Physicsof theEarth andPlanetaryInteriors, 64 (1990)1—20 1ElsevierSciencePublishersBy., Amsterdam

Evidencefor andimplicationsof self-healingpulsesof slipin earthquakerupture

ThomasH. Heaton

U.S. GeologicalSurvey,525 S. WilsonAve.Pasadena;CA 91106(USA.)

(ReceivedApril 27, 1990; revisionacceptedMay23, 1990)

ABSTRACT

Heaton,T.H., 1990. Evidencefor andimplications of self-healingpulsesof slip in earthquakerupture. Phys. Earth Planet.Inter., 64: 1—20.

Dislocationtime historiesof modelsderivedfrom waveformsof sevenearthquakesarediscussed.In eachmodel, dislocationrise times(thedurationof slip for a given pointon thefault) arefoundto be short comparedto theoverall durationof theearthquake(— 10%). However, in manycrack-likenumericalmodelsof dynamicrupture,the slip durationata givenpoint iscomparableto theoverall durationof therupture; i.e. slip ata given point continuesuntil information is receivedthat therupturehasstoppedpropagating.Alternativeexplanationsfor thediscrepancybetweentheshort slip durationsusedto modelwaveformsand the long slip durationsinferred from dynamiccrack models are: (1) the dislocation modelsare unabletoresolvetherelatively slow partsof earthquakeslip and haveseriouslyunderestimatedthedislocationsfor theseearthquakes;(2) earthquakesarecomposedof a sequenceof small-dimension(short duration) eventsthat areseparatedby lockedregions(barriers);(3) ruptureoccursin a narrowself-healingpulseof slip that travels alongthe fault surface. Evidenceis discussedthat suggests that slip durationsare indeed short and that the self-healingslip-pulse model is the most appropriateexplanation.

A qualitativemodel is presentedthatproducesself-healingslip pulses.Thekey featureof themodel is theassumptionthatfriction on the fault surfaceis inversely relatedto the local slip velocity. The model hasthefollowing features:high staticstrengthof materials(kilobarrange), low staticstressdrops(in the rangeof tensof bars),and relatively low frictional stressduringslip (less than severalhundredsof bars).It is suggestedthat thereasonthat the averagedislocation scaleswith faultlength is becauselarge-amplitudeslip pulsesaredifficult to stopandhencetendto propagatelargedistances.This model mayexplainwhy seismicity and ambientstressare low along fault segmentsthat have experiencedlargeearthquakes.It alsoqualitativelyexplainswhy therecurrencetime for largeearthquakesmaybe irregular.

1. Infroduction (at a given point) relativeto the time requiredtorupture the entirefault surface.Evidence is pre-

In this papertheimplications of modelsof the sentedthat the slip duration at a given point isdistribution of slip in time and space that have significantlyshorterthan the time to receiveinfor-beendeducedfrom groundmotion datafrom seven mation aboutthe overallrupturedimensions.Pos-earthquakesare discussed.Severalaspectsof these siblemechanismsto explain this observationareslip models are inconsistentwith standardearth- discussed.A qualitative rupture model is sug-quakerupturemodels,anddirectinspectionof the gestedthatwill causethe fault to heal itself shortlyslip models deducedfrom seismic data leadsto after the passageof the rupture front. The sug-surprisingconclusionsaboutthe natureof rupture gestedhealingmechanismis a dynamicfault fric-dynamics.The centralissueis theduration of slip tion that decreaseswith increasingslip velocity.

2 T.H.HEATON

Theimplicationsof this rupturemodel fora variety 1985 MICHOACANof issuesrelatedto stressand stresschangesasso-ciatedwith earthquakesarethendiscussed. ~ SE NW

One of the most important observationsof 0~earthquakesize is that

zM0ozS

3~2 (1) ~ 60

0where x signifies proportionality, M

0 is the Qseismic moment, and S is the rupture area W

(KanamoriandAnderson,1975).Since Z 120

M0 = ~.tS (2) ~ 30 60 90 120 150

DISTANCE ALONG STRIKE (KM)where D is the dislocationaveragedoverS, thisimplies that Fig. 1. Slip distribution (cm) for the M = 8.1 1985 Michoacan,

Mexico, earthquakederivedby Mendozaand Hartzell (1989)cx (3) from the simultaneousinversion of teleseismic and strong

motionwaveform data.The largedot denotesthehypocenterIf the averagestress drop ~ of an earthquake andthe stippledregion denotestheapproximateregionthat isruptureis assumedto beproportionalto the aver- slippingat a particularinstantin time.

agedislocationdivided by a length dimensionofthe rupture, then

— groundmotions.An alternativeearthquakescalingD model is presentedin which the dynamic features

cx ~—~- (4) of the propagatingrupture are of central impor-tance.

Relations (3) and (4) imply that averagestressdrop is independentof S and thereforeof M0. Acommoninterpretationof thesescalingrelationsis

2. Dislocationrise timesthat the averagefault slip D is determinedby arupturedimensionandthe averagestressdrop,i.e.as a rupturegrows over a fault surface, the final Figures 1—7 show contour maps of the slip

distribution for the preferredmodelsfor theearth-fault slip at any given point is not known untilquakes used in this study. These models were

information reachesthat point about the finaldimensionsof the rupture surface. In most dy-namic rupture models, the slip duration varies 1983 BORAH PEAKfrom point to point on the fault andthe slip timehistorycan itself be fairly complex.However, ex- ~amininationof slip historiesfor dynamicrupture zmodelsfor faults with aspectratios of <2 (Day, 0

1982) yieldsa rough estimateof the averageslip W 5

durationthat is given by z

2%% or,

(5) 5 ib 15 20 25 30 35 40 45 50where 7 is the averageslip duration, ~‘ is the DISTANCE ALONG STRIKE (km)

velocity of the rupture front, and signifies a Fig. 2. Slip distribution(cm) for the M = 7.3 1983 BorahPeak,

very approximateequality. However, as i will ID, earthquakederivedby Mendozaand Hartzell (1988) fromthe inversionof long- and short-periodteleseismicwaveform

show,this estimateof the slip durationfor a point data. The large dot denotes the hypocenterstippled region

on the fault is an order of magnitudelarger than denotesthe approximateregionthat is slippingata particular

that inferred from the modeling of earthquake instantin time.

SELF-HEALING PULSESOF SLIP IN EARTHQUAKE RUPTURE 3

1971 SAN FERNANDO IMPERIAL VALLEY__________________ C ____Sierra Madre Fault

1—Depth3km

~05O

0.U0 -8-~~ 0 4 8 12 6 20 24 28 32 38 40LENGT Ilk”)

Fig. 5. Slip distribution (cm) for the M = 6.2 1984 MorganHill, CA, earthquakederivedby Hartzell and Heaton(1986)

Z from theinversionof strong motionwaveform data.The large

Cl) 1st dot denotesthehypocenterandthestippledregiondenotesthe

time.4km Depth I6~ ypocenter approximateregion that is slipping at a particular instant in

after the passageof therupturefront. This classofDISTANCE ALONG STRIKE (krn) modelsis usuallyreferredto as ‘dislocation mod-

Fig. 3. Slip distribution(m) for theM = 6.5 1971 SanFernando, els’ andslip durationis synonomouswith disloca-CA, earthquakederivedby Heaton (1982) from the modeling tion rise time. The spatial distribution of the slipof strongmotionand teleseismicwaveformsandgeodeticdata.The earthquakewas modeledas a double event and only the amplitudeis variedto give anacceptablematchtofirst eventis usedin this study.Thestippledregiondenotesthe the observedwaveforms.In someof thesemodelsapproximateregion that is slipping at a particular instant in the dislocationrise timeis assumedto be uniformtime. on the rupturesurface,andin othersit is allowed

to vary spatially to improve the quality of the fit.

derivedfrom deterministic modelingof observed In principal, dislocation models can be para-strong motion and teleseismic waveforms. Al- meterizedwith enoughflexibility to encompassallthoughdifferent procedureswereusedto produce possibleslip historieson a fault surface(including

thosethat result from dynamic‘crack-like’ rupturethesemodels,a generalassumptionis that ruptureproceedsalong the fault with an approximatelyconstantrupturevelocity and that the slip occursin some time period (the dislocation rise time) 1986NORTH PALM SPRINGS

MORGAN HILL ~___________________________ az5

00

~1029Z 10

Cl)

12 . 5 10 15 200 3 6 9 12 5 18 21 24 27

LENGTH (km) DISTANCE ALONG STRIKE (km)Fig. 4. Slip distribution (cm) for the M= 6.5 1979 Imperial Fig. 6. Slip distribution(cm) for theM = 6.0 1986 North PalmValley, CA, earthquakederivedby HartzellandHeaton (1983) Springs, CA, earthquakederived by Hartzell (1989) from thefrom the simultaneousinversionof strong motion and tele- inversion of strong motion waveform data. The large dotseismic waveformdata. The largedot denotesthe hypocenter denotesthe hypocenterand the stippled region~denotestheand thestippled region denotestheapproximateregionthat is approximateregion that is slipping at a particularinstant inslippingata particularinstantin time. time.

