Prediction Probabilities From Foreshocks

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 96, NO. B7, PAGES 11,959-11,971, JULY 1, 1991 Predictin Prbabilities Frm Freshcks DUNCAN CARR AGNEW Institute f Gephysics and Planetary Physics, University f Califrnia, La Jlla LUCILE M. JONES U.S. Gelgical Survey, Pasadena, Califrnia When any earthquake ccurs, the pssibility that it might be a freshck increases the prbability that a larger earthquake will ccur nearby within the next few days. Clearly, the prbability f a very large earthquake ught t be higher if the candidate freshck were n r near a fault capable f prducing that very large mainshck, especially if the fault is twards the end f its seismic cycle. We derive an expressin fr the prbability f a majr earthquake characteristic t a particular fault segment, given the ccurrence f a ptential freshck near the fault. T evaluate this expressin, we need: (1) the rate f backgrund seismic activity in the area, (2) the lng-term prbability f a largearthquake n the fault, and (3) the rate at which freshcks precede large earthquakes, as a functin f time, magnitude, and spatial lcatin. Fr this last functin we assume the average prperties f freshcks t mderat earthquakes in Califrnia: (1) the rate f mainshck ccurrence after freshcks decays rughly as t -1, s that mst freshcks are within three days f their mainshck, (2) freshcks and mainshcks ccur within 1 km f each ther, and (3) the fractin f mainshcks with freshcks increases linearly as the magnitude threshld fr freshcks decreases, with 5% f the mainshcks having freshcks with magnitudes within three units f the mainshck magnitude (within three days). We apply ur results t the San Andreas, Hayward, San Jacint, and Imperial faults, using tlie prbabilities f large earthquakes frm the reprt f the Wrking Grup n Califrnia Earthquake Prbabilities (1988). The magnitude f candidate event required t prduce a 1% prbability f a large earthquake n the San Andreas fault within three days ranges frm a high f 5.3 fr the segment in San Grgni Pass t a lw f 3.6 fr the Carriz Plain. Prbably the mst evil feature f an earthquake is its suddenness. It is true that in the vast majrity f cases a severe shck is heralded by a series f preliminary shcks f slight intensity... [but] nly after the havc has been wrught des the memry recall the sinister warnings f hypgene actin. 1. INTRODUCTION C. G. Kntt (198, p. 1) Many damaging earthquakes have been preceded by smaller earthquakes that ccur within a few days and a few kilmeters f the mainshck [e.g., Jnes and Mlnar, 1979]; these are referred t as immediate freshcks. If such freshcks culd be recgnized befre the mainshck, they wuld be very effective fr shrt-term earthquake predictin; but s far n way has been fund t distinguish them frm ther earthquakes. Even withut this, the mere existence f freshcks prvides sme useful predictive capacity. When any earthquake ccurs, the pssibility that it might be a freshck increases the prba- bility that a larger earthquake wili sn happenearby. Fr suthern Califrnia, Jnes [1985] shwed that after any earthquake there is a 6% prbability that a secnd ne equal t r larger than the first will fllw within five days and 1 km f the first. The prbability is much lwer fr a secnd earthquake much larger than the first; fr example, the prbability f an earthquake tw units f magnitude larger is nly.2%. Using Cpyright 1991 by the American Gephysical Unin. Paper number 91JB /91/91JB-191 $5. 11,959 these results, the U.S. Gelgical Survey has issued fur shrtterm earthquake advisries after mderate earthquakes [e.g., Gltz, 1985]. A mre recent study by Kagan and Knpff [1987] develped a mdel fr the clustering f earthquakes which culd indicate areas f space and time in which larger events might fllw smaller nes. The size f these areas depended n the prbability gain, the rati f prbability f an earthquake given the ccurrence f a pssible precursr (such as a freshck) t the prbability in the absence f such a precursr [Kagan and Knpff, 1977; Vere-Jnes, 1978; Aki, 1981]. Fr lw levels f prbability gain, Kagan and Knpff [1987] fund that ne-third f all earthquakes with magnitudes 4 and abve fell within their predicted regins. These results are frm studies f earthquake catalgs; Jnes [1985] used a catalg fr suthern Califrnia, and Kagan and Knpff [1987] used ne fr central Califrnia. As a cnsequence, bth papers give generic results abut pairs f earthquakes, withut much regard fr ther factrs. But it ught t be pssible t d better: the prbability f a very large earthquake shuld be higher if the candidate freshck were t ccur near a fault capable f prducing that mainshck than if it were lcated in an area where we believe such a mainshck t be very unlikely. Mrever, the chance f a candidate earthquake actually being a freshck shuld be higher if the rate f backgrund (nnfreshck) activity were lw. In this study we derive an expressin fr the prbability f a majr earthquake fllwing a pssible freshck near a majr fault frm the basic tenets f prbability thery. This prbability tums ut t depend n the lng-term prbability f the mainshck, the rate f backgrund seismicity alng the fault, and sme assumed characteristics f the relatins between mainshcks and freshcks. We then apply this expressin t

2 11,96 AGNEW AND JONES: PREDICTION PROBABILITIES FROM FORESHOCKS the San Andreas fault system t develp shrt-term prbabilities fr pssible earthquake warnings based n pssible freshcks. 2. MODELS FOR PROBABILITIES FROM FORESHOCKS Because f the nature f seismicity alng majr fault systems such as the San Andreas fault, we have been led t address certain fundamental issues abut the relatinship between freshcks and large earthquakes. These majr faults illustrate in an extreme frm the "maximum magnitude" mdel intrduced by Wesnusky et al. [1983], in which the frequency f the largest earthquakes n a fault zne is much higher than wuld be predicted by the extraplatin f the frequencymagnitude distributin fr backgrund earthquakes. FOr many parts f the San Andreas fault this is a straightfrward cnsequence f the lw level f present-day seismicity. Fr instance, alng the Cachella Valley segment f the San Andreas fault (Figure 4) an extraplatin f present seismicity t higher magnitudes predicts a magnitude 7.5 earthquake every 29 years, whereas the recurrence rate estimated frm slip-rate data is 2-3 years. This behavir implies that the large characteristic earthquakes n a fault zne are nt simply the largest members f the ttal ppulatin f earthquakes there, but are smehw derived frm a different ppulatin. Freshcks t such events can thus reasnably be regarded as als being a separate class f events frm the backgrund earthquakes. A physical mdel that might underlie this is that sme special failure prcess takes place befre characteristic earthquakes, with an enhanced rate f small earthquakes and eventual failure n a large scale bth