Spatial aftershock distribution: Effect of normal stress

Transcription

1 JOURNAL OF GEOPHYSCAL RESEARCH, VOL. 103, NO. B10, PAGES 24,453-24,467, OCTOBER 10, 1998 Spatial aftershck distributin: Effect f nrmal stress Yan Y. Kagan and David D. Jacksn Department f Earth and Space Sciences, University f Califrnia, Ls Angeles Abstract. We study the spatial clustering f shallw aftershck hypcenters with respect t fcal mechanisms f mainshcks. We use the Harvard centrid mment tensr (CMT) glbal catalg, the Preliminary Determinatin f Epicenters (PDE) earthquake list, the Califrnia nstitute f Technlgy/U.S. Gelgical Survey catalg f earthquakes in suthern Califrnia, and a catalg f fcal mechanisms fr all earthquakes since 1850 in suthern Califrnia with magnitude larger than 6. We need t accunt fr pssible systematic bias in hypcenter distributin due t the gemetry f seismgenic znes, especially that f subductin znes. We als select nly strike-slip earthquakes frm the catalgs t investigate aftershck clustering in circumstances mre favrable fr direct bservatin. We cmpare the spatial distributin f hypcenters befre each strng earthquake with the distributin during the first 250 days after the earthquake and fr the time interval extending beynd 250 days. f the frictin cefficient in the Culmb criterin is psitive ne expects that after a strng earthquake, aftershcks and ther earthquakes wuld cncentrate in the directin f the P axis (dilatatinal quadrant) rather than in the directin f the T axis (cmpressin quadrant). Such crrelatins have been pinted ut previusly fr selected earthquakes sequences, but is such crrelatin a general feature f earthquake ccurrence? We study spatial earthquake distributins befre and after each event fr several chices f fcal sphere partitin, cutff magnitude, fcal mechanisms f large events, time perids, distance frm a mainshck, etc. Althugh sme earthquake distributins agree with a nnzer frictin cefficient, thers prduce the ppsite pattern, suggesting that the cncentratin f events alng the P and T axes is due t randm effects. This result implies that the frictin cefficient in the Culmb law is clse t zer. 1. ntrductin frictin cefficient fr rck material in situ is similar t that btained in labratry tests. Recent wrk has shwn several instances f shallw The nrmal stress increases as a result f a strng aftershcks and small earthquakes ccurring preferenearthquake in the directin f the P axis (see mre tially in regins f high stress increments left by earlier belw in sectin 3.2). f the frictin cefficient in the earthquakes [Das and $chlz, 1981; Stein and Liswski, Culmb criterin is psitive, ne expects that after a 1983; Harris and Simpsn, 1992; King et al., 1994; Kastrng earthquake, aftershcks and ther earthquakes gan, 1994; Nalbant et al., 1996; Hdgkinsn et al., 1996; wuld cncentrate in the directin f the dilatatinal Kilb et al., 1997; Deng and Sykes, 1997; Stein et al., quadrant rather than in that f the cmpressin quad- 1997]. Althugh the influence f the stress caused by rant. n ther wrds, the spatial aftershck distributin strng earthquakes n subsequent shcks is strng and wuld be asymmetrical with regard t fcal mechanism bvius, less is knwn abut the details f the spatial f a mainshck. Such a crrelatin fr selected earthdistributin f aftershcks. Sme previus wrk sugquake sequences has been suggested previusly [Das and gests that aftershcks are cncentrated in the dilata- Schlz, 1981; Stein and Liswski, 1983; King et al., rihal quadrant (crrespnding t the directin f the 1994; Stein et al., 1997]; hwever, whether such cr- P axis) rather than in that f the cmpressin quadrant relatin is a general feature f earthquake ccurrence is (crrespnding t the directin f the T axis). This be- nt established. havir is predicted by the standard interpretatin f the There are tw appraches t investigating the Mter- Culmb criterin, if we assume that the value f the shck spatial distributin. The first is t study particular cases f aftershck sequences. T test whether the Cpyright 1998 by the American Gephysical Unin. frictin cefficient is nnzer, ne can either shw that Paper number 98JB the earthquake distributin in dilatatinal quadrants / 98 / 98 JB f a mainshck is different frm that f cmpressinal 24,453