4 T.H. HEATON

1979 COYOTE LAKE models),althoughmostapplicationsof dislocation* Hypoc•nt.v models assumethat the duration of slip is rela-

-a tively homogeneouson the fault surface.Several functional forms for the slip time his-

tory havebeentried, the simplestbeinga simple

ramp of duration 1~.However, more complexfunctional forms havebeentried. For instance,aslip thatgrows as the squareroot of time hasalso

-8 ..~ I.2—~ been used (eg. Beroza and Spudich, 1988), al-.8 thoughslip historiesof this form havenotsatisfac-

-to ~ ~ . torily matchedobservedwaveforms.BecausetheseModel Ni models were not constructed to match high-

I2~ ~ frequencydata, details of the slip history cannot

DISTANCE ALONG STRIKE (km) bederived. Instead,the modelsgive an indicationof how much slip occurswithin a given time afterFig. 7. Slip distnbution from the M = 5.9 1979 Coyote Lake,

CA, earthquakederived by Liu and Helmberger(1983) from the passageof the rupturefront.modelling strong motion waveform data.The stippledregion Table1 gives a comparisonof dislocationrisedenotestheapproximateregionthat is slippingat a particular times 7~derivedfrom thesemodelswith disloca-instantin time. tion rise times that would be expectedif the

TABLE 1Observedruptureparameters

Earthquake Moment I w ‘A Vr 2~&21A Reference

(X1025 (km) (km) (km) (~~_1) ~M7~dyn cm) (s) (s) (s)

19/9/88(M = 8.1) MendozaandMichoacan,Mexico 1500 150 120 = 40 2.6 33 10 5 Hartzell(1989)

28/10/83(M = 7.3) MendozaandBorahPeak 23 40 20 9 2.9 6.5 2.1 0.6 a Hartzell(1988)

9/2/71 (M = 6.5)SanFernando~’ 7 12 14 7 2.8 3.1 1.9 0.8 Heaton(1982)

15/10/79(M = 6.5) HartzellandImperialValley 5 30 10 7 2.6 4.5 1.8 1.0 Heaton(1983)

24/4/84(M = 6.2) HartzellandMorganHill 2.1 20 8 4 2.8 3.0 1.0 0.3 Heaton(1986)

8/7/86(M = 6.0) HartzellNorth PalmSprings 1.8 18 10 5 3.0 3.0 1.1 0.4 C (1989)

6/8/79(M = 5.9) Liu andHelm-CoyoteLake 0.35 6 6 4 2.8 1.4 0.9 0.5 d berger(1983)a From teleseismicdata;maybeas largeas 1 s.b Estimatesarefor thefirst of two sources.C A maximal value, mayhaveactuallybeenshorter.d A maximal value, mayactuallyhavebeenshorter.Bouchon(1982)modeledthis dataassumingan instantaneousrise time.

1, approximaterupturelength; w,approximaterupturewidth; ‘A’ approximatedimensionof themajorasperity; l’, averagerupturevelocity; S, rupturearea(1w); 7, observeddislocationrise time.

SELF-HEALING PULSES OF SLIP IN EARTHQUAKE RUPTURE 5

durationof slip is comparableto the overall rup- however,I simply wish to testthe hypothesisthatture duration of the earthquake.In many in- the slip durationat any point is comparableto thestances,the estimateof the rise time from these time requiredfor overall rupture.modelsis a maximumnumbersince the datawas As can be seen in Table 1, the rise timeslow-passfiltered. Althoughdislocationrise timeis derivedfrom modelingdataare,on average,onlyoneof the most difficult numbersto extractfrom 16% as long as that expected from the overalldeterministicsourcemodels,it would be very dif- ruptureduration(eqn. 5) and 42% as long as thatficult to exceedthe rise time estimatesgiven in expectedfrom the ruptureduration of the domi-Table 1 by a factor of 2. However, in many nant asperity(eqn. 6). Kanamori and Andersoninstances,it would bepossibleto usesignificantly (1975)also reportedcomparablyshort dislocationshorterslip durations.Severalestimatesof char- rise timesbasedon a completelydifferent dataset.acteristic rupturelength dimensionsfor the earth- We are led to the rathersurprisingconclusionthatquakesare also given in Table 1. In many in- only a smallportionof the overall rupturesurfacestancesthe slip distribution is spatially heteroge- is undergoingslip at anygiven point in time. Thisneousand therefore the characteristicdimension is shown graphically in Figs. 1—7; the stippled

‘A of the dominantasperityfor eachearthquakeis bandsrepresentthe approximateareaslipping atalso given, one instant in time. We can imagine that the

Two predictionsof the dislocationrise time are ruptureis actuallyapropagatingpulseof slip thatgiven assumingthat the slip durationis compar- covers only 10% of the rupture surfaceat anyable to the time required for rupture. These point in time.The sizeof the slip in this pulserisespredictionsare basedon the assumptionthat in- and falls as the ruptureproceedsalong the faultformation must reacha given point on the fault surface.aboutthe final dimensionsof the rupturesurface. This model of earthquakeruptureas a propa-In the first it is assumedthat the dislocation gating pulse of slip is very similar to the simplecontinuesuntil information is receivedthat the model introducedby Haskell (1964) 25 yearsagoentire fault has ruptured, in which casethe rise to estimatethe energy radiated by earthquakes.time 7 is approximatelygiven by eqn.(5). In the Haskell foundthat relatively short dislocationrisesecond,the dislocationrise time is assumedto be timeswere necessaryto accountfor the relativelycomparableto the time required to rupture the large radiated energy from earthquakes.Brunedominantasperity,in which casewe would expect (1970),qualitatively introduceda similar self-heal-the dislocationrise time to be approximatedby ing slip-pulse model to explain apparent dis-

2/ crepancies between ‘effective stress’ estimated(6) from thespectralcontentof radiatedseismicwaves

and staticstressdrops.Brunecalled this the ‘par-

Other methodsfor estimatingthe expecteddis- tial stress-drop’and ‘abrupt locking’ model andlocationrise timecanbe devised.Forinstance,the this model hasbeensuggestedto explain the radi-fault width may be the most appropriatedimen- ation from aftershocksof the 1971 SanFernando,sion for long, narrow rupturesurfaces.However, California, earthquake(Tucker and Brune, 1977)for the fault modelsshown,the rupturewidths are andfrom small earthquakesnearAnza,Californiaof the same order as rupture lengths and the (Bruneet al., 1986).widths are larger than the dimensionof the domi- Otherevidenceis now presentedin supportofnant asperity. Furthermore,we might expect the short slip durations.The first is eye witness re-slip durationto behighly variableoverthe rupture ports of the scarp formation of the M7.3, 1983surface,with long rise timesin the centerof faults Borah Peak earthquake.The time required toand short rise timesnear the ruptureboundaries. propagatea rupturefront along the 40-kmlengthA more completeestimatefor expectedslip dura- of this earthquakeexceeded10 s. Wallace(1984)tionswould requirespecificdynamicrupturemod- documentedthe recollections of Lawana Knoxeli for the earthquakesconsidered.At this point, who was about 300 m from the surfacetrace of

6 T.H. HEATON

theearthquake.Accordingto Wallace,“Mrs. Knox t9N

reportedthat the 1- to 1.5-meter-highscarpformed MEXICOin about 1 s. She reportedthat the scarp reached ,150its full height quickly, and that it did not appear Epicenter

to adjust up or down later or oscillate up anddown while reachingits full height”. This shortrise time was independentlycorroboratedby theaccountof two othereyewitnesses(Peltonet al., rN

1984) andis remarkablycloseto the valueusedbyMendozaand Hartzell (1989).

