Determination of the Tectonic Stress Tensor from Slip Along Faults with Coulomb Yield Condition

TECTONICS,VOL. 6, NO. 6, PAGES849-861, DECEMBER1987

DETERMiNATION OF THE TECTONIC STRESS TENSOR FROM SLIP ALONG FAULTS THAT OBEY THE COULOMB YIELD CONDITION

Ze'ev Reches

Departmentof Geology,Hebrew University, Jerusalem

Abstract. A new method to calculate the stress tensor symmetryof the measuredsets, the magnitudeof the dihedral associatedwith slip alonga populationof faultsis derived angleof the conjugatesets, utilization of known friction here.The methodincorporates two constraints:First, the anglesand assumedpore pressure [e.g., Zoback and Zoback, stressesin the slip directionsatisfy the Coulombyield 1980]. Most faultsare obliqueslip andmany fault populations criterion,and second,the slip occursin the directionof cannotbe separatedinto clear,symmetric sets. Therefore some maximumshear stress along the fault. The computations investigationsare basedon determinationof the strainaxes of providethe completestress tensor (normalized by the vertical shorteningand extensionassociated with the individualfaults stress),and evaluationof the meancoefficient of friction and or the entirepopulation [e.g., Arthaud, 1969; Reches, 1976; the meancohesion of the faultsduring the time of faulting. Aydin andReches, 1980]. This approachis equivalentto the The presentmethod is appliedto threefield cases:Dixie P-B-T axesanalysis of focalplane solutions [Angelier, 1984]. Valley, Nevada,Wadi Neqarot,Israel, and Yuli Severalstress inversion methods recently derived assume, microearthquakes,Taiwan. It is shownthat the coefficieetof following Bott [1959], that slip along a fault occursin the frictionof the threecases varies between g < 0.1 to g = 0.8. directionof the resolvedshear stress or, equivalently,normal Further,it is demonstratedthat previousstress inversion to the directionof zero shearstress [e.g., Carey and Bruiner, methodsimplicitly assumezero frictionalong the faults. 1974; Angelier, 1979, 1984; Armijo et al., 1982;Ellsworth, 1982; Gephartand Forsyth,1984; Michael, 1984]. These INTRODUCTION methodsdetermine the orientationsof the principalstress axes which minimizethe angulardeviations between the observed The resolutionof the stateof stressassociated with slip slip directionalong a fault and the directionof the maximum alonga collectionof faults,has been the goal of numerous resolvedshear stress determined from thecalculated principal studiessince the work of Anderson [1942]. Some stressaxes. However, the inversion "..method does not ensure investigationsof fault relatedstresses are based on the thatthe observedfault planesare consistentwith a masonable failure criterion."[Gephart and Forsyth,1984, p. 9314]. To overcomethis inherent difficulty, Gephart and Forsyth [1984] Copyright1987 and Michael [1984] determinedthe reducedstress tensor and by the AmericanGeophysical Union. theninspected whether the failure criterionis satisfiedalong each of the observed faults. PaperNumber 7T0614. A new method to determine the state of stress associated 0278-7407/87/007T-0614510.00 with slipalong faults is presentedhere. The analysisis based $50 Reches:Determination of the TectonicStress Tensor on two constraints:First, the slip alonga fault occursin the Formulation directionof maximumshear stress (as in previousinversion methods),and second,the magnitudesof the shearand normal The knownparameters for eachfault are fault planeattitude, stresseson the fault satisfythe Coulombyield criterion.This orientationof the slip axis and the senseof slip (normalor methodprovides the orientationsand magnitudes of the reverse).These parameters are represented by two unit vectors principalstresses and constrainsthe coefficientof frictionand onenormal to thefault N i, i= 1, 2 and3, andthe second the cohesion of the faults. parallelto theslip axis S i, i=l, 2 and3, whereN i andS i are I presentbelow the theoryand method of computation, thedirectional cosines in an orthogonalcoordinate system, X i applythe methodto threefield casesand compare the present (Figure1). In thepresent paper, X 1 pointsnorthward, X 2 techniquewith previousmethods. pointseastward, and X 3 pointsdownward. The geometricrelations of directionalcosines indicate that