2 24,454 KAGAN AND JACKSON: SPATAL AFTERSHOCK DSTRBUTON quadrants, r calculate stress changes (bth static due t a mainshck and a tectnic stress) in the neighbrhd f the mainshck and cmpare the numbers f aftershcks in the regins f high and lw Culmb stress change. The secnd apprach is t investigate the avibility is impssible t evaluate. Mrever, because f the reprting bias mentined abve, it is nt clear that published case histries represent an bjective sample. n the stacking methd, it is pssible in principle t apply apprpriate statistical tests. Hwever, there is a erage distributin f earthquakes in bth cmpressinal prblem here as well: the prperties f aftershck seand dilatatinal quadrants by stacking distributins fr several earthquakes and using frmal criteria fr earthquake selectin. Frhlich and Willemann [1987] and Michael [1989] reprt similar calculatins. They studied the crrelatin f aftershck hypcenters with respect t fcal mechanisms f mainshcks. We emply this methd in the present paper. Case studies have the advantage f accunting fr quences in three dimensins are nt yet well knwn; thus we have difficulty frmulating the apprpriate null hypthesis [Kagan and Jacksn, 1996]. One natural null hypthesis wuld be t assume that aftershcks and ther small earthquakes arund a main event are distributed symmetrically with regard t its fcal mechanism; that is, there is n difference in the numbers f earthquakes in the directins f the P and T axes. tectnic and ther details f each aftershck sequence. Then we can test whether the difference between the Hwever, it is difficult t avid selectin bias due t particular selectin criteria r a particular chice f the adearthquake numbers in crrespnding directins is due t randm factrs. Such tests wuld be easy t perjustable parameters and assumptins. Mrever, there frm if the statistical distributin fr the numbers culd may exist a reprting bias; that is, results exhibiting a certain pattern are mre likely t be published than thers. Mst f the publicatins listed abve discuss aftershck patterns fr strike-slip earthquakes, since it be described by simple, knwn laws. f each earthquake wuld ccur independently, the Pissn r binmial distributin culd be used fr testing. Unfrtunately, earthquake ccurrence, especially in aftershck is easier t visualize these distributins. Each case f sequences, is highly clustered in space and time. Alaftershck sequences requires a new adjustment f varius cnditins; therefre the ttal number f degrees f freedm in such studies is high and cmparable with the effective number f data pints. Many investigatins f particular aftershck sethugh sme research has been dne in describing and mdelling earthquake tempral clustering [Ogata, 1988; Kagan, 1991; Utsu and Ogata, 1977], the results f these investigatins are nt easily applicable t ur prblem. Mrever, all earthquakes, including freshck, mainquences have been carried ut, especially in recent years shck, and aftershck sequences, are dependent n the [cf. King et al., 1994; Nalbant et al., 1996; Stein et al., same reginal stress pattern. On the ne hand, re- 1997]. They shw that aftershcks tend t cncentrate ginal stress causes earthquakes, but n the ther hand, in areas f inferred high stress levels. Mst investiga- earthquakes release and redistribute stress [Harris and trs nte that results are nt strngly dependent n the Simpsn, 1992; Harris et al., 1995; Kagan, 1994]. value f the frictin cefficient ( ) in the Culmb law This self-rganizatin f stress and earthquakes may intrduce cmplex statistical interrelatinships between (see equatin (1)), implying that the nrmal stress has little, if any, influence n the aftershck distributin seemingly independent events, making statistical test- [King et al., 1994; Kilb et al., 1997]. Hwever, n sys- ing even mre difficult. tematic study has been dne t determine if the value Fr example, earthquake catalgs are ften declus- - 0 is cnsistent with the distributin f aftershcks tered t diminish the influence f aftershck sequences. and ther earthquakes in the fcal znes f large events. n ur case, this wuld delete the earthquakes whse n the secnd (stacking) methd, we cnsciusly prperties are being investigated. Anther frequently avid adjusting cnditins fr each aftershck sequence. used methd f validatin applies a randmized catalg The sequences are nt identified in. a catalg, and every as a cntrl test [Kagan and Jacksn, 1996]. Here we earthquakevent is selected accrding t a frmal rule achieve a similar result by cmparing the premainshck distributin with that f aftershcks. (a cmputer algrithm). Thus several members f the same aftershck sequence may be stacked tgether. Al- Hwever, if we cmpare the results frm different catthugh individual sequences might be separated using algs, their cnsistency can be interpreted as psitive a prescribed algrithm, such subdivisin culd intr- evidence. Each earthquake catalg may have specific duce a bias (the selectin rule) int the interpretatin systematic errrs. Hwever, we analyze all earthquakes f earthquake data. t is nt clear whether the results in a catalg; that is, there is n specific preselectin. n f such an analysis wuld be applicable t sequences such a case, the repeatability f results means that the identified using a different rule. Hw d we establish the validity f btained distribserved regularities are due t the prperties f seismicity, nt t mdel assumptins r methds f data butin prperties? There is n bvius methd fr case prcessing. studies, except lking fr internal cnsistency f the results. Hwever, cnsistency may be illusry since cnditins have been adjusted and mdified in each case. 2. Data As an example f earthquake data, Figure displays Each individual result may be s "sft" that its cred- fcal mechanisms fr the earthquakes in suthern Cal-