Somemay arguethat seismicmodelingis sensi-0

tive only to the highdislocationparticlevelocitiesat the rupturefront andthat seismicwavesare not I I 7°N

103’W tO2W lOI’Wsensitiveto the long-periodslip that occurs afterthe rupturefront passesthrough.However,Beroza DISPLACEMENT(cm)

andSpudich(1988) attemptedto model the 1984 CAL VIL UNI

Morgan Hill earthquakewith just such a model. —‘s ~

Following the model of Kostrov (1966), they used Nort~~—~- North North

a dislocationtime historythat variesas the square ~ ~ ~root of time. Thisproducesa time functionwith avery sharp rise and a very smooth tail. Yet in ~ ~ jt~.

Eost Eastorder to fit the strongmotion records,they wereforced to truncatethe slip function after a mere _._?.~_.!i~. “‘~-~- ~

0.2 s, an even shorterdislocation rise time than—/--~ ~v

that deducedby Hartzell andHeaton(1986). ~t~2-~’ ~

As afinal example,considerthe stronggroundI ________________________________________________________ ________________________________________________________

motions recorded during the 1985 Michoacan o 25 50 io 35 50 20 45 ioearthquake.Andersonet al. (1986) showed that TIME (seconds)

grounddisplacements(relativetoaninertialframe) Fig. 8. Comparisonof grounddisplacements(with respectto

could be deducedfrom the ground acceleration aninertial frame) observedabove the 1985 Michoacanearth-quakewith displacementssynthesizedfrom the rupturemodelrecords.Theyshowedthatobservedstaticsea-level of Mendozaand Hartzell (1989). The observedvertical dis-

changescoincide with the ground displacement placementatCaletade Campos(CAL) coincided with locally

inferredfrom strongmotionrecords.Furthermore, observedsea-levelchanges(Andersonet al., 1986), indicating

thesestatic deformationsoccurredwithin a rela- that the final static displacementwasobtainedwithin the 10-slively short time period, and they cited this as interval inferred from the displacementrecord. The rupture

surfaceis approximately30 km belowthestationsandmost ofevidencein support of the Brune ‘partial stress thedurationof therecord is due to thefinite time requiredfor

drop’ model (short slip duration). Mendoza and the rupture to propagateand for the waves to arrive from

Hartzell (1989) modeled teleseismicbody waves different parts of the fault. If the dislocation rise time was

and the long-periodgroundvelocity (with respect comparable to the rupture time, then the duration of the

to an inertial frame)in the near-sourceregionand recordswould be considerablylonger.

found that this datarequiredshort dislocationrisetimes(< 5 sfor a M8.1 earthquake).Comparisonsof the observeddisplacementswith thosesynthe- stoppedbefore it even begins at UNI which issized by Mendozaand Hartzell (1989) are shown locatednearthe southernendof the rupture.in Fig. 8. A dislocationrise time comparableto RuppertandYomogida(1990)alsoinvestigatedthe rupture time would effectively double the the ground displacementat CAL and they con-length of the synthetic records.In Fig. 8, notice cludedthat it wasbest explainedby a ‘crack-like’that the motion at CAL (nearthe epicenter)has modelin which the slip durationis variableon the

SELF-HEALING PULSES OF SLIP IN EARTHQUAKE RUPTURE 7

rupture surfaceand is controlled by the overall surfaceis free to slip along the fault plane; trac-rupturedimension.In order to producethe short- tions are then applied parallel to the rupturedurationramp-likedisplacementat CAL, Ruppert surfacesuchthat theyreproducesomedistributionand Yomogida (1990) proposebilateral rupture of slip. These tractions are averagedover the(i.e. bothup- and down-dip) on a fault width of rupture surfaceto give an averagestatic stress<40 km which resultsin slip durationsof <6 s drop ~.ofor an earthquake.Sinceall of the equi-and 8-rn dislocationsin the hypocentral region. librium equationsfor stressand strain are linearThis slip durationis roughlycomparablewith that functions of D, the estimateof the static stressdeducedby Mendoza and Hartzell (1989), but drop only dependson the final distributionof slipthey used considerably larger fault dimensions and is independentof how the slip actually oc-andinferredsmallerdislocationsto model a much curred in time. Severalsolutions have beende-largerset of data. nvedfor the slip distribution that would result if

In summary,thereare a numberof observa- the stressdrop is uniform. Eshelby(1957)derivedtions that suggestthat the slip durationat a given the following stressdrop for a circular fault in apoint on a fault is short comparedwith the time Poissonianwhole space.required for information to travel from theboundariesof the rupture. However, becausethe z~a= 0.78~s—~= (7)slip duration must be inferred from interpreta-tions of rupturemodels, I believethat it is notyet Stressdropshavealso beennumericallycomputedpossibleto considerthis as a provenhypothesis. from the averageslip assumingrectangularfaultsIf, in fact, the dislocationrise times are not short at variousdepthsof burial in anelastic half-spacecomparedwith the overall rupturetime for earth- by Parsonset al. (1988) and their resultscan bequakes,then thereare many earthquakemodels summarizedbydeterminedfrom waveform data that haveseri- —

ouslyunderestimatedthe slip for earthquakes.For &~= c,~ (8)the remainderof thispaper,it is assumedthat slip L

durationsare indeedshort relative to the overall whereL is eitherthe fault length 1 or width w andrupture duration. Physical mechanismsthat are C is a constantthat canhavevaluesbetween0.65compatiblewith both a short slip duration and and 2.55 dependingupon the ratio of I to w, thealsoa correlationbetweenaverageslip andoverall rupturedepth,andthe slip direction.rupturedimensionswill be discussedlater. Das (1988) investigated the relationship be-

tweenstatic stressdrop averagedover the rupturesurfaceEsa andaverageslip D for modelsin which3. Sfressin timeand space the stressdrop is spatiallyheterogeneousandcon-

Severalestimatesof stressare now discussed cluded that eqn. (8) is not stronglydependentonthat canbe obtainedfrom the earthquakemodels the spatial distribution of the stressdrop, i.e. it isusedin this study. In large part, thesestressesti- usually valid to approximatethe average stressmatesare obtainedby estimatingstrainvariations drop from a heterogeneousstressdistribution as-(spatialderivativesof displacements).Measuresof suming eqn. (8). When the slip distribution isstresscan vary considerablydependingupon the significantly heteorgeneous,the stress drop maydetails of the slip history; details that are only actuallybe negativeon some partsof therupturepoorly resolved by waveform modeling studies, surface,i.e. an areaof low slip that is surroundedNevertheless,it is possibleto get someideaof the by areasof high slip may havea higher stressafterorder of magnitudeof stressandstrainvariations the rupturethan it did before.implied by the rupturemodelsconsideredin this Staticstressdrop estimatesare givenin Table2paper. for the sevenearthquakes.For simplicity, theaver-

The static stress drop for a rupture can be age stressdrop ~ is calculatedassuminga cir-calculated by first assumingthat the rupture cularrupturein a whole space(eqn.7). The aver-

8 T.H.HEATON

TABLE 2

Estimatesof stress

Earthquake ‘A ‘p 1) Dmax ~ ~°A

(kin) (km) (km) (cm) (cm) (bars) (bars) (bars) (bars)

19/9/88(M = 8.1)Michoacan 134 40 13 238 650 5 29 14 3728/10/83(M= 7.3)Borah Peak 28 9 2 a 82 147 8 30 26 469/2/71 (M = 6.5)SanFernando” 13 7 2 120 250 25 65 40 8415/10/79(M = 6.5)

ImperialValley 17 7 2.7 48 180 8 47 13 5024/4/84 (M 6.2)Morgan Hill 13 4 0.8 38 100 8 45 32 848/7/86 (M = 6.0)N. PalmSprings’ 13 5 1.2~1 26 45 6 18 12 226/8/79(M= 5.9)CoyoteLakee 5 4 1.4’ 46 120 22 48 22 57

Lognormalaverage9.8 37 20.7 49.5a From teleseismicdata,maybeaslargeas3.b Estimatesarefor thefirst of two sources.C ~ assumedto be3.9x10’

1dyn cm2.d A maximal value,may actuallyhavebeensmaller.

o assumedto be3.1x10’~dyn cm2.A maximal value,may actuallyhave beensmaller.

