JOURNALOF GEOPHYSICAL RESEARCH, VOL.102,NO.B8,PAGES17,909-17,919, AUGUST10,1997

Quantifying Anderson's fault types Robert W. Simpson U.S. GeologicalSurvey,Memo Park, California

Abstract. Anderson[1905] explainedthreebasictypesof faulting (normal, strike-slip,and reverse)in termsof the shapeof the causativestresstensorandits orientationrelative to the Earth's surface. Quantitativeparameterscan be definedwhich containinformationaboutboth shapeandorientation[C•ldrier, 1995], therebyofferinga way to distinguishfault-typedomainson plotsof regionalstressfieldsandto quantify,for example,the degreeof normal-faultingtendencies within strike-slipdomains.This paperoffersa geometricallymotivatedgeneralizationof Angelier's [1979, 1984, 1990] shapeparametersq and • to new quantitiesnamedA• andA w. In their simpleforms, A½ variesfrom 0 to 1 for normal, 1 to 2 for strike-slip,and 2 to 3 for reverse faulting,andAwrangesfrom 0 øto 60ø,60øto 120ø,and 120øto 180ø,respectively.After scaling, A½andAwagreeto within 2% (or 1ø),a differenceof little practicalsignificance,althoughAwhas smootheranalyticalproperties.A formulationdistinguishinghorizontalaxesas well as the verticalaxis is alsopossible,yielding an A½rangingfrom-3 to +3 and Awfrom-180 øto +180ø. The geometricallymotivatedderivationin three-dimensional stressspacepresentedheremay aid intuitionandoffersa naturallink with traditionalwaysof plottingyield andfailure criteria. Examplesare given,basedon modelsof Bird [ 1996] andBird and Kong [ 1994], of the useof

Anderson faultparameters A½ andA• forvisualizing tectonic regimes• defined b,.y regional stress fields.

Introduction

Anderson [1905, 1951] postulated a fundamental relation between the three basic fault types and the orientation of the causative

stress tensor

relative

to the Earth's

surface:

new

faults will be normal, strike-slip, or reverse depending on whether the maximum, intermediate, or minimum compressive principal axis, respectively, is most nearly vertical. Almost every introductory geology and structural geology textbook has figures displaying this relationship. Anderson also pointedout that the Earth's surfaceis a free boundary(no shear or normal traction), so that one principal axis of crustal stress tensorswill commonly be close to vertical and the other two axes close to horizontal, especially as the free surface is approached. Although not every crustal stress tensor of interest has this property (because of the effects of topography,heterogenity,and zones of structuralweakness), it is approximated commonly enough to make Anderson's observation a powerful simplification [e.g., Zoback et al., 1989; Zoback, 1992]. Because of the great utility of Anderson's fault types for understandingand characterizingtectonic regimes, geologists have extendedthese types to include transitionalcasessuch as

"strike-slip with normal" [e.g., Philip, 1987; Guiraud et al., 1989] and have devised ways to quantify these types with the help of tensor shape parameters (summarized by Cdldrier [1995, Table 3]). Coblenz and Richardson [1995] and Miiller et al. [1997] have shown the additional usefulness of a

quantitative parameter for averaging regional stressdata and for plotting and visualizing regional stressfields.

Stressfield mapsoften use scaledsymbolsor line segments to show length and orientation of the principal axes of the stresstensor at selectedpoints [e.g., Richardson et al., 1979; Bird, 1996]. Sometimes stress trajectories are plotted to display the orientation of maximum or minimum horizontal axes [e.g., Hansen and Mount, 1990; Philip, 1987, Figure 29; Bird and Li, 1996]. Such maps, although dense with information,are often difficult for the eye to assimilate. Maps that are contouredor colored using a quantitativemeasureof tectonic regime can greatly aid visualization [e.g., Miiller et al., 1997].

