Earthquake Simulation by Restricted Random Walks

Transcription

1 1 Earthqake Simlation by Restricted Random Walks Steven N. Ward Institte of Geophysics and Planetary Physics University of California, Santa Crz Abstract. This article simlates earthqake slip distribtions as restricted random walks. Random walks offer several nifying insights into earthqake behaviors that physically-based simlations do not. With properly tailored variables, random walks generate observed power law rates of earthqake nmber verss earthqake magnitde (the Gtenberg-Richter relation). Criosly b-vale, the slope of this distribtion, not only fixes the ratio of small to law events bt it also dictates diverse earthqake scaling laws sch as mean slip verss falt length and moment verss mean slip. Moreover, b-vale determines the overall shape and roghness of earthqake rptres. For example, mean random walk qakes with b=-1/2 have elliptical slip distribtions characteristic of a niform stress drop on a crack. Random walk earthqake simlators, tned by comparison with field data, provide improved bases for statistical inference of earthqake behavior and hazard. 1. Introdction. Imagine hiking the trace of a new earthqake rptre and measring srface slip offsets at many points along the falt strike (see Figre 1, for examples). This mapping exercise may be akin to a random walk in the sense that the measred offset at position n gives scant hint as to whether the offset at position n+1 will be larger or smaller. Consider then, earthqake slip fnctions (n L) sampled at reglar intervals L as simlated in (n L) = 0 + χ i (0, σ i ) (1) n i=1 Figre 1. Measred dextral displacement along the srface falt trace of the Sakarya Segment, M7.4 Ismit, Trkey earthqake of Agst 17, 1999 (Barka et al., 2002) and the M7.1 Dzce, Trkey earthqake of November 12, 1999 (Akyz et al., 2002). Here, is some small slip vale and the χ(ν,σ) are nitless random variables with mean ν and variance σ 2. Random walk "rptres" (1) start with 0 = and rn a distance L=N L to where (N L) 0 Figre 2. Sketch of an earthqake rptre simlated by a restricted random walk. The rptre starts at l=0 with slip and rns down strike taking random flctations in slip ntil (L) first falls to zero or below. V(2.0) 2/12/04 For sbmission to Blletin Seismological Society of America

2 2 falling between and +. If the random variables χ n are well enogh behaved sch that the central limit theorem holds, then for large L P ( L) () wold be normally distribted with zero mean and variance σ 2 (L) P ( L) () = 2 erf ( 2σ(L) ) (2) Figre 3. Sketch of several nrestricted walks based on the same random variables as the restricted walk in Figre 2. P ( L) (), the probability distribtion of at L from many trials, is generally Gassian for long walks. (Figre 2). The rptre terminates there and the earthqake is finished. Random walks like these, that terminate at a barrier are labeled restricted or absorbing. The extent to which simlation (1) might represent earthqakes hinges pon the selection of the random variables. In a leap of faith, sppose that the synthetic rptres cold indeed be made to reprodce many earthqake behaviors. If so, then random walks wold, at the least, provide simple statistical bases for testing rptre hypotheses. Strong leapers might even imagine random walks leading to insights into earthqake behavior that physicsbased rptre simlations obscre in crtains of complexity. The erf(x) here is the integrated Gassian or error fnction (see Appendix A). The variance of nrestricted walk heights at L is defined by σ 2 (L) = 1 2 P (L) ()d (3) = L 1 L σ 2 (l) dl 0 where σ is the variance increment along the path from 0 to L. For independent χ n in (1) and σ 2 (n L < l < (n + 1) L) = 2 σ n 2 N=L/ σ 2 (L) = 2 2 σ (4) n n=1 Restricted Walks Reaching Length L. When the L 2. Walk Statistics. Unrestricted Walks Reaching Length L. Envision initially, nrestricted random walks with 0 =0 that rn length L as shown in Figre 3. Let P ( L) () be the probability of (L) Figre 4. Probability distribtion P ( L) (, ) of (L) from many restricted walks that reach L (eq 5). P ( L) (, ) is one-sided with positive mean.

