 A multilayer model of time-dependentdeformation following an earthquake on a strike-slipfault Journal of Geophysical Research v 87 p 5409ndash5421

HFPEG (Hayward Fault Paleoearthquake Group) 1999Timing of paleoearthquakes on the northern HaywardfaultmdashPreliminary evidence in El Cerrito CaliforniaUS Geological Survey Open File Report 99-318

Henstock T J Levander A and Hole J A 1997Deformation in the lower crust of the San Andreas faultsystem in northern California Science v 278 p 650ndash653

Feigl K L King R W and Jordan T H 1990 Geo-detic measurement of tectonic deformation in theSanta Maria fold and thrust belt California Journal ofGeophysical Research v 95 p 2679ndash2699

Gill P E Murray W and Wright M H 1981 Practicaloptimization London UK Academic Press 401 p

Grant L B and Sieh K 1994 Paleoseismic evidence ofclustered earthquakes on the San Andreas fault in the

FIG 17 Simulation of Bay Area velocity field as a function of time Each line is separated by roughly 27 years

ESTIMATES OF SLIP RATE 69

Because the far-field lithospheric velocity is drivensolely by the repeating earthquakes Representative examples are given in Figure 8 forvarious times through the earthquake cycle Noticethat when the asthenospheric relaxation time is ofthe order of the earthquake cycle time or longer TtR pound1 the surface velocity distribution is rathersteady with time (Fig 8A) On the other hand if therelaxation time is short compared to the recurrenceinterval TtR gt 1 there is a significant transientwith higher than average strain rates early in thecycle and lower than average strain rates late in thecycle

The Savage-Prescott model involves five param-eters lithospheric thickness H relaxation time tRtime since the last earthquake t recurrence time Tand long-term slip rate on the fault Because thetheory is linear it is straightforward to constructmodels with multiple parallel faults however in thiscase the number of parameters is even larger Forthree faults there are 11 parameters if one assumesthat the elastic thickness and relaxation time are thesame for all faults Note that the model is kinematicwith imposed slip events on the various faults at reg-ular intervals and thus does not account for thestress interactions between the faults

As a starting point I consider the Carrizo Plaindata with some of the parameters constrained byindependent observations The time since the lastgreat earthquake (1857) is t = 136 years To beginwith I assume a recurrence time of T = 160 yearsbased on the average interval found by Grant andSieh (1994) There are three remaining parametersto be estimated from the data tR H and Theoptimal estimates of the non-linear parameters tRand H are found by simple grid search (Fig 9) with

estimated by linear least squares The optimalmodel is found for H = 114 km and tR = 80 yearswith a slip rate of 402 mmyr There is a strongtradeoff between relaxation time and elastic-layerthickness such that models with elastic-layer thick-ness of up to 20 km and relaxation time of 180 yearsfit the data nearly equally well The reason for thetradeoff is as follows Larger values of H cause thedeformation to be spread out in space At the sametime the deformation becomes more diffuse withincreasing tT and TtR (Fig 8) and hence t tR = (tT) times (TtR ) Thus decreasing ttR causes the defor-mation to bemore localized which counters theeffect of increasing H Interestingly the slip ratedoes not vary with H and tR confirming that thisparameter is well constrained by the data (given thatall other parameters are fixed)

The inferred slip rate is however higher thanestimated from stream offsets at Wallace Creek of 33plusmn 3 mmyr (Sieh and Jahns 1984) Constraining theslip rate to 34 mmyr (Fig 10) does not fit the dataas well especially for the stations farthest west ofthe San Andreas The optimal model is H = 114 kmas before but the relaxation time increases to 180years

Relaxation times tR gt 50 years are long com-pared to what has been inferred for the post-1906strain rate transient north of the San Francisco Bayas measured by repeated triangulation surveys(Thatcher 1975a 1975b Kenner and Segall 2000)The relaxation time tR consistent with the post-1906transient can be inferred from the triangulation dataand the spatial derivative of equation (3) It wouldbe most consistent to employ the same Savage-Pres-cott model as used with the Carrizo Plain data how-ever this requires knowing the recurrence time oflarge earthquakes on the north coast segment of theSan Andreas fault An alternate approach employedhere is to remove the estimated secular deformationusing a model (see Kenner and Segall 2001) andthen fit the remaining strain rate data For 60meters of slip in the 1906 earthquake the optimal

smiddot Du Tcurren=

smiddot

smiddot

smiddot

FIG 7 Model of a faulted elastic plate overlying viscoelas-tic half-space The elastic layer has shear modulus micro the vis-coelast ic half-space shear modulus micro and viscosity hEarthquakes occur at a regular interval T such that thefar-field velocity is vyen2

70 PAUL SEGALL

model is H = 19 km with a relaxation time of 17years (Fig 11) substantially less than found fromthe Carrizo Plain interseismic data1

A tentative conclusion of this analysis is that theCarrizo Plain interseismic data require a relaxationtime in excess of 50 years whereas the post-1906triangulation data require relaxation times of 25years or less In order to test whether this discrep-ancy is truly required by the data I estimated theuncertainty in the two relaxation times using a boot-strap resampling procedure As discussed previ-ously the residuals to the best-fitting model wererandomly resampled with replacement and added

to the model predictions for each bootstrap resam-ple This approach seems best for small data setsand ensures that there were data at each observationposition (interseismic data) and time (postseismicdata)

For the post-1906 transient admissible relax-ation times range from 10 years to 30 years (Fig 12)On the other hand the present distribution of defor-mation on the Carrizo Plain demands relaxationtimes in excess of 35 years (Fig 13) Thus we cansafely conclude that there is no single relaxationtime that fits both data sets

There are a number of possible explanations forthis apparent discrepancy One possibility is thatthe model of a fully ruptured elastic plate overlyinga linear Maxwell half-space does not capture all ofthe relevant physics For example nonlinear rheol-ogy may be important The data are consistent withthe expectation that the effective viscosity is low

1 Note that the discrepancy is not due to the correction forinterseismic deformation in the post-1906 data Assuming arecurrence time of 250 years and a slip rate on the SAF of 25mmyr the optimal relaxation time is 228 years with an elas-tic-layer thickness of 167 km

FIG 8 Savage-Prescott model for (A) T tR = 1 and (B) T tR = 5 t is the time since the last earthquake T is theearthquake repeat time and tR is the Maxwell relaxation time In these calculations the earthquakes rupture the entireelastic layer

ESTIMATES OF SLIP RATE 71

early in the seismic cycle when the stresses arelarge and increases late in the cycle when thestresses have largely dissipated In addition recentreflection and refraction results in northern Califor-nia show the San Andreas fault cutting the lowercrust and possibly the Moho (eg Henstock et al1997 Parsons and Hart 1999) Aseismic slip onthe downdip extension of the seismogenic zone (asin Tse and Rice 1986) may take place in additionto a more deeply distributed relation Results fromKenner and Segall (2001) which include viscousrelaxation in a narrow fault zone and more deeplydistributed slip find the spatiotemporal pattern ofdeformation in the post-1906 triangulation data isbest fit with a short relaxation time in the fault zoneand a longer relaxation in the underlyinghalf-space Another possibility is that the thermo-mechanical structure of the lithosphere differsbetween the two areas It is also possible that along-strike variations in fault geometry (the Big Bend) or

slip behavior (the creeping zone) may bias the com-puted relaxation time although this seems lesslikely

Although the apparent relaxation times differbetween the Carrizo Plain and the northern SanFrancisco Bay region the elastic-layer thicknessestimates are consistent in the range of 13ndash20 kmCarrizo Plain slip-rate estimates cluster near theupper bound of 37 mmyr Interestingly the CarrizoPlain data yield recurrence times greater than 160years Whereas the distribution is reasonably broadboth the mean (288 years) and median (277) repeattimes fall between the 1988 working group estimateof 296 years (WGCEP 1988) and the 1995 workinggroup estimate of 206 years (WGCEP 1995) Theaverage coseismic slip for the predicted events isroughly 10 meters There are significant correlationsbetween parameters (between H and tR as dis-cussed previously as well as between tR and T)which means that any additional data that constrain

FIG 9 Comparison of Savage-Prescott model to the SCEC velocity data for the Carrizo Plain A Misfit as a functionof elastic-layer thickness and relaxation time B Best-fitting San Andreas slip rate corresponding to the elastic-layerthickness and relaxation time C Fit to the data Effect of the Hosgri fault is removed employing a dislocation model withslip rate of 3 mmyr and depth of 10 km

72 PAUL SEGALL

some of the parameters will help resolve others aswell

It should be emphasized that the paleoseismicobservations of Grant and Sieh (1994) show an irreg-ular pattern of earthquakes that cannot be explainedby the simple two-dimensional model used here tointerpret the geodetic observations Nevertheless itis interesting that the GPS data appear to have someresolving power for average recurrence interval

San Francisco Bay Data

In order to eliminate any geographic bias in theinterpretation associated with possibly differentcrustal structure and temperature distribution Inext consider the velocity profile across the SanAndreas fault north of the San Francisco Bay asdetermined by repeated GPS observations (Prescottet al 2001) as shown in Figure 14 The GPS sitesare sufficiently close to the post-1906 triangulationnetworks that it is reasonable to assume that the

crustal structure and thermal profile are the samefor both data sets In the following the GPS-derivedvelocities are rotated into fault-parallel and perpen-dicular components Only the fault-parallel motionsare modeled here although as pointed out by Pres-cott et al (2001) and Murray and Segall (2001)there is significant fault-normal motion in the east-ern coast ranges at the western edge of the CentralValley

The model is the same as discussed previouslywith three principal faults the San Andreas Hay-wardndashRodgers Creek and ConcordndashGreen Valleyfaults The parameters are the elastic plate thick-ness H Maxwell relaxation time tR the recurrencetime for great earthquakes on the SAF TSAF the sliprate on the SAF and the recurrence timeTHRC time since past event tHRC and slip rate on the Hayward-Rodgers Creek fault system Thetime since the last great earthquake on the northernSan Andreas fault tSAF is of course known Theeffect of the ConcordndashGreen Valley fault is modeled

smiddotSAFsmiddotHRC

FIG 10 Comparison of Savage-Prescott model to the SCEC velocity data for the Carrizo Plain with slip rate con-strained to 34 mmyr Effect of San GregoriondashHosgri fault has been removed with a dislocation model with slip rate of 3mmyr and depth of 10 km A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

ESTIMATES OF SLIP RATE 73

kinematically with a simple screw dislocationwhich adds two parameters the fault slip rate andlocking depth It is certainly inconsistent to ignoreviscoelastic effects for the ConcordndashGreen Valleyfault but I do so for three reasons (a) much less isknown about this fault (b) earthquakes there arepresumably least well modeled by infinitely longruptures and (c) it keeps the number of parametersreasonably tractable In essence the effect of theConcordndashGreen Valley is removed empirically and Imake no attempt to gain further insights into itsmechanical behavior Future work should addressand hopefully rectify this simplification

A priori bounds on the parameters are listed inTable 1 The lower bound on the lithospheric thick-ness comes from a conservative estimate of thedepth extent of slip in large earthquakes In fact itis more likely that slip in 1906 extended at least to15 km (Matthews and Segall 1993) as it certainlydid in the Loma Prieta earthquake (eg Beroza1991) The bounds on relaxation time come from thepreliminary analysis above (Figs 12 and 13) The

lower bound on TSAF comes from the elapsed timesince 1906 the upper bound from limits on theamount of coseismic slip (see below) The lowerbounds on tHRC and THRC are derived from the his-torical record there is no known earthquake on thenorthern HaywardndashRodgers Creek system since atleast 1775 (Toppozada and Borchardt 1998WGCEP 1999) The lower and upper bounds on sliprates were based loosely on the Northern CaliforniaWorking group (WGCEP 1999 Table 2) The Work-ing Group listed 2 s uncertainties of = 24 plusmn 3mmyr = 9 plusmn 2 mmyr and = 5 plusmn m 3mmyr

The estimation problem is to find the choice ofthe nine parameters in Table 1 that minimize themisfit to two geodetic data sets the present-day spa-tial distribution of deformation in the north SanFrancisco Bay region (Fig 14) and the time-depen-dent decay in strain rate following the 1906 earth-quake (Fig 11) For a non-linear problem such asthis one always needs to be concerned about localminima However preliminary testing with a simu-

smiddotSAFsmiddotHRC smiddotHRC

FIG 11 Post-1906 strain rate from triangulation data in the San Francisco Bay area Model fit to the model withpostseismic deformation only A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

74 PAUL SEGALL

lated annealing algorithm did not show evidence oflocal minima so the remaining analysis was doneusing a Sequential Quadratic Programming methodwith BFGS Hessian updates (eg Gill et al 1981)within Matlab The method allows upper and lowerbounds to be placed on all parameters as well asgeneralized inequality constraints The latter areneeded because tHRC is strictly bounded by tHRC poundTHRC In the context of the Savage and Prescottmodel earthquakes are periodic and it makes nosense for the elapsed time since the last event toexceed the average recurrence time In addition Ihave found it desirable to constrain the maximumcoseismic slip on both the San Andreas and Hay-ward faults This provides constraints of the form

and

Noting that the coseismic slip should be interpretedas an average over the fault in this two-dimensional

model I set meters and

metersIt should be noted that the GPS network does not

extend very far west of the San Andreas fault Forthis reason the data do not adequately constrain the

total motion across the plate boundary A reasonableapproach would be to include data from Pacificplate sites such as Hawaii I chose simply to con-strain the sum of the slip rates on all three faults tobe no greater than 40 mmyr (eg Argus and Gordon2001) This constraint significantly affects the solu-tion estimates without the cumulative slip con-straint tended to put the maximum slip rate allowedon all three faults exceeding the total displacementrate between the Sierra NevadandashGreat Valley blockand the Pacific plate

The optimal solution is found for H ~ 184 km tR~ 345 years TSAF = 280 years ~ 25 mmyrtHRC = 225 years THRC = 4286 years and ~ 7mmyr The ConcordndashGreen Valley fault slips at

~ 8 mmyr below 22 km A surprising resultis that the predicted recurrence time on the SanAndreas is reasonable in the sense of requiring 70-meter slip events every 280 years In comparisonSchwartz et al (1998) suggested that the penulti-mate event on the north coast segment of the SanAndreas occurred in the mid-1600s possibly in theinterval 1632ndash1659 corresponding to a recurrenceinterval of roughly 260 years Also surprising is thatthe time since the last earthquake on the Haywardndash

smiddotSAF TSAFtimes DuSAFmaxpound smiddotHRC THRCtimes DuHRC

maxpound

DuSAFmax

70= DuHRCmax

30=

smiddotSAFsmiddotHRC

smiddotHRC

FIG 12 Bootstrap results for post-1906 data in the San Francisco Bay area A Relaxation time B Elastic-layerthickness (1000 bootstrap resamples)

ESTIMATES OF SLIP RATE 75

Rodgers Creek fault is the minimum allowed 225years A comparison of the model fit to the data setsis shown in Figure 15 Generally speaking the dataare fit quite well with the possible exception of thestrain rate in the first post-1906 epoch The uncer-tainties in the post-1906 strain rates were actuallydecreased by a factor of two to ensure that these dataare fairly weighted compared to the GPS velocitiesDecreasing the uncertainties in the strain rates fur-ther to fit high strain rate in the immediatepost-1906 epoch causes a very significant increasein the misfit to the modern GPS data This is mostlikely a result of the large uncertainties in the trian-gulation data but may also indicate that the physi-cal model is inadequate

How is it that the geodetic data have any capabil-ity of resolving the recurrence interval on the SanAndreas fault Recall that there are three times inthe physical model the time since the last majorearthquake t the asthenospheric relaxation time tRand the recurrence interval T For the San Andreasfault we know tSAF ~93 years a priori and tR is rea-

sonably well resolved by the post-1906 strain ratetransient The estimation is thus free to vary T insuch a way as to optimize the fit to the velocity gra-dients near the San Andreas Constraints on themaximum slip in the earthquake help to furthertighten those constraints

In order to determine how well the data constrainthe model parameters I ran 500 bootstrap resam-ples and optimizations using the best-fitting modelas the starting point of the optimization As beforethe residuals were resampled and added to the pre-dicted data from the optimal model in order toensure that each observation point contributes to thesolution The results of the bootstrap are shown inFigure 16 The posterior distribution for recurrencetime on the San Andreas fault TSAF is quite broadwith a peak near the maximum allowed of (7000mm)(19 mmyr) ~370 years In contrast the poste-rior distribution for recurrence time on the Hay-wardndashRodgers Creek fault THRC is bimodal withprominent peaks near the minimum ~225 years andmaximum allowed value (3000 mm)(7 mmyr)

FIG 13 Bootstrap results for SCEC data along the Carrizo Plain segment of the San Andreas fault A Elastic-layerthickness B Relaxation time Note that 200 years was the upper bound considered C Recurrence time Recurrencetimes in excess of 400 years were not considered as these would lead to excessive coseismic slip D Slip rate

76 PAUL SEGALL

~ 428 years The distributions for tHRC the timesince the most recent HaywardndashRodgers Creekearthquake is strongly concentrated near the mini-

mum allowed range The distribution of slip rates onthe San Andreas and HaywardndashRodgers Creekfaults are bimodal with peaks at 19 mmyr and 25

FIG 14 GPS-derived velocities for the north San Francisco Bay region after Prescott et al (2001) AbbreviationsSAF = San Andreas fault HF = Hayward fault CF = Calaveras fault RGF = Rodgers Creek fault CGVF = ConcordndashGreen Valley fault

TABLE 1 A priori Bounds on Parameters

Parameter Symbol Min Max

Lithospheric thickness km H 10 30

Relaxation time yr tR 10 200

SAF recurrence interval yr TSAF 100 500

Time since last Hayward event yr tHRC 225 690

Hayward recurrence interval yr THRC 225 700

Green valley fault locking depth km d 05 15

SAF slip rate my 0019 0027

Hayward slip rate myr 0007 0013

Green valley slip rate myr 0002 0008

smiddotSAF

smiddotHRC

smiddotCGV

ESTIMATES OF SLIP RATE 77

mmyr (SAF) and 7 mmyr and 13 mmyr (HRC)These results still must be viewed with some cau-tion in that I have not demonstrated that the boot-strap optimizations were immune to local minima

The bootstrap calculations suggest that the con-straint on maximum coseismic slip does play astrong role in the results To explore this further Ivaried the maximum coseismic slip allowed on thetwo faults The results (Table 2) demonstrate that theSan Andreas recurrence interval is not fully con-trolled by the constraint on the maximum coseismicslip In contrast the recurrence interval on the Hay-ward fault is totally controlled by the maximum slipallowed (for a maximum slip of 10 meters the recur-rence interval reaches the upper bound of 700years) Notice also that the time since the last Hay-wardndashRodgers Creek event is always at the mini-mum value allowed (225 years) The fit to the data isslightly better for large infrequent earthquakes

Discussion

What has been learned from the modeling pre-sented here It seems clear that geodetic data

should not be analyzed in a vacuum The resolutionof the data alone is insufficient to address the prob-lems of greatest interest However when combinedwith prior information from geologic studies GPSdata appear to be able to resolve parameters of con-siderable interest such as the average recurrenceinterval between great earthquakes on the SanAndreas fault Using a simple model of a repeatingdislocation rupturing an elastic plate overlying aMaxwell visco-elastic medium I found an averagerecurrence interval of roughly 280 years on thenorth coast segment of the San Andreas faultAlthough the uncertainties in this estimate arelarge the result is in surprisingly good accordancewith recent paleoseismic results (Schwartz et al1998) that suggest an event similar to 1906 in themid 1600s

The results for the HaywardndashRodgers Creek faultare also not unreasonable According to the GPSdata the most recent event here occurred in ~1775regardless of the estimated recurrence time Thisresult is not inconsistent with the paleoseismic find-ing that the most recent event on the Rodgers Creekfault took place in the interval 1670ndash1776

