Physicsof theEarth andPlanetary Interiors, 71(1992)15—35 15ElsevierSciencePublishersB.V., Amsterdam

On thegeometryof an earthquakefault system

Y.Y. Kagan

Instituteof Geophysicsand PlanetaryPhysics,Universityof California, LosAngeles,CA 90024-1567,USA

(Received7 October1991; revisionaccepted28 October1991)

ABSTRACT

Kagan,Y.Y., 1992. On the geometryof anearthquakesystem.Phys.EarthPlanet. Inter., 71: 15—35.

Invariantsof a two-point correlationfunctionof the seismicmoment tensorwereusedto investigateanearthquakefaultsystem. The geometryof the fault systemis significantly different from the standardmodel of an earthquakefault, i.e.coherentruptureon a planarsurface.Contraryto the ‘flat-fault’ model,we seeclearevidencefor non-planarityof the faultsystemgeometry,andobservethat thefocalmechanismsof neighboringeventsmay havevery differentorientation,i.e. theyundergolarge three-dimensionalrotations.Therefore,earthquakedeformationmodels need to be fully three-dimensionaland shouldinclude large rotations.

The spatialbehaviorof the invariantsis approximatelythe samefor earthquakesin different depthintervals: shallow,intermediate,and deep.The temporalbehaviorof the invariantsdiffers only in that shallowearthquakesareclusteredintime, whereasfor deepereventsthe clusteringis muchless pronounced;as soon as we ‘decluster’ shallowseismicity, theinvariants’ temporalpropertiesbecomesimilar for earthquakesof all depths.This demonstratesthat thebasic geometricalpropertiesof earthquakerupture do not depend on depth, and thereforethey are generallyindependentof rheologicalpropertiesof rocks, lithostaticpressure,or thepresenceof a free boundaryfor strongshallowearthquakes.

1. Introduction and Woodhouse, 1984; Kagan and Knopoff,1985b;Kagan,1991b),investigationof the geolog-

It is well establishedthat earthquakefaults are ical structureof earthquakefault zones(Gay andnot exactly planes; rather, planargeometry is a Ortlepp, 1979; Segall and Pollard, 1980; Sibson,first approximation.In reality, earthquakefaults 1986; Pollard and Segall, 1987; Martel and Pol-are complexesof fracture surfaces,or potential lard, 1989; Steinand Yeats, 1989; Scholz, 1990;fracturesurfaces,distributedthree-dimensionally. Segall and Lisowski, 1990), and investigationsMoreover,the fault surfacesand slip vectors of basedon first principlesof continuummechanicsneighboring earthquakesundergo three-dimen- (Rice, 1980;King, 1983;Andrews,1989).Someofsional (3-D) rotations which are often finite and the above-mentionedreseachersalso emphasizedsometimeslarge. Theserotations areobvious on that the distribution of earthquakefaults is fullyfocal mechanism maps (Goter, 1987), as the 3-D with certain predictableconfigurationssuchmechanismsof neighboring eventsmay have a as en echelonfaults,bends,steps,andjogs.Thesevery different orientation. These 3-D rotations studiesuseddifferent methodsof analysisof thehave been documentedfrom studies of earth- complex geometry of earthquakefaulting: de-quakesand their aftershocksequences(see,e.g., scriptive, qualitative, analytical, and determinis-Oppenheimeret a!., 1988; Dietz and Ellsworth, tic.1990), research on distribution of hypocenters Becauseof a significant randomcharacterofand focal mechanismsof earthquakes(Giardini earthquakefault structure,the statisticalmethod

0031-9201/92/$05.OO© 1992 — Elsevier SciencePublishersB.V. All rights reserved

16 yy• KAGAN

of analysisacquiresimportance.As previous re- often impossible in the Earth’s interior, but wesearchhas indicated (Kagan, 1990), a fracture can infer some important featuresof the stresspropagatingthrougha mediumwith small defects field from the way in which earthquakesrelatetois non-planarandis not confinedto one surface. eachother. A considerableand steadily increas-Becauseof the randomcharacterof the defects, ing amount of information is now obtainedrupture geometry is stochastic; propagationof through the inversion of earthquakesource pa-fracture should amplify initial small deviations rameters(Sipkin and Needham,1989; Dziewon-from planarityand3-D rotationsof focal mecha- ski et al., 1991); therefore, even if the stressnisms(King, 1983;Andrews,1989). For example, tensoris not alwaysmeasurable,its effectscanbeAndrews (1989) showedthat a bent earthquake partially deducedfrom the seismicmomentten-fault shoulddevelopan unstabletriple junctionat sor. The influenceof oneearthquakeon anotherthat point. The junction can accommodatefault is importantfor bothpractical andscientific rea-displacementfor a limited numberof events,and sons. In practical terms, theseearthquakeinter-thereafterfault geometrybecomesmorecomplex. actionsareessentialingredientsin any procedureTherefore,not only do the smallfeaturesof frac- usedto estimatethe current earthquakehazardture exhibit randomness,but large-scalegeomet- basedon previousseismicity. Scientifically, theserical componentsare also stochastic,and hence interactionsprovide important information aboutthe complex geometry of earthquakefracture the stresschangesproducedby earthquakes,theshould be studied statistically. Deterministic rheological properties of the crust and mantlemethods,which study the distribution of large andthe triggeringof earthquakes.faults either by observationor by fracture me- Recently,a numberof studieshavebeenpub-chanicsanalysisandthe statisticalmethods,which lishedcomparingthe geometryof deepseismicityconcentratemostly on small, directly unobserv- with that of shallow earthquake occurrenceablefeaturesof the focal zone,shouldbe mutu- (Frohlich, 1987, 1989 (and referencestherein);ally supportivein the studyof an earthquakefault Apperson and Frohlich, 1987; Frohlich andsystem. Willemann, 1987;Michael, 1989). Such investiga-

In our further deliberationit is convenientto tions shouldshowwhetherthe geometryof earth-distinguishtwo geometricalpatterns:the distribu- quakeruptureis influencedby rheologicalprop-tion of hypocenterson a fault or a fault system, ertiesof rocks. In this paperwe call earthquakesand orientationof earthquakefocal mechanisms. above 70 km depth crustal or shallow, earth-Therefore, we designatethe former pattern a quakesin the depthinterval 71—280 km interme-scalar geometry,and the latter structure rota- diate,andthosebelow 280 km deepearthquakes.tional geometry.Forexample,Kagan(1991b)has As the properties of intermediate and deepstudiedthe scalargeometryof earthquakefault- earthquakesare similar in many respects(cf.ing, whereasmy otherpapers(Kagan,1990, 1992) Frohlich, 1989), we will often describeall non-were dedicatedto the rotational distribution of shallow eventsasdeepearthquakes.faults. In general,whereasthe scalar geometry Earlier (Kagan and Knopoff, 1985a,b)we in-has been extensively studied (see references troducedthe one- andtwo-point correlationten-above),the rotational geometryof the fault sys- sors as tools appropriateto describestochastictern hasnot beeninvestigatedasthoroughly,pos- geometryof earthquakefaulting. We havestud-sibly becauseof themethodologicaldifficulties of ied the geometry for a few simple models ofsuchstudies.Wewill usethe term ‘disorientation’ earthquakefaults,for syntheticearthquakefaults,to describethe 3-D stochasticrotationsof earth- aswell asfor catalogsof focal mechanismsof realquakefocal mechanisms. earthquakes(Kagan and Knopoff, 1985a,b).The

Earthquakesinfluenceeachother throughthe use of two-point tensorsallows us to study thestressenvironment;theyaffect both the time and scalar and rotational geometry of earthquakethemechanismof laterearthquakes.Direct stress faults simultaneously.Unfortunately,the qualitymeasurementsare problematic in general and of earthquakecatalogsavailable at the time of