3 3 walk is restricted to positive offsets only and 0 =, then the probability of (L) falling between and + becomes P (L) (, ) = = 2 P (L) ( ) P (L) ( + ) erf + ( 2σ(L) ) erf 2σ(L) ( ) (5) I sketch eqation (5) toward the right in Figre 4. Integrating P (L) (, ) over all vales, gives the probability that the restricted walk rns a distance L or greater P > (, L) = 1 P ( L)(, )d 0 (6) = erf σ(l) ( 2σ(L) ) 2 π The probability that the restricted walk actally ends between L and L+ L P(, L) = L P >(, L) L = 2 π 2 π 2 L σ 2 (L) e 2σ 2 (L) σ(l) L L σ(l) σ 2 (L) L L P > (,L) σ( L) σ(l) L (7) is jst the fraction that reach distance L times a positive nmber less than one. In the restricted walk, the expected vale and variance of the heights at L are E[(L)] = and erf ( 2σ(L) ) 1 Var[(L)] = σ 2 (L) π 2 σ(l) (8a) + 2 π σ(l)e 2σ 2 (L) E[(L)] E[(L)] 2 (8b) (2 π /2)σ 2 (L)=0.43σ 2 ( L) The approximate versions of (6-8) hold for long rptres where σ(l)>>. Restricted walks have a non-zero mean vale with a spread in offsets abot half as large as nrestricted walks of eqal length. Random walk theory (6) predicts an inverse relationship between the lengthsrvivability of earthqake rptres and the degree to which slip varies along strike. To my knowledge, physics-based models of earthqake rptre have not made sch a prediction. In particlar, random walk qakes with large step-to-step variations have less chance of making long rptres than do qakes with small step-to-step variations. Bear in mind, that the variation in earthqake slip along strike, σ 2 (L) is field-measrable, so means exist to test and tne the simlator. Slip variation forms a common thread throgh all predictions below. Restricted Walks Terminating Length L. Consider finally restricted random walks that terminate at L (Figre 5). These walks

4 4 M o = µwlu (12) Figre 5. Points of restricted walks that terminate at L. This class of random walks resembles earthqake rptres of length L. The predicted statistics of slip variation along strike in these rptres provide a key field test for application to real earthqakes. resemble earthqake rptres of length L and I label their offset along strike L (l). Becase the offset vanishes at both ends of the rptre, the variance σ L 2 (l < L) of L (l) increases like (4) initially, bt then tapers to zero at L σ 2 L (l) = σ 2 (l) [ σ 2 (L) σ2 (L ) σ 2 (l)] (9) From (8a), the expected vale of slip at distance l in sch rptres wold be E[ L (l)] π 2 σ L (l) (10) Ths, the mean slip of all earthqakes of length L depends on along-strike slip variability like U ( L) = π / 2 Lσ(L) L 0 [ ] 1/2 σ(l) σ 2 (L) σ 2 (l) dl (11) Seismologists have particlar interest in the seismic moment of earthqakes where µ is the elastic rigidity of the crst, U is the mean slip in the event, and W is the down dip (into the Earth) width of rptres in the region of interest. [W is assmed constant here.] From (11) and (12), the mean moment in all qakes terminating at L wold be M 0 ( L) = µwlu ( L) (13) Once or simlation specifies slip variability σ 2 (L), then M 0 ( L) can be inverted to provide L (M 0 ), the mean rptre length needed to generate a given seismic moment. Back sbstittion of L (M 0 ) into (6) then provides the probability that simlated earthqakes exceed a given moment P > (, M 0 ) 2 π σ(l (M 0 )) 3. Application to earthqakes. (14) Predicted Rate Distribtion. What does this theory really have to do with earthqakes? Sppose that I find a set of random variables χ n sch that standard deviation of slip along the nrestricted walk (4) grows like σ(l) (l / L) q (15) Evalating integral (11) then, provides the mean slip in qakes of length L