TABLE 2 Influence of Maximum Coseismic Slip on Estimated Parameters1

Parameter

Lithospheric thickness H km 150 184 220

Relaxation time tR yr 357 345 340

SAF recurrence interval TSAF yr 220 280 337

Limiting interval yr 290 368 526

Time since last Hayward event tHRC yr 225 225 225

Hayward recurrence interval THRC yr 2857 4286 700

Limiting interval yr 285 428 714

Green Valley fault locking depth d km 11 22 47

SAF slip rate myr 0025 0025 00249

Hayward slip rate myr 0007 0007 00071

Green valley slip rate myr 0008 0008 0008

1Limiting interval is computed using the estimated slip rate for that fault not the minimum slip rate that would yield a longer upper bound

DuSAFmax

55=

DuHRCmax

20=

DuSAFmax

70=

DuHRCmax

30=

DuSAFmax

100=

DuHRCmax

50=

DuSAFmax

smiddotSAFcurren

DuHRCmax

smiddotHRCcurren

smiddotSAF

smiddotHRC

smiddotCGV

78 PAUL SEGALL

(Schwartz et al 1993) and on the northern Hay-ward fault in 1640ndash1776 (HFPEG 1999) The GPSdata alone yield limited information about the aver-age recurrence time on the HaywardndashRodgers Creekfault However adding the reasonable bound thatslip not exceed 2 meters on average yields a recur-rence interval of 285 years The average slip in the1868 Hayward fault earthquake was approximately2 meters (Yu and Segall 1996) In comparisontrenching at the El Cerrito site on the northern Hay-ward fault suggests a recurrence interval of possiblyless than 270 years but no more than 710 years(HFPEG 1999)

It should not need to be stated that earthquakesare neither periodic nor especially in the case of theHaywardndashRodgers Creek faults infinitely long Atwo-dimensional model cannot describe the situa-tion at a given trench site where ruptures mainly tothe north or south of the site cause some offset in the

trench Nevertheless the fact that the results fromsuch a simple model are not unreasonable is encour-aging One useful attribute of the models exploredhere are that they are testable in the sense of predict-ing future behavior Running the model forward intime to the next San Andreas earthquake yields thevelocity distributions shown in Figure 17 Of coursea north HaywardndashRodgers Creek earthquake mayoccur prior to the next San Andreas event

The Maxwell relaxation time is reasonablywell estimated by the data I find an optimal valueof about 35 years which corresponds to a viscos-ity of 16 times 1019 Pa-s It should be noted that theestimates of all of the model parameters is only asgood as the mechanical model used to interpretthe data If as seems quite likely there was after-slip on the San Andreas below the 1906 rupturezone the estimate of the viscosity is some combi-nation of the effects of afterslip and distributed

FIG 15 Fit of model to north San Francisco Bay data Maximum slip on Hayward fault is 30 m and on the SAF is70 m

ESTIMATES OF SLIP RATE 79

relaxation Li and Rice (1987) presented anapproximate method for including post-seismicfault slip at constant resistive stress with a relax-ing asthenosphere in the context of a repeatingperiodic earthquake sequence Combining suchan approach with appropriate constitutive laws forstable fault creep as in Tse and Rice (1986)would presumably provide a more physically real-istic description of fault behavior Clearly three-dimensional effects are also important especiallyfor earthquakes on the Rodgers Creek and Hay-ward faults Finally the effects of non-uniform

elastic properties are known to bias interpreta-tions based on simple homogeneous models Elas-tic moduli increase with depth which has theknown effect of biasing the depth distribution offault slip Similarly elastic properties may varywith lithology across the major fault zones whichthemselves may represent zones of more compli-ant rock Finite elements are certainly the mostgeneral way to compute forward models but arenot yet feasible for the optimization and bootstrapmethods as computed here which literally testedthousands of possible models

FIG 16 Five hundred bootstrap estimates of model parameters fit to the north San Francisco Bay data 90 confi-dence bounds are listed at the top of each panel The vertical line in the histograms for recurrence times TSAF and THRCshow the upper bounds corresponding to the minimum slip rate and maximum allowed coseismic slip respectively Thehorizontal bar in the panel for the San Andreas recurrence time shows the preferred estimate from Schwartz et al (1998)The horizontal bar in the panel for tH time since the last Hayward earthquake shows the range of dates compatible withpaleoseismic estimates of the penultimate HaywardndashRodgers Creek event

80 PAUL SEGALL

Acknowledgments

George Thompson set a standard that geophysi-cal models no matter how elegant must be consis-tent with geologic observations This work wasinspired by that example I would like to thank WillPrescott for discussions about the Bay Area defor-mation measurements and Mark Matthews for manyyears ago introducing me to Bayesrsquo theorem WayneThatcher Mark Murray Jeff McGuire and PeterCervelli provided valuable comments at early stagesin this study I thank Jim Savage Duncan Agnewand Shelley Kenner for their thoughtful reviewsGary Ernst and Simon Klemperer carefully editedthe manuscript

REFERENCES

Argus D F and Gordon R G 2001 Present tectonicmotion across the Coast Ranges and San Andreas faultsystem in central California Geological Society ofAmerica Bulletin v 113 p 1580ndash1592

Beroza G C 1991 Near-source modeling of the LomaPrieta earthquake Evidence for heterogeneous slipand implications for earthquake hazard Bulletin of theSeismological Society of America v 81 p 1603ndash1621

Cohen S C 1982 A multilayer model of time-dependentdeformation following an earthquake on a strike-slipfault Journal of Geophysical Research v 87 p 5409ndash5421

HFPEG (Hayward Fault Paleoearthquake Group) 1999Timing of paleoearthquakes on the northern HaywardfaultmdashPreliminary evidence in El Cerrito CaliforniaUS Geological Survey Open File Report 99-318

Henstock T J Levander A and Hole J A 1997Deformation in the lower crust of the San Andreas faultsystem in northern California Science v 278 p 650ndash653

Feigl K L King R W and Jordan T H 1990 Geo-detic measurement of tectonic deformation in theSanta Maria fold and thrust belt California Journal ofGeophysical Research v 95 p 2679ndash2699

Gill P E Murray W and Wright M H 1981 Practicaloptimization London UK Academic Press 401 p

Grant L B and Sieh K 1994 Paleoseismic evidence ofclustered earthquakes on the San Andreas fault in the

FIG 17 Simulation of Bay Area velocity field as a function of time Each line is separated by roughly 27 years

70 PAUL SEGALL

model is H = 19 km with a relaxation time of 17years (Fig 11) substantially less than found fromthe Carrizo Plain interseismic data1

A tentative conclusion of this analysis is that theCarrizo Plain interseismic data require a relaxationtime in excess of 50 years whereas the post-1906triangulation data require relaxation times of 25years or less In order to test whether this discrep-ancy is truly required by the data I estimated theuncertainty in the two relaxation times using a boot-strap resampling procedure As discussed previ-ously the residuals to the best-fitting model wererandomly resampled with replacement and added

to the model predictions for each bootstrap resam-ple This approach seems best for small data setsand ensures that there were data at each observationposition (interseismic data) and time (postseismicdata)

For the post-1906 transient admissible relax-ation times range from 10 years to 30 years (Fig 12)On the other hand the present distribution of defor-mation on the Carrizo Plain demands relaxationtimes in excess of 35 years (Fig 13) Thus we cansafely conclude that there is no single relaxationtime that fits both data sets

There are a number of possible explanations forthis apparent discrepancy One possibility is thatthe model of a fully ruptured elastic plate overlyinga linear Maxwell half-space does not capture all ofthe relevant physics For example nonlinear rheol-ogy may be important The data are consistent withthe expectation that the effective viscosity is low

1 Note that the discrepancy is not due to the correction forinterseismic deformation in the post-1906 data Assuming arecurrence time of 250 years and a slip rate on the SAF of 25mmyr the optimal relaxation time is 228 years with an elas-tic-layer thickness of 167 km

FIG 8 Savage-Prescott model for (A) T tR = 1 and (B) T tR = 5 t is the time since the last earthquake T is theearthquake repeat time and tR is the Maxwell relaxation time In these calculations the earthquakes rupture the entireelastic layer

ESTIMATES OF SLIP RATE 71

early in the seismic cycle when the stresses arelarge and increases late in the cycle when thestresses have largely dissipated In addition recentreflection and refraction results in northern Califor-nia show the San Andreas fault cutting the lowercrust and possibly the Moho (eg Henstock et al1997 Parsons and Hart 1999) Aseismic slip onthe downdip extension of the seismogenic zone (asin Tse and Rice 1986) may take place in additionto a more deeply distributed relation Results fromKenner and Segall (2001) which include viscousrelaxation in a narrow fault zone and more deeplydistributed slip find the spatiotemporal pattern ofdeformation in the post-1906 triangulation data isbest fit with a short relaxation time in the fault zoneand a longer relaxation in the underlyinghalf-space Another possibility is that the thermo-mechanical structure of the lithosphere differsbetween the two areas It is also possible that along-strike variations in fault geometry (the Big Bend) or

slip behavior (the creeping zone) may bias the com-puted relaxation time although this seems lesslikely

Although the apparent relaxation times differbetween the Carrizo Plain and the northern SanFrancisco Bay region the elastic-layer thicknessestimates are consistent in the range of 13ndash20 kmCarrizo Plain slip-rate estimates cluster near theupper bound of 37 mmyr Interestingly the CarrizoPlain data yield recurrence times greater than 160years Whereas the distribution is reasonably broadboth the mean (288 years) and median (277) repeattimes fall between the 1988 working group estimateof 296 years (WGCEP 1988) and the 1995 workinggroup estimate of 206 years (WGCEP 1995) Theaverage coseismic slip for the predicted events isroughly 10 meters There are significant correlationsbetween parameters (between H and tR as dis-cussed previously as well as between tR and T)which means that any additional data that constrain

FIG 9 Comparison of Savage-Prescott model to the SCEC velocity data for the Carrizo Plain A Misfit as a functionof elastic-layer thickness and relaxation time B Best-fitting San Andreas slip rate corresponding to the elastic-layerthickness and relaxation time C Fit to the data Effect of the Hosgri fault is removed employing a dislocation model withslip rate of 3 mmyr and depth of 10 km

72 PAUL SEGALL

some of the parameters will help resolve others aswell

It should be emphasized that the paleoseismicobservations of Grant and Sieh (1994) show an irreg-ular pattern of earthquakes that cannot be explainedby the simple two-dimensional model used here tointerpret the geodetic observations Nevertheless itis interesting that the GPS data appear to have someresolving power for average recurrence interval

San Francisco Bay Data

In order to eliminate any geographic bias in theinterpretation associated with possibly differentcrustal structure and temperature distribution Inext consider the velocity profile across the SanAndreas fault north of the San Francisco Bay asdetermined by repeated GPS observations (Prescottet al 2001) as shown in Figure 14 The GPS sitesare sufficiently close to the post-1906 triangulationnetworks that it is reasonable to assume that the

crustal structure and thermal profile are the samefor both data sets In the following the GPS-derivedvelocities are rotated into fault-parallel and perpen-dicular components Only the fault-parallel motionsare modeled here although as pointed out by Pres-cott et al (2001) and Murray and Segall (2001)there is significant fault-normal motion in the east-ern coast ranges at the western edge of the CentralValley

The model is the same as discussed previouslywith three principal faults the San Andreas Hay-wardndashRodgers Creek and ConcordndashGreen Valleyfaults The parameters are the elastic plate thick-ness H Maxwell relaxation time tR the recurrencetime for great earthquakes on the SAF TSAF the sliprate on the SAF and the recurrence timeTHRC time since past event tHRC and slip rate on the Hayward-Rodgers Creek fault system Thetime since the last great earthquake on the northernSan Andreas fault tSAF is of course known Theeffect of the ConcordndashGreen Valley fault is modeled

smiddotSAFsmiddotHRC

FIG 10 Comparison of Savage-Prescott model to the SCEC velocity data for the Carrizo Plain with slip rate con-strained to 34 mmyr Effect of San GregoriondashHosgri fault has been removed with a dislocation model with slip rate of 3mmyr and depth of 10 km A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

ESTIMATES OF SLIP RATE 73

kinematically with a simple screw dislocationwhich adds two parameters the fault slip rate andlocking depth It is certainly inconsistent to ignoreviscoelastic effects for the ConcordndashGreen Valleyfault but I do so for three reasons (a) much less isknown about this fault (b) earthquakes there arepresumably least well modeled by infinitely longruptures and (c) it keeps the number of parametersreasonably tractable In essence the effect of theConcordndashGreen Valley is removed empirically and Imake no attempt to gain further insights into itsmechanical behavior Future work should addressand hopefully rectify this simplification

A priori bounds on the parameters are listed inTable 1 The lower bound on the lithospheric thick-ness comes from a conservative estimate of thedepth extent of slip in large earthquakes In fact itis more likely that slip in 1906 extended at least to15 km (Matthews and Segall 1993) as it certainlydid in the Loma Prieta earthquake (eg Beroza1991) The bounds on relaxation time come from thepreliminary analysis above (Figs 12 and 13) The

lower bound on TSAF comes from the elapsed timesince 1906 the upper bound from limits on theamount of coseismic slip (see below) The lowerbounds on tHRC and THRC are derived from the his-torical record there is no known earthquake on thenorthern HaywardndashRodgers Creek system since atleast 1775 (Toppozada and Borchardt 1998WGCEP 1999) The lower and upper bounds on sliprates were based loosely on the Northern CaliforniaWorking group (WGCEP 1999 Table 2) The Work-ing Group listed 2 s uncertainties of = 24 plusmn 3mmyr = 9 plusmn 2 mmyr and = 5 plusmn m 3mmyr

The estimation problem is to find the choice ofthe nine parameters in Table 1 that minimize themisfit to two geodetic data sets the present-day spa-tial distribution of deformation in the north SanFrancisco Bay region (Fig 14) and the time-depen-dent decay in strain rate following the 1906 earth-quake (Fig 11) For a non-linear problem such asthis one always needs to be concerned about localminima However preliminary testing with a simu-

smiddotSAFsmiddotHRC smiddotHRC

FIG 11 Post-1906 strain rate from triangulation data in the San Francisco Bay area Model fit to the model withpostseismic deformation only A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

74 PAUL SEGALL

lated annealing algorithm did not show evidence oflocal minima so the remaining analysis was doneusing a Sequential Quadratic Programming methodwith BFGS Hessian updates (eg Gill et al 1981)within Matlab The method allows upper and lowerbounds to be placed on all parameters as well asgeneralized inequality constraints The latter areneeded because tHRC is strictly bounded by tHRC poundTHRC In the context of the Savage and Prescottmodel earthquakes are periodic and it makes nosense for the elapsed time since the last event toexceed the average recurrence time In addition Ihave found it desirable to constrain the maximumcoseismic slip on both the San Andreas and Hay-ward faults This provides constraints of the form

and

Noting that the coseismic slip should be interpretedas an average over the fault in this two-dimensional

model I set meters and

metersIt should be noted that the GPS network does not

extend very far west of the San Andreas fault Forthis reason the data do not adequately constrain the

total motion across the plate boundary A reasonableapproach would be to include data from Pacificplate sites such as Hawaii I chose simply to con-strain the sum of the slip rates on all three faults tobe no greater than 40 mmyr (eg Argus and Gordon2001) This constraint significantly affects the solu-tion estimates without the cumulative slip con-straint tended to put the maximum slip rate allowedon all three faults exceeding the total displacementrate between the Sierra NevadandashGreat Valley blockand the Pacific plate

The optimal solution is found for H ~ 184 km tR~ 345 years TSAF = 280 years ~ 25 mmyrtHRC = 225 years THRC = 4286 years and ~ 7mmyr The ConcordndashGreen Valley fault slips at

~ 8 mmyr below 22 km A surprising resultis that the predicted recurrence time on the SanAndreas is reasonable in the sense of requiring 70-meter slip events every 280 years In comparisonSchwartz et al (1998) suggested that the penulti-mate event on the north coast segment of the SanAndreas occurred in the mid-1600s possibly in theinterval 1632ndash1659 corresponding to a recurrenceinterval of roughly 260 years Also surprising is thatthe time since the last earthquake on the Haywardndash

smiddotSAF TSAFtimes DuSAFmaxpound smiddotHRC THRCtimes DuHRC

maxpound

DuSAFmax

70= DuHRCmax

30=

smiddotSAFsmiddotHRC

smiddotHRC

FIG 12 Bootstrap results for post-1906 data in the San Francisco Bay area A Relaxation time B Elastic-layerthickness (1000 bootstrap resamples)

ESTIMATES OF SLIP RATE 75

Rodgers Creek fault is the minimum allowed 225years A comparison of the model fit to the data setsis shown in Figure 15 Generally speaking the dataare fit quite well with the possible exception of thestrain rate in the first post-1906 epoch The uncer-tainties in the post-1906 strain rates were actuallydecreased by a factor of two to ensure that these dataare fairly weighted compared to the GPS velocitiesDecreasing the uncertainties in the strain rates fur-ther to fit high strain rate in the immediatepost-1906 epoch causes a very significant increasein the misfit to the modern GPS data This is mostlikely a result of the large uncertainties in the trian-gulation data but may also indicate that the physi-cal model is inadequate

How is it that the geodetic data have any capabil-ity of resolving the recurrence interval on the SanAndreas fault Recall that there are three times inthe physical model the time since the last majorearthquake t the asthenospheric relaxation time tRand the recurrence interval T For the San Andreasfault we know tSAF ~93 years a priori and tR is rea-

sonably well resolved by the post-1906 strain ratetransient The estimation is thus free to vary T insuch a way as to optimize the fit to the velocity gra-dients near the San Andreas Constraints on themaximum slip in the earthquake help to furthertighten those constraints

In order to determine how well the data constrainthe model parameters I ran 500 bootstrap resam-ples and optimizations using the best-fitting modelas the starting point of the optimization As beforethe residuals were resampled and added to the pre-dicted data from the optimal model in order toensure that each observation point contributes to thesolution The results of the bootstrap are shown inFigure 16 The posterior distribution for recurrencetime on the San Andreas fault TSAF is quite broadwith a peak near the maximum allowed of (7000mm)(19 mmyr) ~370 years In contrast the poste-rior distribution for recurrence time on the Hay-wardndashRodgers Creek fault THRC is bimodal withprominent peaks near the minimum ~225 years andmaximum allowed value (3000 mm)(7 mmyr)

FIG 13 Bootstrap results for SCEC data along the Carrizo Plain segment of the San Andreas fault A Elastic-layerthickness B Relaxation time Note that 200 years was the upper bound considered C Recurrence time Recurrencetimes in excess of 400 years were not considered as these would lead to excessive coseismic slip D Slip rate

76 PAUL SEGALL

~ 428 years The distributions for tHRC the timesince the most recent HaywardndashRodgers Creekearthquake is strongly concentrated near the mini-

mum allowed range The distribution of slip rates onthe San Andreas and HaywardndashRodgers Creekfaults are bimodal with peaks at 19 mmyr and 25

FIG 14 GPS-derived velocities for the north San Francisco Bay region after Prescott et al (2001) AbbreviationsSAF = San Andreas fault HF = Hayward fault CF = Calaveras fault RGF = Rodgers Creek fault CGVF = ConcordndashGreen Valley fault

TABLE 1 A priori Bounds on Parameters

Parameter Symbol Min Max

Lithospheric thickness km H 10 30

Relaxation time yr tR 10 200

SAF recurrence interval yr TSAF 100 500

Time since last Hayward event yr tHRC 225 690

Hayward recurrence interval yr THRC 225 700

Green valley fault locking depth km d 05 15

SAF slip rate my 0019 0027

Hayward slip rate myr 0007 0013

Green valley slip rate myr 0002 0008

smiddotSAF

smiddotHRC

smiddotCGV

ESTIMATES OF SLIP RATE 77

mmyr (SAF) and 7 mmyr and 13 mmyr (HRC)These results still must be viewed with some cau-tion in that I have not demonstrated that the boot-strap optimizations were immune to local minima

The bootstrap calculations suggest that the con-straint on maximum coseismic slip does play astrong role in the results To explore this further Ivaried the maximum coseismic slip allowed on thetwo faults The results (Table 2) demonstrate that theSan Andreas recurrence interval is not fully con-trolled by the constraint on the maximum coseismicslip In contrast the recurrence interval on the Hay-ward fault is totally controlled by the maximum slipallowed (for a maximum slip of 10 meters the recur-rence interval reaches the upper bound of 700years) Notice also that the time since the last Hay-wardndashRodgers Creek event is always at the mini-mum value allowed (225 years) The fit to the data isslightly better for large infrequent earthquakes