In this paper, I discuss a geometric approach to quantificationbased on two specific shapeparameters,q (or ß ) and gt, usedby Angelier [1979, 1984, 1990] in methodshe devised for inferring regional stress information from fault slip data. Although the new generalizedparameterspresented here are similar to othersthat have been previouslysuggested, in particular to 0 of Armijo et al. [1982], the geometric derivation offers insights that are not so apparent in more

analyticalapproaches.It shouldbe notedthat not all stress fields of interest will satify the assumptionof a near-vertical principal axes and that the presenceof preexisting planes of weaknesscan lead to faulting types different from Anderson's pure types [e.g., Bott, 1959; Cdldrier, 1995].

Generalizing Angelier's Shape Parameters Parameter Based on • The purposeof a shapeparameteris to convey, in a single quantity, information about the relative magnitudes of the

threeprincipal axesof thetensor.Forexample, if Crl>Cr2>cr 3 This paperis not subjectto U.S. copyright.Publishedin 1997 by the American GeophysicalUnion.

are the magnitudes of maximum, intermediate, and least principal stress, respectively, with compression being

positive, thenq = (0'2-0'3)/(0'1-0'3) is onepossible measure of Papernumber97JB01274.

shape. This definition essentiallycomparesthe magnitudeof 17,909

17,910

SIMPSON: QUANTIFYINGANDERSON'S FAULTTYPES

theintermediate axiso'2 to theotheraxes:if theintermediateThe first term k•l is an isotropictensorand the secondterm axis is closein size to the minimum axis, •p approaches 0; if closein size to the maximumaxis, •p approaches 1; if exactly

k2T• is a nonisotropic tensor.T½contains all theinherent

shapeinformation in T. Conversely,one can reconstructa halfwaybetween,•p = 0.5. Angelierpointsout thatfor many generalstresstensorin diagonalform knowingthe shaperatio

purposes,knowledgeof the shapealone permitsuseful •p,the maximumstressdifferenceo.•-o.3,andthe minimum inferenceseven in the absenceof completeinformationabout compressive stresso.3' NotethatAngelier'sreduction was carriedout without regardto which of the threeprincipalaxes theactualvaluesof o.],o.2,ando.3' In the definition of •p, only the relative sizes of the is vertical,so that •pcontainsno informationon that subject. principalstresses comeintoplay;no particular regardis given to which of the three principal valuesbelongsto the most Generalized Parameter A½ nearly vertical principalaxis. As will be shown,if one A generalization of •pcalledA½canbedevised thatincludes distinguishes the vertical axis during the derivation,one arrivesat a continuouslyvaryinggeneralizedshapeparameter shapeas well as informationaboutthe verticalprincipalstress which containsinformation about fault type, as well as the axisand,thereby,thefaulttype. To calculate A½for a stress shapeof thetensor.Calldrier[1995]usesthewords"intrinsic" tensor,define a right-handedcoordinatesystem(a,•,7) in and "vertical" to distinguishthesetwo kinds of stresstensor stressspace. Let the 7 axis coincidewith the (most nearly) vertical principal axis, and let the a and fi axes be the shape parameter. Angelier's •p arises naturally from his corresponding horizontal principal axes(for the moment,without regardto "reduction" of a generalstresstensor.The reductionis carried their relative magnitudes). Angelier'soriginal•preductioncanbe viewedgeometrically out as follows: Starting with a general stress tensor T expressed in diagonal(principalaxis) form with o.•>o.2>o.3 , in Figure 1. His first step, removal of an isotropic

shiftsa general pointP at (o.a'o.l•'o.r)parallel firstsubtract o.3,the minimum principal stress, fromall component, diagonalelementsandthendividethe new diagonalelements to theline o.a=o./•= o.7 (whichpasses through theorigin by the maximumstress difference (o.•-o.3)'Theendresultis

and the point (1,1,1)) until it interceptsone of the quarter

thatT canberewritten asT = k11+ k2T½wherekl = o.3,I isthe planesthat bound the positiveoctant (point Q in Figure 1, in thisexample thato./•is the minimumprincipal identitymatrix,k2 = o.]-o.3 andthereduced matrixT½ is assuming stresso.3). The shiftedpoint mustlie on one of thesequarter planes, because the smallest componentof the partially

given by

T½=

•p

(1)