THEORY N12+ N22+ N32=1 S12+S22+S32=1 (2) Approach N1S1 + N2S2 + N3S3 = 0

The objectiveis to determinethe stresstensor which fits slip alongall faultsin a measuredgroup. The methodis based The unknownstress components are o11, o22, o33, x23, on the following assumptions: x13,and x12. 1. The slip alonga fault occursin the directionof maximum Application of the Assumptionsto the Known Parameters. GivenS i asthe slip axis and B i is theaxis normal to it on the shearstress or, equivalently,normal to the directionof zero shearstress (as in previousinversion methods). fault plane,then B = N x S, wherex indicatesvector multiplication.By following the stressanalysis of Jeagerand 2. The magnitudesof the shearand normal stresses on the Cook [1969, chap.2] and usingthe geometricrelations of (2), fault satisfythe Coulombyield criterion, assumption1. becomes

Ixl> C + IXo n (1)

Xl wherex is the magnitudeof the shearstress in the slip direction,C is cohesion,!x is coefficientof frictionand o n is the normal stresson the fault. No assumptionis madeabout the age of the faults:They may be new faults,and thenC is the cohesivestrength of the intactrock and g is the coefficient of internalfriction, or they may be old, reactivatedfaults, and then C is the cohesiveresistance to slip and Ix is the coefficient of friction. 3. The slip eventoccurred under relatively uniform conditions: The cohesion and friction of the measured faults canbe representedby their meanvalues, and the faultswere active under a uniform state of stress. The calculationsprovide a set of stresstensors, each one of them being the best fit tensorfor a given coefficientof o slip axis • friction.The anglebetween observed and predicted slip axes is calculatedand is regardedas an estimateof thedegree of misfit Fig. 1. Coordinatesystem and fault orientationdata, of the solution.One selectsthe most suitabletensor by stereographicprojection, lower hemisphere. X 1 points consideringboth a reasonablecoefficient of frictionand a low northward,X 2 pointseastward and X 3 pointsdownward. Ni degreeof misfit.The advantagesof theprocedure are andS i arethe directional cosines of thenormal to thefault and demonstrated below. the slip axis. Reches:Determination of the TectonicStress Tensor 85 '1

(Oll - o33)N1B1+ (022 - o33)N2B2+ x23(N2B3 + B2N3) + [(N1S1 - gN12)]k,[(N2S2 - gN22)] k , [(N2S3+ S2N 3 - x13(N1B3 + B1N3)+ x12(N1B2+ B1N2) =0, (3) 2BN2N3)]k, [(N1S3+ S1N3 - 2gN1N3)]k , [(N1S2+ S1N2 - 2gN1N2)]k and assumption2. becomes

(ø11- ø33)N1S1+ (g22- g33)N2S2+ g23(N2S3+ S2N3)+ wherethe subscripts1 andk indicatethe parameters associated x13(N1S3 + S1N3)+ x12(N1S2+ S1N2) (4) = C + g[(Ol1 - ø33)N12+ (ø22- ø33)N22 + ø33+ withthe first and kth faults, and k rangesfrom 1 toK. The vector D of the unknown stresses has the form 2x23N2N3 + 2x13N1N3 + 2x12N1N2]. (o11 - ø33),(ø22- ø33),x23,x13,x 12 (6) Equation(1) appearshere with an equalitysign to facilitate the formulationof a systemof linearequations. The vector F has the form By writingthese two equationsfor eachof the K faultsin (7) the studiedset and afterrearrangement of (4), oneobtains a 0,0...... (C + go33),(C+ go33).... systemof 2K equations.This systemis the matrix multiplicationA x D = F, whereA is a 2K by 5 matrix,D is where the first K terms are zero and the last K terms are (C + a vector of unknowns stresses with five terms and F is a vector go33). with 2K terms. The matrix A has the form The systemA x D = F is an overdeterminedlinear systemin which A andF are knownfor the measuredfault and slip [N1B111, [N2B2] 1, [(N2B3+ B2N3)]1, [(N1B3+ B1N3)11, orientationsand the selectedg and C. The stressvector D can [(N1B2 + B1N2)] 1 be determinedby linearalgebra methods. The magnitudesand orientationsof theprincipal stresses, o 1, ø2 ando 3, canbe calculatedfrom the stress tensor oij. [N1B1]k,[N2B2]k, [(N2B3 + B2N3)]k , [(N1B3+ B1N3)]k , [(N1B2 + B1N2)]k (5) Computation