Discussion

What has been learned from the modeling pre-sented here It seems clear that geodetic data

should not be analyzed in a vacuum The resolutionof the data alone is insufficient to address the prob-lems of greatest interest However when combinedwith prior information from geologic studies GPSdata appear to be able to resolve parameters of con-siderable interest such as the average recurrenceinterval between great earthquakes on the SanAndreas fault Using a simple model of a repeatingdislocation rupturing an elastic plate overlying aMaxwell visco-elastic medium I found an averagerecurrence interval of roughly 280 years on thenorth coast segment of the San Andreas faultAlthough the uncertainties in this estimate arelarge the result is in surprisingly good accordancewith recent paleoseismic results (Schwartz et al1998) that suggest an event similar to 1906 in themid 1600s

The results for the HaywardndashRodgers Creek faultare also not unreasonable According to the GPSdata the most recent event here occurred in ~1775regardless of the estimated recurrence time Thisresult is not inconsistent with the paleoseismic find-ing that the most recent event on the Rodgers Creekfault took place in the interval 1670ndash1776

TABLE 2 Influence of Maximum Coseismic Slip on Estimated Parameters1

Parameter

Lithospheric thickness H km 150 184 220

Relaxation time tR yr 357 345 340

SAF recurrence interval TSAF yr 220 280 337

Limiting interval yr 290 368 526

Time since last Hayward event tHRC yr 225 225 225

Hayward recurrence interval THRC yr 2857 4286 700

Limiting interval yr 285 428 714

Green Valley fault locking depth d km 11 22 47

SAF slip rate myr 0025 0025 00249

Hayward slip rate myr 0007 0007 00071

Green valley slip rate myr 0008 0008 0008

1Limiting interval is computed using the estimated slip rate for that fault not the minimum slip rate that would yield a longer upper bound

DuSAFmax

55=

DuHRCmax

20=

DuSAFmax

70=

DuHRCmax

30=

DuSAFmax

100=

DuHRCmax

50=

DuSAFmax

smiddotSAFcurren

DuHRCmax

smiddotHRCcurren

smiddotSAF

smiddotHRC

smiddotCGV

78 PAUL SEGALL

(Schwartz et al 1993) and on the northern Hay-ward fault in 1640ndash1776 (HFPEG 1999) The GPSdata alone yield limited information about the aver-age recurrence time on the HaywardndashRodgers Creekfault However adding the reasonable bound thatslip not exceed 2 meters on average yields a recur-rence interval of 285 years The average slip in the1868 Hayward fault earthquake was approximately2 meters (Yu and Segall 1996) In comparisontrenching at the El Cerrito site on the northern Hay-ward fault suggests a recurrence interval of possiblyless than 270 years but no more than 710 years(HFPEG 1999)

It should not need to be stated that earthquakesare neither periodic nor especially in the case of theHaywardndashRodgers Creek faults infinitely long Atwo-dimensional model cannot describe the situa-tion at a given trench site where ruptures mainly tothe north or south of the site cause some offset in the

trench Nevertheless the fact that the results fromsuch a simple model are not unreasonable is encour-aging One useful attribute of the models exploredhere are that they are testable in the sense of predict-ing future behavior Running the model forward intime to the next San Andreas earthquake yields thevelocity distributions shown in Figure 17 Of coursea north HaywardndashRodgers Creek earthquake mayoccur prior to the next San Andreas event

The Maxwell relaxation time is reasonablywell estimated by the data I find an optimal valueof about 35 years which corresponds to a viscos-ity of 16 times 1019 Pa-s It should be noted that theestimates of all of the model parameters is only asgood as the mechanical model used to interpretthe data If as seems quite likely there was after-slip on the San Andreas below the 1906 rupturezone the estimate of the viscosity is some combi-nation of the effects of afterslip and distributed

FIG 15 Fit of model to north San Francisco Bay data Maximum slip on Hayward fault is 30 m and on the SAF is70 m

ESTIMATES OF SLIP RATE 79

relaxation Li and Rice (1987) presented anapproximate method for including post-seismicfault slip at constant resistive stress with a relax-ing asthenosphere in the context of a repeatingperiodic earthquake sequence Combining suchan approach with appropriate constitutive laws forstable fault creep as in Tse and Rice (1986)would presumably provide a more physically real-istic description of fault behavior Clearly three-dimensional effects are also important especiallyfor earthquakes on the Rodgers Creek and Hay-ward faults Finally the effects of non-uniform

elastic properties are known to bias interpreta-tions based on simple homogeneous models Elas-tic moduli increase with depth which has theknown effect of biasing the depth distribution offault slip Similarly elastic properties may varywith lithology across the major fault zones whichthemselves may represent zones of more compli-ant rock Finite elements are certainly the mostgeneral way to compute forward models but arenot yet feasible for the optimization and bootstrapmethods as computed here which literally testedthousands of possible models

FIG 16 Five hundred bootstrap estimates of model parameters fit to the north San Francisco Bay data 90 confi-dence bounds are listed at the top of each panel The vertical line in the histograms for recurrence times TSAF and THRCshow the upper bounds corresponding to the minimum slip rate and maximum allowed coseismic slip respectively Thehorizontal bar in the panel for the San Andreas recurrence time shows the preferred estimate from Schwartz et al (1998)The horizontal bar in the panel for tH time since the last Hayward earthquake shows the range of dates compatible withpaleoseismic estimates of the penultimate HaywardndashRodgers Creek event

80 PAUL SEGALL

Acknowledgments

George Thompson set a standard that geophysi-cal models no matter how elegant must be consis-tent with geologic observations This work wasinspired by that example I would like to thank WillPrescott for discussions about the Bay Area defor-mation measurements and Mark Matthews for manyyears ago introducing me to Bayesrsquo theorem WayneThatcher Mark Murray Jeff McGuire and PeterCervelli provided valuable comments at early stagesin this study I thank Jim Savage Duncan Agnewand Shelley Kenner for their thoughtful reviewsGary Ernst and Simon Klemperer carefully editedthe manuscript

REFERENCES

Argus D F and Gordon R G 2001 Present tectonicmotion across the Coast Ranges and San Andreas faultsystem in central California Geological Society ofAmerica Bulletin v 113 p 1580ndash1592

Beroza G C 1991 Near-source modeling of the LomaPrieta earthquake Evidence for heterogeneous slipand implications for earthquake hazard Bulletin of theSeismological Society of America v 81 p 1603ndash1621

Cohen S C 1982 A multilayer model of time-dependentdeformation following an earthquake on a strike-slipfault Journal of Geophysical Research v 87 p 5409ndash5421

HFPEG (Hayward Fault Paleoearthquake Group) 1999Timing of paleoearthquakes on the northern HaywardfaultmdashPreliminary evidence in El Cerrito CaliforniaUS Geological Survey Open File Report 99-318

Henstock T J Levander A and Hole J A 1997Deformation in the lower crust of the San Andreas faultsystem in northern California Science v 278 p 650ndash653

Feigl K L King R W and Jordan T H 1990 Geo-detic measurement of tectonic deformation in theSanta Maria fold and thrust belt California Journal ofGeophysical Research v 95 p 2679ndash2699

Gill P E Murray W and Wright M H 1981 Practicaloptimization London UK Academic Press 401 p

Grant L B and Sieh K 1994 Paleoseismic evidence ofclustered earthquakes on the San Andreas fault in the

FIG 17 Simulation of Bay Area velocity field as a function of time Each line is separated by roughly 27 years

ESTIMATES OF SLIP RATE 71

early in the seismic cycle when the stresses arelarge and increases late in the cycle when thestresses have largely dissipated In addition recentreflection and refraction results in northern Califor-nia show the San Andreas fault cutting the lowercrust and possibly the Moho (eg Henstock et al1997 Parsons and Hart 1999) Aseismic slip onthe downdip extension of the seismogenic zone (asin Tse and Rice 1986) may take place in additionto a more deeply distributed relation Results fromKenner and Segall (2001) which include viscousrelaxation in a narrow fault zone and more deeplydistributed slip find the spatiotemporal pattern ofdeformation in the post-1906 triangulation data isbest fit with a short relaxation time in the fault zoneand a longer relaxation in the underlyinghalf-space Another possibility is that the thermo-mechanical structure of the lithosphere differsbetween the two areas It is also possible that along-strike variations in fault geometry (the Big Bend) or

slip behavior (the creeping zone) may bias the com-puted relaxation time although this seems lesslikely

Although the apparent relaxation times differbetween the Carrizo Plain and the northern SanFrancisco Bay region the elastic-layer thicknessestimates are consistent in the range of 13ndash20 kmCarrizo Plain slip-rate estimates cluster near theupper bound of 37 mmyr Interestingly the CarrizoPlain data yield recurrence times greater than 160years Whereas the distribution is reasonably broadboth the mean (288 years) and median (277) repeattimes fall between the 1988 working group estimateof 296 years (WGCEP 1988) and the 1995 workinggroup estimate of 206 years (WGCEP 1995) Theaverage coseismic slip for the predicted events isroughly 10 meters There are significant correlationsbetween parameters (between H and tR as dis-cussed previously as well as between tR and T)which means that any additional data that constrain

FIG 9 Comparison of Savage-Prescott model to the SCEC velocity data for the Carrizo Plain A Misfit as a functionof elastic-layer thickness and relaxation time B Best-fitting San Andreas slip rate corresponding to the elastic-layerthickness and relaxation time C Fit to the data Effect of the Hosgri fault is removed employing a dislocation model withslip rate of 3 mmyr and depth of 10 km

72 PAUL SEGALL

some of the parameters will help resolve others aswell

It should be emphasized that the paleoseismicobservations of Grant and Sieh (1994) show an irreg-ular pattern of earthquakes that cannot be explainedby the simple two-dimensional model used here tointerpret the geodetic observations Nevertheless itis interesting that the GPS data appear to have someresolving power for average recurrence interval

San Francisco Bay Data

In order to eliminate any geographic bias in theinterpretation associated with possibly differentcrustal structure and temperature distribution Inext consider the velocity profile across the SanAndreas fault north of the San Francisco Bay asdetermined by repeated GPS observations (Prescottet al 2001) as shown in Figure 14 The GPS sitesare sufficiently close to the post-1906 triangulationnetworks that it is reasonable to assume that the

crustal structure and thermal profile are the samefor both data sets In the following the GPS-derivedvelocities are rotated into fault-parallel and perpen-dicular components Only the fault-parallel motionsare modeled here although as pointed out by Pres-cott et al (2001) and Murray and Segall (2001)there is significant fault-normal motion in the east-ern coast ranges at the western edge of the CentralValley

The model is the same as discussed previouslywith three principal faults the San Andreas Hay-wardndashRodgers Creek and ConcordndashGreen Valleyfaults The parameters are the elastic plate thick-ness H Maxwell relaxation time tR the recurrencetime for great earthquakes on the SAF TSAF the sliprate on the SAF and the recurrence timeTHRC time since past event tHRC and slip rate on the Hayward-Rodgers Creek fault system Thetime since the last great earthquake on the northernSan Andreas fault tSAF is of course known Theeffect of the ConcordndashGreen Valley fault is modeled

smiddotSAFsmiddotHRC

FIG 10 Comparison of Savage-Prescott model to the SCEC velocity data for the Carrizo Plain with slip rate con-strained to 34 mmyr Effect of San GregoriondashHosgri fault has been removed with a dislocation model with slip rate of 3mmyr and depth of 10 km A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

ESTIMATES OF SLIP RATE 73

kinematically with a simple screw dislocationwhich adds two parameters the fault slip rate andlocking depth It is certainly inconsistent to ignoreviscoelastic effects for the ConcordndashGreen Valleyfault but I do so for three reasons (a) much less isknown about this fault (b) earthquakes there arepresumably least well modeled by infinitely longruptures and (c) it keeps the number of parametersreasonably tractable In essence the effect of theConcordndashGreen Valley is removed empirically and Imake no attempt to gain further insights into itsmechanical behavior Future work should addressand hopefully rectify this simplification

A priori bounds on the parameters are listed inTable 1 The lower bound on the lithospheric thick-ness comes from a conservative estimate of thedepth extent of slip in large earthquakes In fact itis more likely that slip in 1906 extended at least to15 km (Matthews and Segall 1993) as it certainlydid in the Loma Prieta earthquake (eg Beroza1991) The bounds on relaxation time come from thepreliminary analysis above (Figs 12 and 13) The

lower bound on TSAF comes from the elapsed timesince 1906 the upper bound from limits on theamount of coseismic slip (see below) The lowerbounds on tHRC and THRC are derived from the his-torical record there is no known earthquake on thenorthern HaywardndashRodgers Creek system since atleast 1775 (Toppozada and Borchardt 1998WGCEP 1999) The lower and upper bounds on sliprates were based loosely on the Northern CaliforniaWorking group (WGCEP 1999 Table 2) The Work-ing Group listed 2 s uncertainties of = 24 plusmn 3mmyr = 9 plusmn 2 mmyr and = 5 plusmn m 3mmyr

The estimation problem is to find the choice ofthe nine parameters in Table 1 that minimize themisfit to two geodetic data sets the present-day spa-tial distribution of deformation in the north SanFrancisco Bay region (Fig 14) and the time-depen-dent decay in strain rate following the 1906 earth-quake (Fig 11) For a non-linear problem such asthis one always needs to be concerned about localminima However preliminary testing with a simu-

smiddotSAFsmiddotHRC smiddotHRC

FIG 11 Post-1906 strain rate from triangulation data in the San Francisco Bay area Model fit to the model withpostseismic deformation only A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

74 PAUL SEGALL

lated annealing algorithm did not show evidence oflocal minima so the remaining analysis was doneusing a Sequential Quadratic Programming methodwith BFGS Hessian updates (eg Gill et al 1981)within Matlab The method allows upper and lowerbounds to be placed on all parameters as well asgeneralized inequality constraints The latter areneeded because tHRC is strictly bounded by tHRC poundTHRC In the context of the Savage and Prescottmodel earthquakes are periodic and it makes nosense for the elapsed time since the last event toexceed the average recurrence time In addition Ihave found it desirable to constrain the maximumcoseismic slip on both the San Andreas and Hay-ward faults This provides constraints of the form

and

Noting that the coseismic slip should be interpretedas an average over the fault in this two-dimensional

model I set meters and

metersIt should be noted that the GPS network does not

extend very far west of the San Andreas fault Forthis reason the data do not adequately constrain the

total motion across the plate boundary A reasonableapproach would be to include data from Pacificplate sites such as Hawaii I chose simply to con-strain the sum of the slip rates on all three faults tobe no greater than 40 mmyr (eg Argus and Gordon2001) This constraint significantly affects the solu-tion estimates without the cumulative slip con-straint tended to put the maximum slip rate allowedon all three faults exceeding the total displacementrate between the Sierra NevadandashGreat Valley blockand the Pacific plate

The optimal solution is found for H ~ 184 km tR~ 345 years TSAF = 280 years ~ 25 mmyrtHRC = 225 years THRC = 4286 years and ~ 7mmyr The ConcordndashGreen Valley fault slips at

~ 8 mmyr below 22 km A surprising resultis that the predicted recurrence time on the SanAndreas is reasonable in the sense of requiring 70-meter slip events every 280 years In comparisonSchwartz et al (1998) suggested that the penulti-mate event on the north coast segment of the SanAndreas occurred in the mid-1600s possibly in theinterval 1632ndash1659 corresponding to a recurrenceinterval of roughly 260 years Also surprising is thatthe time since the last earthquake on the Haywardndash

smiddotSAF TSAFtimes DuSAFmaxpound smiddotHRC THRCtimes DuHRC

maxpound

DuSAFmax

70= DuHRCmax

30=

smiddotSAFsmiddotHRC

smiddotHRC

FIG 12 Bootstrap results for post-1906 data in the San Francisco Bay area A Relaxation time B Elastic-layerthickness (1000 bootstrap resamples)

ESTIMATES OF SLIP RATE 75

Rodgers Creek fault is the minimum allowed 225years A comparison of the model fit to the data setsis shown in Figure 15 Generally speaking the dataare fit quite well with the possible exception of thestrain rate in the first post-1906 epoch The uncer-tainties in the post-1906 strain rates were actuallydecreased by a factor of two to ensure that these dataare fairly weighted compared to the GPS velocitiesDecreasing the uncertainties in the strain rates fur-ther to fit high strain rate in the immediatepost-1906 epoch causes a very significant increasein the misfit to the modern GPS data This is mostlikely a result of the large uncertainties in the trian-gulation data but may also indicate that the physi-cal model is inadequate

How is it that the geodetic data have any capabil-ity of resolving the recurrence interval on the SanAndreas fault Recall that there are three times inthe physical model the time since the last majorearthquake t the asthenospheric relaxation time tRand the recurrence interval T For the San Andreasfault we know tSAF ~93 years a priori and tR is rea-

sonably well resolved by the post-1906 strain ratetransient The estimation is thus free to vary T insuch a way as to optimize the fit to the velocity gra-dients near the San Andreas Constraints on themaximum slip in the earthquake help to furthertighten those constraints

In order to determine how well the data constrainthe model parameters I ran 500 bootstrap resam-ples and optimizations using the best-fitting modelas the starting point of the optimization As beforethe residuals were resampled and added to the pre-dicted data from the optimal model in order toensure that each observation point contributes to thesolution The results of the bootstrap are shown inFigure 16 The posterior distribution for recurrencetime on the San Andreas fault TSAF is quite broadwith a peak near the maximum allowed of (7000mm)(19 mmyr) ~370 years In contrast the poste-rior distribution for recurrence time on the Hay-wardndashRodgers Creek fault THRC is bimodal withprominent peaks near the minimum ~225 years andmaximum allowed value (3000 mm)(7 mmyr)

FIG 13 Bootstrap results for SCEC data along the Carrizo Plain segment of the San Andreas fault A Elastic-layerthickness B Relaxation time Note that 200 years was the upper bound considered C Recurrence time Recurrencetimes in excess of 400 years were not considered as these would lead to excessive coseismic slip D Slip rate

76 PAUL SEGALL

~ 428 years The distributions for tHRC the timesince the most recent HaywardndashRodgers Creekearthquake is strongly concentrated near the mini-

mum allowed range The distribution of slip rates onthe San Andreas and HaywardndashRodgers Creekfaults are bimodal with peaks at 19 mmyr and 25

FIG 14 GPS-derived velocities for the north San Francisco Bay region after Prescott et al (2001) AbbreviationsSAF = San Andreas fault HF = Hayward fault CF = Calaveras fault RGF = Rodgers Creek fault CGVF = ConcordndashGreen Valley fault

TABLE 1 A priori Bounds on Parameters

Parameter Symbol Min Max

Lithospheric thickness km H 10 30

Relaxation time yr tR 10 200

SAF recurrence interval yr TSAF 100 500

Time since last Hayward event yr tHRC 225 690

Hayward recurrence interval yr THRC 225 700

Green valley fault locking depth km d 05 15

SAF slip rate my 0019 0027

Hayward slip rate myr 0007 0013

Green valley slip rate myr 0002 0008

smiddotSAF

smiddotHRC

smiddotCGV

ESTIMATES OF SLIP RATE 77

mmyr (SAF) and 7 mmyr and 13 mmyr (HRC)These results still must be viewed with some cau-tion in that I have not demonstrated that the boot-strap optimizations were immune to local minima

The bootstrap calculations suggest that the con-straint on maximum coseismic slip does play astrong role in the results To explore this further Ivaried the maximum coseismic slip allowed on thetwo faults The results (Table 2) demonstrate that theSan Andreas recurrence interval is not fully con-trolled by the constraint on the maximum coseismicslip In contrast the recurrence interval on the Hay-ward fault is totally controlled by the maximum slipallowed (for a maximum slip of 10 meters the recur-rence interval reaches the upper bound of 700years) Notice also that the time since the last Hay-wardndashRodgers Creek event is always at the mini-mum value allowed (225 years) The fit to the data isslightly better for large infrequent earthquakes