0

7>

13=0

reduced tensor, whether it be o.a,o./•, oro.7,isnowzero.The next step,which corresponds to extractingthe multiplier k2, shiftsthe point along a line passingthroughpoint Q and the

7> 13> o=0 N

Figure1. Oblique viewof thethree-dimensional (o.a,o./•,o.r) stress axissystem in stress space, showing the stepsin Angelier's•preduction leadingto his definitionof shapeparameter •p. It is assumed thatthe

example stress tensor atpoint P started witho.r> o.a> o.?ß Notethata, •, and7areusedasshorthand for stresses o.a,o.&ando.?,Boldface N, S,andR stand forttiethree Anderson faulttypes (normal, strike-slip, and reverse), withthesubscript indicating whichof thehorizontal axes(o.aor o./•)hasthelargervalue.Thesmall

numbers at the cornersof theunit squares indicatethe valuesof A½asdefinedin thetext. PointP locatesthe originalunreduced stress tensor,Q showsits locationafterthefirstreduction step,andR is its locationafter the second step.

SIMPSON: QUANTIFYING ANDERSON'S FAULT TYPES

origin until it intercepts one of the edges of the unit square (point R in Figure 1). It must lie on an edge because the largest componentof the fully reducedtensoris 1. The shape parameter •bis now equal to the distanceof the point R along the edge of the unit squareto the nearestaxis. Regardlessof which of the six edgesthe point R endsup on, •b is always the distance

to the nearest

17,911

Na

N

Q

axis.

0

If we selecta starting pointon the crraxisin Figure1 and defineA½to measure thedistance traversed around theedges of the unit squaresin the directions shown by the small arrows, negative for clockwise distances and positive for

Sa

Sl•

counterclockwise, thenA½will rangefrom-3 to 3. Usingo•,

,/3,7 as shorthand for cra,cr/3,cry, respectively, andCrH to indicate the maximum compressive horizontal stress, the

correspondences between A½and•barelistedin Table1. If A½is to be calculated for stress tensors at twoseparated

%'

Ra : ,,

Ri•

points and stresstrajectories are not available to reveal by

continuitya consistentchoiceof o•and,/3axesfor the pair of points, then the horizontal axis with the larger principal Figure 2. Samestressspacegeometryas Figure 1 but viewed stressat each point (commonly referred to as SHmax ) can be looking downtheline era=crl3 = cry.The Anderson faultarbitrarily assignedto be the o• axis. With this assignment, type parameter A½ is the distance traversed around the

perimeter of thehexagon fromtheprojection of thecryaxis. everytensorhasan A½valuebetween 0 and3, andA½ still The small numbersat the cornersindicate the values of

contains information about the type of faulting to be expected. The relation to •b can be written as an equation: If normal, strike-slip, and reverse types are assigned index numbers n = 0, 1, and 2, respectively(n equals the number of

Other symbolsare the sameas in Figure 1.

principal components largerthancry),then (2)

A½= (n+ 0.5)+(-1)n(•p-0.5)

whereAcrangescontinuously from0 to 1 for normal,1 to 2 for strike-slip, and 2 to 3 for reverse faults. Essentially, the cornersseenin plots of ½ as it rangesthroughthe spectrumof tectonic deformation types [e.g., Philip, 1987, Figure 1] have

Na

N

beenremoved by forcingA½to increase monotonically across thespectrum.This simplified versionof A½ canbe usedto index all the pure and transitional fault types described by Philip [1987], Guiraud et al. [1989], and Calldrier [1995]. There is another geometric interpretation for the

generalized parameter A½. If thestress axissystem in Figure

Sa

1 is viewed looking toward the origin down the line

S•

era=o'/3=crywhichis symmetrically locatedbetween the three axes, it appearsin projection as shown in Figure 2. The

edgesof the unit squares now form a hexagon,andA½is proportional to the distance traversed along the perimeter of

the hexagon. AlthoughA½ has the virtueof preserving information

about both the relative

sizes of the axes and the

faulting domain, the path traced could be smoother.