being a result f it. It is f curse als pssible that n such prcess ccurs; a mderate shck might, depending n the details f stress nearby, trigger nly smaller events (in which case it is a mainshck) r larger nes (making it a freshck), as suggested by Brune [1979]. There wuld then be n innate difference between backgrund events and freshcks; but we believe that it remains fruitful (as will be shwn) t make at least a cnceptual divisin Zer-Dimensinal Mdel Starting frm the assumptin that freshcks are a separate P (F)+P (B) P (F IC )P(C) + P (B) class f earthquake frm backgrund earthquakes, we can set ut a frmal prbabilistic scheme fr finding the prbability f a large shck, given the ccurrence f a pssible freshck. Fr Fr P (B) >> P (F Ic)P (C) this expressin is small (the candidate event is prbably a backgrund earthquake), while fr P (B)=, the expressin becmes equal t ne: any candidate clarity we begin with a "zer-dimensinal" mdel, ignring earthquake must be a freshck. spatial variatins, magnitude dependence, and ther cmplica- The secnd frm f expressin in (4) is a functin f three tins, which will be added in later sectins. With these quantities, which in practice we btain frm very different simplificatins, a numerical example will illustrate the reasn- surces. P (B), the prbability f a backgrund earthquake, ing. Suppse that mainshcks ccur every 5 years (n average), and that half f them have freshcks (defined as being within a day f the mainshck); then we expect a freshck every 1 years. If a cmparable backgrund earthquake ccurs, n average, annually, we get 1 backgrund earthquakes per freshck. If an earthquake ccurs that culd be either ne, we then wuld assume the prbability t be 1-3 that it is a freshck, and s will be fllwed by a mainshck within a day. This is lw, but still far abve the backgrund ne-day prbability f 5.5 x Fr a frmal treatment we begin by defining events (in the prbability-thery meaning f the term): B: A backgrund earthquake has ccurred. F: A freshck has ccurred. C: A large (characteristic) earthquake will ccur. As nted abve, if a small backgrund shck were t happen by cincidence just befre the characteristic earthquake, we wuld certainly class it as a freshck. Thus, B and C cannt ccur tgether: they are disjint. The same hlds true fr B and F: we can have a freshck r a backgrund earthquake, but nt bth. The prbability that we seek is the cnditinal ne f C, given either F r B, because we d nt knw which has ccurred. This is, by the definitin f cnditinal prbability, P (C1F B) = P (C c (F c9b )) (1) P(F B ) Because F and B are disjint, the prbability f their unin is the sum f the individual prbabilities, allwing us t write the numeratr f (1) as P ((C ) (C rag )) = P (C c F ) + P (C ) = P (C c F ) where the disjinmess f C and B eliminates the P (C ) term. Frm the definitin f cnditinal prbability, P(Cc" ) = P(F 1C)P(C) where P (F IC) is the prbability that a mainshck is preceded That we make this divisin des nt mean that there are any by a freshck. Again using the disjinmess f F and B, we characteristics that can distinguish between freshcks and ther earthquakes; indeed, if there were, we wuld nt have had t can write the denminatr as cnsider the secnd mdel abve. We can nly identify freshcks, like aftershcks, by virtue f their assciatin with a P(F B) = P(F) + P(B) (2) larger event; and, as ur pening qutatin suggests, fr freshcks such identificatin can nly be retrspective. Such Because a freshck cannt, by definitin, ccur withut a mainshck, the intersectin f C and F is F, and therefre classificatin by assciatin means that any particular shck might have been classified "incrrectly", and actually have been a backgrund shck that just happened t fall clse t a larger event. In ur present state f knwledge this is unavidable, and it may always remain s. P (F) = P (F c C ) = P (F IC )P(C) We can use (2) and (3) t write (1) as (3) P (C IF ub ) = P (F) = P (C)P(F IC) (4) wuld be fund frm seismicity catalgs fr the fault zne. P (C), the prbability f a characteristic earthquake, wuld be

3 AGNEW AND JONES: PREDICTION PROBABILITIES FROM FORESHOCKS 11,961 fund frm calculatins f the type presented by the Wrking Grup n Califrnia Earthquake Prbabilities [1988]. If we had a recrd f the seismicity befre many such characteristic earthquakes, we culd evaluate P(FIC) (which we shall hereafter call FC) frm it directly. (Fr this simple mdel, Fc is just the fractin f large earthquakes preceded by freshcks.) Of curse, we d nt have such a recrd, and s are frced t make a kind f reverse ergdic assumptin, namely that the time average f Fc ver many earthquakes n ne fault is equal t the spatial average ver many faults. This may nt be true, but it is fr nw the best we can d One-Dimensinal Mdel As a simple extensin t the previus discussin, suppse that we have N "regins" and that Ci, Bi, and Fi dente the ccurrence f an event in the i th regin, with C (fr example) nw being the ccurrence f a large earthquake in any pssible regin. These regins can be sectins f the fault r (as we will see belw) vlumes in a multidimensinal space f all relevant variables. The quantity f interest is nw?(c [FiuBi): we have a candidate freshck in ne regin, and want the prbability f a large earthquake starting anywhere. Assuming that the ccurrences Ci are disjint (the epicente r can nly be in ne place), we then have that the prbability f a freshck in the i th regin can be written as N P (Fi) = FC (i, j)p (Cj) j=l where Fc(i,j)= P (Fi ICj). We may regard Fc as the prbability f a freshck in regin i given a large earthquake in regin j. We call this the precurrent prbability because it refers t the prbability f an event preceding a secnd ne (nt, it shuld be nted, with an implicatin f vilated causality). As a simple example, we culd take Fc(i,j)= ct ii, which wuld imply that large earthquakes are preceded by freshcks nly in the same regin, and even then nly a frac- tin t f them have freshcks N N at all. (5) P (F) = _ _ FC (i, j)p (Cj) (7) i=1 j=l giving us the verall prbability f a freshck smewhere in the ttal regin. This must satisfy P (F)= tp (C), where ct is the fractin f mainshcks with freshcks; this and equatin (7) tgether cnstrain the nrmalizatin f Fc. Next t the prbability level itself, the scially mst interesting quantities wuld seem t be the chance f an alert being a false alarm, and the rate at which false alarms ccur fr a given prbability level. The prbability that an alert is a false alarm is P(( IFi JBi), which is just 1- P(CIFi Bi): if we have a 1% chance f having a mainshck, we have a 9% chance f nt having ne. The rate f false alarms is equivalent t the prbability f a false alarm happening in sme given time, and this is just the prbability that an alert is a false alarm times the prbability f the event that triggers it, namely N _ [ 1 - P (C IF i [.JBi )][P(Bi) + P (Fi)] i=1 As will be shwn in sectin 4, we wuld in practice usually chse the prbability f a mainshck given a small event, P (C IFicJBi) t have a fixed value (e.g., 1%), which we dente by S, fr all regins. This value f S then sets the value f P (B i ) fr the i th regin; frm (6), P (B i ) = P (F i )[(1 - S )/S ], which makes the prbability f a false alarm N (1-S) N N S i=1 S -=.