Discussion

What has been learned from the modeling pre-sented here It seems clear that geodetic data

should not be analyzed in a vacuum The resolutionof the data alone is insufficient to address the prob-lems of greatest interest However when combinedwith prior information from geologic studies GPSdata appear to be able to resolve parameters of con-siderable interest such as the average recurrenceinterval between great earthquakes on the SanAndreas fault Using a simple model of a repeatingdislocation rupturing an elastic plate overlying aMaxwell visco-elastic medium I found an averagerecurrence interval of roughly 280 years on thenorth coast segment of the San Andreas faultAlthough the uncertainties in this estimate arelarge the result is in surprisingly good accordancewith recent paleoseismic results (Schwartz et al1998) that suggest an event similar to 1906 in themid 1600s

The results for the HaywardndashRodgers Creek faultare also not unreasonable According to the GPSdata the most recent event here occurred in ~1775regardless of the estimated recurrence time Thisresult is not inconsistent with the paleoseismic find-ing that the most recent event on the Rodgers Creekfault took place in the interval 1670ndash1776

TABLE 2 Influence of Maximum Coseismic Slip on Estimated Parameters1

Parameter

Lithospheric thickness H km 150 184 220

Relaxation time tR yr 357 345 340

SAF recurrence interval TSAF yr 220 280 337

Limiting interval yr 290 368 526

Time since last Hayward event tHRC yr 225 225 225

Hayward recurrence interval THRC yr 2857 4286 700

Limiting interval yr 285 428 714

Green Valley fault locking depth d km 11 22 47

SAF slip rate myr 0025 0025 00249

Hayward slip rate myr 0007 0007 00071

Green valley slip rate myr 0008 0008 0008

1Limiting interval is computed using the estimated slip rate for that fault not the minimum slip rate that would yield a longer upper bound

DuSAFmax

55=

DuHRCmax

20=

DuSAFmax

70=

DuHRCmax

30=

DuSAFmax

100=

DuHRCmax

50=

DuSAFmax

smiddotSAFcurren

DuHRCmax

smiddotHRCcurren

smiddotSAF

smiddotHRC

smiddotCGV

78 PAUL SEGALL

(Schwartz et al 1993) and on the northern Hay-ward fault in 1640ndash1776 (HFPEG 1999) The GPSdata alone yield limited information about the aver-age recurrence time on the HaywardndashRodgers Creekfault However adding the reasonable bound thatslip not exceed 2 meters on average yields a recur-rence interval of 285 years The average slip in the1868 Hayward fault earthquake was approximately2 meters (Yu and Segall 1996) In comparisontrenching at the El Cerrito site on the northern Hay-ward fault suggests a recurrence interval of possiblyless than 270 years but no more than 710 years(HFPEG 1999)

It should not need to be stated that earthquakesare neither periodic nor especially in the case of theHaywardndashRodgers Creek faults infinitely long Atwo-dimensional model cannot describe the situa-tion at a given trench site where ruptures mainly tothe north or south of the site cause some offset in the

trench Nevertheless the fact that the results fromsuch a simple model are not unreasonable is encour-aging One useful attribute of the models exploredhere are that they are testable in the sense of predict-ing future behavior Running the model forward intime to the next San Andreas earthquake yields thevelocity distributions shown in Figure 17 Of coursea north HaywardndashRodgers Creek earthquake mayoccur prior to the next San Andreas event

The Maxwell relaxation time is reasonablywell estimated by the data I find an optimal valueof about 35 years which corresponds to a viscos-ity of 16 times 1019 Pa-s It should be noted that theestimates of all of the model parameters is only asgood as the mechanical model used to interpretthe data If as seems quite likely there was after-slip on the San Andreas below the 1906 rupturezone the estimate of the viscosity is some combi-nation of the effects of afterslip and distributed

FIG 15 Fit of model to north San Francisco Bay data Maximum slip on Hayward fault is 30 m and on the SAF is70 m

ESTIMATES OF SLIP RATE 79

relaxation Li and Rice (1987) presented anapproximate method for including post-seismicfault slip at constant resistive stress with a relax-ing asthenosphere in the context of a repeatingperiodic earthquake sequence Combining suchan approach with appropriate constitutive laws forstable fault creep as in Tse and Rice (1986)would presumably provide a more physically real-istic description of fault behavior Clearly three-dimensional effects are also important especiallyfor earthquakes on the Rodgers Creek and Hay-ward faults Finally the effects of non-uniform

elastic properties are known to bias interpreta-tions based on simple homogeneous models Elas-tic moduli increase with depth which has theknown effect of biasing the depth distribution offault slip Similarly elastic properties may varywith lithology across the major fault zones whichthemselves may represent zones of more compli-ant rock Finite elements are certainly the mostgeneral way to compute forward models but arenot yet feasible for the optimization and bootstrapmethods as computed here which literally testedthousands of possible models

FIG 16 Five hundred bootstrap estimates of model parameters fit to the north San Francisco Bay data 90 confi-dence bounds are listed at the top of each panel The vertical line in the histograms for recurrence times TSAF and THRCshow the upper bounds corresponding to the minimum slip rate and maximum allowed coseismic slip respectively Thehorizontal bar in the panel for the San Andreas recurrence time shows the preferred estimate from Schwartz et al (1998)The horizontal bar in the panel for tH time since the last Hayward earthquake shows the range of dates compatible withpaleoseismic estimates of the penultimate HaywardndashRodgers Creek event

80 PAUL SEGALL

Acknowledgments

George Thompson set a standard that geophysi-cal models no matter how elegant must be consis-tent with geologic observations This work wasinspired by that example I would like to thank WillPrescott for discussions about the Bay Area defor-mation measurements and Mark Matthews for manyyears ago introducing me to Bayesrsquo theorem WayneThatcher Mark Murray Jeff McGuire and PeterCervelli provided valuable comments at early stagesin this study I thank Jim Savage Duncan Agnewand Shelley Kenner for their thoughtful reviewsGary Ernst and Simon Klemperer carefully editedthe manuscript

REFERENCES

Argus D F and Gordon R G 2001 Present tectonicmotion across the Coast Ranges and San Andreas faultsystem in central California Geological Society ofAmerica Bulletin v 113 p 1580ndash1592

Beroza G C 1991 Near-source modeling of the LomaPrieta earthquake Evidence for heterogeneous slipand implications for earthquake hazard Bulletin of theSeismological Society of America v 81 p 1603ndash1621

Cohen S C 1982 A multilayer model of time-dependentdeformation following an earthquake on a strike-slipfault Journal of Geophysical Research v 87 p 5409ndash5421

HFPEG (Hayward Fault Paleoearthquake Group) 1999Timing of paleoearthquakes on the northern HaywardfaultmdashPreliminary evidence in El Cerrito CaliforniaUS Geological Survey Open File Report 99-318

Henstock T J Levander A and Hole J A 1997Deformation in the lower crust of the San Andreas faultsystem in northern California Science v 278 p 650ndash653

Feigl K L King R W and Jordan T H 1990 Geo-detic measurement of tectonic deformation in theSanta Maria fold and thrust belt California Journal ofGeophysical Research v 95 p 2679ndash2699

Gill P E Murray W and Wright M H 1981 Practicaloptimization London UK Academic Press 401 p

Grant L B and Sieh K 1994 Paleoseismic evidence ofclustered earthquakes on the San Andreas fault in the

FIG 17 Simulation of Bay Area velocity field as a function of time Each line is separated by roughly 27 years

72 PAUL SEGALL

some of the parameters will help resolve others aswell

It should be emphasized that the paleoseismicobservations of Grant and Sieh (1994) show an irreg-ular pattern of earthquakes that cannot be explainedby the simple two-dimensional model used here tointerpret the geodetic observations Nevertheless itis interesting that the GPS data appear to have someresolving power for average recurrence interval

San Francisco Bay Data

In order to eliminate any geographic bias in theinterpretation associated with possibly differentcrustal structure and temperature distribution Inext consider the velocity profile across the SanAndreas fault north of the San Francisco Bay asdetermined by repeated GPS observations (Prescottet al 2001) as shown in Figure 14 The GPS sitesare sufficiently close to the post-1906 triangulationnetworks that it is reasonable to assume that the

crustal structure and thermal profile are the samefor both data sets In the following the GPS-derivedvelocities are rotated into fault-parallel and perpen-dicular components Only the fault-parallel motionsare modeled here although as pointed out by Pres-cott et al (2001) and Murray and Segall (2001)there is significant fault-normal motion in the east-ern coast ranges at the western edge of the CentralValley

The model is the same as discussed previouslywith three principal faults the San Andreas Hay-wardndashRodgers Creek and ConcordndashGreen Valleyfaults The parameters are the elastic plate thick-ness H Maxwell relaxation time tR the recurrencetime for great earthquakes on the SAF TSAF the sliprate on the SAF and the recurrence timeTHRC time since past event tHRC and slip rate on the Hayward-Rodgers Creek fault system Thetime since the last great earthquake on the northernSan Andreas fault tSAF is of course known Theeffect of the ConcordndashGreen Valley fault is modeled

smiddotSAFsmiddotHRC

FIG 10 Comparison of Savage-Prescott model to the SCEC velocity data for the Carrizo Plain with slip rate con-strained to 34 mmyr Effect of San GregoriondashHosgri fault has been removed with a dislocation model with slip rate of 3mmyr and depth of 10 km A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

ESTIMATES OF SLIP RATE 73

kinematically with a simple screw dislocationwhich adds two parameters the fault slip rate andlocking depth It is certainly inconsistent to ignoreviscoelastic effects for the ConcordndashGreen Valleyfault but I do so for three reasons (a) much less isknown about this fault (b) earthquakes there arepresumably least well modeled by infinitely longruptures and (c) it keeps the number of parametersreasonably tractable In essence the effect of theConcordndashGreen Valley is removed empirically and Imake no attempt to gain further insights into itsmechanical behavior Future work should addressand hopefully rectify this simplification

A priori bounds on the parameters are listed inTable 1 The lower bound on the lithospheric thick-ness comes from a conservative estimate of thedepth extent of slip in large earthquakes In fact itis more likely that slip in 1906 extended at least to15 km (Matthews and Segall 1993) as it certainlydid in the Loma Prieta earthquake (eg Beroza1991) The bounds on relaxation time come from thepreliminary analysis above (Figs 12 and 13) The

lower bound on TSAF comes from the elapsed timesince 1906 the upper bound from limits on theamount of coseismic slip (see below) The lowerbounds on tHRC and THRC are derived from the his-torical record there is no known earthquake on thenorthern HaywardndashRodgers Creek system since atleast 1775 (Toppozada and Borchardt 1998WGCEP 1999) The lower and upper bounds on sliprates were based loosely on the Northern CaliforniaWorking group (WGCEP 1999 Table 2) The Work-ing Group listed 2 s uncertainties of = 24 plusmn 3mmyr = 9 plusmn 2 mmyr and = 5 plusmn m 3mmyr

The estimation problem is to find the choice ofthe nine parameters in Table 1 that minimize themisfit to two geodetic data sets the present-day spa-tial distribution of deformation in the north SanFrancisco Bay region (Fig 14) and the time-depen-dent decay in strain rate following the 1906 earth-quake (Fig 11) For a non-linear problem such asthis one always needs to be concerned about localminima However preliminary testing with a simu-

smiddotSAFsmiddotHRC smiddotHRC

FIG 11 Post-1906 strain rate from triangulation data in the San Francisco Bay area Model fit to the model withpostseismic deformation only A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

74 PAUL SEGALL

lated annealing algorithm did not show evidence oflocal minima so the remaining analysis was doneusing a Sequential Quadratic Programming methodwith BFGS Hessian updates (eg Gill et al 1981)within Matlab The method allows upper and lowerbounds to be placed on all parameters as well asgeneralized inequality constraints The latter areneeded because tHRC is strictly bounded by tHRC poundTHRC In the context of the Savage and Prescottmodel earthquakes are periodic and it makes nosense for the elapsed time since the last event toexceed the average recurrence time In addition Ihave found it desirable to constrain the maximumcoseismic slip on both the San Andreas and Hay-ward faults This provides constraints of the form

and

Noting that the coseismic slip should be interpretedas an average over the fault in this two-dimensional

model I set meters and

metersIt should be noted that the GPS network does not

extend very far west of the San Andreas fault Forthis reason the data do not adequately constrain the

total motion across the plate boundary A reasonableapproach would be to include data from Pacificplate sites such as Hawaii I chose simply to con-strain the sum of the slip rates on all three faults tobe no greater than 40 mmyr (eg Argus and Gordon2001) This constraint significantly affects the solu-tion estimates without the cumulative slip con-straint tended to put the maximum slip rate allowedon all three faults exceeding the total displacementrate between the Sierra NevadandashGreat Valley blockand the Pacific plate

The optimal solution is found for H ~ 184 km tR~ 345 years TSAF = 280 years ~ 25 mmyrtHRC = 225 years THRC = 4286 years and ~ 7mmyr The ConcordndashGreen Valley fault slips at

~ 8 mmyr below 22 km A surprising resultis that the predicted recurrence time on the SanAndreas is reasonable in the sense of requiring 70-meter slip events every 280 years In comparisonSchwartz et al (1998) suggested that the penulti-mate event on the north coast segment of the SanAndreas occurred in the mid-1600s possibly in theinterval 1632ndash1659 corresponding to a recurrenceinterval of roughly 260 years Also surprising is thatthe time since the last earthquake on the Haywardndash

smiddotSAF TSAFtimes DuSAFmaxpound smiddotHRC THRCtimes DuHRC

maxpound

DuSAFmax

70= DuHRCmax

30=

smiddotSAFsmiddotHRC

smiddotHRC

FIG 12 Bootstrap results for post-1906 data in the San Francisco Bay area A Relaxation time B Elastic-layerthickness (1000 bootstrap resamples)

ESTIMATES OF SLIP RATE 75

Rodgers Creek fault is the minimum allowed 225years A comparison of the model fit to the data setsis shown in Figure 15 Generally speaking the dataare fit quite well with the possible exception of thestrain rate in the first post-1906 epoch The uncer-tainties in the post-1906 strain rates were actuallydecreased by a factor of two to ensure that these dataare fairly weighted compared to the GPS velocitiesDecreasing the uncertainties in the strain rates fur-ther to fit high strain rate in the immediatepost-1906 epoch causes a very significant increasein the misfit to the modern GPS data This is mostlikely a result of the large uncertainties in the trian-gulation data but may also indicate that the physi-cal model is inadequate

How is it that the geodetic data have any capabil-ity of resolving the recurrence interval on the SanAndreas fault Recall that there are three times inthe physical model the time since the last majorearthquake t the asthenospheric relaxation time tRand the recurrence interval T For the San Andreasfault we know tSAF ~93 years a priori and tR is rea-

sonably well resolved by the post-1906 strain ratetransient The estimation is thus free to vary T insuch a way as to optimize the fit to the velocity gra-dients near the San Andreas Constraints on themaximum slip in the earthquake help to furthertighten those constraints

In order to determine how well the data constrainthe model parameters I ran 500 bootstrap resam-ples and optimizations using the best-fitting modelas the starting point of the optimization As beforethe residuals were resampled and added to the pre-dicted data from the optimal model in order toensure that each observation point contributes to thesolution The results of the bootstrap are shown inFigure 16 The posterior distribution for recurrencetime on the San Andreas fault TSAF is quite broadwith a peak near the maximum allowed of (7000mm)(19 mmyr) ~370 years In contrast the poste-rior distribution for recurrence time on the Hay-wardndashRodgers Creek fault THRC is bimodal withprominent peaks near the minimum ~225 years andmaximum allowed value (3000 mm)(7 mmyr)

FIG 13 Bootstrap results for SCEC data along the Carrizo Plain segment of the San Andreas fault A Elastic-layerthickness B Relaxation time Note that 200 years was the upper bound considered C Recurrence time Recurrencetimes in excess of 400 years were not considered as these would lead to excessive coseismic slip D Slip rate

76 PAUL SEGALL

~ 428 years The distributions for tHRC the timesince the most recent HaywardndashRodgers Creekearthquake is strongly concentrated near the mini-

mum allowed range The distribution of slip rates onthe San Andreas and HaywardndashRodgers Creekfaults are bimodal with peaks at 19 mmyr and 25

FIG 14 GPS-derived velocities for the north San Francisco Bay region after Prescott et al (2001) AbbreviationsSAF = San Andreas fault HF = Hayward fault CF = Calaveras fault RGF = Rodgers Creek fault CGVF = ConcordndashGreen Valley fault

TABLE 1 A priori Bounds on Parameters

Parameter Symbol Min Max

Lithospheric thickness km H 10 30

Relaxation time yr tR 10 200

SAF recurrence interval yr TSAF 100 500

Time since last Hayward event yr tHRC 225 690

Hayward recurrence interval yr THRC 225 700

Green valley fault locking depth km d 05 15

SAF slip rate my 0019 0027

Hayward slip rate myr 0007 0013

Green valley slip rate myr 0002 0008

smiddotSAF

smiddotHRC

smiddotCGV

ESTIMATES OF SLIP RATE 77

mmyr (SAF) and 7 mmyr and 13 mmyr (HRC)These results still must be viewed with some cau-tion in that I have not demonstrated that the boot-strap optimizations were immune to local minima

The bootstrap calculations suggest that the con-straint on maximum coseismic slip does play astrong role in the results To explore this further Ivaried the maximum coseismic slip allowed on thetwo faults The results (Table 2) demonstrate that theSan Andreas recurrence interval is not fully con-trolled by the constraint on the maximum coseismicslip In contrast the recurrence interval on the Hay-ward fault is totally controlled by the maximum slipallowed (for a maximum slip of 10 meters the recur-rence interval reaches the upper bound of 700years) Notice also that the time since the last Hay-wardndashRodgers Creek event is always at the mini-mum value allowed (225 years) The fit to the data isslightly better for large infrequent earthquakes

Discussion

What has been learned from the modeling pre-sented here It seems clear that geodetic data

should not be analyzed in a vacuum The resolutionof the data alone is insufficient to address the prob-lems of greatest interest However when combinedwith prior information from geologic studies GPSdata appear to be able to resolve parameters of con-siderable interest such as the average recurrenceinterval between great earthquakes on the SanAndreas fault Using a simple model of a repeatingdislocation rupturing an elastic plate overlying aMaxwell visco-elastic medium I found an averagerecurrence interval of roughly 280 years on thenorth coast segment of the San Andreas faultAlthough the uncertainties in this estimate arelarge the result is in surprisingly good accordancewith recent paleoseismic results (Schwartz et al1998) that suggest an event similar to 1906 in themid 1600s

The results for the HaywardndashRodgers Creek faultare also not unreasonable According to the GPSdata the most recent event here occurred in ~1775regardless of the estimated recurrence time Thisresult is not inconsistent with the paleoseismic find-ing that the most recent event on the Rodgers Creekfault took place in the interval 1670ndash1776

TABLE 2 Influence of Maximum Coseismic Slip on Estimated Parameters1

Parameter

Lithospheric thickness H km 150 184 220

Relaxation time tR yr 357 345 340

SAF recurrence interval TSAF yr 220 280 337

Limiting interval yr 290 368 526

Time since last Hayward event tHRC yr 225 225 225

Hayward recurrence interval THRC yr 2857 4286 700

Limiting interval yr 285 428 714

Green Valley fault locking depth d km 11 22 47

SAF slip rate myr 0025 0025 00249

Hayward slip rate myr 0007 0007 00071

Green valley slip rate myr 0008 0008 0008

1Limiting interval is computed using the estimated slip rate for that fault not the minimum slip rate that would yield a longer upper bound

DuSAFmax

55=

DuHRCmax

20=

DuSAFmax

70=

DuHRCmax

30=

DuSAFmax

100=

DuHRCmax

50=

DuSAFmax

smiddotSAFcurren

DuHRCmax

smiddotHRCcurren

smiddotSAF

smiddotHRC

smiddotCGV

78 PAUL SEGALL

(Schwartz et al 1993) and on the northern Hay-ward fault in 1640ndash1776 (HFPEG 1999) The GPSdata alone yield limited information about the aver-age recurrence time on the HaywardndashRodgers Creekfault However adding the reasonable bound thatslip not exceed 2 meters on average yields a recur-rence interval of 285 years The average slip in the1868 Hayward fault earthquake was approximately2 meters (Yu and Segall 1996) In comparisontrenching at the El Cerrito site on the northern Hay-ward fault suggests a recurrence interval of possiblyless than 270 years but no more than 710 years(HFPEG 1999)