THE GEOMETRY OF AN EARTHQUAKE FAULT SYSTEM 17

the research(KaganandKnopoff, 1985b),as well both methodsof analysis,statisticaland geologi-as thequantityof datain them,wasnot sufficient cal, shouldbe combined in the study of earth-to characterizestatistically the structureof an quakefault geometry.earthquakefault system, nor were we able to In anotherpaper(Kagan,1992) thestatisticsofinfer any useful information aboutdependenceof focal mechanismrotationsis studieddirectly, andearthquakegeometry on depth of earthquakes, the resultsare moregeometricallyobvious; how-andhenceto studytheinfluenceof rock rheologi- ever,the numberof degreesof freedomfor statis-cal propertieson fault structure.Severalexten- tical distributions in these investigations is sosive catalogsof focal mechanismscoveringlarge largethat it is difficult to obtain reliable resultsregions,and world-wide catalogsof seismic mo- from limited empirical data. In this paper wement tensor inversionsare now available; thus, considercorrelationtensorsof earthquakefocalwe cansignificantlyadvanceour understandingof mechanisms.The advantageof this approachisfault geometricalpatterns, that weareobtainingglobal correlationfunctions

The study of specific casesof the earthquake with relatively few degreesof freedom,and thusfault systemor crack growth has a clear appeal, we canstudycatalogsin moredetail.The correla-as the geometricalpatternsof hypocentersand tion functions are also useful in a statisticalcon-focal mechanismscan be visualized and various tinuum mechanicaldescriptionof finite deforma-approachesandtechniquescanbe tried to inves- tions (Kröner, 1972; Kagan and Knopoff, 1985b,tigate theproblem.Moreover,we canchoosethe and referencestherein).Previously, the progressmost appropriatesystemof coordinates,and in of statisticalcontinuummechanicshasbeenham-almost all casesthe geometryof faults can be pered by an absenceof empirical results relatedreduced to 2-D considerationand easily dis- to the distributionof dislocationdensityandtheirplayed. However, the predictive power of such correlations.The results reported in this paper‘anecdotal’ evidenceis limited, as the reported are important for furthering our theoreticalun-cases might represent somewhat ‘pathological’ derstandingof the earthquakedeformationpro-examples,andthusthe sampleselectionis biased. cess.Moreover, it is difficult to separatethe descrip-tive part of a case studyfrom its analytical, pre-dictive part. 2. Data and analysis

Statistical investigations,on the other hand,havethe advantageof usingunbiasedsamplingof Two instrumental worldwide catalogs havea completecatalogof earthquakes,and thusthe been used in theseinvestigations.The Harvardresultscanbe easilyextrapolated.The disadvan- catalog(Dziewonski et al., 1991, and referencestageof statisticalmethodsis a limited insightinto therein) seemsmost complete and uniform inspecific geometricalpatternsof earthquakefaults: reporting seismic moment tensor solutions. Wewe aredealingwith statisticalaverages,andheavy also useanother(US GeologicalSurvey(USGS))averagingmight alsoscrambleimportantinforma- catalogof momenttensorinversions(Sipkin, 1986;tion obvious evento a casualinspectionof the Sipkin and Needham, 1989, and referencesfaults. As thereis no preferredor specialsystem therein). In thesecatalogswe use the solutionsof coordinateswhen we analyze hundredsor correspondingto a double-couple source.Thethousandsof events, the full 3-D geometrical Harvardcatalogstartson 1 January1977,andthepatternsand 3-D rotations of focal mechanisms USGS cataloglists earthquakesfrom 1 Januaryshouldbe considered.This three-dimensionality 1980. They bothendon 30 April 1990. The Har-greatlyincreasesthe complexity of the mathemat- yardcatalogcontainscloseto 9000 eventswith aical tools we needto usein the investigations.As magnitudeof 5.0 or more (we use earthquakesthesemathematicalmethods are not generally with M~~ 5.0 in this work), whereasthe USGSusedby most geophysicists,in the following sec- cataloghasclose to 1000 eventswith magnitudetion we discussthem in detail.Thus, in principle, of 6.0 or larger.

18 Y.Y. KAGAN

In our other statisticalinvestigationsof earth- and ~2), and C2 and C3 are scalar functions ofquakecatalogs(see,e.g. Kagan,1991a,b),to in- distanceandtime difference.sureuniformity of data,we usuallydeletedfrom a The situationbecomesevenmorecomplicatedcatalogall eventswith magnitudeor seismicmo- when the randomvariable in questionis not ament lower than a certain threshold.Failure to scalar, but a tensor. Then, even the two-pointdo this may result in significant biasescausedby second-ordercorrelationfunction is a fourth-rankinhomogeneityof catalogs.A catalogis expected tensor(Kagan andKnopoff, 1985b):to be moreuniform after removalof weak events.

‘~“1’ijk1(’~’‘i-) = (m

11(r1, ti)nkl(r2, t2)) (3)However, in the statisticalstudiesof seismicmo-menttensorsdescribedin thispaper,we useonly wherem and n areseismicmomenttensors,andorientationof principal axesof focal mechanisms. r = — r1, ‘r = — t1. The two-point, third-orderTherefore,the datawere found to be sufficiently momentis a sixth-ranktensorand the two-point,homogeneous,so that we do not needto remove fourth-ordermomentis a eighth-ranktensor:weak events.

The statisticalanalysisof time—spacedistribu- MJklpq(r, r) = Km1~(r1, t1) mkl( r1, t~)tion of seismicmomenttensorsolutionsis a diffi- Xflpq(T2, t2)) (4a)cult problem. More familiar statistical problemsinvolve scalar quantitiescollected at one point. Mijktpqrs(r, r) = (m~~(r1,tl)mkl(rl, t1)

The momentsof thesequantities,at least lower- Xflpq(T2, t2)n~5(r2,t2)) (4b)order ones,are well known: the first-order mo-ment is an averageor a mean,the second-order In the tensor case,usually severalvariables, in-moment is a variance,etc. For example, for a steadof one, areneededto characterizea singlerandomvariable a moment;as Kaganand Knopoff (1985b) showed,

the correlation two-point tensor for space—timem1 = (a), m2 = (a

2), Var(a) = m2 m~ (1) distribution of earthquake focal mechanisms

needsthreespace—timefunctionsfor its full char-Angularbracketsin (1) indicatea statisticalaver- acterization.age(more detailshavebeengiven by Monin and It canbe shown that, for a homogeneousandYaglom (1971,p. 223)). isotropic field, the correlationtensors(3) or (4)

To investigatea scalarfield, we form one-point should dependon certain invariantswhich arestatistical moments (first-order, second-order, formed from seismic tensorsand a vector con-etc.)which are space—timefunctions(fields). For necting two points. In particular, we select thestationary and homogeneousfields these mo- three bilinear invariants (Monin and Yaglom,mentsareindependentof time andposition.More 1975,section 12.2; KaganandKnopoff, 1985b) toinformation is containedin two- or three-point characterizethe two-point second-ordercorrela-functions: a two-point second-orderfunction is tion tensorof earthquakefocal mechanisms:known as a correlationfunction; similarly, we candefine two-point third-order fields, and so on J~ ((p,m~1p1). (Pk~k1P1))

(Monin andYaglom, 1971,p. 228). For a station- j = ((pmnp)) (5)ary scalarfield, for example,the two-point, sec-ond- andthird-ordermomentsare respectively

C2(r2 — T1, t2 — t~) (a(r1, t1)a(T2, t2)) where m and n are normalizedseismicmomenttensorsof two arbitrary earthquakes;p, = r,/ I r I

C3(T2 — r1, t2 — t1) = (a2(r

1, t1)a(r2, t2)) are the direction cosinesof r; and T is a vector

(2) connectingtwo foci, and its horizontalprojectionis a greatcircle arc (more details are given by

where a scalar function a is measuredat two Kaganand Knopoff (1985b)). In practical terms,points (T1 and T2) and two momentsof time (r1 we calculatethe normalizedseismicmomentten-

