good indicators of regional maximum and minimum compres- sion directions. It is well known, however, that the P and T axes from a sin- gle fault plane solution ...
JOURNAL OF GEOPHYSICAL
RESEARCH, VOL. 89, NO. Btt, PAGES 9305-9320, OCTOBER t0, 1984
An ImprovedMethod for Determiningthe RegionalStress TensorUsing EarthquakeFocal MechanismData: Applicationto the San FernandoEarthquakeSequence JOHN W. GEPHART AND DONALD W. FORSYTH
Departmentof GeologicalSciences,Brown University
The orientationsof fault planesand slip directionsindicatedby a populationof earthquakefocal mechanismscan be used to determinebest fit regional principal stressdirectionsand R = (o:-o•)/(o3-ol), a measure of relativestressmagnitudes,underthe assumption of uniformstressin the sourceregion.This analysis allows for the possibilitythat failure occurson preexistingzones of weaknessof any orientation.In the
inversion we performa gridsearchof stressmodelsto findtheonewhichrequi•res thesmallest totalrotation of all the fault planesthat is neededto matchthe observedand predictedslip directions;the methodallows foi' errorsin orientationsof both the fault planesand slip directions.We have an objectivemeansfor identifying the more likely of the two possiblefault planesfrom each focal mechanismrelativeto a given stress model; thus we do not face the problemof ambiguityof nodal planeswhich plaguesother analysesof this kind. By using a grid searchof stressmodelsrather than a linearizationscheme,we are able to perform a realisticerror analysisand thus establishconfidencelimits for the preferredregional stresses.The method can be usedto investigatepossiblestressinhomogeneities during earthquakesequences by analyzingsubsets of the data population.The techniquehas been applied to 76 eventsfrom the San Fernandoearthquake sequencefor which we havefoundbestfit stresses (plungeand azimuth):o• = 7,187; o: = 27,281; o3= 62,84; andR = 0.65. The averagemisfit betweenthis stressmodeland all the data is about8ø, and all but eight of the aftershockshave misfitsof lessthan 20ø. These valuesare considerablylessthan the uncertaintyof the focal mechanismdeterminations;therefore significantstressinhomogeneitiesare not required by the data. Our analysisdoesnot supportthe suggestionof a changein stresses during the aftershocksequence,as proposedby otherson the basisof an apparentchangein focal mechanisms. INTRODUCTION
stress directions, by which the stress directions might be
In discussions of the significanceof focal mechanismsthe P, B, and T axes are sometimestaken as approximationsof the maximum oi, intermediate o2, and minimum o3 compressive stressdirections.One justificationfor this is that if the principal stressdirectionswere orientedalong the P, B, and T axes, the nodal planes would representorientationsof maximum shear stress,which might be reasonableloci for new fractures.Also, this determinationof principal stressdirectionsdoes not require the identificationof one of the nodal planes as the true fault plane, since the orientationsof the P, B, and T axes are fixed by the focal mechanism,independentof the choice of fault plane. If a number of focal mechanismsare compiled for a seriesof earthquakes in a region, then clusteranalysesmay be performed on the P and T axes to determineestimatesof regionalcompression and tension directions. Many workers, including Zoback and Zoback [1980] and Sbar [1982], while recognizing the iraprecisionof suchdeterminations,suggestthat groupingsof P and T axes from a numberof fault plane solutionsare generally good indicatorsof regional maximum and minimum compres-
inferred. McKenzie [1969] has shown that the maximum com-
pressive stressmay have an orientation anywhere within the dilatationalquadrantof the focal mechanism.In short, the principal stressdirectionsare poorly constrainedby a single fault planesolution. If, however, there are a variety of different focal mechanisms within a region of uniform stress,then both the principalstress directionsand a measureof relative stressmagnitudesmay be determined.This is possiblebecauseon each fault plane, slip occurs in the direction of resolved shear stress [Bott, 1959];
with this constraint,each observationplacesa strongrestriction on the stressesthat generatedthe fault motion. It happensthat each focal mechanismis consistentwith only a relatively limited family of stresstensors.By inspectingthe overlapof families of stresses associated with a number of focal mechanisms, we can
define the range of sti'esseswhich may have acted over the region. Our goal is to find the set of stressesthat is most nearly consistent with all the observed focal mechanisms.