= ' (1- S) ZP(Fi)= / lj l(i)f½(i j)p(cj) where we have used (5). Fr fixed S and Fc this expressin is prprtinal t P (Ci) nly: the rate f false alarms fr a given prbability depends nly n the rate f mainshcks and nt n the rate f backgrund activity. In terms f the simple example at the beginning f sectin 2.1, fixing a prbability level f.1% means that we wuld set the magnitude level f candidate events such that there wuld be 1 backgrund events fr each actual freshck; but the abslute rate f such backgrund earthquakes (and thus f false alarms) is then determined nly by the rate f freshcks, and thus f mainshcks. 3. A MULTIDIMENSIONAL MODEL FOR FORESHOCKS We can then easily revise (4) abve t get the prbability we We nw develp an expanded versin f (5), which cntains seek; simply adding subscripts t the candidate event yields mre variables. The first step is t define ur events mre thrughly: P (Fi) + P (C)P (Bi) P (C IFi LJBi) = (6) B: A backgrund earthquake has ccurred at crdinates P (Bi) + P (Fi) (x+e, Y+e), during the time perid [t,t +15], with magnitude M +g. (All f the quantities e, 5, and g Equatins (5) and (6) are the basic nes we shall use in the mre general case. Equatin (5) shws us hw t cmpute the prbability f a freshck happening in the lcatin f ur candidate earthquake, by summing ver all pssible mainshcks. The use f the precurrent prbability Fc is the key t this apprach; we can (and in the next sectin shall) design it t embdy ur knwledge and assumptins abut the relatin between freshcks and the earthquakes they precede. Having fund the freshck prbability, we then use (6) t find the cnditinal prbability f a large earthquake. F: C: are small and are included because we will be dealing with prbability density functins; as will be seen belw, they cancel frm the final expressin). A freshck has ccurred, with the same parameters as in event B. A majr earthquake will ccur smewhere in the regin f cncern, which we dente by Ac (als using this variable fr the area f this regin). This earthquake will happen during the time perid [t + A, t + A + 5 ], with magnitude between Me and Mc + gc. An imprtant cnsequence f (5) is that we may sum ver all pssible freshcks (again assuming disjinmess) t get We assume that we are cmputing the prbability at sme time in the interval (t + 5, t + A); the pssible freshck has happened, but the predicted mainshck is yet t cme Rate Densities f Earthquake Occurrence We begin by defining a rate f ccurrence fr the backgrund seismicity (in the literature n pint prcesses this wuld be

4 11,962 AGNEW AND JONES: PREDICTION PROBABILITIES FROM FORESHOCKS called an intensity, a term we avid because f existing seismlgical usage). This rate (r, strictly speaking, rate density) we call A(x,y,M); it is such that the prbability f B is x +e Y +e M +g P(B)=i5 I dx I dy l dm A(x,y,M) (8) x -e Y -e M-g A(x, y, M) -- As (x, y ) e- ( c'y)m (9) where [ is 2.3 times the usual b value. (While cmmn rather than natural lgarithms are cnventinal in this area, they lead t messier expressins, and we have therefre nt used them). If [ is cnstant ver a regin f area A, and during a time interval T the cumulative number f earthquakes f magnitude M r greater is given by the usual frmula then, since the expected value f N (M) is N(M) = 1 a-bm (1) E[N(M)] = T l dm Ildx dy As(x,y)e - M M A (11) we have that As = (1a )/(AT) fr As cnstant within the regin. Similarly, we can define a rate density fr the ccurrence f large earthquakes, f(x, y,m,t) = fs(x, y,t)e - '(x'y)m (12) where [ ' is 2.3 times the b value fr these events. In this case, we intrduce a dependence n time t because the ccurrence f large earthquakes is ften frmulated as a renewal prcess [e.g., Nishenk and Buland, 1987], with time being measured relative t the last earthquake. The prbability f C is then regins), we may i n fact make them three-dimensinal r nedimensinal if we s chse, making sure that we adjust the numbers f the integrals in (8) and (13) accrdingly. The nedimensinal mdel is easiest t develp analytical expressins fr, and may be an adequate apprximatin fr the case f a lng fault zne. In this case, f curse, we need t prject the backgrund seismicity (ut t sme distance away) nt the fault zne. By nt making A dependent n the time t we make the ccurrence f backgrund earthquakes int a Pissn prcess Cmputatin f the Freshck Prbability If we assume that at any lcatin the Gutenberg-Richter frequency-magnitude relatin hlds, we may write MC +gc = Ilex I f(x,y,m,t +A) (13) fs = P (C)[ ' (14) Active - ' c(1- e - ' c) Nte that while we have regarded bth A and Ac as twdimensinal regins (and hence als as the areas f such We are nw in a psitin t write the frmal expressin fr the freshck prbability P (F) in the same way as was dne in (5) fr the discrete ne-dimensinal case. In this case, rc becmes a density functin ver all the variables invlved, its value indicating with what relative frequency freshcks with different parameters ccur befre mainshcks With particular Ones. Instead f a single sum, as in (5), we have a multiple integral: t +5 x +e Y +e M +g t +A+ 1 MC +gc P(F)= l t dt x -e l dx Y l -e dy M-g l dm t +A I a,'jiax'ay' M I C ' ec (t, t',x, y,x', y',m,m') fs (x ', y', t')e - '(x"y') ' Of these eight integrals, the last fur are the integratin f the precurrent prbability density times the density f mainshck ccurrence ver the space f pssible mainshcks and are the equivalent f the sum in (5). But this gives nly the rate density fr freshcks, which must in turn be integrated ver the space f the candidate event (the first fur integrals) t prduce the actual prbability P (F). Equatin (15) is clearly quite intractable as it stands. T render it less s, we assume that we can separate the behavirs f P (F) in time, magnitude, and lcatin. This implies the fllwing assumptins: 1' ' des nt depend n x' r y'. 2: Over the range f integratin, fs des nt depend n t'. 3' The functinal frms f the precurrent prbability density fr time, space, and magnitude are independent, s that we can write the functin as the prduct f the marginal distributins: rc = s(x, y,x', y') t(t,t') m(m,m') A C M C Of these assumptins, the third seems the least likely t be where Ac is the area f cncern, i.e., the particular segment f a valid, since the dependence n bth distance and time might be fault. crrelated with the magnitude f either the mainshck r the Fr lack f better infrmatin we wuld usually take fix t be a cnstant, but we culd chse t make it spatially varying. candidate freshck. The mst likely crrelatin, with mainshck magnitude, des nt matter very much, since ur Such variatin culd include increases near fault jgs and termi- range f integratin f this variable is small. natins if we think that rupture nucleatin is mre likely there, These assumptins made, we can divide the integral in (15) r a prprtinality t Ax if we suspecthat backgrund earth- int a prduct f three integrals (in space, time, and magnitude): quakes are (n the average) the likely triggers f large nes (bth issues are discussed in sectin 3.2.2). Fr fix cnstant, t +5 t +A+fi 1 M +g MC +gc we have that P(F) = ( 5) I at I at(i), (t,t) I am I dm t(i) m (m,m')e- 'M' t t+a M-g M C x +e Y +e I dx I * If dx'*'*x(x'y'x"y')nx(x"y') X O-e Y O-e O A C (16)