Figure 3.

Table1. A½and•bCompared AS Range

Faulting Type AxisSize

-3 < A$

0 < A½ o•> [3

1< A• < 2

strike-slip

2 < A½< 3

reverse

•pValue

O'H

surface in the Coulomb criterion. The circle and regular hexagon will not change size as the plane of projection is

A0

a

Where q is Angelier's[1979, 1984, 1990] shapefactorand c•i-iis the maximum

horizontal stress axis.

Same stressspace view as Figure 2, showing

angle •. The distancetraced around the circumferenceof the circle by ß is definedto be the Andersonfault-typeparameter AW. In this projection,the circle could also representa yield surfacecorrespondingto the von Mises criterion, the regular hexagon could represent a yield surface for the Tresca criterion, and the irregular hexagon could representa failure

shifted fromera+ cr + cry= 0 to era+ cr + cry= C because

the first two criteria are independentof the mean stress. The irregular hexagon corresponding to the Coulomb criterion will, however, changesize dependingon C, except when the coefficient of friction equals zero, in which case the Coulomb criterion reduces to Tresca's and the irregular hexagon becomes regular.

17,912

SIMPSON: QUANTIFYING ANDERSON'S FAULT TYPES

Details are given in the appendix. The resultis that

GeneralizedParameterAw A smootherapproachis readily suggestedby this projected view; why not use polar coordinatesin the plane of Figure 3

C1=O';

(tiltedcylindrical coordinates in o.a, o.f,0.7 space) sothatthe

C2=-•0. r 3

COS •'I't =•(O'7--•) / o'r

distance around a unit circle in the projected plane is the measure of shape? This approach turns out to generalize Angelier's secondmeasureof shape gtwhich correspondsto his second method of reduction [e.g., Angellet et al., 1982; Angellet, 1990, Appendix III]. In the first step of this new

(4)

3

sin q•t =• (o.a - O'f )/ O'r aw=•F=ATAN2[O. a-Oo.f,•/-•(o. 7-•)]

reduction, thegeneral pointP = (0.a,o.f,0.7) is projected parallel to theline o.a =o.f = 0.7 untilit reaches theplane where ATAN2 is the FORTRAN arc-tangentfunction with perpendicular to this line that passes through the origin, range from-zrtozrwhich preserves quadrant information. Aw is identicalto Angelier'sshapeparametergt to withina phase

becomingpoint Q in Figure 3. The perpendicularplane has

equation o.a + o.f+ 0.7 = 0. Theprojection of P to thisplane differenceof a multipleof 2zr/3. It also has tectonicregimeis equivalent to converting a general stress tensor to selecting properties similarto AC, although Aw runsfrom -zr to zr (or-180 øto + 180ø)insteadof from-3 to 3 (Figure4).

deviatoric form, which is accomplishedby subtracting the

mean stress • = (o.a+ o.f+ 0.7) / 3 fromtheoriginal diagonal The polar radius o.r is also related to two common elements.

quantities

As before, a general stresstensor T in principal axis form can be written as the sum of a symmetrictensor cll and a

o'r=(2J2) 1/2=N/•'•oct

(5)

nonsymmetric tensorc2Tv/(whichis the deviatoricstress where J2 is the secondinvariantof the deviatoricstresstensor tensor) containingthe essentialshape information'

T = cll + c2Tv/

and •:octis the octahedralshearingstress[e.g., Jaeger, 1969]. Conversely, the original diagonal stress tensor can be recoveredfrom knowledgeof the mean stress •, the second