3 KAGAN AND JACKSON: SPATAL AFTERSHOCK DSTRBUTON 24, Santa R%a -- ' Santa { Cruz Saltn - ' Catalina O0 km an Dieg Clemente Lngitude Figure 1. Fcal mechanisms f earthquakes frm the mdified Ellswrth [1990] list in suthern Califrnia area and majr surface faults. Lwer hemisphere diagrams f fcal spheres are shwn. Symbl size is prprtinal t earthquake magnitude. The striped regins crrespnd t cmpressin quadrants. Mre densely striped "beach-balls" crrespnd t pint surces; extended surces are shwn by a series f less striped symbls. Stripes are cncentrated tward the earthquake fault plane prjectin t indicate the presumed fault plane. ifrnia frm 1932 t 1996 [see Jacksn et al., 1995]. The catalg cvers øN, øW. We use the Ellswrth [1990] histrical/instrumental earthquake catalg with mment magnitude Mw >_ 6.0 as ur initial data set. We have added the fcal mecha- nism slutins f recent earthquakes frm the Harvard catalg [Dziewnski et al., 1998] and frm ther available publicatins. The study has used tw variants f earthquake fault representatin: a pint surce and extended surce mdels. n the latter mdel, we represent each earthquake with Mw _ 6.5 as a finite surce, apprximating the fault area by several pint surces distributed n a rupture plane. Fr finite earthquake surces, we need t reslve the fault plane ambiguity f fcal mechanisms; we d this by assigning the fault plane based n the area tectnics r distributin f aftershcks. There are 206 surces in the extended surce variant f the catalg, and 84 earthquakes in the single- pint versin. Here we call this list the Ellswrth catalg. The suthern Califrnia (Califrnia nstitute f Technlgy/U.S. Gelgical Survey, CT/USGS) earthquake catalg [Hileman et al., 1973; Huttn and Jnes, 1993, and references therein] spans 65 years ( ) f lcal seismicity. t is essentially cmplete fr lcal magnitude M _ 3.0 and cntains 19,781 events. Figure 2 displays all f the 28 strike-slip earthquakes frm the Ellswrth catalg and M _ 3.5 events frm the CT/USGS catalg during We cnsider any earthquake which is rtated (in three dimensins), n mre than 20 ø frm a pure strike-slip fcal mechanism, as a strike-slip event. We use tw glbal catalgs: (1) the catalg f mment tensr inversins cmpiled by the Harvard grup [Dziewnski et al., 1998] which cntains 14,039 slutins frm January 1, 1977 t February 28, Of these,

4 24,456 KAGAN AND JACKSON: SPATAL AFTERSHOCK DSTRBUTON 37 q øø c: øø ø O ø 36 % 35 O ) ø c ( C e ø ø c e % ø ø 8 O Sant a O O Cruz ø e i ø c9 an Dieg Clement km 0% ø Lngitude %0 O -114 Figure 2. Fcal mechanisms f strike-slip earthquakes in suthern Califrnia frm the mdified Ellswrth [1990] catalg and epicenters f -/kfl_ 3.5 events during Symbl size is prprtinal t earthquake magnitude. The striped regins crrespnd t cmpressin quadrants. Stripes in "beach-balls" are cncentrated tward the earthquake fault plane prjectin t indicate the presumed fault plane earthquakes are shallw with a mment magnitude Mw _ 5.8. (2) The Preliminary Determinatin f Epicenters (PDE) wrldwide catalg f earthquakes [U.S. fcal sphere f a mainshck. n ur calculatins, we cnsider earthquakes in the Ellswrth and Harvard catalgs as primary events (mainshcks), whereas earth- Gelgical Survey, 1996] which ends n March 20, quakes in the CT/USGS and the PDE catalg serve T ensure catalg unifrmity, all earthquakes with bdy wave magnitude mb f less than 5.0 have been remved. The catalg cntains 22,568 shallw earthquakes with as secndary events (ften called aftershcks). The lcatin f earthquakes in the mdified Ellswrth and the CT/USGS catalgs is self-cnsistent, that is, magnitude mb _ 5 frm 1977 t T investigate crdinates f centrid psitins and fault planes fr exthe influence f magnitude cutff n ur results, we repeated the calculatins using the PDE catalg with mb _ 4.5. As we see frm sectin 3.3, the results with a different magnitude cutff are similar. tended earthquakes in the frmer catalg are reviewed and revised t agree with the CT/USGS data. Owing t a very large vlume f data, n such adjustment is pssible fr the Harvard versus PDE catalgs. Bth 3. Aftershck Number Distributin catalgs may have different systematic errrs in lcatin f centrids and hypcenters; thus the distributins f 3.1. Fcal Sphere Partitin This paper studies a pssible asymmetry f afteraftershcks arund each fcal mechanism may be biased by these errrs. Hwever, since we average thusands f events ver the entire Earth, systematic effects shuld shck spatial distributin, in dilatatinal and cmpres- be smthed ver as well, s we expect that they wuld sinal quadrants f a mainshck fcal sphere. T reduce the data, we stack the directins t aftershck hypcenters frm the psitin f the main event n a cause nly a general smthing f patterns, nt any systematic change in symmetry f the aftershck distributin. The Harvard earthquakes are als used bth