[(N1S1- gN12)]l,[(N2S2 - gN22)] 1, [(N2S3+ S2N 3 - The computationwas accomplishedin BASIC, usingan 2gN2N3)]1, [(N1S3 + S1N3 - 2gN1N3)]1 , [(N1S2 + S1N2 IBM-PC computer.The main stepsin the programare the - 2gN1N2)]1 following: 1. Selectingthe coefficientof frictiong; I selecta wide

Fig. 2. Tectonicfeatures in the Dixie Valley area,Nevada. Heavy solidlines indicate fault tracesactive in the 1903,1915, and 1954 earthquakes [after Thompson and Burke, 1973]; heavy arrows indicate the orientation of o3, theleast compressive stress axis calculated here (Table 1). 852 Reches:Determination of the TectonicStress Tensor

N N

o o o ø•o•øø____• _ o o

o Norm oShp axis •

OCTAHEDRAL SHEAR/VERTICAL STRESS MISFIT ANGLE 14 12 '

o6 o4 / 7 0.2 0.1 :O.l 05 05 07 0.9 013 01.5 0.7 09

MEAN COHESION STRESS RATIO

0.3

01 0.3 05 07 0.9 0.1 0,3 0.5 07 0.9

FRICTION COEFFICIENT

Fig. 3. Stresstensor analysis to a groupof 22 faultsin Dixie Valley, Nevada.Vertical bars in Figures3c-3f for ILt=0.8indicate the standarddeviations of theselected solutions. (a) Normalsto faultsand slip axes [after Thompson andBurke, 1973; Thompson, written communication, 1980]. Lower hemisphere, equal area projection. (b) Orientationof thethree principal stress axes o 1, ø2 and0 3, calculatedin thepresent method for themarked coefficientsof friction.Lower hemisphere,equal area projection. Solid symbols indicate the orientationsof the principalstresses of the selected solution of ILt=0.8:star for o 1, triangle for o 2 andrhomb for 03; thecircles around theprincipal stress indicate the standarddeviations of orientationsof theprincipal stresses. (c) Normalizedoctahedral shearstress SO (equation (9)). (d) Mean misfit angle between observed and calculated slip axes of allfaults. (e) Calculatedmean cohesion (marked as fractions of 033). (f) Stressratio as function of coefficientof friction. rangefor friction: 0.01 < !.t< 1.0. The meancohesion is set as 4. The six stresscomponents are usedto calculatethe C=0. magnitudesand the orientations of theprincipal stresses o 1, ø2 2. Calculatingthe coefficientsof matrixA (equation(5)) and ando 3 [afterJaeger and Cook, 1969, chap. 2]. vectorF (equation(7)). 5.The stress tensor oij is substituted into(4) for each fault. 3. Solvingthe overdeterminedsystem A x D = F for the five Theprogram calculates forthe k th fault the normal stress, the unknownstress components (the vertical stress 033 scalesthe shearstress in the slip direction,the cohesion magnitudesof all otherstresses). Following Schied [1968], the Ck='r k - la(On)k, and the misfit angle (the angle between the programdetermines the solutionwith the leastsquares method observedslip axis and the expected slip axis). The meanmisfit andprovides thebest fit tensor oij and the mean square error angleand the meancohesion are computed. RMS. 6. The resultsare displayedas plots of theoctahedral shear Reches:Determination of the TectonicStress Tensor $5 3

the maximumshear stress calculated for the faultsby the selectedsolution together with the assumedyield envelopeand the Mohr circles.If the calculatedslip directionalong a fault is the inverseof the observedslip direction,this particular fault is excludedfrom thedata set, and the computation is repeated. 8. The accuracyof the solutionsis estimatedfrom the mean square error.