(3)

invariantJ2,andtheparameter Av/. Asbefore, in theabsence This is accomplished by introducing a new right-handed

of information about stresstrajectories, it is convenientto

(u,v,w)coordinate system in (o.a,o.f,0.7) space withthew assign the maximum horizontal stress to the a axis, so that 0 andn:.Armijoet al. [1982]defineda axisalongtheline o.a=o.f = 0.7 andtheu andv axesin the Av/will varybetween 0 = n:/2-Av/ basedon analyticalrather o.a+ o.f+0.7=0 planesuchthattheu, w and0.7axesare similarparameter coplanar. This (u,v,w) axis system is used to define a tilted than geometricintuition, which varies between-n:/2 and +rd2. cylindrical coordinate system in which the polar-coordinate An interestingparallel existsbetweenthe abovegeometric angle •F is measured counter-clockwise about the w axis derivation of Av/andthetransformations usedto convert color starting at the u axis as shown in Figure 3. The polar radius coordinatesin a three-dimensionalred-green-blue(RGB) color

o.risgiven byo.r=[(O'o• --•)2+(Off _•)2+(O. 7_•)211/2

spaceto hue-saturation-intensity (HSI) values [e.g., Fortner

radial extension

pure normal \

/

\

/

\

• •0

/

\

/ \

/ /

normal/ss

transition

•

o.x• ,._• \

/ /

,•O".,x•X'?\\ .-

0 •'• Alit/

purestrike-slip ....-v •-•-•-2(t--!•

•t

1.5 .............

reverse / sstranst ion

% • •

• //

/

\\

/

\ \ \ \ \

/

•

/

\ \

/

\

pure reverse constriction

Figure 4. A stressspacecomparisonof the deformationcategoriesof Philip [1987] with the values of A½

andAwviewed projected ontothezrplanedefined by o.a + o.tt+ 0.7 = 0. Identical categories (notindicated) are mirrored on the right-hand side for negative valuesof A½ andAw.

SIMPSON: QUANTIFYING ANDERSON'S FAULT TYPES

AS

• •) -- ; -•)

AW

((s•, c•,

of the quantifiedAndersonianfault type and to refer to either or both generically with A, using no subscript. Figure 5 shows the correspondencebetween some typical symbols used to plot stress tensors and the correspondingvalues of the A

0 0ø (1,0.5,0.5) 0.5 30 ø (1, O, -1) parameters.

Table2 andFigure4 summarize therelation between A (A• or A W)andthe variousgeneralized tectonicregimes.Names

used to describe the regimes are taken from Philip [1987]. Note

I

' •!•-

17,913

1

60ø

(0.5, 0.5,-1)

1.5

90 ø

(o, 1,-1)

that

the

transitional

domains

such

as "normal

with

strike-slip" are intended to identify a tectonic style in which both normal and strike-slip faulting are presentrather than to specifically signify oblique slip with components of both normal and strike-slipoffset on a single fault plane. Oblique slip can occur in any of the domains depending on the orientation of a preexisting fault plane relative to the stress field [Wallace, 1951; Bott, 1959; Cdldrier, 1995]. Preexistingplanes of weaknessprobably explain most of the observed cases of mixing of fault-types in the transitional regions. The "pure" statesin Table 2 and Figure 4, ignoring the presenceof the Earth's surfacefor the moment, have stress tensorsimposing pure shearin the engineeringsense,whereas

120 ø (-0.5, 1,-0.5) -+-

,,-+-,,

2.5

150ø

(-1, 1, 0)

3

180 ø (-1, 0.5, 0.5) the radial extension, constrictive, and transitional

Figure 5. A comparison of some typical stress tensor symbols used for plotting stress fields on maps with correspondingvalues of A½ and A•t. Lines without arrows indicate compression direction; dark lines without arrows indicate maximum compressiondirection. Lines with arrows on the ends (also dashed in the second column) indicate tension direction.