THE GEOMETRY OF AN EARTHQUAKE FAULT SYSTEM 19

sor usingthe P- and T-axesof the double-couple that they are independentevents,hencethe zerosolution (Kagan and Knopoff, 1985b, p. 649). If value is never reached).Whenwe evaluatesumsthereare N earthquakesin a catalog,invariants in (5) and (6) while calculatingthe momentfor a(5) arecalculatedfor N(N — 1)/2 pairsof events, residualcatalog,we use the product of weightsThe repeatedindices in (5) imply summation for eachpair of events. Removingan aftershockfrom i = 1 to i 3, thusthe invariantsarescalar from a catalog is equivalent to assigningzeroquantities. J1, J2, and J3 are invariants,as the weight to the event. Thus the procedureis asum-productsin (5) do not dependon the system sophisticatedvariantof deletingaftershocksfromof coordinates.As shown previously(Kagan and a catalogandprocessingthe deciusteredcatalog;Knopoff, 1985b), these invariantswhich depend comparingthe results from both the original andon time interval and distancebetween events declusteredcatalogs, we understandthe biasfully definethe fourth-rankcorrelationtensor(3) which aftershocksintroduceinto our results.for a set of deviatoric earthquake sources. In principle, if the Poisson cluster process,Danilova and Yunga(1990) investigatedthe sta- with a uniform rate of cluster centers,is a suit-tistical propertiesof a sumof independentran- able model of an earthquakeoccurrence,thedomly rotateddouble-couples;their invariant ,c is momentsof a residualcatalogshouldbe equaltosimilar to our J3. a constant,independentof time. In reality, our

We also calculatestandarddeviationsfor each seismicitymodel is notperfect; in Kagan(1991a)of the invariants in (5): we discussvarious approximationsand possible

2 2 2 errors which may contaminateour estimatesof= ([(p~m~~p,)(p,.~n,apt)]) _~t eventdependence.In the presentstudy, the cata-

2 ~2 2 1 LT\ logs are more inhomogeneous,aswe use a rela-= (I(PmnkPk)T ) —J2 .

“ ttvely low magnitudethreshold(see above).This= ([(m,1n~1)12> — J3

2 shouldalso contributeto deficienciesof the after-shock removal. Thus the momentsof a residual

These invariants define the two-point fourth- catalogshouldshow the defectsof our likelihoodorder correlation tensor (4b) (compare Monin model approximationand other influenceswhichandYaglom (1975, p. 69)). In addition to invari- havenotbeenproperlymodeledin the likelihoodants(6), threemoreinvariantscan be formedto procedure. However, even with an imperfectcharacterizethe fourth-ordermoment(4b): residualcatalogwe can evaluatethe influenceof

— /,‘ \. ii \ \ 1 \ aftershocksand other dependenteventson the4 — \~p~m~InJkpkJ[kp~m~JpJ)kpknklpl1J / results.

= ([(p~m1JpJ)(pknk1p1)]. (m~n~1)) (7)

= <(m~~n~~). [(p1m~JnJkpk)]>3. Simulation

We did not studyinvariants(7) in this work.To study dependenceof the momentson the To simulatethe 3-D rotations of focal mecha-

presenceof aftershocksand other dependent nismswe use the establishedcorrespondencebe-events in a catalog,we createresidual (declus- tweennormalizedquaternionsand 3-D rotationstered) catalogs.In processingthesecatalogs,we (Altmann, 1986, Chapter 12; Kagan, 1990). Asfirst calculate the probability of an earthquake Altmann (1986) and Chang et a!. (1990) stated,being independent.This probability is evaluated the quaternionparametrizationof the 3-D rota-usingthe resultsof the likelihood analysisof the tion has many advantages.In fact, quaternionsearthquakecatalog(Kagan,1991a).Theprobabil- are the only rationalmethod to describethe 3-Dity is taken as the weight of each earthquake. rotations;hencethey havebeenused,for exam-Mainshocks,for instance,areassignedthe weight pie, in investigationsof tectonic block rotationsof 1.0, whereas aftershocksusually receive a (Le Pichonet a!., 1973; Thompsonand Prentice,weight close to zero (there is a slight possibility 1987; Changet a!., 1990).As anotherrepresenta-

20 Y.Y. KAGAN

tion of the 3-D rotationwe usehere the rotation andaboutthe axis e by the angle (F (Altmann,1986). —1/2

The simulation of a completelyrandom rota- q = u1(1 + u~+ u~+ u~) for i = 1, 2, 3tion hasbeenconsideredin our previouspublica- (10)tions (Kagan and Knopoff, 1985a; Kagan,1990).The rotation correspondsto selectinga random The orthogonalmatrix which correspondsto thispoint on the surfaceof a 3-D hypersphereor a quaternionsimulatesa Fisher-typerotational dis-unit quaternion,q: tribution for small values of o~(o~~ 0.1). Once

more, we see that for small a-, q0 should beq=q0+q1t+q2j+q3k (8)generallyclose to 1.0; hence,rotationsaresmall.

where q~+ -I- q~+ q~= 1 in the 4-D space. In this case,as u is distributedaccordingto the(Above, we use the mathematicalconvention of Gaussiandistribution, (F is more concentrateddescribing,for example,a conventionalspherein around(F = 0 thanin the caseof (9).a 3-D spaceas a 2-D surface.A hyperspherein a The effects of a 3-D random rotation of a4-D quaternionspaceis thena 3-D surface.)The double-couple earthquakesource are compli-first quaternion’s component (q0) is its scalar catedby the symmetryof the source.For conve-part; q1, q2, and q3 arevector componentsof a nience,we summarizetheseeffects from Kaganquaternion,and correspondto direction cosines (1990, eqn. (3.1)). For (F <90°,the axis of rota-of the rotation axis, the rotation angle being tion can take any position in the positive octantdefinedas (F = 2arccos(q0). x, ~ 0 (i = 1, 2, 3; we usethe Cartesiansystemof

A simulationof a non-uniformrotation is corn- coordinateshere with the N-, P-, and T-axesofplicated becauseof the paucity of appropriate the double-couplesolution). The axesof a dou-modelsof rotation.Onepossiblemodel for simu- ble-couple source have no preferreddirection,lation of non-uniformrotation is to generatethe and by changingthe direction of each axis weuniform distribution mentionedabove, and then obtain severalequivalentsystemsof coordinates.retain only the points on the hyperspherewhich Therefore,becauseof the symmetryof the prob-lie in the interval 1 ~ q0 < 1 — e (Kagan and lem, the distribution for the other octantsis theKnopoff, 1985a), where � is a small positive sameas that of the first. For (F> 90°,the rota-quantity.As the aboveinequality appliesonly to tion axis is restrictedto a part of the octant; forq0, otherquaternioncomponents(q1, q2, and q3) example, for (F = 2arccos(3~”

2) 109.47°,thecan take any admissible value. Thus this case axis is confinedinside the triangle formedby thecorrespondsto a rotation in which theposition of points (x

1 = x3, Xk = 0), where the indices i, j,the rotation axis is uniformly random(i.e. is a and k alternatelytake on the values one, two,randompointon the surfaceof anordinarysphere and three. The maximum value of the rotationin the 3-D space), but the rotation angle (F is anglefor double-coupleis 120°,for any (F > 120°,limited by the inequality it can be shown (Kagan, 1990) that the same

(F rotation canbe accomplishedby anotherrotation1 — � s~cos—<1 (9) with (F ~ 120°.The rotationwith (F = 120°corre-

2 spondsto the axis situated in the centerof theThis meansthat the rotation is uniform in the above-mentionedtriangle, which is at the pointrange0 <(F ~ 2arccos(1— �). xi =x2 =x3. Therefore, whereas a completely

For the purpose of simulation we also use random3-D rotation of arbitraryobjectsinvolvesanother approach(Kagan and Knopoff, 1985a): rotation angles (F as large as 180°,the randomfirst, we generatea 3-D normally distributedran- rotation of double-couplesrequires ~ to varydom variable u (u1, u2, u3) and then calculate from 90°to 120°(Kagan,1990).the unit quaternion To understandthe resultsof earthquakecata-