I~,lengthof therupturepulse(V~T,); D, averagedislocation(7; = 3.5X lO~dyn cm2); Dm

1,,,, peakdislocation; ~, average

staticstressdrop; AOA, staticstressdropin thevicinity of thedominantasperity;~, averagedynamicstressdrop in thevicinity oftherupturepulse; ~OAp, dynamicstressdropof the rupturepulseat thedominantasperity.

agedislocation.D is calculatedfrom the moment the overall changein stressstateon the rupturedestimatesassumingthe givenruptureareaS anda surface.However, it has beenhypothesizedthatrigidity of 3.5 x 1011dyn cm

2. Staticstressdrops slip occurs in a narrow pulse that propagates~aA for the dominant asperity are also given, along the rupturesurface,and therefore we canwhere ~ was calculatedusing the assumption anticipate that there are larger transient stressthat changesin the vicinity of the propagatingslip

pulse.Calculation of thesestresschangesin the4~GA 0.52~s—~ (9) vicinity of the slip pulsemustbe madeby assum-

A ing a detailedmodel of the conditionson the fault

where Dm~ is the peak slip and ‘A is the ap- during slip. As pointedout by Freund(1979), theproximate diameter of the dominant asperity. stress from a propagatingcrack is a function ofEquation (9) assumesuniform stressdrop over a the rupture velocity. Calculation of these stresscircular crackandis similarto eqn.(7) exceptthat variationsfor the modelsused in this paper is ait is computedusing the peak displacementin- formidabletask that is beyondthe scopeof thissteadof the averagedisplacement(Eshelby,1957). study. Furthermore,the detailsof the slip historyOf course,theseare very rough estimatesof the are not sufficiently resolvedby thesemodels tostatic stressdrop andhigherstressdrop estimates warrant suchcalculations.However, it is possibleare expectedfor modelsin which the slip distribu- to obtain some insight into this problem by ex-tion is elongatedin a direction. amiinga verysimplemodel of apulseof slip that

The staticstressdropsprovidesomeestimateof propagatessteadilyalonga fault surface.

SELF-HEALING PULSES OF SLIP IN EARTHQUAKE RUPTURE 9

z in the x-direction andinfinitely wide in the y-di-rection, that propagatessteadily along the x-axis.

-. - ~L ~ ~. The shearstressacross the shpping part of thelocked __________ locked fault is assumedto be umformly equal to Tf As

U+:U~#D /tzy”tzy’t ~ u~u- can be seen in Fig 9 the stress on the faultchangesdramaticallywith respectto the location

(X Vr T) ~ of the slip pulse.Thereis a squareroot singularity

of the dynamicstress changein this model is theat the leading edgeof the slip pulse.OnemeasureD Slip shearstress applied at infinity minus the shear

stresson the slipping strip of the fault which isreferredto as &i,,, given by Freund(1979) as

D ~ocity i~cr.~= = ~ 4~ — (ic)0 where$ is theshear-wavevelocity and l~,= VrT,~iS

______________________________________ the length of the slipping region. If the rupturevelocity is assumedto be 80% of the shear-wave

velocity,then = 0.19,.LD/l~.It shouldbenotedthat this definitionof stressdrop is similar in form

Fault plane sheart~lto to eqn. (8) for static stressdrops but with asmall

valuefor the geometricconstantC. In fact, sinceFreund’s (1979) solution is steadystate,it canbe

solvedasa static problemsuperposed on a movingcoordinate system. As a consequence of thissteady-statecondition, this solution does notradiateseismic wavesto the far field (eventhough

there is an infinitely large stress changeat theFig. 9. Idealizedmodel of Freund(1979) in which a pulseof crack tip). Anotherunusualfeatureof this modelslip of length 1,, in the x-direction and infinitely wide in the is the fact that thestatic stressdrop is zero,i.e. they-direction propagatessteadily at velocity V in the positive .x-direction.A uniform shearstressT = r

0 is appliedat infin- stress at large distancesbehind the propagatmgity and the shearstresson the slippingportion of thefault is slip pulse is the sameas the stressat large dis-assumedto beuniformly equalto a dynamicfriction i~.In this tancesaheadof the slip pulse.Furthermore,noticemodel, theslip is confinedto a pulseartificially, i.e. thereis no that as the rupturevelocity approachesthe shear-physicalcondition that causestheruptureto heal.Becausethe wave velocity, a pulsewith an arbitrarily largeslipproblemis steady-state,the pulsecausesa finite dislocation . . .that propagatesover an infinitely largefault andthusthestatic will propagatewith an arbitranly small dynamicstressdrop is zero. Thedynamicstressdrop in thepulse~ is stress drop z~o,,.Because there are many non-a function of the dislocationand thelength of the slip pulse, physical consequencesof the restrictiveassump-andis givenby eqn.(10). tions of Freund’smodel andbecausethe length of

the estimatedslip pulseI~ is an upperbound inmanyof theslip models,I expectthat the dynamic

Freund(1979)gives the solution for a pulseof stressdrop givenby eqn.(10) gives a lower boundslip that propagatessteadilyat velocity Vr along on the dynamic stressdrops implied by the slipthe x-axis assumingthat a homogeneousshear modelsshownin Figs. 1—7.stress T0 is applied from infinity. In this model Estimatesof the averagedynamic stressdrop(shown in Fig. 9), the fault is assumedto be in the vicinity of the rupturepulseare given inwelded everywhere, except in a strip of length I~, Table2. TheaveragedislocationD is usedin eqn.

10 T.H. HEATON

(10) to calculatethe dynamic stressdrop ~ in 4. Healing of the rupturepulsethe slip pulseaveragedover the rupture,and themaximumdislocationDm~ is used to calculatethe A mechanismfor producinga self-healingrup-stressdrop L~aAPin the slip pulsein the region of ture pulse is now discussed.This beginswith athe dominant asperity. The logarithmically aver- very simple model that does not heal and thenaged(the antilog of the averageof the logarithms) proceedsto a more complicatedmodel that doesdynamic stressdrop in the vicinity of slip pulses heal within some characteristicdislocation rise

is 21 bars, whereasthe averagestatic stress time. First considerthe simple model shown in

drop ~a is only 9.8 bars.Thus, accordingto this Fig. 10. Assumethat a rupturethat is infinitelyvery simplesteady-statemodel,thedynamicstressdrop (the difference betweenthe ambient shearstressand the shear stressat the slip pulse) isseveraltimeslarger than the static stressdrop (inFreund’smodel, the static stressdrop ~.a is zero).