5 5 U ( L) = π π 8 2q = K (q)( L / L) q ( 1+ q 2q ) ( 1 + 4q 2q ) L L q (16) Placing (16) into (13) and inverting for the mean rptre length for qakes with moment M 0, I find L (M 0 ) = = L and (15) gives σ(l (M 0 )) M L 0 (µ W L)K(q) M 0 M 0 K(q) 1 q+1 M 0 M 0 K(q) 1 q+1 q q+1 (17) (18) with M 0 = µ W L. From (14) the probability that a qake grows to moment M o or greater is P > (M 0 ) = 2 [ π M 0K(q) ] q /(q+1) q /(q +1) M 0 (19) Finally, sing the moment-magnitde relation M w =(2/3)[log(M 0 )-9.05], I write (19) as log N > (M W ) = log N > (M W min ) +[ 3 2 q (1+ q) ]( M W M W min ) (20) Eqation (20) specifies a power-law distribtion in the rate of qakes of magnitde greater than M w. That is, random walk simlations (1) with variance (15) prodce earthqakes that obey the Gtenberg Richter relation with b = 3 q 2 (1+ q) or q = 2b 3 + 2b (21) Eqation (21), the first clear tie of random walks with earthqake behaviors, offers initial jstification for that leap in Section 1. Predicted Scaling Laws. From (16) and (13) other earthqake scaling laws too can be written in terms of b-vale U ( L) = M 0 ( L) = U ( M 0 ) = K(q) 2b L 3+2b = K(q)σ( L) L (22) M 0 K(q) K(q) L L 3+ 2b 3 M 0 M b 2b 3 (23) (24) Remarkably, b-vale not only dictates the ratio of large to small earthqakes, bt it also fixes the slope of many of their scaling laws. I know of no physics-based model of earthqake rptre that makes this link. In fact, random walk theory predicts that not only are slopes of the scaling laws interrelated bt so are the leading constants. (K(q) is given in Table 1.) Reprodction of observed scaling law slopes and their constants wold be a strong endorsement of random walk simlations.

6 6 q b U(L) U(M) M(L) K(q) 2-1 L 2 M 0 2/3 3/2-0.9 L 3/2 M /4 L 1 M 0 1/2 1/2-1/2 L 1/2 M 0 1/3 1/5-1/4 L 1/5 M 0 1/6 U peak ( L) U ( L) U field peak (L) U ( L) M 0 L M 0 L 5/ M 0 L M 0 L 3/ M 0 L 6/ Table 1. Earthqake Scaling Laws verss b-vale as predicted from random walk theory. For earthqakes in blk, a b-vale of -1 is nearly niversal. Random walk qakes with b=-1 scale like U L 2 ; M 0 L 3 and U M 0 2/ (25) There is no reason however, that the simlator need reprodce blk qake rates especially if it is intended to model events on a particlar falt. Earthqakes restricted to the vicinity of a particlar falt can have b-vales considerably greater than -1 becase isolated falts often prodce many similar-sized events. Table 1 shows that seismic sorces that vary in b-vale by only a few tenths generate earthqakes that scale qite differently from (25). A dependence of scaling laws on the nderlying b-vale of the earthqake sorce cold be a core confsion in qantifying earthqake behavior. Comparing rptre statistics from mixed poplations of different b-valed sorces wold create a wide variability in observed earthqake scaling. Predicted Slip Fnction Featres. From (9), (10) and (15) the mean shape of earthqake slip fnctions in rptres of length L is E[ L (l)] = 2U peak with l q L 2q U peak = π 8 [ L 2q l 2q ] 1/2 (26) L L q (27) Amazingly, random walk theory says that b- vale also controls the mean shape of earthqake slip fnctions. Figre 6 (left) plots (26) for b=-1/4 to b=-1. Note that the mean slip from b=-1/2 sorces Figre 6. (Left) Mean shape of earthqake slip fnctions verss b-vale of the sorce. (Center) 2-D Stress drops associated with the mean slip fnctions. (Right) Typical realizations of rptres with the given b-vale.

7 7 E[ L (l)] = 2U peak L [ l(l l) ] 1/2 (28) follows exactly the elliptical shape expected from a niform stress drop of τ = µu peak L (29) over a 2-D shear crack of length L (middle Figre 6). Eqation (29) is another indication that fndamental relationships exist between random walks and earthqake physics. As b- vale decreases to b=-1, the crack-like slip fnctions evolve to a dogtail/rainbow appearance (Ward, 1997). In these rptres, stress drops over the concave down part of the rptre (the rainbow) while stress increases over the concave p part (the dogtail). Both dogtail/rainbow and elliptical slip fnctions are observed freqently in the field. Note too that rptres from b<-1/2 sorces are intrinsically skewed in walks (15). Skewness of slip fnctions is a field measrable qantity and this prediction too shold be testable. From (22), the peak mean slip to mean slip ratio U peak (L) U (L) = π 8 K 1 (q) (30) also depends on b-vale bt it is independent of the length of rptre. Peak slip to mean slip ratio might be thoght of as an nrecognized earthqake scaling law. Random walk theory says that slip fnctions from low b-vale earthqake sorces shold be more "peaky" than those from higher b-vale sorces. As mentioned above, high b-vale sorces prodce a larger proportion of characteristic earthqakes than do low b-vale sorces. Redced slip variability for more characteristic sorces makes sense. Peak mean slip to mean slip ratio (30) goes from 1.3 to 2.1 as b vale decreases from -1/2 to -1. Eqation (30) however mst nderestimate field-measred peak slip to mean slip ratios. Smooth shapes (26) represent the slip averaged over all qakes of length L. Individal rptre events randomly pertrb these shapes (see far right Figre 6) and will certainly peak ot higher. From (8b) and (10) the variance in mean peak height is Var[U peak ] = (2 π /2) 2 2 π U 2 peak Likely field-measred peak slip might exceed the mean vale by 2 standard deviations field U peak (2 π /2) 2 U π peak = 1.68U peak so a better, metric of slip variability might be U field peak (L) U ( L) = 1.68 π 8 K 1 (q) (31) Eqation (31) goes from 2.2 to 3.5 as b vale increases from -1/2 to -1. Predictions of slip fnction "peakyness" as measred by the ratio of peak slip to mean are field-testable.