2 2 —1/2 log analysis,we simulatean earthquakefault ofq0 = (1 + u~+ u2 + u3) 20000km length and100 km width. On this fault

THE GEOMETRY OF AN EARTHQUAKE FAULT SYSTEM 21

10000earthquakefoci havebeendistributeduni- the distribution of foci which are spatially uni-formly randomlywith the slip vector parallel to form in the 3-D space.Two possibilitiesare con-the short side of the fault. The hypocenterswere sidered: (a) all focal mechanismsare parallel tothen displacedby a randomly oriented vector eachother (coherent),and(b) half of the sourceswith a Gaussiandistribution for eachof the three (chosenrandomly)haveone orientation,andthevector components.The standarddeviation for remainder have the opposite orientation (i.e.each componentis takento be 15 km, which is diag[1, — 1, 0] and diag[ — 1, 1, 0]). We call thecloseto estimatesof locationerrorsin realearth- latter distribution anticoherent.To obtain a-1quake catalogs (Dziewonski et al., 1991). The from (6) we integrateover a sphere(cf. Kaganfocal mechanismsof syntheticearthquakeswere andKnopoff, 1985b, p. 639):rotated, according to two rotation distributions 2

(9) and (10). This rotation can be visualized, for ([(p1m11p~) (p~~1(~p~)})instance,by taking two vectorsrepresentingtheslip vectorandthe normalvectorto a fault plane, = 1 12ir cos

424 d4 I sin9ada —

and rotating them around an axis taken ran- 4~r~o Jo 105domly, with a small rotation angle.The new seis- (11)mic moment tensor is calculated from rotatedpositionsof the two vectorsusing known formu- where ~ and a aresphericalcoordinates.J

1 haslae. beenalreadyevaluatedin Table 1 of Kagan and

Knopoff (1985b)(J1 = 4/15).Two other examples,shown in the third and

4. Results the last columnsin Table 1, representa planarearthquake fault with coherent sources dis-

4.1. Theoreticalfaults tributed randomlyover a 2-D surface(the stan-dard model of an earthquakefault system),and

We calculate the valuesof the invariants (5) sourcesrotated randomly (incoherent distribu-and(6) for severalsimple cases(Table 1). As (6) tion), respectively.In the latter case,the scalarrepresentsstandarddeviations of (5) they are fault geometry is not relevant,as the invariantsshown with ‘±‘ signs after the valuesof J1, J2, do not depend on the spatial distribution ofand J3. The first two columnsshow thevaluesfor hypocenters.In most caseswe list approximate

TABLE 1

Valuesof tensorinvariantsfor severalmodelsof the earthquakefault system

Invariant Spatiallyuniformdistribution Plane Incoherent

Coherent Anticoherent fault distribution

0.267±0.285 0±0.390 0±0.2667-‘1 0±0

~(1±2(4)1/2) 0±(105)1/2 O±~(?)

0.667±0.298 0 ±0.730 0.5±0.354 0 ±0.3887

/ 1 \ 2(2)Iz~2 1 (34)1/2

4~,i±~5T72) 0±(15)1~’2 2(2)~~ 0± 15 (?)

.13 2±0 0±2 2±0 0±0.8944

22 yy•KAGAN

numericalvaluesof the invariantsand their stan- a natural3-D scatterof sources.Thefirst invari-darddeviations,as well as their exactvalues.The ant values are far from zero (Fig. 1), which isvaluesfor the incoherentdistribution cannot be clearly inconsistentwith theplanefault asa mode!calculated analytically; the entriescited are ob- of the earthquakefault system(see Table1). Thetamed only by simulation.Therefore,the ‘exact’ sources’incoherenceat large range is to be ex-valuesfor a-

1, a-2, and a-3 in the last column of pected,as the correlation of focal mechanismsTable 1 are our estimates(guesses) from the should not exceed the size of major tectonicnumericalresults. plates(2000—4000km). At largerdistances,as

anda-3 testify, the focal mechanisms’orientations4.2. Earthquakecatalogs are in effect independentof eachother, i.e. the

orientationis randomlyrotated.In Fig. 1 we show the dependenceof the Additional detailed analysis of the distance

invariants(5) and(6) on distanceandearthquake curves in Fig. 1 indicates that thesecurves ap-depthinterval for the Harvardcatalogof seismic proach the line distribution (see Kagan andmomenttensorsolutions(Dziewonskiet al., 1991). Knopoff (1985b, last entry in Table 1)), corre-Distance intervals in the figure are subdivided spondingto a normal thruston a linear horizon-into 48 subintervalswith the multiplicative step tal fault. The line distributionyields thevaluesof21~~4.In our plots we display symbolsonly for invariants J1 = = 0, J3 = 2. The distancerangethose intervals for which the numberof earth- where f1 and J2 approachzero increasesfromquakepairsequalsor exceeds10. valuesof about 200—300 km for shallow earth-

Comparingthe curves,we seethat, in general, quakes,to about 600—1000 km for intermediatethe invariants’behavioris similar for earthquakes anddeepevents.This is understandable,asmuchof different depth intervals. This meansthat the seismicity is concentratedin subductingseismicearthquakefracture processon a fault system is belts; the depth interval is smaller for shallowessentiallyindependentof depth. For small dis- earthquakes(0—70 km) than for deepershockstancerange the invariants’valuesapproachval- (71—280or 281—700km), andthereforethecurvesuesof uniform distribution of coherentsourcesin approachthe linear distribution at larger dis-the 3-D space(see Table 1); for largedistances tancesfor deepevents.thevaluesarecloseto valuesof incoherentdistri- The valuesof standarddeviations a-1 and a-2bution of sources.The valuesof invariantscorre- show little dependenceeither on depth or onspondingto thesetwo distributionsareshownon distance:in general,they fluctuatebetweenthethe left and right side of the plots, respectively, valuescorrespondingto the coherent3-D distri-Theseresults can easily be anticipated:earlier bution for small distancesand an incoherenten-(Kagan and Knopoff, 1985b), we noted that at semble of sourcesfor large distancerange.Assmall distancerange earthquakefoci are sur- both setsof valuesare very close (see Table 1),roundedby otherapproximatelyparallel sources. little information can be extracted from theseThis is partly due to location errors,but for the curves.However,a-3 curvesshow a more interest-curvesshownin Fig. 1, the rangeof the invariant ing pattern:at shortrangethe standarddeviationchangeis much largerthan 10—20km, which is an tendsto be zero,whereasJ3 is closeto 2.0. Thisestimate of location errors (Dziewonski et al., is consistentwith the model of coherentsources1991); thus, the causeof the crossovershouldbe randomlydistributedin the 3-D space.

Fig. 1. Dependenceof invariants(a) J1, (b) J2 and(c) .13 on distanceanddepth(Harvardcatalog).Depthrange:circles—0—70km,the numberof eventsN = 6392; crosses—71—280 km, N = 1530; stars—281—700 km, N = 636. Solid lines correspondto valuesofinvariants(5), dottedlines correspondto standarddeviationsfor the invariants(seeeqn.(6)). The lines in the left partof the plotcorrespondto coherentuniformly distributedsourcesfrom Table 1 (first column), and the lines in the right sideof the plotcorrespondto incoherentsourcesfrom Table 1 (lastcolumn).