It should not need to be stated that earthquakesare neither periodic nor especially in the case of theHaywardndashRodgers Creek faults infinitely long Atwo-dimensional model cannot describe the situa-tion at a given trench site where ruptures mainly tothe north or south of the site cause some offset in the

trench Nevertheless the fact that the results fromsuch a simple model are not unreasonable is encour-aging One useful attribute of the models exploredhere are that they are testable in the sense of predict-ing future behavior Running the model forward intime to the next San Andreas earthquake yields thevelocity distributions shown in Figure 17 Of coursea north HaywardndashRodgers Creek earthquake mayoccur prior to the next San Andreas event

The Maxwell relaxation time is reasonablywell estimated by the data I find an optimal valueof about 35 years which corresponds to a viscos-ity of 16 times 1019 Pa-s It should be noted that theestimates of all of the model parameters is only asgood as the mechanical model used to interpretthe data If as seems quite likely there was after-slip on the San Andreas below the 1906 rupturezone the estimate of the viscosity is some combi-nation of the effects of afterslip and distributed

FIG 15 Fit of model to north San Francisco Bay data Maximum slip on Hayward fault is 30 m and on the SAF is70 m

ESTIMATES OF SLIP RATE 79

relaxation Li and Rice (1987) presented anapproximate method for including post-seismicfault slip at constant resistive stress with a relax-ing asthenosphere in the context of a repeatingperiodic earthquake sequence Combining suchan approach with appropriate constitutive laws forstable fault creep as in Tse and Rice (1986)would presumably provide a more physically real-istic description of fault behavior Clearly three-dimensional effects are also important especiallyfor earthquakes on the Rodgers Creek and Hay-ward faults Finally the effects of non-uniform

elastic properties are known to bias interpreta-tions based on simple homogeneous models Elas-tic moduli increase with depth which has theknown effect of biasing the depth distribution offault slip Similarly elastic properties may varywith lithology across the major fault zones whichthemselves may represent zones of more compli-ant rock Finite elements are certainly the mostgeneral way to compute forward models but arenot yet feasible for the optimization and bootstrapmethods as computed here which literally testedthousands of possible models

FIG 16 Five hundred bootstrap estimates of model parameters fit to the north San Francisco Bay data 90 confi-dence bounds are listed at the top of each panel The vertical line in the histograms for recurrence times TSAF and THRCshow the upper bounds corresponding to the minimum slip rate and maximum allowed coseismic slip respectively Thehorizontal bar in the panel for the San Andreas recurrence time shows the preferred estimate from Schwartz et al (1998)The horizontal bar in the panel for tH time since the last Hayward earthquake shows the range of dates compatible withpaleoseismic estimates of the penultimate HaywardndashRodgers Creek event

80 PAUL SEGALL

Acknowledgments

George Thompson set a standard that geophysi-cal models no matter how elegant must be consis-tent with geologic observations This work wasinspired by that example I would like to thank WillPrescott for discussions about the Bay Area defor-mation measurements and Mark Matthews for manyyears ago introducing me to Bayesrsquo theorem WayneThatcher Mark Murray Jeff McGuire and PeterCervelli provided valuable comments at early stagesin this study I thank Jim Savage Duncan Agnewand Shelley Kenner for their thoughtful reviewsGary Ernst and Simon Klemperer carefully editedthe manuscript

REFERENCES

Argus D F and Gordon R G 2001 Present tectonicmotion across the Coast Ranges and San Andreas faultsystem in central California Geological Society ofAmerica Bulletin v 113 p 1580ndash1592

Beroza G C 1991 Near-source modeling of the LomaPrieta earthquake Evidence for heterogeneous slipand implications for earthquake hazard Bulletin of theSeismological Society of America v 81 p 1603ndash1621

Cohen S C 1982 A multilayer model of time-dependentdeformation following an earthquake on a strike-slipfault Journal of Geophysical Research v 87 p 5409ndash5421

HFPEG (Hayward Fault Paleoearthquake Group) 1999Timing of paleoearthquakes on the northern HaywardfaultmdashPreliminary evidence in El Cerrito CaliforniaUS Geological Survey Open File Report 99-318

Henstock T J Levander A and Hole J A 1997Deformation in the lower crust of the San Andreas faultsystem in northern California Science v 278 p 650ndash653

Feigl K L King R W and Jordan T H 1990 Geo-detic measurement of tectonic deformation in theSanta Maria fold and thrust belt California Journal ofGeophysical Research v 95 p 2679ndash2699

Gill P E Murray W and Wright M H 1981 Practicaloptimization London UK Academic Press 401 p

Grant L B and Sieh K 1994 Paleoseismic evidence ofclustered earthquakes on the San Andreas fault in the

FIG 17 Simulation of Bay Area velocity field as a function of time Each line is separated by roughly 27 years

ESTIMATES OF SLIP RATE 73

kinematically with a simple screw dislocationwhich adds two parameters the fault slip rate andlocking depth It is certainly inconsistent to ignoreviscoelastic effects for the ConcordndashGreen Valleyfault but I do so for three reasons (a) much less isknown about this fault (b) earthquakes there arepresumably least well modeled by infinitely longruptures and (c) it keeps the number of parametersreasonably tractable In essence the effect of theConcordndashGreen Valley is removed empirically and Imake no attempt to gain further insights into itsmechanical behavior Future work should addressand hopefully rectify this simplification

A priori bounds on the parameters are listed inTable 1 The lower bound on the lithospheric thick-ness comes from a conservative estimate of thedepth extent of slip in large earthquakes In fact itis more likely that slip in 1906 extended at least to15 km (Matthews and Segall 1993) as it certainlydid in the Loma Prieta earthquake (eg Beroza1991) The bounds on relaxation time come from thepreliminary analysis above (Figs 12 and 13) The

lower bound on TSAF comes from the elapsed timesince 1906 the upper bound from limits on theamount of coseismic slip (see below) The lowerbounds on tHRC and THRC are derived from the his-torical record there is no known earthquake on thenorthern HaywardndashRodgers Creek system since atleast 1775 (Toppozada and Borchardt 1998WGCEP 1999) The lower and upper bounds on sliprates were based loosely on the Northern CaliforniaWorking group (WGCEP 1999 Table 2) The Work-ing Group listed 2 s uncertainties of = 24 plusmn 3mmyr = 9 plusmn 2 mmyr and = 5 plusmn m 3mmyr

The estimation problem is to find the choice ofthe nine parameters in Table 1 that minimize themisfit to two geodetic data sets the present-day spa-tial distribution of deformation in the north SanFrancisco Bay region (Fig 14) and the time-depen-dent decay in strain rate following the 1906 earth-quake (Fig 11) For a non-linear problem such asthis one always needs to be concerned about localminima However preliminary testing with a simu-

smiddotSAFsmiddotHRC smiddotHRC

FIG 11 Post-1906 strain rate from triangulation data in the San Francisco Bay area Model fit to the model withpostseismic deformation only A Misfit as a function of elastic-layer thickness and relaxation time B Fit to the data

74 PAUL SEGALL

lated annealing algorithm did not show evidence oflocal minima so the remaining analysis was doneusing a Sequential Quadratic Programming methodwith BFGS Hessian updates (eg Gill et al 1981)within Matlab The method allows upper and lowerbounds to be placed on all parameters as well asgeneralized inequality constraints The latter areneeded because tHRC is strictly bounded by tHRC poundTHRC In the context of the Savage and Prescottmodel earthquakes are periodic and it makes nosense for the elapsed time since the last event toexceed the average recurrence time In addition Ihave found it desirable to constrain the maximumcoseismic slip on both the San Andreas and Hay-ward faults This provides constraints of the form

and

Noting that the coseismic slip should be interpretedas an average over the fault in this two-dimensional

model I set meters and

metersIt should be noted that the GPS network does not

extend very far west of the San Andreas fault Forthis reason the data do not adequately constrain the

total motion across the plate boundary A reasonableapproach would be to include data from Pacificplate sites such as Hawaii I chose simply to con-strain the sum of the slip rates on all three faults tobe no greater than 40 mmyr (eg Argus and Gordon2001) This constraint significantly affects the solu-tion estimates without the cumulative slip con-straint tended to put the maximum slip rate allowedon all three faults exceeding the total displacementrate between the Sierra NevadandashGreat Valley blockand the Pacific plate

The optimal solution is found for H ~ 184 km tR~ 345 years TSAF = 280 years ~ 25 mmyrtHRC = 225 years THRC = 4286 years and ~ 7mmyr The ConcordndashGreen Valley fault slips at

~ 8 mmyr below 22 km A surprising resultis that the predicted recurrence time on the SanAndreas is reasonable in the sense of requiring 70-meter slip events every 280 years In comparisonSchwartz et al (1998) suggested that the penulti-mate event on the north coast segment of the SanAndreas occurred in the mid-1600s possibly in theinterval 1632ndash1659 corresponding to a recurrenceinterval of roughly 260 years Also surprising is thatthe time since the last earthquake on the Haywardndash

smiddotSAF TSAFtimes DuSAFmaxpound smiddotHRC THRCtimes DuHRC

maxpound

DuSAFmax

70= DuHRCmax

30=

smiddotSAFsmiddotHRC

smiddotHRC

FIG 12 Bootstrap results for post-1906 data in the San Francisco Bay area A Relaxation time B Elastic-layerthickness (1000 bootstrap resamples)

ESTIMATES OF SLIP RATE 75

Rodgers Creek fault is the minimum allowed 225years A comparison of the model fit to the data setsis shown in Figure 15 Generally speaking the dataare fit quite well with the possible exception of thestrain rate in the first post-1906 epoch The uncer-tainties in the post-1906 strain rates were actuallydecreased by a factor of two to ensure that these dataare fairly weighted compared to the GPS velocitiesDecreasing the uncertainties in the strain rates fur-ther to fit high strain rate in the immediatepost-1906 epoch causes a very significant increasein the misfit to the modern GPS data This is mostlikely a result of the large uncertainties in the trian-gulation data but may also indicate that the physi-cal model is inadequate

How is it that the geodetic data have any capabil-ity of resolving the recurrence interval on the SanAndreas fault Recall that there are three times inthe physical model the time since the last majorearthquake t the asthenospheric relaxation time tRand the recurrence interval T For the San Andreasfault we know tSAF ~93 years a priori and tR is rea-

sonably well resolved by the post-1906 strain ratetransient The estimation is thus free to vary T insuch a way as to optimize the fit to the velocity gra-dients near the San Andreas Constraints on themaximum slip in the earthquake help to furthertighten those constraints

In order to determine how well the data constrainthe model parameters I ran 500 bootstrap resam-ples and optimizations using the best-fitting modelas the starting point of the optimization As beforethe residuals were resampled and added to the pre-dicted data from the optimal model in order toensure that each observation point contributes to thesolution The results of the bootstrap are shown inFigure 16 The posterior distribution for recurrencetime on the San Andreas fault TSAF is quite broadwith a peak near the maximum allowed of (7000mm)(19 mmyr) ~370 years In contrast the poste-rior distribution for recurrence time on the Hay-wardndashRodgers Creek fault THRC is bimodal withprominent peaks near the minimum ~225 years andmaximum allowed value (3000 mm)(7 mmyr)

FIG 13 Bootstrap results for SCEC data along the Carrizo Plain segment of the San Andreas fault A Elastic-layerthickness B Relaxation time Note that 200 years was the upper bound considered C Recurrence time Recurrencetimes in excess of 400 years were not considered as these would lead to excessive coseismic slip D Slip rate

76 PAUL SEGALL

~ 428 years The distributions for tHRC the timesince the most recent HaywardndashRodgers Creekearthquake is strongly concentrated near the mini-

mum allowed range The distribution of slip rates onthe San Andreas and HaywardndashRodgers Creekfaults are bimodal with peaks at 19 mmyr and 25

FIG 14 GPS-derived velocities for the north San Francisco Bay region after Prescott et al (2001) AbbreviationsSAF = San Andreas fault HF = Hayward fault CF = Calaveras fault RGF = Rodgers Creek fault CGVF = ConcordndashGreen Valley fault

TABLE 1 A priori Bounds on Parameters

Parameter Symbol Min Max

Lithospheric thickness km H 10 30

Relaxation time yr tR 10 200

SAF recurrence interval yr TSAF 100 500

Time since last Hayward event yr tHRC 225 690

Hayward recurrence interval yr THRC 225 700

Green valley fault locking depth km d 05 15

SAF slip rate my 0019 0027

Hayward slip rate myr 0007 0013

Green valley slip rate myr 0002 0008

smiddotSAF

smiddotHRC

smiddotCGV

ESTIMATES OF SLIP RATE 77

mmyr (SAF) and 7 mmyr and 13 mmyr (HRC)These results still must be viewed with some cau-tion in that I have not demonstrated that the boot-strap optimizations were immune to local minima

The bootstrap calculations suggest that the con-straint on maximum coseismic slip does play astrong role in the results To explore this further Ivaried the maximum coseismic slip allowed on thetwo faults The results (Table 2) demonstrate that theSan Andreas recurrence interval is not fully con-trolled by the constraint on the maximum coseismicslip In contrast the recurrence interval on the Hay-ward fault is totally controlled by the maximum slipallowed (for a maximum slip of 10 meters the recur-rence interval reaches the upper bound of 700years) Notice also that the time since the last Hay-wardndashRodgers Creek event is always at the mini-mum value allowed (225 years) The fit to the data isslightly better for large infrequent earthquakes

Discussion

What has been learned from the modeling pre-sented here It seems clear that geodetic data

should not be analyzed in a vacuum The resolutionof the data alone is insufficient to address the prob-lems of greatest interest However when combinedwith prior information from geologic studies GPSdata appear to be able to resolve parameters of con-siderable interest such as the average recurrenceinterval between great earthquakes on the SanAndreas fault Using a simple model of a repeatingdislocation rupturing an elastic plate overlying aMaxwell visco-elastic medium I found an averagerecurrence interval of roughly 280 years on thenorth coast segment of the San Andreas faultAlthough the uncertainties in this estimate arelarge the result is in surprisingly good accordancewith recent paleoseismic results (Schwartz et al1998) that suggest an event similar to 1906 in themid 1600s

The results for the HaywardndashRodgers Creek faultare also not unreasonable According to the GPSdata the most recent event here occurred in ~1775regardless of the estimated recurrence time Thisresult is not inconsistent with the paleoseismic find-ing that the most recent event on the Rodgers Creekfault took place in the interval 1670ndash1776

TABLE 2 Influence of Maximum Coseismic Slip on Estimated Parameters1

Parameter

Lithospheric thickness H km 150 184 220

Relaxation time tR yr 357 345 340

SAF recurrence interval TSAF yr 220 280 337

Limiting interval yr 290 368 526

Time since last Hayward event tHRC yr 225 225 225

Hayward recurrence interval THRC yr 2857 4286 700

Limiting interval yr 285 428 714

Green Valley fault locking depth d km 11 22 47

SAF slip rate myr 0025 0025 00249

Hayward slip rate myr 0007 0007 00071

Green valley slip rate myr 0008 0008 0008

1Limiting interval is computed using the estimated slip rate for that fault not the minimum slip rate that would yield a longer upper bound

DuSAFmax

55=

DuHRCmax

20=

DuSAFmax

70=

DuHRCmax

30=

DuSAFmax

100=

DuHRCmax

50=

DuSAFmax

smiddotSAFcurren

DuHRCmax

smiddotHRCcurren

smiddotSAF

smiddotHRC

smiddotCGV

78 PAUL SEGALL

(Schwartz et al 1993) and on the northern Hay-ward fault in 1640ndash1776 (HFPEG 1999) The GPSdata alone yield limited information about the aver-age recurrence time on the HaywardndashRodgers Creekfault However adding the reasonable bound thatslip not exceed 2 meters on average yields a recur-rence interval of 285 years The average slip in the1868 Hayward fault earthquake was approximately2 meters (Yu and Segall 1996) In comparisontrenching at the El Cerrito site on the northern Hay-ward fault suggests a recurrence interval of possiblyless than 270 years but no more than 710 years(HFPEG 1999)

It should not need to be stated that earthquakesare neither periodic nor especially in the case of theHaywardndashRodgers Creek faults infinitely long Atwo-dimensional model cannot describe the situa-tion at a given trench site where ruptures mainly tothe north or south of the site cause some offset in the

trench Nevertheless the fact that the results fromsuch a simple model are not unreasonable is encour-aging One useful attribute of the models exploredhere are that they are testable in the sense of predict-ing future behavior Running the model forward intime to the next San Andreas earthquake yields thevelocity distributions shown in Figure 17 Of coursea north HaywardndashRodgers Creek earthquake mayoccur prior to the next San Andreas event

The Maxwell relaxation time is reasonablywell estimated by the data I find an optimal valueof about 35 years which corresponds to a viscos-ity of 16 times 1019 Pa-s It should be noted that theestimates of all of the model parameters is only asgood as the mechanical model used to interpretthe data If as seems quite likely there was after-slip on the San Andreas below the 1906 rupturezone the estimate of the viscosity is some combi-nation of the effects of afterslip and distributed

FIG 15 Fit of model to north San Francisco Bay data Maximum slip on Hayward fault is 30 m and on the SAF is70 m

ESTIMATES OF SLIP RATE 79

relaxation Li and Rice (1987) presented anapproximate method for including post-seismicfault slip at constant resistive stress with a relax-ing asthenosphere in the context of a repeatingperiodic earthquake sequence Combining suchan approach with appropriate constitutive laws forstable fault creep as in Tse and Rice (1986)would presumably provide a more physically real-istic description of fault behavior Clearly three-dimensional effects are also important especiallyfor earthquakes on the Rodgers Creek and Hay-ward faults Finally the effects of non-uniform

elastic properties are known to bias interpreta-tions based on simple homogeneous models Elas-tic moduli increase with depth which has theknown effect of biasing the depth distribution offault slip Similarly elastic properties may varywith lithology across the major fault zones whichthemselves may represent zones of more compli-ant rock Finite elements are certainly the mostgeneral way to compute forward models but arenot yet feasible for the optimization and bootstrapmethods as computed here which literally testedthousands of possible models

FIG 16 Five hundred bootstrap estimates of model parameters fit to the north San Francisco Bay data 90 confi-dence bounds are listed at the top of each panel The vertical line in the histograms for recurrence times TSAF and THRCshow the upper bounds corresponding to the minimum slip rate and maximum allowed coseismic slip respectively Thehorizontal bar in the panel for the San Andreas recurrence time shows the preferred estimate from Schwartz et al (1998)The horizontal bar in the panel for tH time since the last Hayward earthquake shows the range of dates compatible withpaleoseismic estimates of the penultimate HaywardndashRodgers Creek event

80 PAUL SEGALL

Acknowledgments

George Thompson set a standard that geophysi-cal models no matter how elegant must be consis-tent with geologic observations This work wasinspired by that example I would like to thank WillPrescott for discussions about the Bay Area defor-mation measurements and Mark Matthews for manyyears ago introducing me to Bayesrsquo theorem WayneThatcher Mark Murray Jeff McGuire and PeterCervelli provided valuable comments at early stagesin this study I thank Jim Savage Duncan Agnewand Shelley Kenner for their thoughtful reviewsGary Ernst and Simon Klemperer carefully editedthe manuscript

REFERENCES

Argus D F and Gordon R G 2001 Present tectonicmotion across the Coast Ranges and San Andreas faultsystem in central California Geological Society ofAmerica Bulletin v 113 p 1580ndash1592

Beroza G C 1991 Near-source modeling of the LomaPrieta earthquake Evidence for heterogeneous slipand implications for earthquake hazard Bulletin of theSeismological Society of America v 81 p 1603ndash1621

Cohen S C 1982 A multilayer model of time-dependentdeformation following an earthquake on a strike-slipfault Journal of Geophysical Research v 87 p 5409ndash5421

HFPEG (Hayward Fault Paleoearthquake Group) 1999Timing of paleoearthquakes on the northern HaywardfaultmdashPreliminary evidence in El Cerrito CaliforniaUS Geological Survey Open File Report 99-318