5 KAGAN AND JACKSON: SPATAL AFTERSHOCK DSTRBUTON 24,457 as primary and secndary events: that is, we stack the distributins f all the earthquake hypcenters in the system f crdinates frmed by each fcal mechanism in turn. Fr the fcal mechanisms btained n the basis f far-field elastic wave inversin, the plane slutins are ambiguus: the fault plane and the auxiliary plane cannt be distinguished. As we discussed earlier [Kagan, 1992, p. 308; Kagan and Jacksn, 1994], the gemetry f subductin znes makes it statistically mre prbable that the fault plane passes thrugh the N and T-P vectrs. Owing t the fcal mechanism symmetry [Kagan, 1992], nly tw ctants crrespnding t P and T axes need be cmpared in the infinite elastic space. n an elastic half-space, the stress distributin symmetry is preserved t a large degree fr a strike-slip earthquake. Fr earthquakes having a general rientatin, there is n symmetry in a half-space. Hwever, studying full spherical distributin f hypcenters is impssible because f insufficient data. Hence we illustrate stress nly frm strike slip earthquakes, and cmpare the hypcenter distributins in nly tw ctants. N axis N axis Each ctant f the fcal sphere is subdivided int 55 spherical triangles and quadrilaterals with equal area (cnsequently cvering equal slid angles). Rakhmanv et al. [1994] discuss partitining a sphere int equal-area cells, having a minimum diameter: that is, quadrilaterals which are as clse t spherical squares as pssible. Mrever, these cells are easy t represent and display, since their bundaries are circles f cnstant spherical latitude and lngitude. $aff and Kuijlaars [1997, Figure 3] give an example f the partitin. Everett [1997] discusses varius partitins (tesselatins) f a sphere in tw and three dimensins emplyed in gesciences. Hwever, as described abve, the fcal mechanism sphere has a certain symmetry [Kagan, 1992]; thus we cmpare the distributins in neighbring ctants. Under such restrictins, the partitin algrithm T ajxls! prpsed by Rakhmanv et al. [1994] yields a cell pattern similar t that used by Kagan [1994]. Figure 3. Partitin f a sphere ctant. Lambert equal- Figure 3a shws the distributin f partitin cells in area prjectin is used t plt cells. Whereas the area ne ctant. Since fr sme earthquake catalgs, the is preserved under this prjectin, distances and the number f aftershcks is t small fr a display in Fig- shape f the cells are defrmed. (a) Equal-area regular partitin. (b) Partitin int semilunes. ure 3a, we als use a mre carse partitin f a sphere ctant. n particular, we add all the first cells adjining the fault plane in ne f such partitins (crrespnd Stress Distributin fr a Test Earthquake ing t all right-hand cells in Figure 3a) and then add all secnd cells, etc. Thus the last new cell cnsists f A cmmnly accepted mdel suggests that the nly ne cell n the extreme left part f Figure 3a. Fig- Culmb failure stress change (rs[schlz, 1990] cntrls ure 3b demnstrates anther ctant partitin int 10 earthquake ccurrence: semilunes. (r s - 7' + P(rn, (1) Figure 4 displays the fcal plane f the 1992 Jshua Tree, Califrnia, earthquake and epicenter psitins fr where r is the incremental shear stress n a fault plane, ML _ 3.5 earthquakes during the first 250 days after p is a static (psitive) cefficient f frictin, and (r. is that event. Prjectins f the semilunes frm Figure 3b a nrmal stress change (psitive(r. crrespnds t relare als shwn, demnstrating the cunting algrithm. ative extensin). Labratry experiments [Segall, 1991]

6 24,458 KAGAN AND JACKSON: SPATAL AFTERSHOCK DSTRBUTON Santa Cruz : Sa Catalina 33 San Dieg km Lngitude Figure 4. Epicenters f ML _ 3.5 events 250 days after the Jshua Tree, Califrnia, earthquake f Prjectin f the lune partitin f the fcal sphere (see Figure 3b) is als shwn yield the value f/ fr mst rck materials. Since it is pssible that fluid pressure reduces the nrmal stress n a fault, the u value shuld be cnsidered as an effective (apparent) frictin cefficient [Deng and Sykes, 1997]. Therefre, accrding t equatin (1), an earthquake wuld be mre likely t ccur when v--f Ueyn_ 0. This equatin assumes that we knw the stress and the directin f the fault plane f an earthquake. Hwever, in mst cases, especially fr small earthquakes, the fault plane directin is unknwn. Thus we als calculate the Culmb failure stress fr average values f the stress f = J2 - lrl, (2) where z is the first stress invariant crrespnding t the average istrpic stress and J2 is the secnd invariant f the deviatric stress crrespnding t average deviatric (shear) stress [Jaeger and Ck, 1979; Kagan, 1994]. f the frictin cefficient in the Culmb criterin is nnzer, ne shuld expect [cf. King et al., 1994, Figure 2a] that after a strng earthquake, aftershcks and ther earthquakes wuld cncentrate in the directin f the P axis (dilatatinal quadrant) rather than in that f the T axis (cmpressin quadrant). n Plate 1 we display the invariants fr stress change due t a test earthquake M The stress calculatins fr an istrpic half-space are perfrmed, using Okada's [1992] apprach. Fr cmparisn we superimpse the suthern Califrnia map ver the stress pattern. The first invariant exhibits a familiar quadruplar pattern [Chinnery, 1963, Figure 3]. We assume the value f the frictin cefficient is u We cannt determine Therefre the abslute stress level in situ. we are t infer the stress value frm the num- ber f usually small earthquakes registered in a neighbrhd f larger events. T prject the stress changes n a fcal sphere f the reference events, we assume that the number f these secndary earthquakes are prprtinal t the incremental stress. There is a prblem in cmparisn f the Culmb stress level with the number f secndary events, since the event number is a nnnegative quantity, whereas stress is a tensr the cmpnents f which culd be bth negative and psitive. T make the cmparisn pssible, we subtract the minimum value f the stress frm the stress level. The btained nnnegative stress values in a distance interval km frm the test earthquake are then prjected nt tw sphere ctants crrespnding t the P and