5 AGNEW AND JONES: PREDICTION PROBABILITIES FROM FORESHOCKS 11, Functinal Frms fr the Freshck Density T evaluate the integrals in (16), we need t knw the three precurrent prbability densities (I)t, (I)s, and (I)m. Our expressins fr these incrprate ur knwledge and assumptins abut freshcks. In the fllwing sectins, we describe in sme detail what is knwn abut the tempral, spatial, and mag- nitude dependences f freshcks. Frm these data, we find functins fr the relevant (I); these functins must include bth where tw is the ttal time windw within which we admit the actual dependence n the variables and a nrmalizatin. preceding earthquakes t be freshcks. This then gives The nature f the nrmalizatin can be seen if we imagine t +{3 t +A+ 51 extending the range f the first fur integrals in (15) t cver all ln[1 + b /(A + c)] pssible freshcks (hwever we chse t define them); the dt ' (I) t ( t, t') = b ln[1 + tw/( 5 + c )] resulting? (F) must then be equal t ctp (C), where (x is, as fr the ne-dimensinal mdel, the fractin f mainshcks preceded by freshcks. In deriving ur expressins we have aimed fr = bb (/X, 8 ) (2) simplicity rather than attempting t find a functin that can be shwn t be statistically ptimal. where, with an eye t future simplificatins, we have separated ut the 5 term. Nte that (17) predicts a finite rate fr all Time. Mst freshcks ccur just befre the times, whereas the assumptin f a limited time windw mainshck. An increase in earthquake ccurrence abve the autmatically frces the rate t fall t zer beynd sme time; backgrund rate has nly been seen fr a few days [Jnes, we can easily mdify (I)t t allw fr this. 1984; 1985; Reasenberg, 1985] t a week [Jnes and Mlnar, Lcatin. Freshcks nt nly ccur clse in time t 1979] befre mainshcks. Fr 26% f Califrnian mainshcks, the mainshck, but are als nearby in space. Jnes and Mlnar the freshcks are mst likely t ccur within 1 hur f the [1979] fund that epicenters f mainshcks (M > 7) and their mainshck; the rate f freshck ccurrence befre mainshcks freshcks in the Natinal Earthquake Infrmatin Center (Figure 1) varies with the t - type behavir als seen in Omri's (NEIC) catalg were almst all within 3 km f each ther, law fr aftershcks [Jnes, 1985; Jnes and Mlnar, 1979]. apprximately the lcatin errr fr the NEIC catalg. Jnes This variatin can be well fit by the functin that Reasenberg [1985], with the mre accurate lcatins f the Califrnia Instiand Jnes [1989] fund fr Califrnia aftershck sequences: tute f Technlgy (Caltech) catalg, fund that epicenters f mainshcks (M _> 3) and their freshcks were almst all within Nt 1 km f each ther; this result als held fr freshcks f (I)t(t,t') = (17) t'-t +c M > 5 mainshcks within the San Andreas system [Jnes, 1984] if relative relcatins were used. Even the largest where t is the freshck time and t' the mainshck time; c is a cnstant, fund by Reasenberg and Jnes [1989] t be 2 s fr aftershcks. The relevant integral frm (16) is then freshcks (M > 6 at Mammth Lakes and Superstitin Hills) have had epicenters within 1 km f the epicenters f their mainshcks. t +5 t +A+$1 t t+a dt' (I)t(t, t') = 5Nt In[ 1 + ts /(A + c )] (18) where we have assumed that 5(the uncertainty f the time f 3OO 25O '½ 2-15,- 1 z Time between Freshck and Mainshck, Hurs Fig. 1. The number f freshck-mainshck pairs recrded in suthern Califrnia versus the time between freshck and mainshck in hurs fr freshcks M > 2. and mainshcks M > 3. recrded between 1932 and the candidate earthquake) is small. The nrmalizatin is determined by the requirement that t + +tw t +b dt' t(t,t') = 1 (19) We have assembled a data set f sequences with high-quality lcatins t examine the dependence f the distance between freshcks and mainshcks n the magnitudes f the earthquakes. This data set includes all freshck-mainshck pairs with mfre _> 2.5 and Mmain -> 3. recrded in suthern Califrnia since 1977 (the start f digital seismic recrding), and several sequences relcated in special studies, with relative lcatin accuracy f at least 1 km. Figure 2 shws the distance between freshck and mainshck versus magnitude f the mainshck (2a) and magnitude f the freshck (2b). The epicentral separatin between freshck and mainshck des nt crrelate strngly with either magnitude. Rather, the data seem t grup int tw classes: freshcks that are essentially at the same site as their mainshck (<3 km) and freshcks that are clearly separated frm their mainshcks. Only freshcks t larger mainshcks (mmain-> 5.) OCCur at greater epicentral distances (5-1 km). Of these spatially separate freshcks sme (but nt all) ruptured twards the epicenter f the mainshck (the rupture znes are shwn by the vals in Figure 2). The greatest reprted distance between freshck and mainshck epicenters is 8.5 km; the greatest reprted distance between freshck rupture zne and mainshck epicenter is 6.5 km. It wuld therefre seem that, whatever ther behavir (I)s may have, it can be taken t be zer fr distances greater than 1 km. It is pssible (and allwed fr in ur chice f variables fr (I)s) fr freshcks t be preferentially lcated in sme sectins

6 11,964 AGNEW AND JONES: PREDICTION PROBABILITIES FROM FORESHOCKS (a) 15 EP1- ß, (b) I _1 I ," 1981 ( I...,,;,,,,?,, Mainshck Magnitude E e1 ß, 198 u ; ,,,m,,,,... I- I... I... I Freshck Magnitude city. Fr example, the Calaveras fault in central Califrnia has a relatively high rate f backgrund activity and n freshcks. Freshcks and mainshcks thus clearly ccur clse tgether in space, within 1 km f each ther in all reslvable cases -but shw n ther clear dependence n lcatin. We therefre have made s depend nly n p, the distance between candidate freshck and pssible mainshck epicenters (p = [(x-x') 2 + (y_y,)2],/2). The cnditin fr s t be prperly nrmalized is I laxay I ldx'dy'rbs(x, y,x', y')f s(x', = Fig. 2. Distance between freshck and mainshck epicenters versus the (a) magnitude f the mainshck and (b) magnitude f the freshck fr XO-e AC freshck-mainshck sequences (freshcks M > 2.5 and mainshcks M _> 3.) recrded in the Caltech catalg between 1977 and s(x) 2e -- 2els(x) Sequences that have been relcated in special studies are als pltted 1-pw2/4Ac (23) and include 1966 Parkfield, 1968 Brreg Muntain, 197 Lytle Creek, 1972 Bear Valley, 1975 Haicheng (M = 7.3), 1975 Galway Lakes where we have