THE GEOMETRY OF AN EARTHQUAKE FAULT SYSTEM 23

0.35

(a)

025

10° iO~ 10 iO~ 104 10’

0.7 . - —---

(b) —

1O~ 1O~ 10~ i0~ 1’

(c)

10° 10’ 102 10~ t0~ 10’

Distancekm

24 Y.Y. KAGAN

For shallow earthquakes,a-3 in the distance ham, 1989). Although the curves exhibit a much

rangeof 100—1000km is largerthanexpectedfor higher level of random fluctuationsbecauseofa purely randomly disoriented distribution of the smallersizeof the catalog,thegeneralbehav-sources(see Fig. 1(c)); therefore we cannotcx- ior of the invariants is similar to that shown inplain the a-3 value by randomrotations due to Fig. 1(c). Comparisonof the plots confirms theerrors and natural stochasticfactors. Moreover, proposition(seesection2) that the dataare suffi-the value of the third invariant is non-zerofor ciently uniform even if we retain all the weakthis distancerange;hence,focal mechanismsare eventsin the catalog.In addition,we seethat if astill correlated.In fact, when a-3 first reachesthe catalogcontainsfewerthana fewhundredevents,value 0.9 correspondingto the incoherentdistri- the statisticalfluctuationsare so large that littlebution at the distance50—70 km, the coherence information canbe extractedfrom the invariants.of sourcemechanisms,as revealedby the third Shallow earthquakesin the Harvard cataloginvariant, is still strong:J3 1.1—1.2. Such a pat- provide yet anotheropportunity to test the de-tern of f3 and a-3 can be explainedby the pres- pendenceof invariants on the size of earth-enceof a certain deterministiccomponentin the quakes:the numberof eventsis largeenoughforrotationaldistribution of sources.The rotationof comparisonof two functionswith M~� 5.0 andsources,as a result of sucha deterministiccorn- M~� 6.0 (Fig. 3). Again, curves for largereventsponent, should be relatively large to yield the exhibit largerstatisticalfluctuations,but it is ob-valueof a-3, which is larger than is found for the viousthat J1, J2, andtheir standarddeviationsdocompletelydisorientedcase.Among the models not dependon M~.However, the third invariantin Table 1, such a value of a-3 can be obtained seemsto dependon magnitude,albeitweakly: f3only by admitting a certain amount of antico- is generally larger for the larger magnitudeherencyin the sourcepattern, threshold,whereasa-3 is marginally lower.

For all distances,shallow earthquakespossess Figure 4 displaysthe invariants’ dependencelarger J3 values, and hence larger correlation on the time interval betweenevents for shallowbetweenfocal mechanisms,than do deepevents, earthquakesin the Harvardcatalog.The time inThe valuesof the third invariant do not decrease the figuresis subdividedinto 24 subintervalswithwith distanceas fast as f1 and ~‘2 values; for the multiplicative step 21/2. We show threesetsdistancesof 200—300 km J1 and J2 of shallow of curves:for short-rangeinteraction(distanceReventsare close to zero,whereasJ3 values(ap- betweenhypocentersis less than 156.25 1cm), forproximately0.6) continueto indicate a significant medium-range(156.25km � R <625 km), andforcoherence between these mechanisms. This the remainderof earthquakepairs (R> 625 km).sourcecorrelation is caused,probably, by global The values of J1, J2, and J3 for small timetectonic forces acting on a boundary between intervals and small distancesare consistentwithcrustal plates. One might argue that a visual the model of coherentsourcesdistributed uni-inspectionof focal mechanismmaps(similar to fonnly randomly in the 3-D space.Long-termthosecompiledby Goter (1987)) leadsto similar valuesof invariantsfor small distanceare closerconclusions: earthquakesare concentrated in to the values of a linear distribution (J1 = =

seismicbelts, their mechanismsexhibiting a gen- 0, J3 = 2). However, the third invariant and itseral mutual correlation, although large random standarddeviation show the declining level ofrotations of mechanismsare also evident.How- coherencefor large time intervals (f3 0.8),ever, our measurementshave the advantageof whereasa-3 is significantly larger(a-3 1.1) thanquantifying theseobvious features of the fault the value correspondingto randomsourcerota-geometry. tion (a-3 0.89; see Table 1). This is another

In Fig. 2 we display the third invariant’s de- indication of the presenceof rotationsof shallowpendenceon distanceand earthquakedepth in- earthquakefocal mechanismswhich might be interval for the USGS catalogof seismic moment excessof purely random rotations;furthermore,tensorsolutions (Sipkin, 1986; Sipkin and Need- suchrandomrotations yield f3 = 0.

THE GEOMETRY OF AN EARTHQUAKE FAULT SYSTEM 25

.4. unpin uupInnul, pnnpupuIr 1111~1u1 ilium

15

100 101 102 10~ 10~ 105

Distancekm

Fig. 2. Dependenceof the third invariant J3 on distanceand depth(USGS catalog).Depth range:circles—0—70km, N = 659;

crosses—71—280km, N 202; stars—281--700km, N = 66. Line-typesare the sameas in Fig. 1.

Plots similar to Fig. 4 are obtained for other section3. Figure 6 demonstratesthe invariants’depthintervals.As might be expected,the invari- dependenceon distancebetweenevents,andFig.antsandtheir standarddeviationsfor deepearth- 7 describesthe dependenceon degreeof mecha-quakesarelargelyindependentof time. For t —~ ~ nismdisorientation.Because,for simulatedfaults,the values of all of the invariants are approxi- the value of the third invariant dependsonly onmately the samefor all depthranges.In Fig. 5 we the rotation of focal mechanisms,its curves aredisplaydependenceof the third invarianton time displayedonly in Fig.7.for intermediate(Fig. 5(b)) and deep(Fig. 5(c)) Four different geometricalconfigurationscanevents,whereasin Fig. 5(a) the declusteredcata- be modeledwith the help of the sypthetic faultlog of shallow events(see section 2) is used to describedin section3: (a) a planarcoherentfault,produce the plot. In these plots the invariant (b) a coherentfault zoneof finite thickness,(c) avalues are independentof time for all depth partially incoherentplanarfault, and (d) a par-rangesincluding shallow earthquakes.This shows tially incoherentfault zoneof finite thickness.Tothat shallow seismicity is different from deeper simulatethe partial incoherence,we rotate focalearthquakesin only one respect: the shallow mechanismsusing (10). The value of a-~usedinearthquakesareclusteredin time. As soonas we (10) is 0.15.declusterthe catalogof shallow events,the cata- For model(a) only the secondinvariantshowslog’s geometrical propertiesbecome similar to any interesting features (Fig. 6(b)): the curvesthoseof catalogsof intermediateanddeepearth- exhibit a clearcrossoverat a distanceof 100 kin,quakes. the width of the fault, from the planar to the

linear geometry(see Table 1). The samecurves4.3. Syntheticcatalogs for scheme(b) show the presenceof threescales:

for distancesup to 20 km the valuesof J1, J2, a-1,In Figs. 6 and 7 we display the invariants’ and a-2 correspondto the 3-D spatiallyuniform

behaviorfor the synthetic catalogsdescribedin distribution, and for distances20 km <R < 100

26 Y.Y. KAGAN

km the values for the first invariant are in the the range20 km <R < 100km, andfinally reachesintermediaterange.However, a2 increasesin this 1.5—1.6 for largerdistances.rangebeforestartingto decrease,andif we calcu- As shownearlier(Kaganand Knopoff, 1985b),late the ratio of J2 for model (a) vs. (b), it the influenceof rotation on invariants is simple:changesfrom 1.3—1.4in the closerangeto 1.25 in the 3-D rotation causesa decreaseof invariants

1.8 ~

(a)

1.2 000

I o.~-

().6

+ * *c~++’’~..~~**+++ ~100 10’ 102 iO~ 10~ 1O~

Distance km

(b) —

1.4 -

1.2

O 0~ 00 0~ 1 0 000 00.0 00 00 0

00 ~ 00~

0.8- 0

0.6 0

0.4 00 +3 * +~ ~ +~

3*0* * * ** *++*+*+ ft *

0.: t1~+_*.:/~s:,:*. *,***0***0

—0.2 !~,!]!UIIl~!!l,!IIl!Il~l!IItI~!!!!t

10” tO’ 102 101 10~ 105

Distancekin

Fig. 3. Dependenceof invariants J1, J2, and J3 on distancefor the completeHarvard catalogof shallow earthquakes.(a)MagnitudethresholdM~� 5.0, N = 6392; (b) magnitudethresholdM~� 6.0, N= 1457. Invariants:circles—f3crosses—f2stars—J1. Line-typesare thesameasin Fig. 1.