7 KAGAN AND JACKSON: SPATAL AFTERSHOCK DSTRBUTON 24,459 E i epmg rl

8 24,460 KAGAN AND JACKSON: SPATAL AFTERSHOCK DSTRBUTON a a Nrmalized Earthquake Numbers fr M7 test Nrmalized Earthquake Numbers fr M7 test b Nrmalized Earthquake Numbers fr M7 test Nrmalized Earthquake Numbers fr M7 test Nrmalized Earthquake Numbers fr M7 test Plate 2. Distributin n a sphere quadrant f stress level fr a test earthquake. (a) Stress cmpnent ern (nrmal stress). (b) Stress cmpnent - (shear stress). (c) Culmb stress Nrmalized Earthquake Numbers fr M7 test Plate 3. Distributin n a sphere quadrant f stress invariants fr a test earthquake. (a) Stress invariant 1 (average nrmal stress). (b) Stress invariant J2 (average shear stress). (c) Culmb average stress. T axes. We display the distributins fr tw relevant stress cmpnents and the Culmb stress (equatin 1) in Plate 2. Plate 3 shws similar distributins fr stress invariants and the resulting Culmb stress (equatin 2). Again, we take the frictin cefficient u Bth plts (Plates 2 and 3) shw that if the distributin f aftershcks fllws the Culmb stress criterin (equatins r 2), there shuld be mre secndary events cncentrated in the directin f the P and nt the T axis.

9 KAGAN AND JACKSON: SPATAL AFTERSHOCK DSTRBUTON 24,461 Table la. Shallw Earthquakes in PDE Catalg Occurring Befre an Earthquake in Harvard Catalg Numbers f Eart es in Cells f Figure 3a N Axis Axes T + P P T- P T T + P Table lb. Earthquakes in PDE Catalg Occurring Days After Earthquake in Harvard Catalg With Nrmalized Values f Table 1 a Subtracted Difference in Nrmalized Numbers f Farthquakes in Cells f Figure 3a N Axis Sum Axes T + P P T- P T T + P Table lc. Earthquakes in PDE Catalg Occurring >250 Days After Earthquake in Harvard Catalg With Nrmalized Values f Table 1 a Subtracted Difference in Nrmalized Numbers f Eart es in Cells f Figure 3a N Axis Sum Axes T + P P T- P T T + P 3.3. Number Distributin fr Earthquakes T establish the symmetry r asymmetry f earthquake distributins, we calculate the number f times an aftershck hypcenter is prjected int the cells, as shwn in Figure 3. These cells subdivide the fcal sphere f a mainshck r a reference event. We need t take int accunt a pssible systematic bias in hypcenter distributin due t the gemetry f seismgenic znes, especially t that f sub-