Henstock T J Levander A and Hole J A 1997Deformation in the lower crust of the San Andreas faultsystem in northern California Science v 278 p 650ndash653

Feigl K L King R W and Jordan T H 1990 Geo-detic measurement of tectonic deformation in theSanta Maria fold and thrust belt California Journal ofGeophysical Research v 95 p 2679ndash2699

Gill P E Murray W and Wright M H 1981 Practicaloptimization London UK Academic Press 401 p

Grant L B and Sieh K 1994 Paleoseismic evidence ofclustered earthquakes on the San Andreas fault in the

FIG 17 Simulation of Bay Area velocity field as a function of time Each line is separated by roughly 27 years

74 PAUL SEGALL

lated annealing algorithm did not show evidence oflocal minima so the remaining analysis was doneusing a Sequential Quadratic Programming methodwith BFGS Hessian updates (eg Gill et al 1981)within Matlab The method allows upper and lowerbounds to be placed on all parameters as well asgeneralized inequality constraints The latter areneeded because tHRC is strictly bounded by tHRC poundTHRC In the context of the Savage and Prescottmodel earthquakes are periodic and it makes nosense for the elapsed time since the last event toexceed the average recurrence time In addition Ihave found it desirable to constrain the maximumcoseismic slip on both the San Andreas and Hay-ward faults This provides constraints of the form

and

Noting that the coseismic slip should be interpretedas an average over the fault in this two-dimensional

model I set meters and

metersIt should be noted that the GPS network does not

extend very far west of the San Andreas fault Forthis reason the data do not adequately constrain the

total motion across the plate boundary A reasonableapproach would be to include data from Pacificplate sites such as Hawaii I chose simply to con-strain the sum of the slip rates on all three faults tobe no greater than 40 mmyr (eg Argus and Gordon2001) This constraint significantly affects the solu-tion estimates without the cumulative slip con-straint tended to put the maximum slip rate allowedon all three faults exceeding the total displacementrate between the Sierra NevadandashGreat Valley blockand the Pacific plate

The optimal solution is found for H ~ 184 km tR~ 345 years TSAF = 280 years ~ 25 mmyrtHRC = 225 years THRC = 4286 years and ~ 7mmyr The ConcordndashGreen Valley fault slips at

~ 8 mmyr below 22 km A surprising resultis that the predicted recurrence time on the SanAndreas is reasonable in the sense of requiring 70-meter slip events every 280 years In comparisonSchwartz et al (1998) suggested that the penulti-mate event on the north coast segment of the SanAndreas occurred in the mid-1600s possibly in theinterval 1632ndash1659 corresponding to a recurrenceinterval of roughly 260 years Also surprising is thatthe time since the last earthquake on the Haywardndash

smiddotSAF TSAFtimes DuSAFmaxpound smiddotHRC THRCtimes DuHRC

maxpound

DuSAFmax

70= DuHRCmax

30=

smiddotSAFsmiddotHRC

smiddotHRC

FIG 12 Bootstrap results for post-1906 data in the San Francisco Bay area A Relaxation time B Elastic-layerthickness (1000 bootstrap resamples)

ESTIMATES OF SLIP RATE 75

Rodgers Creek fault is the minimum allowed 225years A comparison of the model fit to the data setsis shown in Figure 15 Generally speaking the dataare fit quite well with the possible exception of thestrain rate in the first post-1906 epoch The uncer-tainties in the post-1906 strain rates were actuallydecreased by a factor of two to ensure that these dataare fairly weighted compared to the GPS velocitiesDecreasing the uncertainties in the strain rates fur-ther to fit high strain rate in the immediatepost-1906 epoch causes a very significant increasein the misfit to the modern GPS data This is mostlikely a result of the large uncertainties in the trian-gulation data but may also indicate that the physi-cal model is inadequate

How is it that the geodetic data have any capabil-ity of resolving the recurrence interval on the SanAndreas fault Recall that there are three times inthe physical model the time since the last majorearthquake t the asthenospheric relaxation time tRand the recurrence interval T For the San Andreasfault we know tSAF ~93 years a priori and tR is rea-

sonably well resolved by the post-1906 strain ratetransient The estimation is thus free to vary T insuch a way as to optimize the fit to the velocity gra-dients near the San Andreas Constraints on themaximum slip in the earthquake help to furthertighten those constraints

In order to determine how well the data constrainthe model parameters I ran 500 bootstrap resam-ples and optimizations using the best-fitting modelas the starting point of the optimization As beforethe residuals were resampled and added to the pre-dicted data from the optimal model in order toensure that each observation point contributes to thesolution The results of the bootstrap are shown inFigure 16 The posterior distribution for recurrencetime on the San Andreas fault TSAF is quite broadwith a peak near the maximum allowed of (7000mm)(19 mmyr) ~370 years In contrast the poste-rior distribution for recurrence time on the Hay-wardndashRodgers Creek fault THRC is bimodal withprominent peaks near the minimum ~225 years andmaximum allowed value (3000 mm)(7 mmyr)

FIG 13 Bootstrap results for SCEC data along the Carrizo Plain segment of the San Andreas fault A Elastic-layerthickness B Relaxation time Note that 200 years was the upper bound considered C Recurrence time Recurrencetimes in excess of 400 years were not considered as these would lead to excessive coseismic slip D Slip rate

76 PAUL SEGALL

~ 428 years The distributions for tHRC the timesince the most recent HaywardndashRodgers Creekearthquake is strongly concentrated near the mini-

mum allowed range The distribution of slip rates onthe San Andreas and HaywardndashRodgers Creekfaults are bimodal with peaks at 19 mmyr and 25

FIG 14 GPS-derived velocities for the north San Francisco Bay region after Prescott et al (2001) AbbreviationsSAF = San Andreas fault HF = Hayward fault CF = Calaveras fault RGF = Rodgers Creek fault CGVF = ConcordndashGreen Valley fault

TABLE 1 A priori Bounds on Parameters

Parameter Symbol Min Max

Lithospheric thickness km H 10 30

Relaxation time yr tR 10 200

SAF recurrence interval yr TSAF 100 500

Time since last Hayward event yr tHRC 225 690

Hayward recurrence interval yr THRC 225 700

Green valley fault locking depth km d 05 15

SAF slip rate my 0019 0027

Hayward slip rate myr 0007 0013

Green valley slip rate myr 0002 0008

smiddotSAF

smiddotHRC

smiddotCGV

ESTIMATES OF SLIP RATE 77

mmyr (SAF) and 7 mmyr and 13 mmyr (HRC)These results still must be viewed with some cau-tion in that I have not demonstrated that the boot-strap optimizations were immune to local minima

The bootstrap calculations suggest that the con-straint on maximum coseismic slip does play astrong role in the results To explore this further Ivaried the maximum coseismic slip allowed on thetwo faults The results (Table 2) demonstrate that theSan Andreas recurrence interval is not fully con-trolled by the constraint on the maximum coseismicslip In contrast the recurrence interval on the Hay-ward fault is totally controlled by the maximum slipallowed (for a maximum slip of 10 meters the recur-rence interval reaches the upper bound of 700years) Notice also that the time since the last Hay-wardndashRodgers Creek event is always at the mini-mum value allowed (225 years) The fit to the data isslightly better for large infrequent earthquakes

Discussion

What has been learned from the modeling pre-sented here It seems clear that geodetic data

should not be analyzed in a vacuum The resolutionof the data alone is insufficient to address the prob-lems of greatest interest However when combinedwith prior information from geologic studies GPSdata appear to be able to resolve parameters of con-siderable interest such as the average recurrenceinterval between great earthquakes on the SanAndreas fault Using a simple model of a repeatingdislocation rupturing an elastic plate overlying aMaxwell visco-elastic medium I found an averagerecurrence interval of roughly 280 years on thenorth coast segment of the San Andreas faultAlthough the uncertainties in this estimate arelarge the result is in surprisingly good accordancewith recent paleoseismic results (Schwartz et al1998) that suggest an event similar to 1906 in themid 1600s

The results for the HaywardndashRodgers Creek faultare also not unreasonable According to the GPSdata the most recent event here occurred in ~1775regardless of the estimated recurrence time Thisresult is not inconsistent with the paleoseismic find-ing that the most recent event on the Rodgers Creekfault took place in the interval 1670ndash1776

TABLE 2 Influence of Maximum Coseismic Slip on Estimated Parameters1

Parameter

Lithospheric thickness H km 150 184 220

Relaxation time tR yr 357 345 340

SAF recurrence interval TSAF yr 220 280 337

Limiting interval yr 290 368 526

Time since last Hayward event tHRC yr 225 225 225

Hayward recurrence interval THRC yr 2857 4286 700

Limiting interval yr 285 428 714

Green Valley fault locking depth d km 11 22 47

SAF slip rate myr 0025 0025 00249

Hayward slip rate myr 0007 0007 00071

Green valley slip rate myr 0008 0008 0008

1Limiting interval is computed using the estimated slip rate for that fault not the minimum slip rate that would yield a longer upper bound

DuSAFmax

55=

DuHRCmax

20=

DuSAFmax

70=

DuHRCmax

30=

DuSAFmax

100=

DuHRCmax

50=

DuSAFmax

smiddotSAFcurren

DuHRCmax

smiddotHRCcurren

smiddotSAF

smiddotHRC

smiddotCGV

78 PAUL SEGALL

(Schwartz et al 1993) and on the northern Hay-ward fault in 1640ndash1776 (HFPEG 1999) The GPSdata alone yield limited information about the aver-age recurrence time on the HaywardndashRodgers Creekfault However adding the reasonable bound thatslip not exceed 2 meters on average yields a recur-rence interval of 285 years The average slip in the1868 Hayward fault earthquake was approximately2 meters (Yu and Segall 1996) In comparisontrenching at the El Cerrito site on the northern Hay-ward fault suggests a recurrence interval of possiblyless than 270 years but no more than 710 years(HFPEG 1999)

It should not need to be stated that earthquakesare neither periodic nor especially in the case of theHaywardndashRodgers Creek faults infinitely long Atwo-dimensional model cannot describe the situa-tion at a given trench site where ruptures mainly tothe north or south of the site cause some offset in the

trench Nevertheless the fact that the results fromsuch a simple model are not unreasonable is encour-aging One useful attribute of the models exploredhere are that they are testable in the sense of predict-ing future behavior Running the model forward intime to the next San Andreas earthquake yields thevelocity distributions shown in Figure 17 Of coursea north HaywardndashRodgers Creek earthquake mayoccur prior to the next San Andreas event

The Maxwell relaxation time is reasonablywell estimated by the data I find an optimal valueof about 35 years which corresponds to a viscos-ity of 16 times 1019 Pa-s It should be noted that theestimates of all of the model parameters is only asgood as the mechanical model used to interpretthe data If as seems quite likely there was after-slip on the San Andreas below the 1906 rupturezone the estimate of the viscosity is some combi-nation of the effects of afterslip and distributed

FIG 15 Fit of model to north San Francisco Bay data Maximum slip on Hayward fault is 30 m and on the SAF is70 m

ESTIMATES OF SLIP RATE 79

relaxation Li and Rice (1987) presented anapproximate method for including post-seismicfault slip at constant resistive stress with a relax-ing asthenosphere in the context of a repeatingperiodic earthquake sequence Combining suchan approach with appropriate constitutive laws forstable fault creep as in Tse and Rice (1986)would presumably provide a more physically real-istic description of fault behavior Clearly three-dimensional effects are also important especiallyfor earthquakes on the Rodgers Creek and Hay-ward faults Finally the effects of non-uniform

elastic properties are known to bias interpreta-tions based on simple homogeneous models Elas-tic moduli increase with depth which has theknown effect of biasing the depth distribution offault slip Similarly elastic properties may varywith lithology across the major fault zones whichthemselves may represent zones of more compli-ant rock Finite elements are certainly the mostgeneral way to compute forward models but arenot yet feasible for the optimization and bootstrapmethods as computed here which literally testedthousands of possible models

FIG 16 Five hundred bootstrap estimates of model parameters fit to the north San Francisco Bay data 90 confi-dence bounds are listed at the top of each panel The vertical line in the histograms for recurrence times TSAF and THRCshow the upper bounds corresponding to the minimum slip rate and maximum allowed coseismic slip respectively Thehorizontal bar in the panel for the San Andreas recurrence time shows the preferred estimate from Schwartz et al (1998)The horizontal bar in the panel for tH time since the last Hayward earthquake shows the range of dates compatible withpaleoseismic estimates of the penultimate HaywardndashRodgers Creek event

80 PAUL SEGALL

Acknowledgments

George Thompson set a standard that geophysi-cal models no matter how elegant must be consis-tent with geologic observations This work wasinspired by that example I would like to thank WillPrescott for discussions about the Bay Area defor-mation measurements and Mark Matthews for manyyears ago introducing me to Bayesrsquo theorem WayneThatcher Mark Murray Jeff McGuire and PeterCervelli provided valuable comments at early stagesin this study I thank Jim Savage Duncan Agnewand Shelley Kenner for their thoughtful reviewsGary Ernst and Simon Klemperer carefully editedthe manuscript

REFERENCES

Argus D F and Gordon R G 2001 Present tectonicmotion across the Coast Ranges and San Andreas faultsystem in central California Geological Society ofAmerica Bulletin v 113 p 1580ndash1592

Beroza G C 1991 Near-source modeling of the LomaPrieta earthquake Evidence for heterogeneous slipand implications for earthquake hazard Bulletin of theSeismological Society of America v 81 p 1603ndash1621

Cohen S C 1982 A multilayer model of time-dependentdeformation following an earthquake on a strike-slipfault Journal of Geophysical Research v 87 p 5409ndash5421

HFPEG (Hayward Fault Paleoearthquake Group) 1999Timing of paleoearthquakes on the northern HaywardfaultmdashPreliminary evidence in El Cerrito CaliforniaUS Geological Survey Open File Report 99-318

Henstock T J Levander A and Hole J A 1997Deformation in the lower crust of the San Andreas faultsystem in northern California Science v 278 p 650ndash653

Feigl K L King R W and Jordan T H 1990 Geo-detic measurement of tectonic deformation in theSanta Maria fold and thrust belt California Journal ofGeophysical Research v 95 p 2679ndash2699

Gill P E Murray W and Wright M H 1981 Practicaloptimization London UK Academic Press 401 p

Grant L B and Sieh K 1994 Paleoseismic evidence ofclustered earthquakes on the San Andreas fault in the

FIG 17 Simulation of Bay Area velocity field as a function of time Each line is separated by roughly 27 years

ESTIMATES OF SLIP RATE 75

Rodgers Creek fault is the minimum allowed 225years A comparison of the model fit to the data setsis shown in Figure 15 Generally speaking the dataare fit quite well with the possible exception of thestrain rate in the first post-1906 epoch The uncer-tainties in the post-1906 strain rates were actuallydecreased by a factor of two to ensure that these dataare fairly weighted compared to the GPS velocitiesDecreasing the uncertainties in the strain rates fur-ther to fit high strain rate in the immediatepost-1906 epoch causes a very significant increasein the misfit to the modern GPS data This is mostlikely a result of the large uncertainties in the trian-gulation data but may also indicate that the physi-cal model is inadequate

How is it that the geodetic data have any capabil-ity of resolving the recurrence interval on the SanAndreas fault Recall that there are three times inthe physical model the time since the last majorearthquake t the asthenospheric relaxation time tRand the recurrence interval T For the San Andreasfault we know tSAF ~93 years a priori and tR is rea-

sonably well resolved by the post-1906 strain ratetransient The estimation is thus free to vary T insuch a way as to optimize the fit to the velocity gra-dients near the San Andreas Constraints on themaximum slip in the earthquake help to furthertighten those constraints

In order to determine how well the data constrainthe model parameters I ran 500 bootstrap resam-ples and optimizations using the best-fitting modelas the starting point of the optimization As beforethe residuals were resampled and added to the pre-dicted data from the optimal model in order toensure that each observation point contributes to thesolution The results of the bootstrap are shown inFigure 16 The posterior distribution for recurrencetime on the San Andreas fault TSAF is quite broadwith a peak near the maximum allowed of (7000mm)(19 mmyr) ~370 years In contrast the poste-rior distribution for recurrence time on the Hay-wardndashRodgers Creek fault THRC is bimodal withprominent peaks near the minimum ~225 years andmaximum allowed value (3000 mm)(7 mmyr)

FIG 13 Bootstrap results for SCEC data along the Carrizo Plain segment of the San Andreas fault A Elastic-layerthickness B Relaxation time Note that 200 years was the upper bound considered C Recurrence time Recurrencetimes in excess of 400 years were not considered as these would lead to excessive coseismic slip D Slip rate

76 PAUL SEGALL

~ 428 years The distributions for tHRC the timesince the most recent HaywardndashRodgers Creekearthquake is strongly concentrated near the mini-

mum allowed range The distribution of slip rates onthe San Andreas and HaywardndashRodgers Creekfaults are bimodal with peaks at 19 mmyr and 25

FIG 14 GPS-derived velocities for the north San Francisco Bay region after Prescott et al (2001) AbbreviationsSAF = San Andreas fault HF = Hayward fault CF = Calaveras fault RGF = Rodgers Creek fault CGVF = ConcordndashGreen Valley fault

TABLE 1 A priori Bounds on Parameters

Parameter Symbol Min Max

Lithospheric thickness km H 10 30

Relaxation time yr tR 10 200

SAF recurrence interval yr TSAF 100 500

Time since last Hayward event yr tHRC 225 690

Hayward recurrence interval yr THRC 225 700

Green valley fault locking depth km d 05 15

SAF slip rate my 0019 0027

Hayward slip rate myr 0007 0013

Green valley slip rate myr 0002 0008

smiddotSAF

smiddotHRC

smiddotCGV

ESTIMATES OF SLIP RATE 77

mmyr (SAF) and 7 mmyr and 13 mmyr (HRC)These results still must be viewed with some cau-tion in that I have not demonstrated that the boot-strap optimizations were immune to local minima

The bootstrap calculations suggest that the con-straint on maximum coseismic slip does play astrong role in the results To explore this further Ivaried the maximum coseismic slip allowed on thetwo faults The results (Table 2) demonstrate that theSan Andreas recurrence interval is not fully con-trolled by the constraint on the maximum coseismicslip In contrast the recurrence interval on the Hay-ward fault is totally controlled by the maximum slipallowed (for a maximum slip of 10 meters the recur-rence interval reaches the upper bound of 700years) Notice also that the time since the last Hay-wardndashRodgers Creek event is always at the mini-mum value allowed (225 years) The fit to the data isslightly better for large infrequent earthquakes

Discussion

What has been learned from the modeling pre-sented here It seems clear that geodetic data

should not be analyzed in a vacuum The resolutionof the data alone is insufficient to address the prob-lems of greatest interest However when combinedwith prior information from geologic studies GPSdata appear to be able to resolve parameters of con-siderable interest such as the average recurrenceinterval between great earthquakes on the SanAndreas fault Using a simple model of a repeatingdislocation rupturing an elastic plate overlying aMaxwell visco-elastic medium I found an averagerecurrence interval of roughly 280 years on thenorth coast segment of the San Andreas faultAlthough the uncertainties in this estimate arelarge the result is in surprisingly good accordancewith recent paleoseismic results (Schwartz et al1998) that suggest an event similar to 1906 in themid 1600s

The results for the HaywardndashRodgers Creek faultare also not unreasonable According to the GPSdata the most recent event here occurred in ~1775regardless of the estimated recurrence time Thisresult is not inconsistent with the paleoseismic find-ing that the most recent event on the Rodgers Creekfault took place in the interval 1670ndash1776

TABLE 2 Influence of Maximum Coseismic Slip on Estimated Parameters1

Parameter

Lithospheric thickness H km 150 184 220

Relaxation time tR yr 357 345 340

SAF recurrence interval TSAF yr 220 280 337

Limiting interval yr 290 368 526

Time since last Hayward event tHRC yr 225 225 225

Hayward recurrence interval THRC yr 2857 4286 700

Limiting interval yr 285 428 714

Green Valley fault locking depth d km 11 22 47

SAF slip rate myr 0025 0025 00249

Hayward slip rate myr 0007 0007 00071

Green valley slip rate myr 0008 0008 0008

1Limiting interval is computed using the estimated slip rate for that fault not the minimum slip rate that would yield a longer upper bound