FIELD EXAMPLES J The new methodis appliedto threecases: A setof 22 surfacesof normalfaults with slip striationsmeasured in Dixie .: : Valley, Nevada [Thompsonand Burke, 1973;Thompson, : ß •. : : .',<. i : C::O• ..'.i'•:" .:.": writtencommunication, 1980] (Figures2 and 3), a setof 16 right-lateraland left-lateralfaults with slickensidesmeasured in theWadi Neqarot,southern Israel (Figures 4 and5) [Eyal and Reches,1983] and 17 faultswith their 17 auxiliaryplanes determinedfrom aftershocksof the Yuli earthquake,Taiwan (Figure6) lAngelief, 1984]. /i//;// Thompsonand Burke [1973] studiedthe rate and directionof spreadingat the Dixie Valley area,Nevada (Figures 2 and3a). They determineda spreadingdirection of N55øW-S55øEfrom slickensidegrooves on old faultsin the bedrock,which is consistentwith the spreadingdirection of the recent 7 earthquakes.Zoback and Zoback [1980] showedthat the Fig. 4A. Simplifiedtectonic map of Sinai-Israelwith the slickensidegrooves measured in Dixie Valley agreewith the stresstrajectories of theSyrian Arc stressfield [afterEyal and regionaldistribution of stressdirections determined from Reches, 1983]. earthquakes,hydrofracturing and fault slipdata in northcentral Nevada. stresses,the meanmisfit angle,the meancohesion, the stress Eyal andReches [1983] determinedthe paleostress ratio, and the orientationsof the principalstresses as functions trajectoriesin Sinai-Israelsubplate by analyzingthe regional of the coefficient of friction. patternsof mesostructures.They detectedtwo regional 7. A Mohr diagramis plottedusing the normalstress and paleostressfields: One field is associatedwith the SyrianArc

Majorfault

•o•

Fig. 4B. Tectonicfeatures in theproximity of theWadi Neqarot station [after Bartov, 1974]. 8514 Reches:Determination of the TectonicStress Tensor

N N

ß o

ß O

b 0'•0':•' ""'".10

OCTAHEDRAL SHEAR / VERTICAL STRESS MISFIT ANGLE

o16

•10

• 06

•04 0 15 I i I 0 013 05 ol.? 019 0 3 0 5 0.7 0.9

MEAN COHESION STRESS RATIO

g 10

_ • 5

z o o

0ø_5 0 0.3 0 5 0 7 0.9 013 015 07 0.9 FRICTION COEFFICIENT

Fig. 5. Stresstensor analysis to a groupof 16 faultsin WadiNeqarot, Israel. Vertical bars in Figures5c-5f for g=0.6 indicatethe standard deviations of theselected solutions. (a) Normalsto faultsand slip axes [after Eyal and Reches,1983]. Lower hemisphere, equal area projection. (b) Orientation of thethree principal stress axes C•l, c•2 and c•3, calculatedin thepresent method for themarked coefficients of friction.Lower hemisphere, equal area projection. Solidsymbols indicate the orientations of the principal stresses of the selected solution of g=0.6:star for c• 1, trianglefor c• 2 andrhomb for c•3; the circles around the principal stress indicate the standard deviations of orientationsof the principal stresses. (c) Normalizedoctahedral shear stress SO (equation (9)). (d) Mean misfit angle betweenobserved and calculated slip axes of all faults. (e) Calculated mean cohesion (marked as fractions ofc•33). (f) Stress ratio as function of coefficient of friction. deformation(Senonian to Miocenein age)(Figure 4A), andthe regionof faults,flexures, and domes of the SyrianAre system, secondis associatedwith slipalong the Dead Seatransform southof the Ramonstructure (Figure 4B), yet the stresstensor (Mioceneto Recentin age).Eyal andReches [1983] foundthat determinedfor thisstation is in accordwith theregional field thepalcostress fields are uniform and essentially independent of (seebelow). local structures. This observation is demonstrated here for one Seventeenfocal planesolutions have been determined for of their 130 stations,where 16 small wrench faults were aftershocksof Yuli earthquakein easternTaiwan [Yu andTsai, measuredin Cenomanianrocks in Wadi Neqarot,southern 1982]. Angelier [1984] citedtheir data and determinedthe Israel (Figure5a). This stationis locatedwithin a complex reduced stress tensor for these focal solutions. The 17 focal Reches:Determination of the TectonicStress Tensor 855