Dynamicstressdrops L~.aAPare also calculated slipping regionin the vicinity of the dominantasperities(Table 2) CX - v~T)usingeqn. (10) togetherwith the estimatedlengthof the slip pulse and the maximum slip in theasperity. Unfortunately,the assumptionsused to 0 Slipderive eqn.(10) are severelytaxedfor this calcula-tion, i.e. the length of the slip I~, is, on average,only onethird the averagedimensionof the domi-

Slipnantasperity.Whereasin Freund’smodel,the slip ~ Velocityis assumedto be infinitely wide along the rupture bfront and it is assumedto propagateuniformly aninfinite distancein the x-direction(therebyresult -_____________________

ing in zerostatic stressdrop). In the vicinity of the ___________________________________________dominantasperity,applicationof eqn.(10) yields Fault planean averagedynamic stressdrop z~aAPof 50 bars, Shearstresswhereasthe static stressdrop L~aA averages37bars. However, it is important to recognize thatthesenumbers are basedon very crude models.Until more realistic numerical models are avail-able to estimatethe dynamic stressesimplied by = tfrupturemodelssuch asthoseshownin Figs.1—7, I L~0.canonly speculatethat the dynamicstressdrop inthe region of the dominant asperitiesis greaterthan the static stressdrop, but less than severaltimes as large. Of coursethe stressjust aheadof Fig. 10. Idealized model of a rupture (infinitely wide in the

the leadingedgeof the rupturecanbemuch larger y-direction) propagatingat a velocity V~in the x-direction.Fractureinitiateswhen the stressat the cracktip exceedstheand presumablyis limited by the overall strength staticstrengthof thefault ; andthe stresson thefault behind

of the materialsin the fault zone. Furthermore,if the crack tip is someconstantfrictional level i~.The stress

the rupturepropagatessmoothly, then thereis no before the initiation of rupture is r0 and the stressafter the

far-field radiated energy and thus the ultimate ruptureis ~ ~ giving a stressdropof ~a — r~.This type

amplitudeof the stresschangein the vicinity of of rupture is describedby Kostrov(1966) and the dislocationcontinuesto grow as %/V,t — x and doesnot heal until infor-the crack front cannotbe deducedfrom observa- mationis receivedthat therupturefront hasstoppedpropagat-tions of radiatedhigh-frequencywaves. ing.

SELF-HEALING PULSESOF SLIP IN EARTHQUAKE RUPTURE 11

longin the y-directionis propagatinguniformly at must be considered.Dynamic rupture modelsina rupturevelocity Vr in the x-direction.The shear which the rupture heals before the rupturestresson the fault ‘r~~is assumedto be ‘r0 at a terminatesare relatively rare. Yoffe (1960) pre-large distancein front of the ruptureand r~at a sentedsuch a model for a plane-strainextensionallarge distancebehind the rupture, giving a static fracture,but the healingcondition wasartificiallystressdrop ~a of ‘

1b—’Ti, In the regionaheadof the imposedand severalkey featuresof Yoffe’s modelrupture,the shearstress;~,is assumedto be given are physically unrealistic. Similarly, Freund’sby crack theory and to be less than somecritical (1979)model discussedin the last section(Fig. 9)strength;.In the regionbehindthe rupturefront, featuresa self-healingpulseof slip thatpropagatesthe shearstressis equal to a friction stress ‘rf that steadily along a fault. However, the healing isis constantin time oncethe fault hasruptured.If merely introducedas a boundarycondition andthis model applies to earthquakes,then the fact thereis no physicalprocessincluded to determinethat averagestatic stressdrops are only tens of whenthe healingshouldoccur.barsallowsus to concludethat the frictional stress Day (1982) constructedfinite-element models‘r1 has an amplitude that is close to the ambient of dynamicrupture on finite faults having stress

shearstress.Furthermore,experimentalstudiesof boundaryconditionson the fault that are similarrock friction indicate that the coefficient of fric- to those in Fig. 10, i.e. the shear stress ‘re., is

tion for a sliding surfaceis within severalpercent limited to be a constantvalue Tf oncerupturehasof the static coefficient of friction (Dieterich, occurred.Day also addedthe following condition;1979). if the coefficient of friction is somewhere if the shearstresson the fault becomesless than ;near 0.6 (typical for rocks in laboratory experi- then the rupture heals itself. With this type ofments), then we conclude that for the model in boundary condition, Day shows that, in someFig. 10, all of the shearstressesplotted would be circumstances,the rupture can heal itself beforein the kilobar range, and any variations in the the ruptureprocessterminates.In particular,for astresswould be relativelysmall comparedwith the long, narrow ruptureof width w (the rupture isambienttectonicshearstress. confinedboth aboveand below), Day shows that

ThedislocationD is zero aheadof the cracktip the dislocation rise time approachesa value ofand in the regionbehind the crack tip it is given W/2Vr as the rupturepropagationapproachesaby steadystate. In some ways it is surprising that

Day’s (1982) simple model produces a healingD =f(t~cr)~Ikt— X (11) time that is so short, since the rupturehealswhile

wherethe functionf dependsupon the stressdrop the leading edge of the rupture is only half of aandthe elasticpropertiesof the medium.The slip fault width away. In a pseudo-staticcrack prob-velocity b(Vt — x) is given by lem, the slip would continueto grow as the rup-

ture front extends.S. Day (personalcommunica-— f(L~G)Vr ‘12~ tion, 1989) attributes this rapid healing to over-— 2v/l’c.t — .~ ‘ ‘ shootof the slip (with respect to the staticprob-

lem) causedby the momentumof the sidesof theThe slip on the fault continuesto increaseuntil fault during the ruptureprocess.Therapid healingthe growth of the rupture is arrested.Models of in Day’s (1982) model only appearsfor steadythis type producedislocation rise times that de- rupture along a long, narrow fault. When Daypend upon the final dimensionsof the rupture assumeda squarefault with a radially spreadingsurface.A review of thesetypesof modelscanbe rupturefront, the dislocationrise time was corn-found in Aki and Richards(1980,chap.15). parableto the rupturetime.

If dislocation rise times are short compared Archuleta and Frazier (1978) constructeddy-with the overall duration of the rupture, as is narnicfinite elementrupturemodelsfor a verticalsuggestedfrom the modeling of seismic waves, semi-circular fault in a half-space and stressthen some other model for the rupture process boundarycondtionson the fault identical to those

12 T.H. HEATON

just described.They show that if the hypocenterislocatedat the bottomedgeof the fault, thenshort

slip durationscanoccur for pointswherethefaultbreaks the free surfaceof the half space.This

Z ~, short duration is attributed to overshoot of the

slip with respectto static equilibrium as the dy-locked ~i / lçcked naimc rupture breaksthe free surface However,

thisshort slip durationis only seenin specialcases- of ArchuletaandFrazier’smodels,andin all cases

s/,pp,ng — that they considered,points interior to the ruptureCX VrT)—*. boundarieshavelong slip durations.

Spatially complex distributions of stress andO Slip strength introduce shorter length scales and it

shouldbe possibleto constructhypotheticalmod-els that haveshort slip durationsfor at leastsomepoints interior to the boundariesof the rupture

Slip surface.However,if the averageslip in suchmod-o Velocity els is determinedby the overall dimensionof therupture surface, then averageslip durations are

________________________________________ o expectedto be long, despite the introduction of— — shorter length scales. Chen et al. (1987) show

‘kts,~.itoakbori numerical examplesof the slip history for a dy-namic fault model in which the stress drop isspatially heterogeneous(but positive everywherein the rupturesurface).Although the slip historybecomescomplexin their models,the durationofslip is still comparableto the overall duration of

______________________ to~ the rupture.

____________ ‘ ~ ~ ~ Q-’iO~o2O One way to make the slip duration short at a— Faultplanesheart2~, given point is to introduce barriers.That is, an——Frictionalstresst~ .., earthquakeconsistsof a number of short-dura-

Fig. 11. Idealizedmodel of a self-healingrupture pulse(in- tion, crack-like ruptureson small rupture areasfinitely wide in they-direction)propagatingata velocity of Vr that are separatedby locked regions (Aki, 1979;in the x-direction. Fractureinitiates when the stressat the Papageorgiouand Aki, 1983a,b). Barrier modelscrack tip exceedsthe staticstrengthof thefault ; (— k bar) differ from the heterogeneousstressdrop modelsand then dropsto a shding fnctional level that is inverselyproportionalto thedislocation particlevelocity D. Dislocation of Chen et al. (1985) in that thereare actuallyparticle velocities are high and sliding friction is very low locked patchesremainingthroughoutthe rupture(<200 bars)neartherupturefront. As thedislocationparticle surface. Large shearstress accumulatesin thesevelocity decreasesaway from the rupture front, the sliding locked patchesas the rupturepropagatesaroundfriction increasesand the fault heals itself. In this simple them effectively resultingin large negativestressmodel, the strengthof the fault returns to a high level im- ‘ .