8 8 We have seen now that random walk theory makes many specific predictions abot the behaviors of earthqake rptres. The b- vale controls not only the scaling law constants bt it also affects the "look and feel" of the slip distribtion. I believe that correctly mimicking the look and feel of earthqake rptres is critically important both in the tning of the simlator and in assring its practical se. 4. Design of the Walks- Nmerical Reslts. Basic Walk. The analytical reslts above provide a good fondation for what to expect from random walk simlations. Let's now compte (1) to visalize slip fnctions and to verify the predictions. As a first step, I fix step distance along strike ( L=100 m), rigidity (µ= 3x10 10 Nm), and down-dip falt width (W=15,000 m). Secondly, I need to find a set of zero mean random variables χ n sch that the smmed variance along the walk (5) eqals (15). Many random variables have eqal variances and zero mean and all shold make qakes that satisfy the scaling relations (22-24). The choice may however, inflence the look and feel of the slip fnctions in sbtle ways. The simplest approach assmes that χ n are normally distribted with zero mean bt with a variance that increases with distance l=n L along strike as χ n = N(0, 2qn (2 q 1)/2 ) (32) For ncorrelated χ n, the variance along the nrestricted walk (4) is Figre 7. (Left) Featres of restricted random walk qakes with b=-1, -3/4, -1/2 and -1/4. The solid lines smmarize 1 million random walk qakes. The dashed lines are expected scaling laws (22-24). (Panel A) Nmber verss Magnitde The slope of these crves is the b-vale. (Panel B) Mean slip verss rptre length. (Panel C) Mean slip verss Moment. (Panel D) Mean Length verss Moment. (Right) Featres of restricted random walk qakes with b=-1. Inclined dashed lines in the bottom panels are the Slip verss Moment relation of Wells and Coppersmith (1994) and the Length verss Moment findings of Kagan (2002). Flat dashed line in Panel C graphs peak height/mean height ratio verss earthqake size.

9 9 σ 2 (l = n L) = 2q 2 n i 2q 1 i=1 2 (l / L) 2q (33) as needed by (15). The only free parameters remaining in the simlation are b-vale and slip increment. Figre 7 (left) smmarizes statistics of one million random walks with b= -1/4, -1/2, - 3/4 and -1. I adjsted slip increment in each case to prodce abot 1-meter of mean slip on falts of 80 km length. The dashed lines are the predicted scaling laws (22)-(24) and the expected power-law behaviors are all well matched. The right half of Figre 7 isolates the behaviors for b=-1 sorces. The inclined dashed lines in the bottom two panels graph the independently observed earthqake scaling relations of Wells and Coppersmith (1994) 2.3 and jst a bit less than expected from (31). Figre 8 plots a selection of the random walk rptres tablated in Figre 7. The qakes have magnitde 7 to 7.5 drawn from b= -1/2, -3/4 and -1 sorces. Yo can see that rptres from lower b-vale sorces (center and right) are progressively more peaky and skewed than those from b=-1/2 sorces. For comparison, the bottom row of Figre 8 plots the observed slip distribtions from Figre 1 scaled somewhat in horizontal and vertical dimensions. Althogh the logu (m) = 0.6logM 0 (Nm) (35) and Kagan (2002) log L(km) = 0.315log M 0 ( Nm) 4.27 (36) Sorces with b=-1 reprodce both of these relations in slope and intercept. The flat dashed line in Panel C (right) plots the peak height to mean height ratio of the simlated rptres, binned as a fnction of earthqake moment. As expected, the peak height to mean height ratio depends on b-vale bt is independent of earthqake size. Compted ratios for b= -1/4, -1/2, -3/4 and -1 are 2.0, 2.0, Figre 8. Earthqake rptres 7.0<M w <7.5 as simlated by a basic random walk (1) with (32) with b=-1/2, -3/4 and -1.