10 24,462 KAGAN AND JACKSON: SPATAL AFTERSHOCK DSTRBUTON ductin znes. The subductin znes accunt fr mst with that during the first 250 days after an earthquake earthquakes in glbal catalgs [cf. Grss and Kisslinger, (Table lb) and the interval extending beynd 250 days 1994]. T crrect fr the pssible influence f cmplex (Table lc). T facilitate the cmparisn, we subtracted fault gemetry, we usually cmpare tw distributins: the nrmalized preearthquake numbers f PDE events the ne preceding a mainshck (preshcks) and the distributin f earthquakes fllwing a mainshck. We as- (Table la) frm that f pstearthquake values. The subtractin results are displayed in Tables lb and lc, sume that the influence f the half-space bundary as where we als shw hrizntal and vertical sums f the well as ther gelgic structures wuld be cancelled in numbers. f the frictin cefficient in the Culmb crisuch a cmparisn. We subdivide earthquakes after a mainshck int tw grups: thse in the first 250 days terin is nnzer, ne shuld expect that aftershcks and ther earthquakes wuld cncentrate mre in the (labelled belw as the "first" r "early" aftershcks)and directin f the P axis (dilatatinal quadrant) than in thse in the interval extending beynd 250 days ("late" aftershcks). We select secndary events accrding t their distance frm a mainshck. n ne mde, we accunt fr all the earthquakes within 100 km radius f a mainshck. n the secnd mde, the distance t a mainshck epithat f the T axis (cmpressin quadrant). Hwever, neither. Tables lb nr lc exhibits the hypcenter migratin in cmparisn with Table la. The difference between preearthquake (Table la) and pstearthquake distributins displayed in Tables lb and lc seems t have an equal chance f being psitive r negative in center is scaled accrding t the empirical frmula: the directin f bth axes. Ri - ri10 (M- )/2, (3) where ri are lwer and upper distance limits fr a magnitude 6 earthquake, taken t be 5 and 10 km in this study. We exclude the spatially clse aftershcks frm the calculatins; these aftershck psitins may be influenced by unknwn randm fluctuatins f earthquake fault gemetry. We list belw varius parameters and data which have been used in the investigatins: (1) catalgs are referred t as Ellswrth-CT/USGS, Harvard-Harvard, Harvard-PDE; (2) fcal mechanisms are strike-slip events nly, r all earthquakes; (3) earthquake surce representatin (nly fr the Ellswrth catalg) is pint surce, r extended surce; (4) distance range is km fr all events, r 5/10 km scaled (see equatin 3); and (5) distributin is n a sphere in 2-D (Figure 3a), r n sphericalunes in 1-D (Figure 3b). Table shws the distributin f the numbers f hypcenters in a crdinate system frmed by the T, N, and P axes f an earthquake fcal mechanism (the TNP crdinate system, see Figure 3). We use shallw strike-slip events Mw _ 5.8 frm the Harvard catalg as generating earthquakes (mainshcks) and shcks mb _ 5.0 frm the PDE catalg as test (ptential aftershcks) events. As mentined in sectin 2, a strike-slip event is taken as an earthquake fcal mechanism rtated n mre than 20 ø frm a pure strike-slip. The distance between hypcenters is less than 100 km. The actual numbers f earthquakes in the PDE catalg preceding the mainshcks in the Harvard catalg are displayed in Table la. T make these distributins mre amenable t cmparisn, we nrmalize the numbers, s that the ttal number in all cells in a table like Table la equals 11,000. Therefre, if the distributin f hypcenters in the TNP system were unifrmly randm, all numbers in Table la wuld be apprximately equal t 100. We cmpare the spatial distributin f the hypcen- ters befre (Table la) a strng earthquake (Mw _ 5.8) Plate 4 displays the earthquake number distributins fr strike-slip earthquakes in the Ellswrth catalg. Hwever, in cntrast t Figure 2, we use ex- tended surce representatin fr strike-slip earthquakes in Plate 4. The distributins in Plate 4a and Plate 4c are largely symmetric with regard t the fault plane (the middle line in the plts), whereas the distributin f earthquake numbers fr a time delay f less than 250 days exhibits a "htspt"(maximum) in the apprximate directin f the T axis. This htspt can be traced t ne earthquake sequence, December 4, 1948, ML = 6.0 Desert Ht Springs (Thusand Palms, Califrnia) event [Hileman et al., 1973]. The aftershck sequence f its large subsequent earthquake (May 2, 1949, ML = 5.8) accunts fr this cncentratin f earthquakes. This example illustrates the advantages and the drawbacks f ur apprach. f we analyze each aftershck sequence separately, such "accidental" distributin peaks can easily be excluded. Hwever, hw can ne specify the selectin criteria f earthquake se- quences? t wuld be difficult t avid a selectin bias. Mrever, since there are nly 28 ML _ 6 strike-slip earthquakes in ur sample, the statistical fluctuatins are very large. n ther examples belw, the numbers f earthquakes are significantly higher; as a result such ht spts are less prminent. Plates 4d and 4e display the difference f the nr- malized numbers between early aftershcks (time delay 250 days) and the distributin f earthquakes be- fre a mainshck and a similar difference fr later sec- ndary earthquakes (delay > 250 days).. Bth pictures are rughly symmetric with regard t the fault plane prjectin; the early aftershcks shw a pattern ppsite t that expected frm the Culmb frmula. As we explained earlier, this pattern is the result f the influence f ne earthquake sequence. Figure 5 displays tw versins f 1-D earthquake number distributin: accrding t Figure 3a summed diagnally, as described earlier, and accrding t Figure 3b, that is, in 9 ø lunes. The negative angle in the