Circles in the second column, dashed for

tension, indicate the relative size of the vertical component. The tensors shown are deviatoric,

and the last column

represents the relative size of the vertical, maximum horizontal, and minimum horizontal axes. Symbols in the first column are variations on ones used by Richardson et al. [1979]; those in the second column are variations on ones usedby Bird [1996]. and Meyer, 1997]. This suggeststhat various colors in the RGB color cube could be used to display a volume in

(•a,•/•,o?,) space, or thatin theplaneof Figure3, color

states have

stress tensors which correspond to simple extension or contraction in the engineering sense, still ignoring the Earth's

surface.

Yield

Conditions

and Failure

The angular polar coordinate • in Figure 3 contains information about the tectonic regime. The radial polar

coordinateor, which equalsthe distancefrom Q to O, contains information about the magnitude of the deviatoric stressesand is closely related to several classic measuresof

failure.

The projection of stress space onto the plane

oa +o/• +o r =0, sometimes referred to as then plane,is commonlyusedin discussions of yield and failure [e.g., Fung, 1965, p. 142]. ComparingFigure 4 with Figure 34 of Jaeger [1969], we see that the circle correspondsto a yield surface matchingthe yon Mises criterion and the hexagon matchesa yield surfacein Tresca's criterion.

huescouldbe usedto represent faultingdomains(Aw) and saturation to convey information about deviatoric stress magnitudesand proximity to failure (Or).

Comparisonof A½and A•

Table 2. Comparisonof Parameters andDeformationTypes

A½ A•

•p

7

0

0

0

0

_oo

+n/2

radial extension

n/6

1/2

-3

+n/3

purenormal

•/3

0

-1

+•/6

normal/strike-slip transition

0

0



It may not appearat first sight that much has been gainedin

thedefinition of A Wthatcouldjustifytheadditional algebraic 0.5 complexity.

However, measuring shape by the distance



aroundthe circumference of a unit circleis a moresymmetric 1 and continuousoperation than using the distance around the



perimeter ofahexagon aswasdone forA½.Consequently, the 1.5 •/2

1/2

A wparametercan be expressedas a single analytical expression of the deviatoric stresses, rather than as a collection of separate expressions which must be pieced

purestrike-slip

2

2•/3 0

+1 •/6

strike-slip/reverse transition

5•/6

1/2

+3

-•/3

purereverse

•

0

+oo

-•/2

constriction



togetheras in the definitionof A½. In practicalterms,

however, if A•tisscaled torunfrom-3 to3,then it differs by 2.5 only2% at mostfromA½whencalculated for thesametensor. 3 If A½is scaled todegrees, thenthemaximum angular difference betweenthe two parametersif they are set equal is -1ø or less. This reflects the fact that a hexagon is a fair approximationto a circle for some purposes. Becauseof this similarity, it may

Deformation Types



Here •p is from Angelier [1979, 1984, 1990], y is from Calldrier

[1995], y=•tan(A•/-n:/2),0is fromArrnijo et al. [1982], O= rc/2-A•/. Deformation typesarefromPhilip [1987];typesin

be convenient to useeitherA• or Awfor practicalestimation anglebracketsdescriberanges.

17,914

SIMPSON:QUANTIFYINGANDERSON'S FAULTTYPES

In the von Mises criterion,yield is assumedto occurwhen

(0'2--0'3) 2+(0'3--0'1) 2+(0'1--0'2) 2=20'•

{Jr (in MPa)

(6)

where0'0 is a constant of thematerial.Thiscanbe shownto be equivalentto requiringthat [Jaeger, 1969,p. 93] "yield takeplacewhenthe elasticstrainenergyof distortion reaches a value characteristic of the material" or that the octahedral

shearingstress•'oct(which is the shearstresson a plane equally inclined to all of the principal axes) exceedsa threshholdvalue [Nadai, 1950] which in termsof 0'0 equals

x/'•0'0/3.Tresca's condition, whichproduces a hexagonal yieldsurface, assumes thatyieldwill occurwhenthemaximum shearingstress 0'mxsh =(0'1-0'3)/2 at a point exceedsa threshholdvalue [Jaeger, 1969].