10 10 simlator has yet to be tned in any way, many of the synthetic slip fnctions cold be mistaken for real. Alternative Walk. As was said, many possible random variables satisfy (15). Any selection eqivalent to basic walk (32) however, prodces earthqake slip fnctions whose potential step to step variations track predictably at each step n. To model earthqakes, this featre may be too restrictive. Althogh the possibilities are still being explored, sppose instead, that the n in (32) is replaced by a random qantity with eqal expectation. That is, make the variance of the random variable a random variable itself. From (8a) and (15) simlations reveal that in the mean, the expected rptre shape is E[ L (l)] = 2U peak l q L 2q [ L l ] q (38) Alternative walks (37) satisfy the same scaling relations (20, 22-24) as basic walks (32) bt with a different K(q) K(q) = π 2 2 (1+ q) (39) 8 (2 + 2q) E[(l = n L)] π 2 σ(l) π 2 nq ths E and therefore 2 π (n L) 1/ q n χ n = N(0, 2q{ 2 π n 1 }(2 q 1 )/2q ) (37) shold be a candidate. Potential slip variation in the n-th step now scales with the offset at n- 1. For b<-1/2, step-to-step variations are large where the offset is large and visa versa. For b=-1/2 (i.e. q=1/2), (37) redces to (32). An explicit expression for the variance along the path (like eqation 15) is not available for the alternative walk; however, compter Figre 9. Earthqake rptres 7.0<M w <7.5 as simlated by alternative random walk (37) with b= -3/4 and -1 Note that the slip fnctions are more "peaky" (peak slip to mean slip ratios: 2.8 and 4.5 respectively) bt that they are nskewed in the mean.

11 11 More importantly, becase slip (hence slip variance) tapers to zero at both ends of the rptre, offsets (38) are nskewed for any b- vale. This may have advantage when fitting real data. Figre 9 shows a selection random walk earthqakes of magnitde 7 to 7.5 from sorces with b= -3/4 and -1 sing alternative (37). While these walks are slightly peakier than those in Figre 8, they look realistic. In tning simlations to fit real earthqake observations, blends of the basic and alternative walks might be most sitable. 5. Application: Paleoseismic Rptre Correlation Even if randomness is a stand in for nknown physics, restricted walk rptres seem capable of mimicking observed earthqake rates, scaling properties, and "look and feel". Granted this, what prpose can the simlations serve? A typical sitation in paleoseismic work is finding that: 1) Earthqake A broke throgh Site A D 1 ± 1 years ago; 2) that Earthqake B broke throgh Site B D 2 ± 2 years ago; and that 3) the age limits D 1 ± 1 and D 2 ± 2 overlap. The paleoseismic correlation problem amonts to deciding whether rptre events A and B are distinct or are one in the same. The decision often strongly inflences earthqake hazard estimates. In the past, the correlation decision largely hinged on the probability that the age dates were in fact eqal. Recently, Biasi and Weldon (2004) proposed a means to improve correlations by blending age overlap probabilities with rptre srvival probabilities. If event offset U can be measred at either Site 1 or 2 and the distance between them is known, then a properly tned random walk simlator can assist in the decision. The statistical simlator views earthqake correlation as a Gambler's Rin problem where slip plays the role of money and each km along strike plays the role of a dice toss. The simlator tells s the probability that the rptre will reach from site A to Site B before its slip is played ot. For instance, a paleo earthqake rptre is fond to have two meters of slip at Wrightwood California. What is the probability that slip rns to zero before the rptre covers the 30 km distance soth to Pallet Creek? Comptationally, rptre srvival calclations se the simlator parameters (, L, W, b or q, and χ n ) tned to the falt in qestion. The only difference being that 0 in (1) is set to the slip vale U>> observed at one of the paleoseismic sites. After many simlations, the fraction of rptres that reach distance L are tablated. Alternatively, if the variance accmlation in the walk σ(l) is specified, then eqation (6) gives the srvival probability analytically. For the basic walk, the rptre srvival crves are P > (U,L) = erf = erf U 2 (L/ L) q U ( 2σ(L) ) = erf K(q) U ( 2 U (L)) (40)