defined Is in a parallel way t It; the depen- (M = 5.2), 1979 Hmestead, 198 Livermre, 1981 Westmreland, dence n x cmes thrugh the dependence n the value f s 1985 Kettleman Hills, 1986 Chalfant Valley, and 1987 Superstitin near the candidate earthquake. Hills. Fr the three freshck sequences with knwn rupture znes the distance range f freshck rupture zne t the mainshck epicenter is Magnitude. The functinal frm fr m(m,m') is shwn by the elngated vals; the circles inside these shw the distance prbably the least certain part f ec. Plts f the difference in fr the freshck epicenter. freshck and mainshck magnitudes with a unifrm magnitude threshld fr freshcks and mainshcks [e.g., Jnes, 1985] shw the magnitude difference t be a negative expnential distributin. Hwever, t cnsider all pssible freshcks t a f majr faults. Jnes [1984] suggested that freshcks are given mainshck, the cmpleteness threshld fr the freshcks mre cmmn at areas f cmplicatin alng faults; this wuld shuld be much lwer than fr the mainshcks. A bivariate plt require that either P(C) r s (r bth) be larger at such f freshck and mainshck magnitudes fr all recrded places. An increase f P (C) wuld be in accrdance with the freshcks in suthern Califrnia (Figure 3) suggests that fr ntin that epicenters f mainshcks are mstly at such pints any given narrw range f mainshck magnitude, freshck [King and Na7 elek, 1985; Bakun et al., 1986). While this magnitudes clse t that f the mainshck are mre cmmn; seems like a valid refinement, in practice differentiating between hwever, fr the larger mainshck magnitudes f interest here, the many pssible cmplex sites and the "smth" parts f the the (admittedly sparse) data suggesthat all freshck magnifault wuld requiring gridding at the kilmeter scale, a level f tudes are equally likely fr given mainshck magnitude. detail that des nt seem justified by ur present level f Because f the simplicity f this last assumptin, we have knwledge. One further chice wuld be t make s prprused it here by making (I) m cnstant; we set (I) m (m,m t) Nm, tinal t the lcal rate f backgrund activity As, thus asserting a nrmalizing factr. The nrmalizatin f (I) m is in general set that mst mainshcks with freshcks ccur in areas with high by backgrund seismicity. The data n freshcks t mderate / earthquakes in Califrnia [Jnes, 1984] des nt supprthis: while the fractin f earthquakes with freshcks des vary by I I (I)m (m, m ') dm rim' = ( I e- 'V 'dm' regin, it des nt appear t be related t backgrund seismi- A C A C I l dx 'dy' f s (X', y ') (21) A C which in general can be dne nly numerically, even fr s cnstant and s having a simple dependence n p. If, hwever, we make the simplificatin, mentined in sectin 3.1, f making ur spatial integrals ne-dimensinal (with Ac then being the length f the fault), assume s cnstant, and make s cnstant fr p <_ Pw and zer fr larger p, we find that s is 1 Pw (1 - Pw 2/4Ac ) if p <_ Pw if p > Pw (22) We use Pw = 1 km t agree with the data presented abve. Then, prvided that the lcatin x f the candidate earthquake s mre than a distance Pw frm an end f the fault zne and that s (x') is cnstant ver a distance 2pw, the integral needed in (16) is x +e I I (24)

7 AGNEW AND JONES: PREDICTION PROBABILITIES FROM FORESHOCKS 11,965 Suthern Califrnia M +p MC +PC dm '(I) m (M, M')e-lY' ' = M -g M C 2 p Nm e - ' c 1 - e -13'PC = 2 p Im (Mc, Pc ) (26) g where we have assumed!-t small, and again separated it ut frm the rest f the expressin Mainshck Prbability We nw can cmbine the integrals in (18), (23), and (26) int (16) t get the freshck prbability: P(F) = 45pelthlm "%. '. '. d. 7'. a'. M(Main) Fig. 3. The number f freshck-mainshck pairs in half unit f magnitude bins fr the magnitudes f freshck and mainshck. Data included all M _> 2. freshcks and M _> 3. mainshcks recrded between 1932 and 1987 in suthern Califrnia. Equatin (24) says that if we lk befre all mainshcks with magnitudes greater than MB fr freshcks abve a cutff magnitude f M, we find that a fractin f the mainshcks have freshcks. Nte that we have chsen t nrmalize (I) t and s t integrate t 1, s (I) m cntains the infrmatin abut the ttal fractin f mainshcks with freshcks. Making (I) m cnstant implies that the fractin f mainshcks preceded by freshcks will increase as the magnitude threshld fr freshcks decreases. This is cnsistent with reprted freshck activity, since the data suggest that freshcks are relatively cmmn befre majr strike-slip earthquakes. Jnes and Mlnar [1979] fund that 3% f the M > 7. earthquakes ccurring utside f subductin znes were preceded by freshcks in the NEIC catalgue (M > ) and almst 5% had freshcks M > 2 reprted in the literature. Jnes [1984] shwed that half f the M > 5. strike-slip earthquakes in Califrnia were preceded by M > 2. freshcks. (Freshcks were less cmmn n thrust faults.) Fr (I) m cnstant and equal t Nm, (24) implies that Nm = (25) 1 + [ '(MB-M ) 9. Slving the integral in (8) fr the backgrund event gives P (B) = 45 }.t e As (x )e - a4 We substitute these values f the backgrund and freshck prbabilities int (6) t btain: Is/tim P (C IF t B ) - (27) Is It Im + As (x )e - M The candidatearthquakerrrs 8, e, and }.t have canceled ut. Fr making calculatins, it is als useful t set It equal t 1 (slve fr the prbability in a fixed time interval) and (fr the case f a linear fault) take Is in (23) t be equal t g2s(x). If we take g2s t be cnstant and cmbine (14) and (26), we find that the dependence n Mc and Itc cancels ut, and we are left with (Nm P (C )/Ac P (C IF twb ) = (28) (NmP(C)/Ac5 ) + As (x)e- a4 4. APPLICATION TO THE SAN ANDREAS FAULT SYSTEM, CALIFORNIA We nw have an expressin fr the cnditinal prbability f a characteristic earthquake n a fault segment given the ccurrence f an earthquake that is either a backgrund event r a freshck. T evaluate this, we need the lng-term prbability f the characteristic mainshck (the terms invlving the actual magnitude f the characteristic earthquake have canceled ut), the length f the fault segment, and the rate density f backgrund seismicity fr that segment. T shw hw this wrks, we nw apply this t the San Andreas fault system in Califrnia, because the lng-term prbabilities fr characteristic The data presented by Jnes [1984], with MB = 5. and M = 2., gave equal t.5 fr strike-slip earthquakes. Adpting this value, with a [ ' f 2.3, gives Nm =.15. A cnsequence f taking (I) m cnstant is then that all earthquakes shuld have freshcks within 6.5 units f magnitude f the mainshck. earthquakes that we need have been estimated fr the majr faults f this system, the San Andreas, Hayward, San Jacint and Imperial faults. This was first dne by Lindh [1983] and Sykes and Nishenk [1984], and mre recently by the Wrking Grup n Califrnia Earthquake Prbabilities [1988], hereafter Hlding (I) m cnstant fr all M wuld f curse lead t the referred t as WGCEP-88. absurd result that mre than 1% f mainshcks have Our divisin f the fault int segments and ur values f freshcks within, say, 8 magnitude units. Fr the smaller range f magnitudes cnsidered here a cnstant (I) m des nt present any difficulties. The integral needed fr (16) is then P (C) fr each segment cme largely frm WGCEP-88. One exceptin is that the lengths f the Suthern Santa Cruz Muntains and the San Francisc Peninsula segments have been altered t match the rupture zne f the 1989 Lma Prieta earth-