________ _______________________ ________ __________________ 27

(a)

0,35 /•---. ....-..-•.••.-.

03

00: ~. . ~..

-0.05

10’s 10~ 10-’ io° to! ioz 10~ 1040.7

(b)

j ::

0~.i5~.2 10-’ 10° 10’ 102 10~

(c)1.8 - -

1.2 ~ ~

I ,,

C10~1 10.2

10-i 100 10’ 102 10~ 104Time days

Fig. 4. Dependenceof invariants(a) J1, (b) ‘2, and (c) J3 on time and distance for the completeHarvard catalogof shallowearthquakes.Solid lines correspondto thevaluesof invariants(5), dashedlines correspondto the standarddeviationsof invariants(seeeqn. (6)). Distancerange:circles—R� 156.25 km; crosses—156.25km <R � 625 km; stars—R>625km.

28 2 ,‘ Y.Y. KAGAN

(a)

1.5 -

i I -

05

0

10-2 10-’ 100 10’ 102 10~ 1O~

(b)

1.5 - - -

0 - 0 —

~°i~0.3 10.2 10” 10° 10’ 102 1O~ 104

2—

(c)

1.5 - - - - -- -

~0i5~~i 10-i 10-’ 10° 10’ 102 10’ 10~

Timedays

Fig. 5. Dependenceof the third invariant .13 on time anddistancefor theresidualHarvard catalogof (a)shallowearthquakes,(b)intermediateevents, and (c) deep earthquakes.Distancerange: circles—R� 156.25 kin; crosses—156.25km <R � 625 km;stars—R> 625 km. Line-typesare the sameasin Fig. 4.

THE GEOMETRY OF AN EARTHQUAKE FAULT SYSTEM 29

J1, J2, and J3 by the samefactor.However,as we nificantly non-zero. Similar changesare evidentseefrom Fig. 6, the influenceof disorientationon for smaller ranges. It is clear, in general, thatthe standarddeviations Cr1, U2, and a3 is more evenvery simple geometryof a fault systemcancomplicated.For example,a1 was zero for large producerather complicatedbehaviorof correla-distances,but after the rotation, a1 becomessig- tion invariants.Figure 7 confirms this conclusion

0 3 - riii’,r —i— i- .— r, ri,? — -—i -ri—i r r I - —rr-r rfl

(a) ~,

0.25

4

0.2

0.15 ~o ‘~~°ono~ :.0

0.1-

0.05 - + 0~.

*

0 - **.........pfl I

— — — — i U - — I — I .1 1_i - — I — 1~ .1. LLt~_ -—. -A. __L_.A A— L C _L - __L__L._J_.i_I. C

10° 10’ 102 10~ 10~ 105

Distancekm

0.~ ~ Yifl C —_ C — — - r m C ri

(b)

0.6

0.5

0.4

00 0 0

~ 0.3 . .. * 4*1-45) *

!/2

0.2 *

°~o •~•

0.1

— i_Si -‘ r *IC.. . I~ I •~•d-.~ ~_i ~-.

10° 10’ 102 l0~ 10~ 105

Distancekm

Fig. 6. Dependenceof the invariants (a) f, and (b) ‘2 on distance for simulated catalogs.Errors: x -signs—no errors;circles—rotation;stars—location;crosses—rotationand location.Line-typesarethe sameas in F~.1.

30 0.2 I I I I * I Y.Y. KAGAN

(a)

40 60 80 100 120 140 160180

0.7 —— I i

(b)

0 20 40 60 80 100 120 140 160 1802 I I I I I

(c}1.8 - - -- -

1.6

1.4 _ _

12

.~ 1 - - -

08

0.6

0.4 -

0.2 - _

024060 80 100 120 140160 180

Maximum rotationangle(degrees)

Fig. 7. Dependenceof theinvariants(a)J~,(b) .J~,and(c) j3 on themaximumrotationanglefor simulateqcat~1pgs:circles—linear

distribution (distanceR= 25000 km) crosses—planardistribution (distanceR= 39 1 km) stars—volumedistrib~ittqi~~1istanceR 13.8 km). Line-typesare thesameasin Fig. 1.

THE GEOMETRY OF AN EARTHQUAKE FAULT SYSTEM 31

as the invariant values are plotted against the med here. The results reported by Kagan andmaximumrotation angle ~~max(seeeqn.(9)). Simi- Knopoff (1985b), basedon the analysisof simu-lar resultsare obtainedfrom a Fisher-typerota- latedearthquakefaults andlocal catalogsof focaltional distribution (eqn. (10)). mechanisms,suggestthat in the beginningof an

As explainedin section 3, the maximumrota- earthquakerupture the fault zone is reasonablytion anglefor a double-couplecannotexceed120° well approximatedby a planardislocation. Thefor any orientationof the rotation axis, and for curves in Fig. 4 also support this interpretation:most of the axis orientation,the angle t1 should althoughshort-termshort-rangevaluesof J

1 seembe less than or equalto 90°.Therefore,it is not to suggesta 3-D volume-like distribution of rup-surprisingthat J1, J2, and J3 decreaseto zeroat tures, thevaluesof J2 and J3 indicatethe pres-a t1 value of about90—100°.The larger rotation enceof an almost coherentensembleof earth-anglescorrespondto almost randomdisorienta- quake foci, consistentwith a planar earthquaketion. Dependingon the distanceand, therefore, zone, possibly perturbedby location errors (seeon the prevalent fault geometry, the standard also Fig. 1(c)). However, if we declusterour cata-deviationsshow a complicatedpatternof behav- log of shallow events or use deep earthquakeior. data, it becomesclear that the earthquakefault

The third invariant (Fig. 7(c)) curvesare sim- systemis highly non-planar.The valuesof stan-pier to analyze, as they are independentof the dard deviationssuggestthat thereare very largefault scalar geometry. The comparisonbetween 3-D rotations of focal mechanismsof evencloseJ3 curves in Figs. 4(c) and 7(c) indicatesthat the earthquakes,and for distanceslarger than 100randomrotation causedby seismicmomentinver- km theserotationsarecomparablein their effectsion doesnot exceed20°.The aboveupper limit to completely randomrotationsof double-coupleis inferred, if we assignall source disorientation sources.However,the comparisonof resultsfromat time intervals 10_2_10_1days to randomer- synthetic and real earthquakecatalogssuggestsrors; most probably there are disorientations that the disorientationalone cannot explain thecausedby natural causesas well. Comparingthe rotationalfault geometry.curvesin Fig. 7 with the correspondingcurvesin Mathematicalsolutions for a crack propaga-Figs. 1, 4, and 5 showsthat, for small and espe- tion indicate an inherentinstability of a crack-cially medium distances,the relationbetweenJ3 pathwith regardto initial small deviationsof theand a3 cannotbe explainedby randomrotations cracksurfacefrom the original crack-plane(Rice,of sources:as a rule, in Fig. 7(c), the value of a3 1980, p. 589). Thesebranchingcrackspropagateis too high if we match the values of .13 in the at anglesof 60—70°to the original crack surface,simulatedand real catalogs.This confirms our hence the rotations are large. Andrews (1989)earliersuggestionthat a significant deterministic and Kagan (1990) also suggestedthat the largerotationof sourcesoccursin the earthquakefault rotations are common in propagationof brittlesystem.To obtain a value of a3 of the order of shearrupture. In most of the abovepapers(see0.8—0.9 (see Figs. 1—4), the rotation angle of also referencesin the Introduction), small or in-some of thesesourcesshouldbe at least 60—70° finitesimaldeformationof rockmaterialhasbeen(Fig. 7(c)). investigated.However, the formationof a mature

earthquakefault systemovertime periodsof mil-lions of yearsshould involve large deformation

with total displacementswhich are comparable5. Discussion with the size of the region.The earthquakefault

systemshould stabilize over such time periodsThe widely acceptedmodel of a shallow earth- into a statistically stationary,but evolving, self-

quakesourceregion is a plane-likevolume where organized (Bak and Tang, 1989; Chen et al.,a shearfailure occurs.This model is inconsistent 1991), geometricallycomplexpattern.Geologicalwith the valuesof the two-point invariantsexam- observations(see, for instance, Thompsonand