11 KAGAN AND JACKSON: SPATAL AFTERSHOCK DSTRBUTON 24,463 Table 2. Asyrnmetry in Distributin f Earthquake Numbers fr Varius Catalgs Cmputatin Mde Catalgs Dist- Orient- Repres- Preshcks < 250 > 250 ance atin entatin days days Ellswrth--CT un ss ext ' Ellswrth--CT var ss ext ' Ellswrth--CT un gs ext * Ellswrth--CT vat gs ext 0.001' Ellswrth--CT un ss pt ' Ellswrth--CT war gs pt ' Harvard--Harvard un ss pt ' Harvard--Harvard un gs pt ' Harvard--Harvard war gs pt * Harvard--PDE5.0 un ss pt ' Harvard--PDE5.0 un gs pt * Harvard--PDE5.0 war gs pt * Harvard--PDE4.5 un ss pt ' Harvard--PDE4.5 un gs pt ' Harvard--PDE4.5 war gs pt * Distance limits are un, unifrm km fr all events; and war, distance limits variable, dependent n magnitude (equatin 3). Surce rientatins are ss, strike-slip surce; and gs, surce f arbitrary rientatin. Surce representatins are pt, pint surce; ext, extended surce. PDE5.0, PDE catalg, mb _> 5.0. PDE4.5 - PDE catalg, mb _> 4.5. * Maximum values. abscissa axis crrespnds t a sectr assciated with the P axis. Mre aftershcks in this part f the diagrams presumably crrespnd t the nnzer value f /. There is little difference between these plts; thus we use nly the semilune partitin (Figure 3b) in the later diagrams. Figure 6 shws the differences in the nrmalized numbers between shrt-term aftershcks and earthquakes befre a mainshck. A similar difference fr lng-term secndary earthquakes is als displayed. Only a few f the btained diagrams can be shwn here, but all f them display a general symmetry f earthquake distributin with regard t the P and T axes. There is significant randm fluctuatin f the curves but n cnsistent increase f the earthquake numbers in the directin f the P axes. Such an increase shuld be strnger fr shrt-term aftershcks than fr earthquakes ccurring after a delay f 250 days. Table 2 summarizes the results f the cmputatins. The values displayed in the last three clumns are the rati r - (Np - NT)/(Np + NT), (4) where Np and N - are the numbers f secndary earthquakes in the P and T quadrant, respectively. Therefre the psitive values f r signify that there are mre earthquakes in the P quadrant, as might be expected frm the Culmb fracture criteria (equatins 1 and 2) with / > 0. Hwever, mst r values in Table 2 are negative. f the aftershcks were t cncentrate in the P quadrants, the maximum r values wuld appear in the clumn crrespnding t clse events (< 250 days). The results fr the PDE catalg with different magnitude cutff are similar. Hence the threshld change des nt influence significantly the spatial aftershck distributin. Althugh the number f the maximums is slightly higher fr this clumn (six maximum values) than ther clumns, this is prbably the result f randm fluctuatins. There is n bvius preference fr a maximum in this time frame. 4. Discussin We have investigated aftershck and secndary earthquake distributins by stacking the hypcenter lcatins n a fcal sphere f several hundred and even thusands f strng earthquakes. The prblem we address is whether the distributin exhibits asymmetry with regard t the fcal P and T axes as predicted by Culmb equatins (1) and (2) with a nnzer value f the cefficient f frictin/. The study is based n fur earthquake catalgs prcessed in different cmbinatins. We mdify several parameters, such as hypcentral distance, fcal mecha.- nism type, and representatin f the surce regin. The search presents a difficult methdlgical prblem, since we need t decide whether the particulars f an bserved distributin are due t search parameters r t randm fluctuatins f seismicity. Cmplex features f seismic-

12 ß.. ß 24,464 KAGAN AND JACKSON: SPATAL AFTERSHOCK DSTRBUTON 12000,,,, 600 a '... i : ' :" '...:... :' k' : '1 : ' ß -d Angle P-axis Directin Angle T-axis 10000! ' /..! ß : b i! % : ß ß! : \"i ' ß" "" ' ß ! : :/// i ' : : --.f, : ß... :... " '.... ;,. x : ß! /: ß, : - : \ / : / : \ : ,..,...,.:.;.\... i'-"i '"-,-,,, i ' - -';-' - ) Angle c 3O 20.i ! i i i i i i. i...,,..., i > '\ b.!.!' iv,,, P-axis Directin Angle T-axis 45O i i i! i!! i ' ' ' '! 'A ' '! : ' C 4O u$ ß /.-... / \ ß ' \ x / '" -10 xx // ' i! - 1 O ; -; ; i. 100 P-axis Direct:n Angle T-axis -40 P- is Directin Angle T-axis Figure 5. Distributin f earthquake numbers befre an earthquake (slid line), up t 250 days after Figure 6. Distributin f differences f nrmalized (dash dtted line), and mre than 250 days after the event (dashed line). The distance limits are 100 km frm the initial event. (a) Earthquake numbers in the CT/USGS catalg fr strike-slip events in the mdified Ellswrth catalg; the numbers are calculated by summatin f cells in Figure 3a (see text). (b) Earthearthquake numbers in 9 ø lunes. The numbers befre an earthquake are subtracted frm thse after the event: mmediate aftershcks (up t 250 days after, slid line), and lng-term aftershcks (mre than 250 days after the event, dashed line) are displayed. The distance limits are 100 km frm the initial event. (a) Earthquake quake numbers in the CT/USGS catalg fr strike-slip numbers in the CT/USGS catalg fllwing all _M_> 6 events in the Ellswrth catalg; the numbers are in 9 ø lunes (Figure 3b). (c) Earthquake numbers in the PDE catalg fr strike-slip events in the Harvard catalg; the numbers are in 9 ø lunes (Figure 3b). events in the mdified Ellswrth catalg. (b) Earthquake numbers in the PDE catalg fllwing all events in the Harvard catalg. (c) Earthquake numbers in the Harvard catalg.