DuSAFmax

55=

DuHRCmax

20=

DuSAFmax

70=

DuHRCmax

30=

DuSAFmax

100=

DuHRCmax

50=

DuSAFmax

smiddotSAFcurren

DuHRCmax

smiddotHRCcurren

smiddotSAF

smiddotHRC

smiddotCGV

78 PAUL SEGALL

(Schwartz et al 1993) and on the northern Hay-ward fault in 1640ndash1776 (HFPEG 1999) The GPSdata alone yield limited information about the aver-age recurrence time on the HaywardndashRodgers Creekfault However adding the reasonable bound thatslip not exceed 2 meters on average yields a recur-rence interval of 285 years The average slip in the1868 Hayward fault earthquake was approximately2 meters (Yu and Segall 1996) In comparisontrenching at the El Cerrito site on the northern Hay-ward fault suggests a recurrence interval of possiblyless than 270 years but no more than 710 years(HFPEG 1999)

It should not need to be stated that earthquakesare neither periodic nor especially in the case of theHaywardndashRodgers Creek faults infinitely long Atwo-dimensional model cannot describe the situa-tion at a given trench site where ruptures mainly tothe north or south of the site cause some offset in the

trench Nevertheless the fact that the results fromsuch a simple model are not unreasonable is encour-aging One useful attribute of the models exploredhere are that they are testable in the sense of predict-ing future behavior Running the model forward intime to the next San Andreas earthquake yields thevelocity distributions shown in Figure 17 Of coursea north HaywardndashRodgers Creek earthquake mayoccur prior to the next San Andreas event

The Maxwell relaxation time is reasonablywell estimated by the data I find an optimal valueof about 35 years which corresponds to a viscos-ity of 16 times 1019 Pa-s It should be noted that theestimates of all of the model parameters is only asgood as the mechanical model used to interpretthe data If as seems quite likely there was after-slip on the San Andreas below the 1906 rupturezone the estimate of the viscosity is some combi-nation of the effects of afterslip and distributed

FIG 15 Fit of model to north San Francisco Bay data Maximum slip on Hayward fault is 30 m and on the SAF is70 m

ESTIMATES OF SLIP RATE 79

relaxation Li and Rice (1987) presented anapproximate method for including post-seismicfault slip at constant resistive stress with a relax-ing asthenosphere in the context of a repeatingperiodic earthquake sequence Combining suchan approach with appropriate constitutive laws forstable fault creep as in Tse and Rice (1986)would presumably provide a more physically real-istic description of fault behavior Clearly three-dimensional effects are also important especiallyfor earthquakes on the Rodgers Creek and Hay-ward faults Finally the effects of non-uniform

elastic properties are known to bias interpreta-tions based on simple homogeneous models Elas-tic moduli increase with depth which has theknown effect of biasing the depth distribution offault slip Similarly elastic properties may varywith lithology across the major fault zones whichthemselves may represent zones of more compli-ant rock Finite elements are certainly the mostgeneral way to compute forward models but arenot yet feasible for the optimization and bootstrapmethods as computed here which literally testedthousands of possible models

FIG 16 Five hundred bootstrap estimates of model parameters fit to the north San Francisco Bay data 90 confi-dence bounds are listed at the top of each panel The vertical line in the histograms for recurrence times TSAF and THRCshow the upper bounds corresponding to the minimum slip rate and maximum allowed coseismic slip respectively Thehorizontal bar in the panel for the San Andreas recurrence time shows the preferred estimate from Schwartz et al (1998)The horizontal bar in the panel for tH time since the last Hayward earthquake shows the range of dates compatible withpaleoseismic estimates of the penultimate HaywardndashRodgers Creek event

80 PAUL SEGALL

Acknowledgments

George Thompson set a standard that geophysi-cal models no matter how elegant must be consis-tent with geologic observations This work wasinspired by that example I would like to thank WillPrescott for discussions about the Bay Area defor-mation measurements and Mark Matthews for manyyears ago introducing me to Bayesrsquo theorem WayneThatcher Mark Murray Jeff McGuire and PeterCervelli provided valuable comments at early stagesin this study I thank Jim Savage Duncan Agnewand Shelley Kenner for their thoughtful reviewsGary Ernst and Simon Klemperer carefully editedthe manuscript

REFERENCES

Argus D F and Gordon R G 2001 Present tectonicmotion across the Coast Ranges and San Andreas faultsystem in central California Geological Society ofAmerica Bulletin v 113 p 1580ndash1592

Beroza G C 1991 Near-source modeling of the LomaPrieta earthquake Evidence for heterogeneous slipand implications for earthquake hazard Bulletin of theSeismological Society of America v 81 p 1603ndash1621

Cohen S C 1982 A multilayer model of time-dependentdeformation following an earthquake on a strike-slipfault Journal of Geophysical Research v 87 p 5409ndash5421

HFPEG (Hayward Fault Paleoearthquake Group) 1999Timing of paleoearthquakes on the northern HaywardfaultmdashPreliminary evidence in El Cerrito CaliforniaUS Geological Survey Open File Report 99-318

Henstock T J Levander A and Hole J A 1997Deformation in the lower crust of the San Andreas faultsystem in northern California Science v 278 p 650ndash653

Feigl K L King R W and Jordan T H 1990 Geo-detic measurement of tectonic deformation in theSanta Maria fold and thrust belt California Journal ofGeophysical Research v 95 p 2679ndash2699

Gill P E Murray W and Wright M H 1981 Practicaloptimization London UK Academic Press 401 p

Grant L B and Sieh K 1994 Paleoseismic evidence ofclustered earthquakes on the San Andreas fault in the

FIG 17 Simulation of Bay Area velocity field as a function of time Each line is separated by roughly 27 years

76 PAUL SEGALL

~ 428 years The distributions for tHRC the timesince the most recent HaywardndashRodgers Creekearthquake is strongly concentrated near the mini-

mum allowed range The distribution of slip rates onthe San Andreas and HaywardndashRodgers Creekfaults are bimodal with peaks at 19 mmyr and 25

FIG 14 GPS-derived velocities for the north San Francisco Bay region after Prescott et al (2001) AbbreviationsSAF = San Andreas fault HF = Hayward fault CF = Calaveras fault RGF = Rodgers Creek fault CGVF = ConcordndashGreen Valley fault

TABLE 1 A priori Bounds on Parameters

Parameter Symbol Min Max

Lithospheric thickness km H 10 30

Relaxation time yr tR 10 200

SAF recurrence interval yr TSAF 100 500

Time since last Hayward event yr tHRC 225 690

Hayward recurrence interval yr THRC 225 700

Green valley fault locking depth km d 05 15

SAF slip rate my 0019 0027

Hayward slip rate myr 0007 0013

Green valley slip rate myr 0002 0008

smiddotSAF

smiddotHRC

smiddotCGV

ESTIMATES OF SLIP RATE 77

mmyr (SAF) and 7 mmyr and 13 mmyr (HRC)These results still must be viewed with some cau-tion in that I have not demonstrated that the boot-strap optimizations were immune to local minima

The bootstrap calculations suggest that the con-straint on maximum coseismic slip does play astrong role in the results To explore this further Ivaried the maximum coseismic slip allowed on thetwo faults The results (Table 2) demonstrate that theSan Andreas recurrence interval is not fully con-trolled by the constraint on the maximum coseismicslip In contrast the recurrence interval on the Hay-ward fault is totally controlled by the maximum slipallowed (for a maximum slip of 10 meters the recur-rence interval reaches the upper bound of 700years) Notice also that the time since the last Hay-wardndashRodgers Creek event is always at the mini-mum value allowed (225 years) The fit to the data isslightly better for large infrequent earthquakes

Discussion

What has been learned from the modeling pre-sented here It seems clear that geodetic data

should not be analyzed in a vacuum The resolutionof the data alone is insufficient to address the prob-lems of greatest interest However when combinedwith prior information from geologic studies GPSdata appear to be able to resolve parameters of con-siderable interest such as the average recurrenceinterval between great earthquakes on the SanAndreas fault Using a simple model of a repeatingdislocation rupturing an elastic plate overlying aMaxwell visco-elastic medium I found an averagerecurrence interval of roughly 280 years on thenorth coast segment of the San Andreas faultAlthough the uncertainties in this estimate arelarge the result is in surprisingly good accordancewith recent paleoseismic results (Schwartz et al1998) that suggest an event similar to 1906 in themid 1600s

The results for the HaywardndashRodgers Creek faultare also not unreasonable According to the GPSdata the most recent event here occurred in ~1775regardless of the estimated recurrence time Thisresult is not inconsistent with the paleoseismic find-ing that the most recent event on the Rodgers Creekfault took place in the interval 1670ndash1776

TABLE 2 Influence of Maximum Coseismic Slip on Estimated Parameters1

Parameter

Lithospheric thickness H km 150 184 220

Relaxation time tR yr 357 345 340

SAF recurrence interval TSAF yr 220 280 337

Limiting interval yr 290 368 526

Time since last Hayward event tHRC yr 225 225 225

Hayward recurrence interval THRC yr 2857 4286 700

Limiting interval yr 285 428 714

Green Valley fault locking depth d km 11 22 47

SAF slip rate myr 0025 0025 00249

Hayward slip rate myr 0007 0007 00071

Green valley slip rate myr 0008 0008 0008

1Limiting interval is computed using the estimated slip rate for that fault not the minimum slip rate that would yield a longer upper bound

DuSAFmax

55=

DuHRCmax

20=

DuSAFmax

70=

DuHRCmax

30=

DuSAFmax

100=

DuHRCmax

50=

DuSAFmax

smiddotSAFcurren

DuHRCmax

smiddotHRCcurren

smiddotSAF

smiddotHRC

smiddotCGV

78 PAUL SEGALL

(Schwartz et al 1993) and on the northern Hay-ward fault in 1640ndash1776 (HFPEG 1999) The GPSdata alone yield limited information about the aver-age recurrence time on the HaywardndashRodgers Creekfault However adding the reasonable bound thatslip not exceed 2 meters on average yields a recur-rence interval of 285 years The average slip in the1868 Hayward fault earthquake was approximately2 meters (Yu and Segall 1996) In comparisontrenching at the El Cerrito site on the northern Hay-ward fault suggests a recurrence interval of possiblyless than 270 years but no more than 710 years(HFPEG 1999)

It should not need to be stated that earthquakesare neither periodic nor especially in the case of theHaywardndashRodgers Creek faults infinitely long Atwo-dimensional model cannot describe the situa-tion at a given trench site where ruptures mainly tothe north or south of the site cause some offset in the

trench Nevertheless the fact that the results fromsuch a simple model are not unreasonable is encour-aging One useful attribute of the models exploredhere are that they are testable in the sense of predict-ing future behavior Running the model forward intime to the next San Andreas earthquake yields thevelocity distributions shown in Figure 17 Of coursea north HaywardndashRodgers Creek earthquake mayoccur prior to the next San Andreas event

The Maxwell relaxation time is reasonablywell estimated by the data I find an optimal valueof about 35 years which corresponds to a viscos-ity of 16 times 1019 Pa-s It should be noted that theestimates of all of the model parameters is only asgood as the mechanical model used to interpretthe data If as seems quite likely there was after-slip on the San Andreas below the 1906 rupturezone the estimate of the viscosity is some combi-nation of the effects of afterslip and distributed

FIG 15 Fit of model to north San Francisco Bay data Maximum slip on Hayward fault is 30 m and on the SAF is70 m

ESTIMATES OF SLIP RATE 79

relaxation Li and Rice (1987) presented anapproximate method for including post-seismicfault slip at constant resistive stress with a relax-ing asthenosphere in the context of a repeatingperiodic earthquake sequence Combining suchan approach with appropriate constitutive laws forstable fault creep as in Tse and Rice (1986)would presumably provide a more physically real-istic description of fault behavior Clearly three-dimensional effects are also important especiallyfor earthquakes on the Rodgers Creek and Hay-ward faults Finally the effects of non-uniform

elastic properties are known to bias interpreta-tions based on simple homogeneous models Elas-tic moduli increase with depth which has theknown effect of biasing the depth distribution offault slip Similarly elastic properties may varywith lithology across the major fault zones whichthemselves may represent zones of more compli-ant rock Finite elements are certainly the mostgeneral way to compute forward models but arenot yet feasible for the optimization and bootstrapmethods as computed here which literally testedthousands of possible models

FIG 16 Five hundred bootstrap estimates of model parameters fit to the north San Francisco Bay data 90 confi-dence bounds are listed at the top of each panel The vertical line in the histograms for recurrence times TSAF and THRCshow the upper bounds corresponding to the minimum slip rate and maximum allowed coseismic slip respectively Thehorizontal bar in the panel for the San Andreas recurrence time shows the preferred estimate from Schwartz et al (1998)The horizontal bar in the panel for tH time since the last Hayward earthquake shows the range of dates compatible withpaleoseismic estimates of the penultimate HaywardndashRodgers Creek event

80 PAUL SEGALL

Acknowledgments

George Thompson set a standard that geophysi-cal models no matter how elegant must be consis-tent with geologic observations This work wasinspired by that example I would like to thank WillPrescott for discussions about the Bay Area defor-mation measurements and Mark Matthews for manyyears ago introducing me to Bayesrsquo theorem WayneThatcher Mark Murray Jeff McGuire and PeterCervelli provided valuable comments at early stagesin this study I thank Jim Savage Duncan Agnewand Shelley Kenner for their thoughtful reviewsGary Ernst and Simon Klemperer carefully editedthe manuscript

REFERENCES

Argus D F and Gordon R G 2001 Present tectonicmotion across the Coast Ranges and San Andreas faultsystem in central California Geological Society ofAmerica Bulletin v 113 p 1580ndash1592

Beroza G C 1991 Near-source modeling of the LomaPrieta earthquake Evidence for heterogeneous slipand implications for earthquake hazard Bulletin of theSeismological Society of America v 81 p 1603ndash1621

Cohen S C 1982 A multilayer model of time-dependentdeformation following an earthquake on a strike-slipfault Journal of Geophysical Research v 87 p 5409ndash5421

HFPEG (Hayward Fault Paleoearthquake Group) 1999Timing of paleoearthquakes on the northern HaywardfaultmdashPreliminary evidence in El Cerrito CaliforniaUS Geological Survey Open File Report 99-318

Henstock T J Levander A and Hole J A 1997Deformation in the lower crust of the San Andreas faultsystem in northern California Science v 278 p 650ndash653

Feigl K L King R W and Jordan T H 1990 Geo-detic measurement of tectonic deformation in theSanta Maria fold and thrust belt California Journal ofGeophysical Research v 95 p 2679ndash2699

Gill P E Murray W and Wright M H 1981 Practicaloptimization London UK Academic Press 401 p

Grant L B and Sieh K 1994 Paleoseismic evidence ofclustered earthquakes on the San Andreas fault in the

FIG 17 Simulation of Bay Area velocity field as a function of time Each line is separated by roughly 27 years

ESTIMATES OF SLIP RATE 77

mmyr (SAF) and 7 mmyr and 13 mmyr (HRC)These results still must be viewed with some cau-tion in that I have not demonstrated that the boot-strap optimizations were immune to local minima

The bootstrap calculations suggest that the con-straint on maximum coseismic slip does play astrong role in the results To explore this further Ivaried the maximum coseismic slip allowed on thetwo faults The results (Table 2) demonstrate that theSan Andreas recurrence interval is not fully con-trolled by the constraint on the maximum coseismicslip In contrast the recurrence interval on the Hay-ward fault is totally controlled by the maximum slipallowed (for a maximum slip of 10 meters the recur-rence interval reaches the upper bound of 700years) Notice also that the time since the last Hay-wardndashRodgers Creek event is always at the mini-mum value allowed (225 years) The fit to the data isslightly better for large infrequent earthquakes

Discussion

What has been learned from the modeling pre-sented here It seems clear that geodetic data

should not be analyzed in a vacuum The resolutionof the data alone is insufficient to address the prob-lems of greatest interest However when combinedwith prior information from geologic studies GPSdata appear to be able to resolve parameters of con-siderable interest such as the average recurrenceinterval between great earthquakes on the SanAndreas fault Using a simple model of a repeatingdislocation rupturing an elastic plate overlying aMaxwell visco-elastic medium I found an averagerecurrence interval of roughly 280 years on thenorth coast segment of the San Andreas faultAlthough the uncertainties in this estimate arelarge the result is in surprisingly good accordancewith recent paleoseismic results (Schwartz et al1998) that suggest an event similar to 1906 in themid 1600s

The results for the HaywardndashRodgers Creek faultare also not unreasonable According to the GPSdata the most recent event here occurred in ~1775regardless of the estimated recurrence time Thisresult is not inconsistent with the paleoseismic find-ing that the most recent event on the Rodgers Creekfault took place in the interval 1670ndash1776

TABLE 2 Influence of Maximum Coseismic Slip on Estimated Parameters1

Parameter

Lithospheric thickness H km 150 184 220

Relaxation time tR yr 357 345 340

SAF recurrence interval TSAF yr 220 280 337

Limiting interval yr 290 368 526

Time since last Hayward event tHRC yr 225 225 225

Hayward recurrence interval THRC yr 2857 4286 700

Limiting interval yr 285 428 714

Green Valley fault locking depth d km 11 22 47

SAF slip rate myr 0025 0025 00249

Hayward slip rate myr 0007 0007 00071

Green valley slip rate myr 0008 0008 0008

1Limiting interval is computed using the estimated slip rate for that fault not the minimum slip rate that would yield a longer upper bound

DuSAFmax

55=

DuHRCmax

20=

DuSAFmax

70=

DuHRCmax

30=

DuSAFmax

100=

DuHRCmax

50=

DuSAFmax

smiddotSAFcurren

DuHRCmax

smiddotHRCcurren

smiddotSAF

smiddotHRC

smiddotCGV

78 PAUL SEGALL

(Schwartz et al 1993) and on the northern Hay-ward fault in 1640ndash1776 (HFPEG 1999) The GPSdata alone yield limited information about the aver-age recurrence time on the HaywardndashRodgers Creekfault However adding the reasonable bound thatslip not exceed 2 meters on average yields a recur-rence interval of 285 years The average slip in the1868 Hayward fault earthquake was approximately2 meters (Yu and Segall 1996) In comparisontrenching at the El Cerrito site on the northern Hay-ward fault suggests a recurrence interval of possiblyless than 270 years but no more than 710 years(HFPEG 1999)

It should not need to be stated that earthquakesare neither periodic nor especially in the case of theHaywardndashRodgers Creek faults infinitely long Atwo-dimensional model cannot describe the situa-tion at a given trench site where ruptures mainly tothe north or south of the site cause some offset in the

trench Nevertheless the fact that the results fromsuch a simple model are not unreasonable is encour-aging One useful attribute of the models exploredhere are that they are testable in the sense of predict-ing future behavior Running the model forward intime to the next San Andreas earthquake yields thevelocity distributions shown in Figure 17 Of coursea north HaywardndashRodgers Creek earthquake mayoccur prior to the next San Andreas event

The Maxwell relaxation time is reasonablywell estimated by the data I find an optimal valueof about 35 years which corresponds to a viscos-ity of 16 times 1019 Pa-s It should be noted that theestimates of all of the model parameters is only asgood as the mechanical model used to interpretthe data If as seems quite likely there was after-slip on the San Andreas below the 1906 rupturezone the estimate of the viscosity is some combi-nation of the effects of afterslip and distributed

FIG 15 Fit of model to north San Francisco Bay data Maximum slip on Hayward fault is 30 m and on the SAF is70 m