32 ~‘x. KAGAN

Prentice,1987;Changet al., 1990;andreferences earthquakesshouldbeconsiderablydifferent.Thetherein) indicate the presenceof largerotations presentfinding might indicatethat despiteessen-of microplatesand tectonic blocks in regionsof tial differencesin mechanicaland elasticproper-largetectonicdeformationsand earthquakefault ties of rocksat thesedepths,an earthquakerup-zones. It can be expectedthat focal mechanisms ture processis controlled by some generalgeo-of earthquakeswhich contributeto the formation metricalandmechanicalfeaturesof solidsdefor-of the seismogenicregions also undergolarge mation. This would imply that geometryof therotations. earthquakefault systemhas universalproperties

The above geometricalpatterns are average which shouldbe observednot only for variouspropertiesof the earthquakefault system, and seismic regionsbut possibly for failures of rockareusuallythe effect of a crustalslab descending specimensaswell.into the mantle.Therefore,seismic regionswith Most of the deformationat depthsexceedingdifferent deformationregimesmay exhibit other 70 km shouldbe affectedby processesother thangeometrical properties. Accumulation of focal brittle failure,for example,creepandplasticflowmechanismdatashouldsoonmake it possibleto (Frohlich, 1989).This might explainwhy the frac-testthis hypothesis. tal dimensionof deepearthquakehypocentersis

Therearesignificant differencesbetweenshal- significantlysmallerthan the shallow earthquakelow anddeepseismicity: (a) temporalbehaviorof dimension(Kagan,1991b),andthereforea set ofearthquakesequences—shalloweventsdisplay a hypocentersof deepearthquakesforms a signifi-patternof fore- and aftershocks,whereasdeep cantly sparsernetwork than do shallow events.earthquakesareusuallysolitary events(Frohlich, However,the plastic deformationat thesedepths1987, 1989;Kagan,1991a);(b) spatialdistribution doesnot significantly influence the geometricalof hypocenters—arefractal for bothshallow and patternof earthquakefocal mechanisms.deepseismicity (Kagan, 1991b), but the dimen- Frohlich (1989) arguedthat a plastic flow ofsion for shallow events is higher than that for rocksanda largelithostaticpressureshouldpre-deepearthquakes(2.2vs. 1.6). vent the formation and fusion of cracksat large

There are also someindicationsof differences depths;crack dynamics is an acceptedexplana-betweenintermediateanddeepearthquakes:they tion for earthquakefailure at shallow depths.havea different numberof aftershocks(Frohlich, Therefore, Frohlich (1989) reasonedthat deep1987, 1989), the temporal decay of aftershock earthquakesare causedby a processwhich issequencesseemsto be different for theseevents essentially different from that responsiblefor(Kagan,1991a),andtheir distribution with depth shallow events. However, we might reversethisvaries (Frohlich, 1989). Frohlich (1989)provided argument, to imply that the current standardargumentsin favor of the hypothesisthat deep modelsof earthquakefractureare fundamentallyearthquakesaredueto a fundamentallydifferent deficient in the way they describethe complexdeformationmechanismfrom that responsiblefor geometryof faulting or mechanicalprocessestak-shallow events,althoughhe found little evidence ing placein the fault system.that individual deepearthquakesdeviatesignifi- The classicaldislocationtheory, with a planarcantlyfrom the standarddouble-couplemodelfor surfaceof rupture, is clearly an over-simplifica-shallow events.Therefore,it is surprisingto dis- tion evenfor an individual earthquake(or a Se-cover that shallow and deepeventssharea simi- quenceof earthquakes),andespeciallyfor a sys-lar geometricalpatternof earthquakefocal mech- tern of interactingearthquakefaults.A model ofanisms. the fault systemshouldbe fully 3-D, with finite

The independenceof invariants from depth 3-D rotations of focal mechanisms.The role ofrepresentsa ratherimportantresult;judging from cracksand especiallyplanaror tensile cracksinsignificantly different conditionsat depthschang- an earthquakefracture also needs to be re-ing from 10 to 650 km, one would expect that evaluated.The planarfault model might be suit-correlationsof focal mechanismsof neighboring able to predict earthquakesusing a foreshock—

33THE GEOMETRY OF AN EARTHQUAKE FAULT SYSTEM

mainshock—aftershocksequence;however,if we relatively rare eventswhich occur on ‘hidden’were to forecastanothersequenceof eventsor faults (Stein and Yeats, 1989) or have ‘unusual’anothermainshock,a full 3-D considerationof focal parameters.the fault systemis necessary.We feel that alter-natives to the planar dislocationmodel or, in-

6. Conclusionsdeed,to any fixed-geometrymodel of earthquakerupture,are long overdue,andwe hope that the (1) We havestudiedthe spatial andtemporalresults reported here will help in the proposal invariants of a correlation function for seismicand developmentof such theories.The study of moment tensor solutions. The earthquakefaultthe stochasticgeometryof earthquakefaulting systemdiffers strongly from a single-planefaulttakesan additional importance,as it provides a which is the preferredmodel for earthquakefocalnecessarylink between an empirical statistical zones:the valuesof invariantsand their standardanalysis of seismicity and physical models of deviations indicate a rather high level of 3-Dearthquakerupturebasedon the nonlinearelas- rotation of focal mechanisms.Even a completelyticity theory and the continuum mechanicsof random rotation of double-couplescannot ex-defectsin rocksandother disorderedmaterials, plain the high values of standarddeviations,so

The resultsof this studymight also havemore we can assumethat in addition to the mecha-practicaleffects. In California, for example,seis- nisms’ disorientation,significantdeterministicro-micity is dominatedby strike-slip eventson the tationsare necessaryto model the fault interac-San Andreasfault and faults parallel to the San tion. Some of the theserotations shouldbe atAndreas.However, out of about 12 eventswith least60—70°.SometheoreticalinvestigationsandM

5 � 6.0 occurring in California since 1971, four geologicalstudiesof earthquakefault zonessug-(one-third)haveasignificantvertical sourcecorn- gest that suchlargerotations arecommon.ponent. Among the latter events are the San (2) The spatial behavior of the invariants isFernando earthquakeof 1971, the Coalinga approximatelythe samefor earthquakesin differ-earthquakeof 1983, the PalmSpringsearthquake ent depth intervals: shallow, intermediate,andof 1986,andthe LomaPrietaearthquakeof 1989. deep.The only differencebetweenthe behaviorTheseevents exhibit a thrust-type deformation of the invariants in time is that shallow earth-which is equivalent to the strong rotation of a quakesareclustered,whereasclusteringof deeperprevalent strike-slip earthquakesource mecha- events is much less pronounced;as soonas wenism.Our resultsshow thatsuchrotations are the ‘decluster’ shallow seismicity,the invariants’tern-rule ratherthan the exception,andtheyoccur at poral propertiesbecomesimilar for earthquakesboth shallow and greatdepths.When moredata of all depths.This demonstratesthat the basicbecome available, it will be feasible to extend geometricalpropertiesof an earthquakerupturetheoretical calculations, such as in Table 1, to do not dependon depth,andthereforeare inde-morecomplicatedgeometriesof fault systems,as pendent of rheological properties of rocks orwell as to evaluate additional correlation func- lithostatic pressure.We speculatethat this is ations such as (7). This should allow usfurther to manifestationof the universal geometricalpat-differentiategeometricalpatternsof earthquake ternsof brittle deformationof solids.occurrenceand to developappropriatemodelsofan earthquakefault systemcombiningdetermin-istic and stochasticcomponents.Focal mecha- Acknowledgmentsnisms of past earthquakesshould enable us todefine quantitatively future earthquakerisk de- This researchwassupportedin partby a grantpendingon the fault andslip orientation.We can from the NationalScienceFoundation,EAR 88-thenuse theseresults to makequantitative pre- 04883. I am grateful to L. Knopoff and D.D.dictionnot only of thoseearthquakeswhichoccur Jacksonof UCLA for their valuablecomments.Ion known faults but to calculateprobabilities of thank Dr. J. Dunphy of the USGSfor sendinga

34 Y.Y. KAGAN

catalog of seismic moment tensor solutions in Kagan,Y.Y., 1991a. Likelihood analysisof earthquakecata-computer-readableform. This paper is Publica- logs. Geophys.J. mt., 106: 135—148.