N N

/t *

ß ß

o Normel • 0 SlipAxis(LowerH)',,,,,,. ß $hpAx•s (Upper H) • and Normal --

OCTAHEDRAL SHEAR/VERTICAL STRESS MISFIT ANGLE

60- • •o

50 -

40 -

• 05 30

•0 0 o 02 O4 05 0 012 0.4I 0 I6 , C

MEAN COHESION STRESS RATIO

0 • o,•

• -•o •1b•ø •

• - 30 *' •o-40 OI O OI 2 O14 O1.$ f O O.2 0.4 O.6 FRICTION COEFFICIENT Fig. 6. S•ess tensor•alysis to a groupof 17 faults•d 17 auxili•y plies of microseismicevents associated with Yuli e•quake, easternTaiwan. Vertical b•s in Fig•es •-6f for g=0.1 indicatethe s•ndard deviationsof the selfted solutions.(a) Norm•s to faul• and slip•es [afterAngelier, 1984]. Lower hemisphere,equal •ea projection.(b) Orientationof •e thr• principals•ess •es o1, ø2 and03, c•culatedin thepresem meted forthe m•ked c•fficien• of &iction.Lower hemisphere,equ• •ea projection.Solid sym•ls indicate•e orientationsof theprincipal s•esses of theselfted solutionof p=0.8:st• for S1, •ianglefor S2 •d rhombfor S3;•e circles around•e pfincip• s•essindicate the smnd•d deviationsof orientationsof theprincipal s•esses. (c) Normalized ocmhe&•she• s•essS o (equmion(9)). (d) Me• misfitangle between observed and c•culated slip axes of all faults.(e) C•culatedme• cohesion(m•ked as&actions of 033).(0 S•essratio as function of coefficientof friction. solutionsprovide the orientationsof 17 fault planesand their cohesionC=0. The solutionsare presentedby six diagrams 17 auxiliaryplanes. Twelve of the focal solutionsindicate which showthe variationsof the followingparameters with reverseslip, and five indicateoblique strike slip. the coefficient of friction: 1. The originalfault slip data (Figures3a, 5a and 6a).

ParameterInvestigation 2. The orientationsof the principal stressaxes (Figures3b, 5b and 6b). The solutionsfor eachcase were calculated for 0.01 < g < 3. The normalized octahedralshear stress, 1.0,vertical stress o33=1 (the scaling stress), and mean So=[(ø 1-ø2)2+(ø2-ø3)2+(ø3 -01)2]1/2/o33 (9) 85 6 Reches:Determination of the TectonicStress Tensor

which is a measure of the three dimensional stress differences (Figures3c, 5c and 6c). S O increases monotonously and nonlinearlywith !xin all threecases. 4. The meanmisfit angle between observed and expected slip axesof the faults(Figures 3d, 5d and 6d). This angleincreases nonlinearlywith Ix. 5. The calculatedmean cohesionalong the faults (Figures 3e, 5e and 6e). 6. The stressratio (Figures3f, 5f and60,

0 = (c•2-c•3)/ (c•1-c•3) (10)

Thisparameter has been used in previousstress determinations[e.g., Angelier, 1984; Michael 1984].

Selectionof a PreferredSolution.

Figures3, 5 and6 presentthe solutionsof the threefield casesfor 0.1 < Ix < 1.0 and C=0, andone has to selecta preferredsolution from this range. A preferredsolution includesthree components: (1) The frictioncoefficient Ix is as closeas possible to Ix--0.8[Byedee, 1978]. (2) The mean misfitangle is as smallas possible.(3) The meancohesion is slip resisting,C > 0. Dixie ValleyFaults. The faultsof Dixie Valley display steadybehavior: Orientations of theprincipal axes are stable (Figure3b), themean misfit angle is between7 ø and8 ø for all frictioncoefficients (Figure 3d), meancohesion is slip resistingfor all frictioncoefficients within the statistical confidencelimits (Figure3e), andall principalstresses are compressive.The steadybehavior of the solutionsreflects the simpleand clear pattern of thisgroup of faults(Figure 3a). I selecta solutionwith the commonlyused friction coefficient,Ix = 0.8 [Byerlee,1978]; this solution is markedin Figure 3b and listedin Table 1. WadiNeqarot Faults. The numerical solution forWadi Neqarotfaults differs from the solution for Dixie Valleyfaults. Theaxes of theprincipal stresses c•1 andc• 2 rotatesignificantly withIx (Figure 5b): Axis Ixl rotatesfrom plunging 13 ø toward 115ø for Ix=0.1,to plunging51 ø toward299 ø for Ix=i.0, axis ø2rotates in aninverted manner, whereas axis ø3 maintains approximatelythe same orientation for all Ixvalues. The mean cohesionis slipresisting for all frictionvalues (Figure 5e). Misfit anglesare relatively small (below 19 ø) for Ix< 0.6 and theyincrease faster for Ix> 0.6 (Figure5d). I selecthere Ix = 0.6 asthe preferred solution (dark symbols in Figure5b andTable 1), becausethe solutions for Ix < 0.6 fit the databetter than solutions with Ix > 0.6 (asreflected by thesmaller misfit angle in Figure5d), andon theother hand, Reches:Determination of the TectonicStress Tensor