ESTIMATES OF SLIP RATE 79

relaxation Li and Rice (1987) presented anapproximate method for including post-seismicfault slip at constant resistive stress with a relax-ing asthenosphere in the context of a repeatingperiodic earthquake sequence Combining suchan approach with appropriate constitutive laws forstable fault creep as in Tse and Rice (1986)would presumably provide a more physically real-istic description of fault behavior Clearly three-dimensional effects are also important especiallyfor earthquakes on the Rodgers Creek and Hay-ward faults Finally the effects of non-uniform

elastic properties are known to bias interpreta-tions based on simple homogeneous models Elas-tic moduli increase with depth which has theknown effect of biasing the depth distribution offault slip Similarly elastic properties may varywith lithology across the major fault zones whichthemselves may represent zones of more compli-ant rock Finite elements are certainly the mostgeneral way to compute forward models but arenot yet feasible for the optimization and bootstrapmethods as computed here which literally testedthousands of possible models

FIG 16 Five hundred bootstrap estimates of model parameters fit to the north San Francisco Bay data 90 confi-dence bounds are listed at the top of each panel The vertical line in the histograms for recurrence times TSAF and THRCshow the upper bounds corresponding to the minimum slip rate and maximum allowed coseismic slip respectively Thehorizontal bar in the panel for the San Andreas recurrence time shows the preferred estimate from Schwartz et al (1998)The horizontal bar in the panel for tH time since the last Hayward earthquake shows the range of dates compatible withpaleoseismic estimates of the penultimate HaywardndashRodgers Creek event

80 PAUL SEGALL

Acknowledgments

George Thompson set a standard that geophysi-cal models no matter how elegant must be consis-tent with geologic observations This work wasinspired by that example I would like to thank WillPrescott for discussions about the Bay Area defor-mation measurements and Mark Matthews for manyyears ago introducing me to Bayesrsquo theorem WayneThatcher Mark Murray Jeff McGuire and PeterCervelli provided valuable comments at early stagesin this study I thank Jim Savage Duncan Agnewand Shelley Kenner for their thoughtful reviewsGary Ernst and Simon Klemperer carefully editedthe manuscript

REFERENCES

Argus D F and Gordon R G 2001 Present tectonicmotion across the Coast Ranges and San Andreas faultsystem in central California Geological Society ofAmerica Bulletin v 113 p 1580ndash1592

Beroza G C 1991 Near-source modeling of the LomaPrieta earthquake Evidence for heterogeneous slipand implications for earthquake hazard Bulletin of theSeismological Society of America v 81 p 1603ndash1621

Cohen S C 1982 A multilayer model of time-dependentdeformation following an earthquake on a strike-slipfault Journal of Geophysical Research v 87 p 5409ndash5421

HFPEG (Hayward Fault Paleoearthquake Group) 1999Timing of paleoearthquakes on the northern HaywardfaultmdashPreliminary evidence in El Cerrito CaliforniaUS Geological Survey Open File Report 99-318

Henstock T J Levander A and Hole J A 1997Deformation in the lower crust of the San Andreas faultsystem in northern California Science v 278 p 650ndash653

Feigl K L King R W and Jordan T H 1990 Geo-detic measurement of tectonic deformation in theSanta Maria fold and thrust belt California Journal ofGeophysical Research v 95 p 2679ndash2699

Gill P E Murray W and Wright M H 1981 Practicaloptimization London UK Academic Press 401 p

Grant L B and Sieh K 1994 Paleoseismic evidence ofclustered earthquakes on the San Andreas fault in the

FIG 17 Simulation of Bay Area velocity field as a function of time Each line is separated by roughly 27 years

78 PAUL SEGALL

(Schwartz et al 1993) and on the northern Hay-ward fault in 1640ndash1776 (HFPEG 1999) The GPSdata alone yield limited information about the aver-age recurrence time on the HaywardndashRodgers Creekfault However adding the reasonable bound thatslip not exceed 2 meters on average yields a recur-rence interval of 285 years The average slip in the1868 Hayward fault earthquake was approximately2 meters (Yu and Segall 1996) In comparisontrenching at the El Cerrito site on the northern Hay-ward fault suggests a recurrence interval of possiblyless than 270 years but no more than 710 years(HFPEG 1999)

It should not need to be stated that earthquakesare neither periodic nor especially in the case of theHaywardndashRodgers Creek faults infinitely long Atwo-dimensional model cannot describe the situa-tion at a given trench site where ruptures mainly tothe north or south of the site cause some offset in the

trench Nevertheless the fact that the results fromsuch a simple model are not unreasonable is encour-aging One useful attribute of the models exploredhere are that they are testable in the sense of predict-ing future behavior Running the model forward intime to the next San Andreas earthquake yields thevelocity distributions shown in Figure 17 Of coursea north HaywardndashRodgers Creek earthquake mayoccur prior to the next San Andreas event

The Maxwell relaxation time is reasonablywell estimated by the data I find an optimal valueof about 35 years which corresponds to a viscos-ity of 16 times 1019 Pa-s It should be noted that theestimates of all of the model parameters is only asgood as the mechanical model used to interpretthe data If as seems quite likely there was after-slip on the San Andreas below the 1906 rupturezone the estimate of the viscosity is some combi-nation of the effects of afterslip and distributed

FIG 15 Fit of model to north San Francisco Bay data Maximum slip on Hayward fault is 30 m and on the SAF is70 m

ESTIMATES OF SLIP RATE 79

relaxation Li and Rice (1987) presented anapproximate method for including post-seismicfault slip at constant resistive stress with a relax-ing asthenosphere in the context of a repeatingperiodic earthquake sequence Combining suchan approach with appropriate constitutive laws forstable fault creep as in Tse and Rice (1986)would presumably provide a more physically real-istic description of fault behavior Clearly three-dimensional effects are also important especiallyfor earthquakes on the Rodgers Creek and Hay-ward faults Finally the effects of non-uniform

elastic properties are known to bias interpreta-tions based on simple homogeneous models Elas-tic moduli increase with depth which has theknown effect of biasing the depth distribution offault slip Similarly elastic properties may varywith lithology across the major fault zones whichthemselves may represent zones of more compli-ant rock Finite elements are certainly the mostgeneral way to compute forward models but arenot yet feasible for the optimization and bootstrapmethods as computed here which literally testedthousands of possible models

FIG 16 Five hundred bootstrap estimates of model parameters fit to the north San Francisco Bay data 90 confi-dence bounds are listed at the top of each panel The vertical line in the histograms for recurrence times TSAF and THRCshow the upper bounds corresponding to the minimum slip rate and maximum allowed coseismic slip respectively Thehorizontal bar in the panel for the San Andreas recurrence time shows the preferred estimate from Schwartz et al (1998)The horizontal bar in the panel for tH time since the last Hayward earthquake shows the range of dates compatible withpaleoseismic estimates of the penultimate HaywardndashRodgers Creek event

80 PAUL SEGALL

Acknowledgments

George Thompson set a standard that geophysi-cal models no matter how elegant must be consis-tent with geologic observations This work wasinspired by that example I would like to thank WillPrescott for discussions about the Bay Area defor-mation measurements and Mark Matthews for manyyears ago introducing me to Bayesrsquo theorem WayneThatcher Mark Murray Jeff McGuire and PeterCervelli provided valuable comments at early stagesin this study I thank Jim Savage Duncan Agnewand Shelley Kenner for their thoughtful reviewsGary Ernst and Simon Klemperer carefully editedthe manuscript

REFERENCES

Argus D F and Gordon R G 2001 Present tectonicmotion across the Coast Ranges and San Andreas faultsystem in central California Geological Society ofAmerica Bulletin v 113 p 1580ndash1592

Beroza G C 1991 Near-source modeling of the LomaPrieta earthquake Evidence for heterogeneous slipand implications for earthquake hazard Bulletin of theSeismological Society of America v 81 p 1603ndash1621

Cohen S C 1982 A multilayer model of time-dependentdeformation following an earthquake on a strike-slipfault Journal of Geophysical Research v 87 p 5409ndash5421

HFPEG (Hayward Fault Paleoearthquake Group) 1999Timing of paleoearthquakes on the northern HaywardfaultmdashPreliminary evidence in El Cerrito CaliforniaUS Geological Survey Open File Report 99-318

Henstock T J Levander A and Hole J A 1997Deformation in the lower crust of the San Andreas faultsystem in northern California Science v 278 p 650ndash653

Feigl K L King R W and Jordan T H 1990 Geo-detic measurement of tectonic deformation in theSanta Maria fold and thrust belt California Journal ofGeophysical Research v 95 p 2679ndash2699

Gill P E Murray W and Wright M H 1981 Practicaloptimization London UK Academic Press 401 p

Grant L B and Sieh K 1994 Paleoseismic evidence ofclustered earthquakes on the San Andreas fault in the

FIG 17 Simulation of Bay Area velocity field as a function of time Each line is separated by roughly 27 years

ESTIMATES OF SLIP RATE 79

relaxation Li and Rice (1987) presented anapproximate method for including post-seismicfault slip at constant resistive stress with a relax-ing asthenosphere in the context of a repeatingperiodic earthquake sequence Combining suchan approach with appropriate constitutive laws forstable fault creep as in Tse and Rice (1986)would presumably provide a more physically real-istic description of fault behavior Clearly three-dimensional effects are also important especiallyfor earthquakes on the Rodgers Creek and Hay-ward faults Finally the effects of non-uniform

elastic properties are known to bias interpreta-tions based on simple homogeneous models Elas-tic moduli increase with depth which has theknown effect of biasing the depth distribution offault slip Similarly elastic properties may varywith lithology across the major fault zones whichthemselves may represent zones of more compli-ant rock Finite elements are certainly the mostgeneral way to compute forward models but arenot yet feasible for the optimization and bootstrapmethods as computed here which literally testedthousands of possible models

FIG 16 Five hundred bootstrap estimates of model parameters fit to the north San Francisco Bay data 90 confi-dence bounds are listed at the top of each panel The vertical line in the histograms for recurrence times TSAF and THRCshow the upper bounds corresponding to the minimum slip rate and maximum allowed coseismic slip respectively Thehorizontal bar in the panel for the San Andreas recurrence time shows the preferred estimate from Schwartz et al (1998)The horizontal bar in the panel for tH time since the last Hayward earthquake shows the range of dates compatible withpaleoseismic estimates of the penultimate HaywardndashRodgers Creek event

80 PAUL SEGALL

Acknowledgments

George Thompson set a standard that geophysi-cal models no matter how elegant must be consis-tent with geologic observations This work wasinspired by that example I would like to thank WillPrescott for discussions about the Bay Area defor-mation measurements and Mark Matthews for manyyears ago introducing me to Bayesrsquo theorem WayneThatcher Mark Murray Jeff McGuire and PeterCervelli provided valuable comments at early stagesin this study I thank Jim Savage Duncan Agnewand Shelley Kenner for their thoughtful reviewsGary Ernst and Simon Klemperer carefully editedthe manuscript

REFERENCES

Argus D F and Gordon R G 2001 Present tectonicmotion across the Coast Ranges and San Andreas faultsystem in central California Geological Society ofAmerica Bulletin v 113 p 1580ndash1592

Beroza G C 1991 Near-source modeling of the LomaPrieta earthquake Evidence for heterogeneous slipand implications for earthquake hazard Bulletin of theSeismological Society of America v 81 p 1603ndash1621

Cohen S C 1982 A multilayer model of time-dependentdeformation following an earthquake on a strike-slipfault Journal of Geophysical Research v 87 p 5409ndash5421

HFPEG (Hayward Fault Paleoearthquake Group) 1999Timing of paleoearthquakes on the northern HaywardfaultmdashPreliminary evidence in El Cerrito CaliforniaUS Geological Survey Open File Report 99-318

Henstock T J Levander A and Hole J A 1997Deformation in the lower crust of the San Andreas faultsystem in northern California Science v 278 p 650ndash653

Feigl K L King R W and Jordan T H 1990 Geo-detic measurement of tectonic deformation in theSanta Maria fold and thrust belt California Journal ofGeophysical Research v 95 p 2679ndash2699

Gill P E Murray W and Wright M H 1981 Practicaloptimization London UK Academic Press 401 p

Grant L B and Sieh K 1994 Paleoseismic evidence ofclustered earthquakes on the San Andreas fault in the

FIG 17 Simulation of Bay Area velocity field as a function of time Each line is separated by roughly 27 years

80 PAUL SEGALL

Acknowledgments

George Thompson set a standard that geophysi-cal models no matter how elegant must be consis-tent with geologic observations This work wasinspired by that example I would like to thank WillPrescott for discussions about the Bay Area defor-mation measurements and Mark Matthews for manyyears ago introducing me to Bayesrsquo theorem WayneThatcher Mark Murray Jeff McGuire and PeterCervelli provided valuable comments at early stagesin this study I thank Jim Savage Duncan Agnewand Shelley Kenner for their thoughtful reviewsGary Ernst and Simon Klemperer carefully editedthe manuscript

REFERENCES

Argus D F and Gordon R G 2001 Present tectonicmotion across the Coast Ranges and San Andreas faultsystem in central California Geological Society ofAmerica Bulletin v 113 p 1580ndash1592

Beroza G C 1991 Near-source modeling of the LomaPrieta earthquake Evidence for heterogeneous slipand implications for earthquake hazard Bulletin of theSeismological Society of America v 81 p 1603ndash1621

Cohen S C 1982 A multilayer model of time-dependentdeformation following an earthquake on a strike-slipfault Journal of Geophysical Research v 87 p 5409ndash5421

HFPEG (Hayward Fault Paleoearthquake Group) 1999Timing of paleoearthquakes on the northern HaywardfaultmdashPreliminary evidence in El Cerrito CaliforniaUS Geological Survey Open File Report 99-318

Henstock T J Levander A and Hole J A 1997Deformation in the lower crust of the San Andreas faultsystem in northern California Science v 278 p 650ndash653

Feigl K L King R W and Jordan T H 1990 Geo-detic measurement of tectonic deformation in theSanta Maria fold and thrust belt California Journal ofGeophysical Research v 95 p 2679ndash2699

Gill P E Murray W and Wright M H 1981 Practicaloptimization London UK Academic Press 401 p

Grant L B and Sieh K 1994 Paleoseismic evidence ofclustered earthquakes on the San Andreas fault in the

FIG 17 Simulation of Bay Area velocity field as a function of time Each line is separated by roughly 27 years

ESTIMATES OF SLIP RATE 81

Carrizo Plain California Journal of GeophysicalResearch v 99 p 6819ndash6841

Jackson D D 1979 The use of a priori data to resolvethe non-uniqueness in linear inversion GeophysicalJournal of the Royal Astronomical Society v 57 p137ndash157

Johnson H O Agnew D C and Wyatt F K 1994Present-day crustal deformation in Southern Califor-nia Journal of Geophysical Research v 99 p23951ndash23974

Kenner S J and Segall P 2000 Postseismic deforma-tion following the 1906 San Francisco earthquakeJournal of Geophysical Research v 105 p 13195ndash13209

______ 2001 Lower crustal structure in northern Cali-fornia Implications from strain-rate variations follow-ing the 1906 San Francisco earthquake Journal ofGeophysical Research in review

Lehner F K Li V C and Rice J R 1981 Stress diffu-sion along rupturing plate boundaries Journal of Geo-physical Research v 86 p 6155ndash6169

Li V C and Rice J R 1987 Crustal deformation ingreat California earthquake cycles Journal of Geo-physical Research v 92 p 11533ndash11551

Lyzenga G A Raefsky A and Mulligan S G 1991Models of recurrent strike-slip earthquake cycles andthe state of crustal stress Journal of GeophysicalResearch v 96 p 21623ndash21640

Matsursquoura M Jackson D D and Cheng A 1986 Dis-location model for aseismic crustal deformation at Hol-lister California Journal of Geophysical Research v91 p 12661ndash12674

Matthews M V and Segall P Est imation ofdepth-dependent fault slip from measured surfacedeformation with application to the 1906 San Fran-cisco earthquake Journal of Geophysical Research v98 p 12153ndash12163

Menke W 1989 Geophysical data analysis Discreteinverse theory San Diego CA Academic Press 1989

Minster J B and Jordan T H 1987 Vector constraintson western US deformation from space geodesy neo-tectonics and plate motions Journal of GeophysicalResearch v 92 p 4798ndash4804

Murray M H and Segall P 2001 Modeling broadscaledeformation in northern California and Nevada fromplate motions and elastic strain accumulation Geo-physical Research Letters in press

Nur A and Mavko G 1974 Postseismic viscoelasticrebound Science v 183 no 4121 p 204ndash206

Parsons T and Hart P E 1999 Dipping San Andreasand Hayward faults revealed beneath San FranciscoBay California Geology v 27 p 839ndash842

Prescott W H Savage J C Svarc J L and ManakerD 2001 Deformation across the PacificndashNorth Amer-ica plate boundary near San Francisco CaliforniaJournal of Geophysical Research v 106 p 6673ndash6682

Pollitz F F and Sacks I S 1992 Modeling of postseis-mic relaxation following the great 1857 earthquakesouthern California Bulletin of the SeismologicalSociety of America v 82 p 454ndash480

Reches Z Schubert G and Anderson C 1994 Model-ing of periodic great earthquakes on the San Andreasfault effects of nonlinear crustal rheology Journal ofGeophysical Research v 99 p 21983ndash22000

Rundle J B 1986 An approach to modeling present-daydeformation in southern California Journal of Geo-physical Research v 91 p 1947ndash1959

Savage J C 1990 Equivalent strike-slip earthquakecycles in half-space and lithosphere-asthenosphereEarth models Journal of Geophysical Research v 95p 4873ndash4879

Savage J C and Burford R O 1970 Accumulation oftectonic strain in California Bulletin of the Seismolog-ical Society of America v 60 p 1877ndash1896

Savage J C and Prescott W H 1978 Asthenospherereadjustment and the earthquake cycle Journal ofGeophysical Research v 83 p 3369ndash3376

Schwartz D P Pantosti D Hecker S Okumara KBudding K E and Powers T 1993 Late Holocenebehavior and seismogenic potential of the RodgersCreek fault zone Sonoma County California Califor-nia Division of Mines and Geology Special Publica-tion v 113 p 393ndash398

Schwartz D P Pantosti D Okumara K Powers T Jand Hamilton J C 1998 Paleoseismic investigationsin the Santa Cruz mountains California Implicationsfor recurrence of large-magnitude earthquakes on theSan Andreas fault Journal of Geophysical Research v103 p 17985ndash18001

Sieh K E and Jahns R H 1984 Holocene activity ofthe San Andreas fault at Wallace Creek CaliforniaGeological Society of America Bulletin v 95 p 883ndash896

Tarantola A and Valette B 1982 Generalized non-lin-ear inverse problems solved using least squares crite-rion Reviews of Geophysics and Space Physics v 20p 219ndash232

Thatcher W 1975a Strain accumulation and releasemechanism of the 1906 San Francisco earthquakeJournal of Geophysical Research v 80 p 4862ndash4872

______ 1975b Strain accumulation on the northern SanAndreas fault zone since 1906 Journal of GeophysicalResearch v 80 p 4873ndash4880

_______ 1983 Nonlinear strain buildup and the earth-quake cycle on the San Andreas fault Journal of Geo-physical Research v 88 p 5893ndash5902

Toppozada T R and Borchardt G 1998 Re-evaluationof the 1836 ldquoHayward faultrdquo earthquake and the 1838San Andreas fault earthquake Bulletin of the Seismo-logical Society of America v 88 p 140ndash159

Tse S T and Rice J R 1986 Crustal earthquake insta-bility in relation to the depth variation of frictional slip

82 PAUL SEGALL

properties Journal of Geophysical Research v 91 p9452ndash9472

Williams S D P Svarc J L Lisowski M and PrescottW H 1994 GPS-measured rates of deformation in thenorthern San Francisco Bay region California 1990ndash1993 Geophysical Research Letters v 21 p 1511ndash1514

WGCEP (Working Group on California Earthquake Prob-abilities) 1988 Probabilities of large earthquakesoccurring in California on the San Andreas fault USGeological Survey Open File Report 88-398

_______ 1995 Seismic hazards in southern CaliforniaProbable earthquakes1994ndash2024 Bulletin of theSeismological Society of America v 85 p 379ndash439

______ 1999 Earthquake probabilities in the San Fran-cisco Bay Region 2000ndash2030mdashA summary of find-ings US Geological Survey Open File Report 99ndash517

Yu E and Segall P 1996 Slip in the 1868 HaywardEarthquake from the analysis of historical triangula-tion data Journal of Geophysical Research v 101 p16101ndash16118