8 11,966 AGNEW AND JONES: PREDICTION PROBABILITIES FROM FORESHOCKS Suthern San Andreas Fault M>1.8 Declustered 36 ø Parkfield Chlame MAGS 35 Carriz MAGNITUDES ß.+ 34 ø Mjave Sail. aim Springs ca 2 KM 33 ø 12 ø 119 ø 118 ø 117 ø 116 ø Fig. 4. A map f M _> 1.8 declustered earthquakes lcated within 1 km f the suthern San Andreas fault recrded in the Caltech catalg between 1977 and 1987 and (fr Parkfield) the CALNET catalg between 1975 and quake (A. Lindh, persnal cmmunicatin, 199). We tk P (C) t be cnstant alng each segment; as nted in sectin 3.2.1, we have nt tried t include the pssibility that nucleatin pints (and higher values f P (C)) are mre likely at pints f cmplicatin. We have als nt altered the distributin f P (C) t accunt fr any pssible relatinship between nucleatin pint and level f backgrund activity. The rate density fr the backgrund seismicity is determined frm the micrearthquakes recrded between 1977 and 1987 by the Caltech/U.S. Gelgical Survey Suthern Califrnia Seismic Netwrk [Given et al., 1988] fr suthern Califrnia and between 1975 and 1989 by CALNET, the U.S. Gelgical Survey Central Califrnia Seismic Netwrk (P. Reasenberg, persnal cmmunicatin, 199), fr nrthern Califrnia. Backgrund seismicity can be defined in many ways; it is imprtant in this applicatin that it be defined in the same way as the freshcks will be. Because freshcks can be up t 1 km frm their mainshck (Figure 2), backgrund seismicity up t 1 km frm the surface trace f the San Andreas fault is included in the backgrund rate. Anther issue is hw t handle tempral clustering in the catalg. We assume that if an earthquake f M = 6 (fr instance) were t ccur n the suthern San Andreas fault with an aftershck sequence, we will nly evaluate the prbability that the M = 6 earthquake is a freshck, and nt individually determine the prbabilities that the M = 6 and each f its aftershcks is a freshck and then sum them. Fr cnsistency we therefre want t determine the backgrund seismicity using a catalg frm which aftershck sequences and swarms have been remved. In such a declustered catalgue, sequences are recgnized by sme algrithm and replaced in the catalgue with ne event at the time f the largest earthquake in the sequence, which is given a magnitude equivalent t the summed mment f all the earthquakes in the sequence. T prduce ur declustered catalgs, we used the algrithm f Reasenberg [1985]. The resulting backgrund seismicity within 1 km f the faults is shwn in Figures 4-6. It is clear frm these that the rate f backgrund seismicity can vary significantly within the fault segments defined by WGCEP-88. Fr example, the Cachella Valley segment f the San Andreas includes the active regin arund Desert Ht Springs (including a M = 6.5 event in 1948) and a very quiet regin (near the Saltn Sea) where the largest earthquake in 55 years has been M = 3.5. T accunt fr this variatin, we have divided sme f the WGCEP-88 segments int smaller regins, which are shwn in Figures 4-6 and listed in Table 1. Table 1 prvides the data needed fr each segment. T use (28) we als need the time perid 81, which we set t 3 days (1.9 x 15 s), t match the recent usage f the U.S. Gelgical Survey and the Califmia's Gvernr's Office f Emergency Services in issuing earthquake advisries. Alert levels fr such advisries are defined t crrespnd t certain prbabilities; the magnitudes f earthquakes needed t trigger thse alert levels can then be cmputed frm (28), and are als given in Table 1. Figure 7 shws the prbability as a functin f the magnitude f

9 AGNEW AND JONES: PREDICTION PROBABILITIES FROM FORESHOCKS 11,967 Nrthern San Andreas Fault System Deelustered ø Pi MAGS 39 ø O Cast MAGNITUDES ß ø N rth Hayward 3.+ [ ) Suth Hayward Peninsula 37 ø Lma P 5 KM I,,,.l,,,d..,.&,d ø 123 ø 122 ø 121 Fig. 5. A map f M > 1.8 declustered earthquakes lcated within 1 km f the San Jacint fault recrded in the Caltech catalg between 1977 and the candidate earthquake fr each segment. mine shrt-term prbabilities with ur methdlgy and given We have treated the Parkfield segment in tw different ways. these in Table 1 as Middle Muntain prbabilities. These In the Table 1 listing fr Parkfield, we treat it in the same way remain lwer than the Bakun et al. results; fr example, a magas the ther segments, regarding the freshck as equally likely nitude 1.5 shck gives a prbability f.1% frm ur methanywhere alng the segment, and taking P (C) frm WGCEP- dlgy and.68% (Level D alert) accrding t Bakun et al. 88. These assumptins give shrt-term prbabilities much lwer As with the lng-term prbabilities f majr earthquakes, than thse estimated by Bakun et al. [1987] fr the Parkfield these shrt-term freshck-based prbabilities are better seen as earthquake predictin experiment. Bakun et al. [1987] used a a means f ranking the relative hazard frm different sectins f smewhat different methdlgy and als used different the faults than as highly accurate abslute estimates. The prassumptins in tw areas: their value f P (C) is 1.5 times that babilities are as uncertain as the data used t calculate them, f WGCEP-88, and they assume that the freshck will be which in sme cases are uncertain indeed. Fr example, the lcated in a small regin under Middle Muntain, making a values f P (C) fund by WGCEP-88 are up t a factr f 4 smaller area fr defining backgrund seismicity. (Their assump- larger than thse fund by Davis et al. [1989]; this wuld lead tin that 5% f Parkfield mainshcks will be preceded by t similarly large differences in the shrt-term prbabilities. freshcks agrees with ur chice in sectin 3.2.3). Fr a better The relative shrt-term prbabilities fr different segments cmparisn we have used the Bakun et al. assumptins t deter- shwn in Table 1 and Figure 7 are strngly affected by bth the

10 San Jacint and Imperial Faults M>I.6 Declustered an Bernardin 34 ø MAGS MAGNITUDES ø 6.+ San Dieg Imperial 2O KM h,,...,i... I 117 ø 116 ø 115 ø Fig. 6. A map f M > 1.5 declustered earthquakes lcated within 1 km f the nrthern San Andreas and Hayward faults recrded in the CALNET catalg between 1975 and TABLE 1. Parameters and Magnitude Levels Segment Length, Pc a b A s, [3 Magnitude fr km /3 days events/km s.1% 1% 1% San Andreas Fault Mecca x x Palm Springs x x San Grgni x x San Bernardin x x Mjave x x Tejn x x Carriz x x Chlame x x Parkfield x x Middle Muntain x x Lma Prieta x x Peninsula x x Nrth Cast x x Pint Arena x x San Jacint Fault San Bernardin x x San Jacint x x Anza x x Brreg x x Hayward Fault Nrth Hayward x x Suth Hayward x x Imperial Fault Imperial x x