mediatelyafter therupture. ~ is thestressbefore theinitation dropsin the barners.Although bamermodelscanof ruptureandT1 is thefinal stressafterrupturehasterminated, produce short slip durations,it appearsthat slip isyielding a static stressdrop t~o= — ~ which may be very relatively smoothly distributed in many of thelow (= 10 bars)dependingon the final length of the rupture. earthquake models shown in Figs. 1—7, and yet,The averagestressdroplocally within the rupturepulsemaybe the durationof slip is short.considerably larger (= 100 bars) with dislocation particle .

velocitesin the rangeof im si. ThedislocationamplitudeD Are therephysicallyreasonableways to rapidlyand thelength of the slip pulse1,, maybea sensitivefunction heal the rupturebefore information arrives thatof the friction law duringslip. the rupturehas stoppedpropagating?The stress

SELF-HEALING PULSES OF SLIP IN EARTHQUAKE RUPTURE 13

distribution in the slipping regionmay bethe key. tide velocity, whenusedin conjunctionwith eqn.Brune(1970,1976)suggestedthat earthquakerup- (13), forces ‘r1 to actuallygo infinitely negativeatture maybeanalogousto the propagationof dislo- the rupture front. Thus, when thinking of thecations in crystals where a localized slip propa- behaviorof thesemodelsat the rupturefront, thegates through a crystal lattice (for example,see squareroot singularity shouldbe takento meanNabarro, 1967). Brune further suggestedthat a that strainsmaybecomeverylarge,but theymustself-healingrupturepulsewould result if the fric- befinitely bounded.In particular,the shearstresstional stress ‘r1 is limited to some relatively low on the fault cannotexceedthe yield strengthofvalueimmediatelybehind the rupturefront but is the fault, anddynamicfriction; cannotbe nega-then allowed to assumerelatively high values at tive.some finite distancebehind the rupturefront. A S. Day (personalcommunication, 1989) hassimple way to accomplishthis is to assumethat constructedfinite elementmodelsof dynamicrup-the friction stressr1 is inverselyrelatedto the slip ture assumingthe velocity-dependentfriction lawvelocity D at any point on the fault. A slip-veloc- given by eqn.(13). He reportsthat sucha frictionity-weakeningfriction law for faultswassuggested law can producea self-healingpulse of slip thatby Burridgeand Knopoff (1967)who numerically propagatesalong a rupture surface,although thesimulatedearthquakeruptureby assumingthat a stablepropagationof suchpulsesoccursonly forfault could be modeled as a systemof discrete specificrangesof the constanta whichdeterminesmasses,coupledby springs, and sliding over a the degreeof slip velocity weakening.rigid surface.Their simulationsdid produceself- Is thereany physical basisfor a dynamicfric-healing,propagatingrupturepulses,althoughthey tion of the typeassumedin eqn. (13)? Onesimpledid notemphasizethe significanceof suchbehav- hypothesisis that the friction on the fault is re-ior. lated to the normal stressacrossthe fault surface

Onesimplehypothesiswould be to assumethat ;~. In idealizedplanarfault models, the normalthe frictional stresson the fault ; is approxi- stress ‘r~ is constantacrossthe fault throughoutmatedby the rupture process.That is, the fault plane is

— — D ~13~ assumedto be a perfectnodefor P-waves.How-—; a ‘ ~‘ ever, if there are geometricirregularities in the

where ; is a static friction and a is a material rupture surface or spatial variations in elasticconstant.A simplemodel of this type is shownin propertiesin the vicinity of the rupture surface,Fig. 11. In this model,slip only occurs if the shear then the fault planewill no longer be a perfectstress on the fault ;.,, = ‘Tf (D), andif ;~<r1 (D), node for P-waves. In fact, observationsof thethen the fault is locked. As was the casein the high-frequencyP-wavesin the near-sourceregionprevioussimplemodel (Fig. 10), the dislocationis usually yield a fairly isotropic P-wave radiationassumedto vary as ~/Vrt— x in the regionbehind pattern(Liu and Helmberger,1985). If therearethe rupturefront andthusthe dislocationvelocity large-amplitudevariationsin the normal stress;~has an inverse squareroot singularity. The dy- in the ruptureregion, thesewill almost certainlynamic friction on the fault ‘r1 dropsdramatically havesomeeffecton theeffectivefrictional strengthin the regionimmediatelybehindthecracktip and of the fault. Sincethe amplitudeof high-frequencythen increasesas the squareroot of distancebe- radiatedwavesis proportionalto the dislocationhind the rupturefront. The rupturehealsitself at velocity D, it might be expectedthat the dynamica distance behind the rupturefront when ‘r1 friction is a function of the slip velocity.exceedsthe ambientstresson the fault. As is the Bruneet al. (1989, 1990) havestudiedsponta-case with most crack models, the square root neousstick-slip failure in foam rubber and havesingularitiesat the rupture front result in some documentedthat variations in normal stressonnon-physicalartefactssuchas infinite strains(de- the slipping surfaceare responsiblefor slip-veloc-rived from infinitesimal strain approximations), ity-weakeningfriction. They found that whenthininfinite particle velocities,and evenmoreinfinite stripsof foamrubberaremountedto rigid surfacesparticleaccelerations.Theinfinite dislocationpar- andthen forcedto slide, the sliding friction is the

14 T.H. HEATON

sameas the static friction. However, when the The constanta, which controls the slip velocitysurroundingmediumis entirely foam rubber,then weakening in the friction law (eqn. 13), has asignificant normal stressesare observed in the fundamentaleffect on the rupture. If a is large,vicinity of the sliding region and the apparent then the length /~of the rupturepulseis largeandsliding friction is reduced.Tolstoi (1967), Oden so is the dislocation.If a becomessmall, then theand Martins (1985)havearguedpersuasivelythat dynamicfriction increasesand rupturestops.Thedynamicvariationsin normal stressare of primary length of the slip pulseand sizeof the dislocationimportancein determiningthe dynamicbehavior in the slip pulserisesandfalls asit proceedsdownof dry friction in metals. the fault, dependingon the relativeamplitude of

If irregularities occur on a very small scale, the ambientstress ‘r0 and the dynamic friction Tf

then it may bethat high-intensity, high-frequency (which itself dependson the dislocationparticlecompressionalwavesmay exist in the immediate velocity). Becausethe slip velocity and the dy-vicinity of propagatingrupture pulses in earth- namic friction are dependent on each other,quakes.Melosh (1979) discussesa model of this mathematicalsolutionsfor the slip are likely to betype that he calls ‘acousticfluidization’. He shows rather unstable with respect to assumedinitialthat very high-frequencyacousticwavesmaybe a conditions.It is conceivablethat thestaticstrengthreasonable mechanism for producing slip- ; andthe ambientstress‘~ may bothbe relativelyvelocity-weakeningbehaviorthat dramaticallyre- homogeneousalonga rupturethat hasarelativelyducesfrictional stressduring fault rupture.If such irregularslip distribution,i.e. the slip distributioncompressionalwaves are to significantly reduce may be controlledby detailsof the behaviorof thethe confining stresson the rupturesurface, then dynamic frictional strength in the region of thetheir intensity would haveto be some significant slip pulse.It might bedifficult to recognizewhichfraction of the confining stress (on the order of segmentsof a fault may have large slips just bykilobars for seismogenicdepths).If this explana- looking at physicalsamplesof fault material.tion is appropriate,then it may only be valid for The amplitudesof various stressesfor the self-rupturesthatpropagatewithin specifiedrangesof healingslip pulsemodel of Fig. 11 are now specu-confining pressureand dislocationparticle veloci- latedupon.Thestatic friction on the fault ; couldties. be quite high, perhapsin the kilobar range,which

Sibson (1973) and Lachenbruch(1980) discuss is the inferred strengthof materialsat confininganalternativemodelfordecreasingfrictional stress stressesappropriatefor seismogenicdepths.Yetduring rupture. They proposethat frictional stress since the dynamicfriction during rupture ‘r1 maymay be dramatically reducedduring rupture by be relatively low (perhapsin the rangeof severalthe expansionof pore fluid causedby frictional hundredbars), the work done againstfriction dur-heating.They suggestthat frictional stresscould ing rupturemaybe relativelylow, thusexplainingbereducedfrom the rangeof kilobars to hundreds the relatively low heat flow seenin the vicinity ofof barsas pore fluids rapidly heatduring fault the most active faults (Brune et al. 1969;slip. Lachenbruch(1980) suggeststhat the pore Lachenbruch and Sass, 1973, 1980; Richards,fluid pressuremay rapidly dissipateby the propa- 1976). The static stressdrop, which is measuredgation of hydrofractures thereby causing re- from the overall fault dimensions,may be muchstrengtheningof the fault. The time requiredfor lower still (in the rangeof tensof bars).If propa-this re-strengtheningcould rangefrom secondsto gation of a rupturepulseof a given slip amplitudeyearsdependingupon the characteristicsof the is arrestedin a relatively short distance,then afluid flow, relatively high static stressdrop results. If that

samerupturepulse travelsa long distance,thenarelatively low static stressdrop results.