N N o• 0øO

0 0

Fig. 7. Stresstensor solutions for two subsets(a andb), eachwith 17 planesof microseismicevents associated with the Yuli earthquake,eastern Taiwan. Each figure includes the normals to faults,the slip axes,and the orientationof thethree principal stress axes S 1, S2,and S 3 forg=0.1. Lower hemisphere, equal area projection. Data from Angelier [1984].

the coefficientof frictionis relativelyclose to the experimental angleis 24ø for g=0.1, 30ø for g=0.3, andincreases faster for resultsof Byeflee [ 1978].This processof somewhat > 0.3 (Figure6d). The solutionsfor g > 0.05 yield negative "subjective"selection is discussedbelow. meancohesion, namely, slip supportingcohesion (Figure 6e); MicroseismicEvents of Yuli Earthquake.Solutions were suchsolutions should be rejected.Thus both the relatively determinedfor threesubsets of theseevents: all 34 planes, largemisfit anglefor g > 0.1 andthe negativecohesion for g includingfaults and auxiliaryplanes, and two subsetseach > 0.05 suggestthat the preferred solution is in the range0.0 < with 17 planes.I calculatedthe setof 34 planesfor g < 0.1. I selectg=0.1 (dark symbolsin Figure 6b and Table comparisonwith the resultsof Angelier [1984], and the other 1) simplybecause I searchfor frictionas largeas possible. two subsetsfor investigationof the solutionstability. The resultsfor the othertwo subsets,each with 17 planes, The solutionsfor the34 planesof Yuli microearthquakes arepresented in Figure7. The orientationof thea 1 axisis providea restrictedrange for thepreferred solution: The misfit similarin all threesubsets (Figures 6b and7), whereasthey

• 40 80r • 20 •) •e 16 ß ß o20 ! .-- • ,o0 • I 2o•,.0 •• I I U3 20 30 40 50 60 3C 50 70 90 I10 95 100 105 I10 li5 120 NORMAL STRESS ( % of vertical stress)

a b c

Fig. 8. Mohr diagramsof the selectedsolutions. Solid circles are the predictedmaximum shear stress and the normalstress acting on eachfault; the large circles are the Mohr circlesof thedetermined principal stresses (Table 1); solidlines are the yield envelope ß = C + gan usedin thederivation (equation (7)); dashedlines are the linear regressionyield envelope.(a) Dixie Valley faults,C=0 andg=0.8. (b) Neqarotfaults, C=0 andg=0.6. (c) Yuli microearthquakes,C=0 andg=0.1. 858 Reches:Determination of the TectonicStress Tensor

differin theorientations of theo 2 ando 3 axes.The orientationsof o 2 ando 3 interchangepositions in a plane normalto the Ol axis(Figure 7).

The Yield Condition

The relations between the calculated stressesand the assumed Coulombyield condition(equation (1)) is examinedhere. Figure8 displaysMohr diagramsof the selectedsolutions of the threefield cases.It includesplots of the maximumshear stressversus the normal stressfor each individual fault, Mohr circlesfor the bestfit principalstresses, the yield envelopes usedin thecalculations (x = 0.8 on for theDixie Valley case, x = 0.6 on forthe Neqarot case and x = 0.1o n forthe Yuli earthquakescase) and the yield envelopecalculated by linear regressionfor the shearand normal stresses. Goodagreement appears between calculated and expected yield conditionsfor Dixie Valley (Figure8a) andWadi Neqarot (Figure8b), and only fair agreementappears for Yuli earthquakes(Figure 8c). The goodagreement is apparentin the distributionof the calculatedstresses (solid circles in Figures 8a and 8b) alongthe selectedyield envelopes(solid lines in Figures8a and 8b), thegood correlation coefficients found in thelinear regression calculations of theyield envelopes (dashed linesin Figures8a and8b), andthe approximatetangential relationshipbetween the Mohr circlesand the yield envelopes. The agreementis only fair for the Yuli earthquakeswhere stressesfor mostfaults are closeto the yield envelopes(solid circlesin Figure8c) andthe slopeof theselected yield envelope,g=0.1, is essentiallythe sameas the slopeof the regressionenvelope.