Changing coordinate systemin stressspaceas described in

the appendix,a generalstresspoint (0'a,0'/•,0'y) in

rectilinear coordinates becomes (0'r,W,x/•-•) in therelated cylindricalcoordinate system.It is not difficultto showthat

0'r=•-•'oct =[csc{mod(•,-•) +'•}]X/-•0.mxsh (7) 0-1

A½'

= NORMAL

1-2 = STRIKE-SLIP 2-3

= REVERSE 0

10

KM

Figure7. Contours of 0'r--'•J-'•'oct calculated using theslip model of Wald et al. [1996]. The values are not likely to be accurate,based as they are on a choice of elastic parameters

(given in the text) which are not representative of heterogeneous near-surfacematerials. However, the locations of the contours probably give a qualitatively correct impressionof the locusof largeststaticstresschanges.

where mod is the modulusfunction. Thus 0'r is equal to a multiple of the octahedralstressand approximatesa multiple of the maximum shearingstress,becausethe factor dependent on ß (in brackets)varies by only 15% from 1.0 to 1.15 over its range. This factor reflectsthe changein distancefrom the origin of points on the sides of a hexagon as one travels around the perimeter. As Jaeger [1969, p. 97] points out, projecting yield

surfaces ontothe 0'a+0'/•+0'7=0 planeonlyworksfor

0

criteria which do not dependon the mean normal stress. This is not true of the Coulomb-Navier failure criterion, commonly applied to explain frictional faulting of rocks. This criterion has a faceted conical shapein stressspace [e.g., Sokolovskii, 1965, p. 7, Figure 3; Scott, 1985, Figure 5; Scholz, 1990, Figure 3.1] unlessthe coefficient of friction equals zero, so

10 KM

Figure 6. Contoursof A½ for the staticstresschangesat the Earth'ssurfacepredictedusinga modelof Wald et al. [1996] thatviewedin a slicecutby plane0'a+ 0'/•+ 0'7.=const,it for the 1994 Northridge,California,earthquake.It is assumed will appearas the symmetricbut irregularhexagonin Figure4. here that at the surface these changeswere larger than the

magnitudeof the existingregionalfield; at depthone would needto add thesechangesto an existingstressfield. (See text for details.) The stress changes were calculated using dislocationsin an elastichalf-space[Okada, 1992]. The grid showsthe separatedislocationelementsof the model. Short lines indicate the maximum horizontal compressive stress direction.

Beach balls

at selected

locations

show the

orientation of the stress tensor; they present redundant

The size of the hexagondependson the slice, and the degreeof irregularity depends on the coefficient of friction: if the coefficient is zero, the Coulomb-Navier

criterion becomes

Tresca's criterion, the hexagon becomes regular, and the conical shapebecome a cylinder. For the von Mises criterion (circle), the proximity to failure is exactly relatedto the distanceof a point from the origin in

Figure 4.

For the Tresca criterion (hexagon) and for the

informationbut are plottedbecausethey help to distinguish Coulomb-Navier criterion (irregular hexagon), especially for the tectonic

domains.

low values of coefficient of friction, the distance from the

SIMPSON: QUANTIFYING ANDERSON'S FAULTTYPES

160 ø

(a)

170 ø

0

1800

o.s

1700

1

160 ø

1500

1.s

17,915

1400

130 ø

2

120 o

2.5

110 ø

3

A•

I

NORMAL 160 ø

(b)

\•

%,

!

170 ø

1600

/

..-v'l •

.t'

/

,,,

•-..--_-, "' • ,/_'-_'._ -,

/'

I .. x

Q,--.- -,. ' -\-

.,,.

..

,, • •.

xx

.

160 ø

REVERSE

150 ø

140 ø

130 ø

I 120 ø

110 ø

.. ./