11 AGNEW AND JONES: PREDICTION PROBABILITIES FROM FORESHOCKS 11, Shrt-term Prbabilities: Suthern Sn Andreas i i i I-1 I-I Mecca 1ø1 A ' ' Palm Springs OSan Grgni San Bernardin 1 -i--- ' x,,, + Mjave ' Tejn x Carriz Magnitude f Candldte Event Shd-term Prbabilities: Nrthern San Andreas I-I I-I Chlame 11 1 : //// A A Parkfield OMiddle Muntain Lama Prieta I I Peninsula Nrth Cast x x Pint Arena 1-1, Magnitude f Candidate Event Fig. 7. The prbability that an earthquake f given magnitude is a freshck t a characteristic mainshck, pltted against magnitude fr each fault segment listed in Table 1. Shwn are results fr the (a) suthern San Andreas, (b) nrthern San Andreas, and (c) ther faults f the San Andreas system. lng-term prbability P (C) and the rate f backgrund seismicity. Outside f the Parkfield and "Middle Muntain" segments, which have very high P (C), the highest shrt-term prbabilities are frm the Carriz and Chlame segments; even thugh the 3-year prbability is nly 1% n the Carriz segment, the backgrund seismicity there is almst nnexistent. The San Francisc Peninsula and the San Bernardin Muntain segments bth have a 3-year prbability f 2%, but the prbabilities in the San Grgni subregin are much lwer than thse near San Francisc because f higher backgrund seismicity. At high magnitudes, the lwest prbabilities are fr the Pint Arena segment because f its very lw [5, which may be a result f catalg incmpleteness at lw magnitudes. The pssibility f the next Parkfield earthquake triggering a larger earthquake n the Chlame segment has been much discussed. Our prcedure gives a magnitude 6 in Chlame a 52% chance f being a freshck t a characteristic mainshck n that segment; but this result cmes frm the lw backgrund rate fr the Chlame segment itself. Since this rate predicts a magnitude 6 shck every 14 years, nt every 22 years as at Parkfield, this high prbability des nt apply t a pssible Parkfield trigger. We can, hwever, use (3) f ur zerdimensinal mdel t rughly estimate the prbability that a Parkfield earthquake will be a freshck t a larger earthquake at Chlame. The WGCEP-88 prbability f a Chlame earthquake is 3% in 3 years, while the backgrund rate fr

12 11,97 AGNEW AND JONES: PREDICTION PROBABILITIES FROM FORESHOCKS Shrf-ferrn lo 2 Prbabilities: Sn Jacinf/Hayward/Irnperial D I-I Sn Bernardin 1ø1 Sn --- Anza dacint <... -(> Brreg lo I I N. Hayward --- S. Hayward x x Imperial lo Magnitude f Candidate Event Fig. 7. (cntinued) Parkfield mainshcks (and hence candidate Chlame freshcks) mainshck [Jnes and Lindh, 1987]. If that relatinship were is ne every 21.7 years. T determine the shrt-term prbabil- parameterized, ec and the integratin in (15) culd include ity with (3) we need t assume a value fr? (F IC), the rate at which Chlame mainshcks have Parkfield mainshcks as freshcks. If we assume that, as fr an average magnitude 7 variables describing the difference in fcal mechanisms; thus nrmal- r thrust-faulting earthquakes wuld be given a lwer prbability f being a freshck t a San Andreas mainshck. earthquake, 15% f Chlamearthquakes are preceded by mag- If any ther characteristics are recgnized as being mre cmnitude 6 freshcks, then the shrt-term prbability f a Ch- mn in freshcks than backgrund earthquakes (such as lame earthquake after a Parkfield earthquak is 3%. At the number f aftershcks), we can rigrusly include this infrmather extreme, if we assume that 5% f Chlamearthquakes tin in ur cmputatin f the cnditinal prbabilities. are preceded by Parkfield earthquakes (the nly type f Anther directin t g is in imprving ur estimates fr the freshck it has is Parkfield earthquakes), the prbability precurrent prbability beynd the rather simple frms described becmes 1%. As discussed in sectin 2.2 abve, the rate f false alarms abve. Cnsiderable wrk has been dne in the last few years n hw t estimate multivariate density functins, which is predepends n the backgrund prbability fr the characteristic cisely the prblem at hand [Silverman 1986]. An bvius quesearthquake. A cumulative false-alarm rate fr the whle San Andreas fault is thus dminated by the cntributin frm Parkfield, fr which a 1% prbability level ccurs every 8.4 years. By cmparisn,, fr the Cachella Valley segment f the San Andreas fault the false alarm rate fr a 1% prbability is nce every 63 years. Fr.1% it is nce every 5.5 mnths, but this prbability level is nly 9 times the backgrund ne. tin is whether the estimated densities differ significantly between regins; if s, this culd reflect significant differences in the nucleatin and triggering f large earthquakes. Of curse, nthing in the derivatins f sectin 2 is specific t freshcks; this prcedure can be used fr any ptential earthquake precursr. Equatin (6) shws that what is needed is a lng-term mainshck prbability P (C), a rate fr backgrund 5. DISCUSSION events P (B), and a precurrent prbability ec, which wuld in many cases just be the fractin f mainshcks with precursrs. At present, these data are nt available fr any precursr but The prcedure develped here can be made mre general freshcks. Fr instance, the backgrund rate f creep events than has been apprpriate fr the abve applicatin t the San can be determined fr sme sectins f the San Andreas fault Andreas fault. As discussed in sectins 3 and 4, we culd include a different dependence f t n time r make? (C) system, but we have almst n data n the fractin f mainshcks preceded by such events. include infrmatin abut the mst likely epicenters fr the There have been a number f earlier papers n estimating the mainshck (such as fault jgs r terminatins). Anther exten- prbabilities f earthquakes in the presence f precursrs sin wuld be t set P(C) frm an extraplatin f the [Kagan and Knpff 1977; Vere-Jnes 1978; Guagenti and frequency-magnitude relatin; while a vilatin f the maximum-magnitude mdel, this wuld allw applicatin f this Scircc 198; Aki 1981; Andersn 1982; Grandri et al. 1984]. Mst f these take a slightly different definitin f events frm technique t many mre regins. The greatest flexibility cmes the nes we have used. Rather than distinguishing between frm the precurrent prbability density ec, since we can, as backgrund events (independent f large earthquakes) and prethe data warrant, alter this functin t include ddi[inal data cursrs (always fllwed by a large earthquake), these papers types. Fr example, there is evidence t suggest 'that mst assume that all pssible precursrs fall int ne class f events, freshcks have fcal mechanismsimilar t that f their with sme prbability f a pssible precursr nt being fllwed