5. Consequencesof self-heatingrupture . . .In the slip pulse model, fnctional heating is

There are some interesting implications that controlledby dynamicfriction duringruptureandresult from a model of the typeshown in Fig. 11. not by the static friction ;. Heat flow measure-

SELF-HEALING PULSESOF SLIP IN EARTHQUAKE RUPTURE 15

mentsapparentlyconstrainthe dynamic friction regions of large slip, then in the region of smallof the San Andreas fault to be less than several slip, the stressafter the earthquake; may actu-hundredbars. Furthermore,the very rough esti- ally be higherthan the stressbeforer~.In the casemate of dynamicstressdrops&~from the earth- of the Morgan Hill earthquake,therewas an ex-quakesconsideredin this paper(Table2) leadsme tensiveareaof small slip betweentwo regionsofto guessthat the ambientstressis usuallyno more largeslip (Fig. 5). Although behaviorof this typethan severalhundredbarshigherthan the average can be understoodin the context of the simpledynamic friction. Therefore, the ambient shear model of Fig. 10, it seemsto imply a very hetero-stresson the SanAndreasfault may be expected geneousdistribution of either ambient or fric-to be less than400 bars.Zobacket al. (1987)also tional stress.However, this behaviorseemsto fol-infer relatively low shear stresseson the San low naturally from the slip pulsemodel showninAndreas fault becausemany independentmea- Fig. 11. The slip pulse, which was less than asurementsimply that the maximumcompression kilometerwide, propagatedfrom north to southataxis is nearlyperpendicularto the fault. Of course, a velocity of about2.8km ~ The sizeof theslipheat flow measurementsdo not constrainthe dy- pulse was initially large, then diminished as itnamic frictional stressvalues for relatively low propagatedsouthwardand then grew relativelyslip-rate faults. The existenceof large mountains large againat the south end of the rupture.Theis convincingevidencethat theambientshearstress small slip in the regionbetweenthe asperitiesdidin the crust is in the rangeof a kilobar at some not grow large when the secondasperitybrokelocalities (Jeffreys,1959). becausethe fault hadalreadybecomelocked.

Once a slip pulse has initiated, the ambient If indeedthe slip pulse model is appropriate,stress TO necessaryto support the propagationof then the ambient stressalong many active faultslarge slip pulsesmay beconsiderablyless than the may be much lower (perhapsby a factor of 0.20)static friction ;. Thus, sectionsof the fault that than the stressnecessaryto initiate rupture andsupport largeslip pulseswould tend to be at an the problemof earthquakeprediction becomesaambientstressthat is low comparedto the stress problem of predicting when and where large slipnecessaryto initiate rupture. This may explain pulses will propagate.Or restating the problem,why sectionsof the fault that havelargecoseismic we must know how to predict when a slip pulseslips tend to havelow seismidity levels (the stress will stoppropagating(this problemis discussedinis too low to initiate rupture). moredetailby Brune,1979).Sincethe static stress

We see, then, that the slip pulse model allows drop ~ is small comparedwith other stressesfor a large variation betweenthe different mea- involved in this model, prediction schemesthatsures of stress in the earthquakeprocess.The dependon static stress drop estimates(for in-static strengthof the fault zone; may be in the stance the time-predictable or slip-predictablekilobar range; the ambient shear stress in the models;ShimazakiandNakata,1980)may notberegion of the earthquakemay be lower but still appropriate.Even after a large earthquake,therequite high (perhapsin the rangeof hundredsof may be sufficient ambient stress to support thebarson veryactivefaults to a kilobar on relatively propagationof anotherslip pulse.As an example,inactive faults); the dynamic stressdifferencesin considerthe apparent6-m slips that occurredonthe regionof the rupturemaybe relativelyhigh (in the San Andreas fault at Pallett Creek in boththe range of tens to hundredsof bars in the 1812 and 1857, a mere45 years apart(Salyards,slipping part of the fault to kilobars at the crack 1989). Alternatively, a region may experienceatip); the sliding friction during rupture ‘I~~may be long interseismicperiod simply becauseno sliprelatively low (less than several hundredbars); pulses of large enoughsize propagatedinto theand the final changein ambientstress(the static region (some intereventperiods at Pallett Creekstressdrop ~a) may be very low (in the rangeof exceed300 years; Salyards,1989). If this is true,tens of bars). theneventhe approximatepredictionof the time

If thereis a regionof smallslip surroundedby of large earthquakesmay be difficult. Kagan and

16 T.H. HEATON

Jackson(1990)report that matureseismicgapsas geneouspropertiesthat allow slip pulsesto propa-defined by McCann et al. (1979) had fewer large gate large distances.Therefore,it is conceivableearthquakesin the period from 1979 to 1988 than that the static strengthof faults and the dynamicrecently filled seismicgaps. stressesduring rupturemay be relativelyindepen-

In the introductionof this paper,it wasnoted dent of geologicslip rate.that the averagefault slip D correlateswell with Although the model that has beenpresentedthe squareroot of the rupture area for a wide may imply that it will be very difficult to predictvariety of earthquakes(Kanamori andAnderson, the time of largeearthquakes,it also implies that1975). If earthquakeruptureshaveapproximately it should be possibleto predict (at least statisti-constant ratios of width and length, then this cally) the overall size of an earthquakesoonaftercorrelationimplies that averagestatic stressdrops the rupturehas initiatedandwell beforethe rup-

are approximatelyindependentof the rupture ture stops.That is, sincethereis a well-establisheddimension. However, Scholz (1982) argues that correlationbetweenaveragedislocationand faultaverage dislocations correlate well with fault length, and because the final dislocation islengths, even for faults in which the apparent reachedquickly after the passageof the rupturerupture width is small comparedto the length. front, it shouldbe possibleto recognizethe dislo-This observation is evidence that long, narrow cationsize quickly. This would allow a statisticalrupturestend to havehigher static stressdrops. prediction of the rupture dimensionbefore theHowever, in the slip-pulsemodel proposedhere, rupture stops. This principle may be useful inthe most natural explanationfor a correlation real-timeearthquakewarningsystemsthat arede-betweenaverage dislocationand rupture length signedto provideveryshort-termpredictions(sec-(regardlessof the rupture width) is that pulses ondsto tens of seconds)of stronggroundshakingwith very large slip tend to propagatelarge dis- for sites located at large enough distancesfromtances.If a very large slip pulse develops on a the epicentralregion(Heaton,1985).

long, narrow fault (e.g. the SanAndreas),it mayrun for a very large distance. This problem isdiscussedat some length by Bodin and Brune 6. Nucleationof rupture(1990).