ACCURACY OF 75t]E SOLUTIONS

The leastsquares solution of the overdeterminedlinear systemprovides the stresstensor (equation (6)), which minimizes the residual vector

R=AxD-F

whereA is a matrixcalculated from thefault slipdata (equation(5)), D is a vectorof the componentsof the stress tensor(equation (6)) and F is a vectorcalculated from the verticalstress, friction, and cohesion (eq. 7). For a setof K faultsthe vector R has2K terms,from rl tor2k. The accuracy of the solutionis representedby the rootof themean square error, Reches:Determination of theTectonic Stress Tensor 859

TABLE 3. Comparisonof Presentand Previous Results

Mean Mean Orientationsb Stress FrictionMisfit a o1 o3 Ratio

Dixie Valley Selectedsolution (present) 0.8 7.8 71/137 18/307 0.08 Specialsolution (present) 0.01 7.0 81/224 1/123 0.39 Michael [1984] 5.6 tO 124 0.48 Yuli earthquakes Selectedsolution (presen0 0.1 24 11/135 66/255 0.08 Specialsolution (present) 0.01 23 9/135 77/276 0.16 Angelier[1984] (first) 30 5/121 82/249 0.2 Angelier[1984] (second) 26 2/131 81/028 0.0

a In degrees. b Plunge(in degrees)/Trend(in degrees).

RMS= [ ( • r2k ) / 2K} 1/2 meancohesion in thesecases hardly depends on the friction coefficients(Figures 3e and5e). The mean cohesion of the whererk is the residual ofthe kth fault [Scheid, 1968, chap. Yuli earthquakesfor g=0.1is slightlynegative, C = (-0.042+ 28]. TheRMS is regardedas an estimate of thestandard 0.027)033(Figure 6e). The difference between this value and deviation of the best fit solution. theinitial cohesionC=0 suggeststhat a bettersolution for the Themagnitudes and orientations of theprincipal stress axes Yuli earthquakesshould probably be with g < 0.05. dependnon linearly on F: Thesolutions for F 1 = F + RMS andfor F 2 = F - RMSare not symmetrical with respect to the DISCUSSION solutionfor F, andthus three sets of solutionsare calculated, forF, F1 , andF 2, forthe selected friction. The results of the Comparisonwith Previous Methods threesolutions are shownin Figures3, 5, and6 andlisted in Table 2. Two of thefield examples were analyzed previously, Dixie The deviationsof themagnitudes of theprincipal stresses Valleyby Michael[1984] and Yuli earthquakesby Angelier rangesfrom 1% to 17%of thevertical stress 033, with mean [ 1984]. Table 3 liststhe previous results, the selected deviationof 5.8% (Table2). Thusthe magnitudes of the solutionsof thepresent method and solutions of thepresent principalstresses of thebest fit solutionsare good estimates of methodfor g=0.01. the"true" magnitudes in thethree field cases. As in thepresent study, Michael [1984] assumed that the Theangular deviations of theaxes orientations of Dixie resistanceto slipalong the faults can be representedby a mean Valleyand Wadi Neqarot range from 7.7 ø to 20.9ø (Table2 and cohesionand a mean friction coefficient. His formulation, Figures3b and5b). For the Yuli microearthquakescase, the whichis basedon theassumptions of uniformshear stress angulardeviations for o 1 axisare up to 15.2ø, and o 2 ando 3 alongthe slipping faults, is similarto thepresent formulation interchangetheir positions in a planenormal to o 1 (Figure (compareequations 3 and 4 of Michael,1984, with present (5) 6b). Thisinterchange of stressaxes is alsoreflected in the and(6)). The resultsof Michaelfor theDixie Valley caseare essentiallyidentical magnitudes of o 2 ando 3 (Table2) andin in agreementwith the results of thesolution for g=0.01 (Table the solutionsfor the two subsetsof this group(Figure 7). 3). Theagreement is goodfor theorientations of theprincipal Thus,o2 -• o3 forthe case of Yuli earthquakes, and0-•O. stresses[see also Figure 4 in thework by Michael,1984] and Figures3e and5e indicatethat the mean cohesion for Dixie theangle of misfit,and fair for thestress ratio (Table 3). Valleyand Wadi Neqarot may be regarded as zero, in agreement Michael'sresults are only roughly similar to theselected with the initial selectedcohesion. Further, it appearsthat the solutionof thepresent method for g=0.8(Tables 2 and3). 860 Reches:Determination of the Tectonic StressTensor