Kagan,Y.Y., 1991b. Fractal dimensionof bottle fracture. J.tlOfl 3644of the Instituteof GeophysicsandPlan- NonlinearSci 1’ 1—16etaryPhysics,Universityof California, Los Ange- Kagan,Y.Y., 1992. Correlationsof earthquakefocal mecha-

les. nisms. Geophys.J. mt., submitted.Kagan,Y.Y. andKnopoff, L., 1985a.The first-orderstatistical

moment of the seismic moment tensor. Geophys. J.R.Astron. Soc., 81: 429—444.

References Kagan,Y.Y. andKnopoff, L., 1985b. The two-point correla-tion function of the seismicmomenttensor.Geophys.JR.

Altmann, S.L., 1986. Rotations, Quaternions and Double Astron.Soc., 83: 637—656.Groups.ClarendonPress,Oxford, 317 pp. King, G., 1983. The accommodationof large strainsin the

Andrews, D.J., 1989. Mechanicsof fault junctions. J. Geo- upper lithosphereof the Earth and other solids by self-phys. Res., 94: 9389—9397. similar fault systems:the geometricalorigin of b-value,

Apperson,K.D. andFrohlich, C., 1987. Therelationbetween Pageoph,121: 761—815.Wadati—Benioff zone geometryand P, T and B axesof Kröner, E., 1972. StatisticalContinuumMechanics.Springer,intermediateand deep focus earthquakes.J. Geophys. New York, 157 pp.Res., 92: 13821—13831. Le Pichon, X., Francheteau,J. and Bonnin, J., 1973. Plate

Bak, P. and Tang, C., 1989. Earthquakesas a self-organized Tectonics.Elsevier,London,300 pp.critical phenomenon.J. Geophys.Res., 94: 15635—15637. Martel, S.J. and Pollard, D.D., 1989. Mechanicsof slip and

Chang,T., Stock,J. andMolnar, P., 1990. Therotationgroup fracture along small faults and simple strike-slip faultin plate tectonicsand the representationof uncertainties zonesin granitic rock. J. Geophys.Res., 94: 9417—9428.of plate reconstruction.Geophys.J. mt., 101: 649—661. Michael, A.J., 1989. Spatial patternsof aftershocksof shallow

Chen, K., Bak, P. and Obukhov, S.P., 1991. Self-organized earthquakesin California andimplications for deepfocuscriticality in a crack-propagationmodel of earthquakes, earthquakes.J. Geophys.Res., 94: 5615—5626.Phys.Rev.,A, 43: 625—630. Monin, AS. and Yaglom, AM., 1971. StatisticalFluid Me-

Danilova, MA. and Yunga, S.L., 1990. Statisticalproperties chanics,Vol. 1. MIT Press,Cambridge,MA, 769 pp.of matricesin problemsof earthquakefocal mechanisms Monin, AS. and Yaglom, AM., 1975. StatisticalFluid Me-interpretation.Jzv. Akad. Nauk SSSR,Phys.Solid Earth, chanics,Vol. 2. MIT Press,Cambridge,MA, 874 pp.26(2): 137—142. Oppenheimer,D.H., Reasenberg,P.A. and Simpson, R.W.,

Dietz, L.D. andEllsworth, W.L., 1990. TheOctober 17, 1989, 1988. Fault plane solutions for the 1984 Morgan Hill,Loma-Prieta,California, earthquakeandits aftershocks— California, earthquakesequence:Evidencefor state ofgeometryof the sequencefrom high-resolutionlocations, stresson the Calaverasfault. J. Geophys.Res., 93: 9007—Geophys.Res. Lett., 17: 1417—1420. 9026.

Dziewonski, A.M., Ekstrom, G., Woodhouse,J.H.and Zwart, Pollard, D.D. and Segall,P., 1987. Theoreticaldisplacements0., 1991. Centroid-momenttensorsolutionsfor January— andstressesnearfracturesin rocks: with applicationstoMarch, 1990. Phys.EarthPlanet. Inter., 65: 197—206. faults, joints,veins,dikes,and solution surfaces.In: BK.

Frohlich, C., 1987. Aftershocks and temporal clustering of Atkinson (Editor), FractureMechanicsof Rock.Academicdeepearthquakes.J. Geophys.Res., 92: 13944—13956. Press,London,pp. 277—349.

Frohlich, C., 1989. The natureof deep-focusearthquakes. Rice, JR., 1980. The mechanicsof earthquakerupture. In:Annu. Rev.EarthPlanet. Sci., 17: 227—254. AM. Dziewonski and E. Boschi (Editors), Physics of

Frohlich, C. and Willemann, R.J., 1987. Statisticalmethods Earth’s Interior, Proc. lot. School Phys. ‘Enrico Fermi’,for comparingdirectionsto the orientationsof focalmech- CourseLXXVIII. North Holland, Amsterdam,pp. 555—anismsandWadati—Benioffzone.Bull. Seismol.Soc. Am., 649.77: 2135—2142. Scholz,C.H., 1990. The Mechanicsof EarthquakesandFault-

Gay, NC. andOrtlepp, W.D., 1979. Anatomyof a mining-in- ing. CambridgeUniversityPress,Cambridge,439 pp.ducedfault zone. Geol.Soc. Am. Bull., 90: 47—58. Segall, P. and Lisowski, M., 1990. Surface displacementsin

Giardini,D. andWoodhouse,J.H.,1984. Deepseismicityand the 1906 San Francisco and 1989 Loma Prieta earth-modesof deformation in Tonga subductionzone. Nature, quakes.Science,250: 1241—1244.307: 505—509. Segall,P. and Pollard, D.D., 1980. Mechanicsof discontinu-

Goter, S.K., 1987. Global distributionof moment-tensorfocal ous faults. J. Geophys.Res., 85: 4337—4350.mechanisms,1981—1985.World Map. US Geol. Survey, Sibson,RH., 1986. Ruptureinteractionwith fault jogs. In: S.Denver,CO. Das,J. Boatwright andC.H. Schulz(Editors),Earthquake

Kagan,Y.Y., 1990. Randomstressand earthquakestatistics: Source Mechanics. Am. Geophys.Union Monogr., 37,spatialdependence.Geophys.J. Int., 102: 573—583. 157—167.

THE GEOMETRY OF AN EARTHQUAKE FAULT SYSTEM 35

Sipkin, S.A., 1986. Estimationof earthquakesourceparame- Stein, R.S. and Yeats, R.S., 1989. Hidden earthquakes.Sci.ters by the inversionof waveform data:global seismicity, Am., 260(6): 48—57.1981—1983.Bull. Seismol.Soc. Am., 76: 1515—1541. Thompson,R. andPrentice,M.J., 1987.An alternativemethod

Sipkin, S.A. and Needham,R.E., 1989. Moment-tensorsolu- of calculating finite plate rotations. Phys. Earth Planet.tionsestimatedusingoptimal filter theory: global seismic- Inter., 48: 78—83.ity, 1984—1987.Phys.Earth Planet. Inter., 57: 233—259.