For faults of a given aspectratio (length to The slip pulsemodel shownin Fig. 11 assumeswidth ratio), the static stressdrop is controlledby a running rupture pulse.Since in this model nothe distancethat a slip pulse propagates.If for slip pulse will propagateunless a slip pulse al-somereason, large slip pulses are arrestedafter readyexists,themodel clearlybegs thequestionofrelatively short rupture lengths, then high static how the rupturepulse starts in the first place. Astressdrops would result. If the samesize slip different classof modelsis necessaryto trigger thepulsedoesnotstop, thena lower static stressdrop initial slip instability (for example;Rice andTse,would result, evenif the dynamicbehavioralong 1986;Okubo, 1989).At what point (or earthquakethe slip pulse was approximatelythe same.This size) does the physics that controls the rupturesuggestsan interesting interpretationof the ob- change from the nucleationprocess(for whichservation that geologic slip ratesinverselycone- inertia is not important)to the slip pulseprocesslate with static stressdrop (Kanamori and Allen, (whereinertiais veryimportant)?Presumably,this1986).That is, faults with thehighestgeologicslip transitionis relatedto the sizeof the rupture,andratestend to haveearthquakeswith relativelylow we would expect that the slip pulse model doesstatic stress drops. If we assumethe slip pulse not apply to earthquakesthat are small enough.model, then this implies that slip pulses with a How small is that? Unfortunately, detailed nip-given dislocationtend to travel largerdistanceson ture modelshavenot yet beendevelopedto studyvery activefaults than they do on relatively mac- the relationshipbetweenrupturetime anddisloca-

‘live faults. Perhapsfaults with high geologic slip tion rise time for earthquakessmaller than aboutratestend to be long faults with relatively homo- magnitude5. However, the overall seismic radia-

SELF-HEALING PULSES OF SLIP IN EARTHQUAKE RUPTURE 17

tion from earthquakesdownto at leastmagnitude 7. Dislocation particle velocities3 seemsto be self-similar(assumingcertainscalingrelationships)with that of largerevents(for exam- Using the valuesfor dislocationsandrisetimesple; Bruneet at., 1986; Frankel andWennerberg, given in Tables 1 and 2 to computedislocation1989). Although this doesn’t prove anything, it particlevelocitiesresultsin the conclusionthat thedoessuggestthat the rupturepulsemodel may be logarithmically averagedvelocity for a particleonapplicablefor relativelysmall earthquakesaswell. the fault is 43 cm s~and in the regions of

Presumablysections of faults that primarily asperitiesthis averageis about103 cms~(assum-creepdo notsupportthepropagationof slip pulses. ing that the motion is symmetricwith respecttoFor example,considerthe creepingsectionof the the fault). Thesevalues are averagesthroughoutSan Andreasfault in central California. Whereas the duration of the slip pulse, and the particlethe locked Carrizo Plains sectionof the fault is velocities may be considerably higher (perhapsdevoid of seismicity, the creeping sectionhas a 10—20 m s~)in the immediate vicinity of thehigh seismicity rate. One explanationis that the rupturefront.Carrizo Plainssegmentsupportsthe propagation These dislocation particle velocity estimatesof large slip pulses that maintain the ambient havedisturbingengineeringconsequencesfor somestressat a relatively low level.However, the creep- classesof structureslocated very close to largeing sectionof the fault evidentlydoesnot support earthquakes.For example,dislocations of 7 mlarge slip pulses, thereby allowing the ambient were observed for some sections of the Sanstress to increaseto a level sufficient to cause Andreas fault during the 1906 San Franciscocreepand the nucleationof numeroussmall rup- earthquake.This implies that someregions nearturesthat don’t propagatevery far. the fault shifted their position with respect to an

A seriousproblemstill existswith this interpre- inertial frameby at least3.5 m in severalseconds.tation. Laboratory measurementsindicate that This may prove to be an engineeringproblemforcreepgenerallydoesnot initiateuntil shearstresses somestructures.In particular,base-isolatedbuild-are a substantialfraction of the confining stress ings are usuallydesignedwith the assumptionthat(typically 0.4 to 0.6; J.H. Dieterich,personalcorn- short-period(<5 s) groundmotionsareless thanmunication). Since the notion of low dynamic 50 cm.friction in this model is not relevant to the prob-lem of fault creep, I cannot claim that shearheatingon creepingsectionsof the San Andreas 8. Conclusionsfault is low becauseof low dynamicfriction. Al-thoughthereis a tendencyfor the heat flow to be It hasbeendemonstratedthat slip durationsforsomewhathigh in the generalregion surrounding a given point on a fault (dislocation rise times)the central creeping sectionof the San Andreas deducedfrom many rupturemodelsderivedfromfault (Lachenbruchand Sass,1980),the heat flow seismic waveform data are significantly shorteris not localizedto the fault andis nothigh enough thanthe timerequiredto notify pointson the faultto be compatiblewith kilobar shearstresslevels that the rupturehasstopped.Theseslip durationson the creepingsectionof the fault (Brune et al., are about15% (or perhapsless)as long as the slip1969). High fluid pore pressuresthat reducethe durationsthat resultfrom dynamiccrack-likerup-effectiveconfining stresscanbe invoked to assert ture models.One explanationis that long-periodthat the yield strengthof the creepingfault may partsof the fault slip historyareunresolvedfromonly be hundredsof bars. Unfortunately, I know thesemodels, which would imply that they haveof little corroboratingevidencefor this somewhat seriouslyunderestimatedthe total slip for earth-ad hocexplanation. quakes. However, evidence has been presented

18 T.H.HEATON

that the slip durationsare indeedshort. Several occursin a shorttime periodafter the passageof apotential modelsresult in short slip durations.In rupture front (often referred to as dislocationthe first, earthquakesare composedof a sequence models) havebeen very successfulfor matchingof small dimension (short duration) events that waveforms over a broad range of frequencies.are separatedby locked regions(barriers). How- However, they have often beencriticized on theever,in at leastsomeof thedislocationmodelsfor basis that they violate basic physical constraints.specific earthquakes,it appearsthat the slip is In this paper,a mechanismhas been suggestedrelatively smoothly distributed(no barriers),and that would produce the types of slip pulses thatyet, the duration of slip is short. In the second seem to fit waveform data so well. While themodel, rupture occurs in a narrow self-healing hypothesizedslip-pulsemodel presentedhere ispulseof slip that travelsalongthefault surface.In admittedlyspeculative,it doesseemto provide athis model, the frictional strengthon the fault is solution for a first-orderproblemthat any modelhigh everywhereexcept in the immediatevicinity of rupture must ultimately deal with. It is nowof the rupturepulse. It is suggestedthat the fric- clear that the ambientshearstressand dynamictional strengthof the fault is inversely related to friction on the San Andreas fault are low com-the dislocationparticle velocity in the slip pulse, pared to the confining pressureat seismogenicperhapscausedby intense compressionalwaves depths.Realisticrupturemodelsmust satisfy thisthat would tend to decreaseconfining stresses constraint.locally in theregion of the slip pulse.

It is suggestedthat this dislocation-velocity-weakening,slip-pulse model may allow for the Acknowledgementsfollowing features: the static strength of faultmaterialsat seismogenicdepthsmay be relatively Becauseof the speculativenatureof this study,high (on the order of kilobars); the ambientstress I soughtthe opinion of many of my colleagues.Iin the vicinity of active faults may be any value particularly thank William Ellsworth who sug-betweenseveral hundredbars and a kilobar (de- gested several key ideas concerningthe impor-pendingon the size and frequencyof occurrence tanceof rupturepropagation,andJim Brunewhoof the slip pulsesthat propagateon the fault); the generouslysharedseveralkey ideas aboutslidingfrictional stressduringslip maybelessthanseveral friction. I also thankJim Knowles,Jim Rice,Jimhundredbars (therebygeneratingrelatively little Dieterich, Hiroo Kanamori,Paul Okubo,Stephenheat); the average dynamic stress drop in the Hartzell, StevenDay, andTom Hanks who maderegion of the slip pulseis on theorder of 50—100 valuablecommentsandwho bear no responsibil-bars; the averagestatic stressdrop may be rela- ity if this model turns out to beincorrect. I thanktively low (tens of bars)and is controlled by the WayneThatcher,JackBoatwright,Jim Brune,Joefinal length that a slip pulsewill propagate;large- Andrews,Steve Wesnousky,Dave Boore,and anamplitude slip pulses tend to propagatelong dis- anonymous referee for their reviews of thetances;regionsthat do not efficiently propagate manuscript.Finally, I thank Carlos Mendozaforslip pulsesmay accumulatehigh ambientstresses providing the displacementtime series for theand may be regions that nucleatemany small Michoacanearthquake.earthquakes;and regions that do support thepropagationof largeslip pulsesmaintain relativelylower ambientstressesand would rarely produce Referencesrupture nucleation. This model would tend tode-emphasizethe importanceof static stressdrop Aki, K., 1979. Characterizationof barrierson an earthquake

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