13 KAGAN AND JACKSON: SPATAL AFTERSHOCK DSTRBUTON 24, Nrmalized Earthquake Numbers (befre) -2O Nrmalized Difference f Earthquake Numbers (<250d-befre) 0 2O 40 6O Nrmalized Earthquake Numbers (<250d) 0 20 Nrmalized Difference f Earthquake Numbers (>250d-befre) Plate 4. Smthed distributin n a sphere quadrant f nrmalized earthquake numbers in the CT/USGS catalg fr strike-slip events in the mdified Ellswrth catalg. (a) Earthquakes in the CT/USGS catalg befre the Ellswrth event. (b) Earthquakes in the CT/USGS catalg less than 250 days after the Ellswrth event. (c) Earthquakes in the CT/USGS catalg mre than 250 days after the Ellswrth event. (d) Difference between the nrmalized numbers in Plate 4b minus numbers in Plate 4a. (e) Difference between the nrmalized numbers in Plate 4c minus numbers in Plate 4a Nrmalized Earthquake Numbers (>250d) ity, such as a multidimensinal character and the presence f extensive aftershck sequences, make the applicatin f standard statistical tests difficult and their results ambiguus. Hwever, if the value f the frictin cefficient/z is high (f the rder ), we shuld see a cnsistent pattern in all the earthquake distributins studied. Therefre we are lking fr cnsistency in the distributins t prve a nnzer value f the frictin ceificient. The distributins f earthquakes n a fcal sphere indicate that it is symmetric with regard t the P and T axes. Such symmetry suggests either that the Culmb frictin ce cient is clse t zer r that earthquake faults and aftershcks rerganize themselves in a pat- tern similar t that which wuld be btained if t - 0. These results are nt incnsistent with the idea that (1) stre ss changes due t strng earthquakes cause af- tershck sequences and (2) aftershcks ccur preferentially where shear stress has been increased. Our results shw that the effective ce cient f frictin is small, indicating that nrmal stress has little r n cnsistent influence n aftershck ccurrence. This result culd simp!ify studies f hw cseismic stress changes future earthquakes, because shear stress changes has the same effect n the auxiliary and fault planes. f the ceiticien

14 24,466 KAGAN AND JACKSON: SPATAL AFTERSHOCK DSTRBUTON f frictin is zer, the Culmb stress can be calculated fr a target earthquake withut having t reslve the fcal plane ambiguity. Sme publicatins suggesthat the value f/ is small (less than 0.2) [Reasenberg and Simpsn, 1992; Kagan, 1994; Bird, 1996]. Other authrs cnclude that earthquake stress triggering des nt depend n the assumed value f/z r weakly depends n/z [King et al., 1994; Deng and Sykes, 1997, p. 9877]. This insensitivity implies that earthquake initiatin either des nt r nly weakly depends n the value f the nrmal stress. Varius hyptheses explaining this phenmenn have been ffered [Deng and Sykes, 1997]. The results reprted here nce again cnfirm the hypthesis that earthquake ccurrence is independent f the nrmal stress value: a result valid fr varius tectnic envirnments and earth- quake fault gemetries. 5. Cnclusins Shallw aftershcks d nt cncentrate in the dilatatinal quadrant. The difference between preearthquake and pstearthquake distributins has an equal chance t be psitive r negative in the directin f T and P axes. This result implies that either the frictin cefficient in the Culmb law is clse t zer, r bth tectnic and earthquake static stress are self-rganizing int a pattern that mimics - 0. Acknwledgments. We appreciate supprt frm the Natinal Science Fundatin thrugh grant EAR We als appreciate partial supprt frm the suthern Califrnia Earthquake Center (SCEC). SCEC is funded by NSF Cperative Agreement EAR and USGS Cperative Agreements A0899 and 1434-HQ- 97AG The authrs thank F. E. Leader f UCLA fr help with clr plts, C. Lmnitz f the Natinal Autnmus University f Mexic, an annymus reviewer, and Assciate Editr J. Gmberg f the University f Memphis fr very useful reviews. The SCEC cntributin number is 426. References Bird, P., Cmputer simulatins f Alaskan netectnics, Tectnics, 15, , Chinnery, M. A., The stress changes that accmpany strikeslip faulting, Bull. 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15 KAGAN AND JACKSON: SPATAL AFTERSHOCK DSTRBUTON 24,467 Saff, E. B., and A. B. J. Kuijlaars, Distributing many pints n a sphere, Math. ntell., 19, 5-11, Schlz, C. H., The Mechanics t' Earthquakes and Faulting, 439 pp., Cambridge Univ. Press, New Yrk, Segall, P., Fault mechanics, U.S. Natl. Rep. nt. Unin Ged. Gephys , Rev. Gephys., 29, , Stein, R. S., and M. Liswski, The 1979 Hmestead Valley earthquake sequence, Califrnia: Cntrl f aftershcks and pstseismic defrmatin, J. Gephys. Res., 88, , Stein, R. S., G. C. P. King, and J. Lin, Stress triggering f the 1994 M Nrthridge, Califrnia, earthquake by its predecessrs, Science, 265, , Stein, R. S., A. A. Barka, and J. H. Dieterich, Prgressive failure n the Nrth Anatlian fault since 1939 by earthquake stress triggering, Gephys. J. nt., 128, , U.S. Gelgical Survey, Preliminary determinatin f epicenters (PDE), mnthly listings, U.S. Dept. f nter., Natl. Earthquake nf. Cent., Denver, Cl., Jan Utsu, T., and Y. Ogata, Statistical analysis f seismicity, in ASPE Sœtware Library, vl. 6, edited by J. H. Healy, V.. Keilis-Brk, and W. H. K. Lee, pp , nt. Assc. f Seisml. and Phys. f the Earth's nter. and Seisml. Sc. Am., E1 Cerrit, CA, D. D. Jacksn and Y. Y. Kagan, Department f Earth and Space Sciences, University f Califrnia, Ls Angeles, CA ( djacksn@ucla.edu; ykagan@ucla.edu) (Received August 20, 1997; revised February 13, 1998; accepted February 18, 1998.)