12 12 Figre 10. Rptre srvivability crves for the basic walk with b=-1 (right colmn Figre 8). Crves show the probability that a rptre with offset U (0.1m to 4m) at Site A reaches Site B given the distance L between them. Figre 10 plots rptre srvival probabilities for the basic walk with b=-1 for U=0.1, 0.5, 1, 2, and 4 meters. For instance, sppose the distance between the two paleoseismic sites is L=25 km, then a rptre with a 10 cm offset at Site A has only 20% chance of reaching Site Figre 11. Rptre srvivability crves for the alternate walk with b=-1 (right colmn Figre 9). Srvival probability is a bit less for these more peaky rptres than those of Figre 10. B. A rptre with a 1/2 m offset however has a 85% chance of breaking both sites. The last of (40) interprets rptre srvival in terms of the mean slip U ( L) in qakes of length L. If the offset U at Site A is large compared to U ( L), then the rptre will most likely reach the distance L to Site B. No need to be a rocket scientist to nderstand this conclsion. Figre 11 plots rptre srvival crves for the alternate walk with b=-1. For L=25 km, a rptre with a 10 (50) cm offset at Site A has only 10% (35%) chance of reaching Site B. The added peakiness in the alternative walk redces srvival probability by abot half. 6. Conclsions Many seismologists are working to constrct physically-based earthqake simlators. Physical simlators hold considerable appeal, however they may become so complex that mch of the modeling effort is expended in finding a physical basis for essentially random behavior. If certain aspects of earthqake behaviors are random to the extent that real data can constrain them, then a more practical approach may be to embrace the randomness whatever its physical origin. This article begins to develop and to apply random walk rptre simlations to earthqake isses and to assemble a catalog of observed earthqake slip fnctions with which to test the simlator. Statistical simlators like these, are intended to complement physical simlators in those applications where they may be better sited. Once calibrated against observed slip fnctions, statistical simlators

13 13 can serve as a scientific tool to aid in the nderstanding of other characteristics of earthqakes that are not easily, or not yet, measred. Restricted random walk (1) with standard deviation (6) generates earthqake sets that reprodce observed power law rates of earthqake nmber verss magnitde as well as diverse earthqake scaling laws sch as mean slip verss falt length and moment verss mean slip. By tying together earthqake rates, earthqake scaling laws, and earthqake slip shape and variation throgh b-vale, hmble random walks hint at a nified theory of earthqake behavior. rptre length, rptre width, rptre area, and srface displacement, Bll. Seism. Soc. Am., 84, Appendices A. Error Fnction The error fnction erf(x) smoothly varies from 0 at x=0 to 1 at x= (Figre A1). Also, erf(1)=0.842, erf(2)=0.995, and erf(-x)=erf(x). For small x,.erf( x) ~2x/π. References. Akyz, H. S. et. al Srface rptre and slip distribtion of the 12 November 1999 Dzce Earthqake (M7.1), North Anatolian Falt, Bol, Trkey, Bll. Seism. Soc. Am., 92, Barka, A., et. al., Srface rptre and slip distribtion of the 17 Agst 1999, Ismit Earthqake (M7.4) North Anatolian Falt, Bll. Seism. Soc. Am., 92, Biasi G, and R. Weldon R., Estimating srface rptre length and magnitde of paleoearthqakes from point observations of rptre displacement, (preprint, to be sbmitted). Kagan, Y. Y Bll. Seism. Soc. Am., 92, Ward, S. N., Dogtails verss Rainbows: Synthetic earthqake rptre models as an aid in interpreting geological data, Bll. Seism. Soc. Am., 87, Wells, D.L. and K. J. Coppersmith, New empirical relationships among magnitde, Figre A1. The error fnction. B. Slip variability and rptre srvivability Random walk (1) with 0 = can be written as a sm or a prodct of terms n = + χ i (0, σ i ) n i=1 n = 1 + i-1 χ i (0,σ i ) i=1 n = 1 + χ i (0, σ i ) i=1 i-1 n = χ i (1, σ i ) i-1 i=1 The walk terminates at the i-th step when (A1) (A2)

14 14 χ i (0, σ i ) < 1 i-1 If χ is normally distribted, the probability of termination is 1 N (x : 0, σ i )dx i-1 = 1 (A3) 2 1 erf i-1 2σ i For a given offset n-1, the larger the step to step variability σ i, the higher the chance of termination. This association gives rise to the inverse relation between slip variability and rptre srvivability characterized by b-vale.