13 AGNEW AND JONES: PREDICTION PROBABILITIES FROM FORESHOCKS 11,971 by an earthquake. (Fr example, Andersn [1982] cmputes the prbabilities f a precursr being useful r useless). Fr seismicity, a divisin int backgrund and precursry events appears t be a better apprximatin t the likely physics. Mst f these papers als deal with the case (nt discussed here) f hw pssible multiple precursrs culd increase the cnditinal prbability abve that fr a single precursr. The discussin abve suggests that this will usually be a mt pint, since nly rarely d we have the infrmatin needed t estimate the cnditinal prbabilities. With the exceptin f the wrk f Kagan and Knpff [1987] and (in part) Andersn [1982], there des nt seem t have been much cnsideratin f any multidimensinal cases f the kind described in sectin 3. The Kagan and Knpff study is clsest t the apprach presented here, thugh the functinal frm emplyed by them is derived frm a fracture mechanics mdel, whereas urs is mre purely empirical. The mdels als differ cnsiderably in their specificatin f lngterm prbability. In the Kagan and Knpff mdel, this is given by a Pissn rate derived frm the frequency-magnitude relatin (1), whereas here it can be independent f that. As nted in sectin 2, such independence appears t be a mre satisfactry representatin f the seismicity f an active fault zne. 6. CONCLUSIONS We have shwn that the prbability that an earthquake that ccurs near a majr fault will be a freshck t the characteristic mainshck depends n the rate f backgrund earthquake activity n that segment, the lng-term prbability f the mainshck, and the rate at which the mainshcks are preceded by freshcks, which we call the precurrent prbability. Assuming certain reasnable frms fr the density functin f this prbability (as a functin f time, lcatin, and magnitude) Acknwledgements. We thank the members f the Wrking Grup n Shrt-term Earthquake Alerts fr the Suthern San Andreas Fault, especially Brad Hager and Dave Jacksn, fr raising sme f the issues that led t this paper. We have benefited greatly frm reviews by Andy Michael, Dave Jacksn (again), A1 Lindh, and especially Mark Mathews. We als thank Paul Reasenberg fr cmments and fr prviding the declustered CALNET data. Preparatin f this paper was in part supprted by U.S. Gelgical Survey grant G1763. REFERENCES Thatcher, Parkfield, Califrnia, earthquake predictin scenaris and respnse plans, U.S. Gelg. Surv. Open-file Rep , Bakun, W. H., G. C. P. King, and R. S. Cckerham, Seismic slip, aseismic slip, and the mechanics f repeating earthquakes n the Calaveras fault, Califrnia, in Earthquake Surce Mechanics (Gephysical Mngraph 37), edited by C. Schlz, pp , American Gephysical Unin, Washingtn D.C., Brune, J. N., Implicatins f earthquake triggering and rupture prpagatin fr earthquake predictin based n premnitry phenmena, J. Gephys. Res., 84, , Davis, P.M., D. D. Jacksn, and Y. Kagan, The lnger it has been since the last earthquake, the lnger the expected time till the next?, Bull. Seisml. Sc. Amer., 79, , Given, D. D., L. K. Huttn, L. Stach, and L. M. Jnes, The Suthern Califrnia Netwrk Bulletin, January-June, 1987, U.S. Gel. Surv. Open-file Rep , Gltz, J., The Parkfield and San Dieg earthquake predictins: a chrnlgy, Special Reprt by the Suthern Califrnia Earthquake Preparedness Prject, Ls Angeles, Calif, Grandri, G., E. Guagenti, and F. Pertti, Sme bservatins n the prbabilistic interpretatin f shrt-term earthquake precursrs, Earthq. Eng. Struct. Dynam., 12, , Guagenti, E.G. and F. Scircc, A discussin f seismic risk including precursrs, Bull. Seisml. Sc. Amer., 7, , 198. Jnes, L. M., Freshcks ( ) in the San Andreas System, Califrnia, Bull. Seisml. Sc. Amer., 74, , Jnes, L. M., Freshcks and time-dependent earthquake hazard assessment in suthern Califrnia, Bull. Seisml. Sc. Amer., 75, , Jnes, L. M. and A. G. Lindh, Freshcks and time-dependent cndi- tinal prbabilities f damaging earthquakes n majr faults in Califrnia, Seisml. Res. Letters, 58, 21, Jnes, L. M. and P. Mlnar, Sme characteristics f freshcks and their pssible relatinship t earthquake predictin and premnitry slip n faults, J. Gephys. Res., 84, , Kagan, Y. and L. Knpff, Earthquake risk predictin as a stchastic prcess, Phys. Earth Planet. Int., 14, 97-18, Kagan, Y. and L. Knpff, Statistical shrt-term earthquake predictin, Science, 236, , King, G. C. P. and J. Na'belek, Rle f fault bends in the initiatin and terminatin f rupture, Science, 228, , Kntt, C. G., The Physics f Earthquake Phenmena, 278 pp., Clarendn Press, Oxfrd, 198. Lindh, A. G., Preliminary assessment f lng-term prbabilities fr large we have fund an expressin fr the shrt-term prbability that an earthquake is a freshck, and applied it t the faults f the San Andreas system. Because the rate f freshcks befre earthquakes alng selected segments f the San Andreas fault system mainshcks is assumed t be the same fr all segments, the in Califrnia, U.S. Gelg. Surv. Open-File Rep , pp. 1-15, differences in shrt-term prbabilities between segments arise frm differences in backgrund rate f seismicity and in lng- Nishenk, S. P. and R. Buland, A generic recurrence interval distributin term prbabilities. The backgrund rates are mre variable fr earthquake frecasting, Bull. Seisml. Sc. Amer., 77, , between regins and lead t larger variatins in shrt-term prbabilities. Fr the San Andreas fault the tw extremes are the Reasenberg, P., Secnd-rder mment f Central Califrnia seismicity, , J. Gephys. Res., 9, , nearly aseismic Carriz Plain, where a 1% prbability fr a Reasenberg, P. A. and L. M. Jnes, Earthquake hazard after a mainshck characteristic earthquake wuld be fund fr a magnitude 3.6 in Califrnia, Science, 243, , candidate event, and the highly seismic San Grgni regin, Silverman, B., Density Estimatin, 175 pp., Chapman and Hall, Lndn, where it wuld take a magnitude 5.3 t reach this level. Aki, K., A prbabilistic synthesis f precursry phenmena, in Earthquake Predictin: An Internatinal Review, edited by P. Richards, pp , American Gephysical Unin, Washingtn D.C., Andersn, J. G., Revised estimates fr the prbabilities f earthquakes fllwing the bservatin f unreliable precursrs, Bull. Seisml. Sc. Amer., 72, , Bakun, W. H., K. S. Breckenridge, J. Bredeheft, R. O. Burfrd, W. L. Ellswrth, M. J. S. Jhnstn, L. Jnes, A. G. Lindh, C. Mrtensn, R. J. Mueller, C. M. Pley, E. Releffs, S. Schulz, P. Segall, and W. Sykes, L. and S. P. Nishenk, Prbabilities f ccurrence f large platerupturing earthquakes fr the San Andreas, San Jacint, and Imperial faults, Califrnia , J. Gephys. Res., 89, , Vere-Jnes, D., Earthquake predictin-a statistician's view, J. Phys. Earth, 26, , Wesnusky, S. G., C. H. Schlz, K. Shimazaki, and T. Matsuda, Earth- quake frequency distributin and faulting mechanics, J. Gephys. Res., 88, , Wrking Grup n Califrnia Earthquake Prbabilities, Prbabilities f large earthquakes ccurring in Califrnia n the San Andreas fault, U.S. Gel. Surv. Open-File Rep , D.C. Agnew, Institute f Gephysics and Planetary Physics, University f Califrnia, San Dieg, La Jlla, CA L. M. Jnes, U.S. Gelgical Survey, 525 S. Wilsn Avenue, Pasadena, CA 9116 (Received August 17, 199; revised January 14, 1991; accepted January 18, 1991.)