Angelier[1984] calculatedthe reduced stress tensor for the magnitudesof the threeprincipal stresses (in relationto the focal solutionsof the Yuli microearthquakesby usinghis verticalstress). As the absolutemagnitude of the verticalstress inversiontechniques. His results,including the orientationsof maybe estimatedfrom depth of faulting,the new method theprincipal stresses, the stress ratio 0 andthe angle of yieldsthe absolutemagnitude of the tectonicstress tensor. misfit,are in goodagreement with thesolution of thepresent The stresstensor determined here for Dixie Valley (Table 1) methodfor g=0.01 (Table 3). is in goodagreement with thelocal fault pattern (Figure 2) and The inversionmethods of Angelierand Michael do not it fits localand regional extension directions of centralNevada considerthe frictioncoefficient and the cohesionand they [Thompsonand Burke, 1973;Zoback and Zoback, 1980]. providea single,special solution. On theother hand, the Similarly,the stresstensor of Wadi Neqarot(Table 1) fits the presentmethod provides a collectionof permissiblesolutions. regionalSyrian Arc deformationof the Sinai-Israelsubplate The goodagreement found between the present solutions for (Figure4) [Eyal & Reches,1983]. Furthermore, the present g=0.01 (solutionfor g=0 is notpermitted for numerical methodprovides results similar to the stressinversion results reasons)and theresults of theprevious methods suggests that of Michael [ 1984] (Dixie Valley faults)and Angelier [ 1984] thesesolutions implicitly consider the caseof g=0. This (Yuli Microearthquakes).However, the results are in good suggestionis expected.Previous methods minimize the angle agreementonly whenthe present method considers g•-0, thus of misfit, and, as demonstratedhere, the smallestangle of suggestingthat previous techniques are valid for casesof very misfitis indeedfor g=0 (Figures3d, 5d and 6d). low coefficient of friction. Futureimprovement of the methodwill incorporate Solution Selection Procedure variationsin the pore-fluidpressure and consideration of non uniform cohesion and friction. While applyingthe present method to field casesI searched for solutions which include coefficients of friction as close as Acknowledgements.This studywas initiateda few yearsago possibleto measuredrock frictionand relatively small angles whenAtilla Aydin and I tried to resolve the stateof stress of misfit.This apparently"subjective" selection procedure associatedwith faultingfrom strainconsiderations and indicatesthe utilizationof two independentconstraints: the symmetryarguments. The studywith Atilla Aydin and yield criterionand the slip in maximumslip direction. stimulatingdiscussions with Paul Delaney significantly One mayuse an "objective"selection procedure which is contributedto thepresent work. GeorgeThompson kindly basedon one constraint,for example,minimizing the angleof providedsome of hisand Dennis Burke's unpublished data on misfit (similarlyto Gephartand Forsyth [ 1984], Angelier faultsin Dixie Valley, Nevada.John Gephart, Andy Michael [1984] andMichael [1984]). However,minimizing the angle andan annonymouscritic reviewed the manuscript and made of misfit providesno controlon the frictioncoefficient, and in manyhelpful comments. I benefitedfrom conversationswith thepresent field examplesit wouldlead to nonrealistic JohnGephart, Mary-Lou Zoback, Marie Jackson-Delaneyand solutionsof g=0 (Figures3, 5 and 6). Thereforethe selected JacquesAngelier. The studywas supported by a grantfrom solutionreflects the choicebetween two options:smallest the CentralResearch Fund of the HebrewUniversity, misfit anglewith a nonrealisticzero frictionor somewhat Jerusalem. largermisfit anglewith reasonablefriction coefficient. I prefer REFERENCES the second.

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