A numberof workershave used inversetechniquesfor calculating the stresstensorfrom field observationsof the orientation of striations on exposed fault surfaces [Carey and Brunier, sion directions. 1974; Carey, 1976, 1979; Angelier, 1975, 1979a,b; Angelier It is well known, however, that the P and T axes from a single fault planesolutionmay vary significantlyfrom the principal and Goguel, 1978; Angelier et al., 1981a,b; Etchecopar et al., stressdirections [McKenzie, 1969; Raleigh et al., 1972]. In a 1981; Armijo et al., 1982]. Ellsworth and Zhonghuai [ 1980] body of rock under stress,slip can occuron preexistingzones have extendedthe methodto apply to earthquakefocal mechaof weaknessin a variety of orientationsrelative to the principal nisms; in their analysisthey must select one of the two nodal stresses.This means that a fault plane which coincideswith a planesfrom each focal mechanismas the true fault plane. In all zone of weaknessmay bear no simple geometricrelation to the of theseapplications,the orientationof the fault plane is taken as perfectly known, and the inverse method involves minimizing, in a least squaressense,the componentof shearstressperCopyright1984by theAmericanGeophysical Union. pendicularto the observedslip direction(or, equivalently,minimizing the sineof the anglebetweenthe observedand predicted Papernumber4B0901. 0148-0227/84/004B-0901
$05.00
slip directions)by adjustingthe orientationand relative magni9305
9306
GEPHART AND FORSYTH: REGIONAL STRESSESFROM FOCAL MECHANISMS
the hydrostaticor uniform pressurecomponentof stress;since this componentis not part of the stressdeviator(doesnot act to deform a body), we have no control on this value from our analysis.Also, sincewe are dealingonly with the directionsof slip on planes, we are fundamentallyconsideringonly stress ratios and not absolutemagnitudesof stress;this further reduces the numberof valueswe can achieveby one. In the end, with our approachwe can hope to solve for only four parameters, which collectivelyrepresenttwo thirds of all the informationin the stress tensor. It is convenient
Fig. 1. Equal-areaplot illustratingCartesiancoordinatesystemsfor principalstre'•;•;es and fault plane geometry.The assumedfault plane is solid and the auxiliary plane is dashed.The unprimedcoordinateaxes are parallelto the principalstressdirections.The primedcoordinateaxes dependon the focal mechanism: x•'= normalto fault plane,x2'= parallel to the intersectionof the nodal planes(B axis), x3'= normal to auxiliaryplane.If t•*•cdashedgreatcircleis takenas the fault planeand the solid one as t'i:•:'auxiliary plane, then the x•' and x3' axes are interchanged.The •: atrix [3 is comprisedof the angle cosinesbetween these two setsof axes, as shown.
for us to describe our results
(four parameters)as three principal stressdirectionsand the scalar quantityR, definedbelow, which specifiesthe magnitudeof o2 relative to Ol and 03. We are trying to find the principalstressdirectionsand this measure of relative stress magnitudes from a number of observed focal mechanisms. In order to do this, we must first
describe the family of stresstensorsthat is consistentwith a given fault plane geometry (a single fault plane and its slip direction).The following is a brief discussionof this problem. McKenzie [1969] and Bott [1959] have consideredthis matter at somelengthandpresentdifferentformulations. Considertwo setsof Cartesiancoordinatesas shownin Figure 1: an unprimedset fixed by the principalstressdirectionsand a
primedset fixed by the observedfault planegeometry(Xl' nortudeof the (uniform)principalstresses. This is not the appropri- mal to the plane, x2' parallel to the plane and normal to the x3' parallelto the slipdirection; herewe tempoate minimizationfor earthquakefocal mechanismdatabecauseit slipdirection, implicitly assumesthat the only errors are in the measurement rarily assumethat we know which of the nodal planesis the of the directionof slip on the plane, whereasthereis often sub- fault plane). The assumptionthat slip on the plane is in the stantialuncertaintyin the orientationof the fault plane as well. directionof the resolvedshearstressor, equivalently,that there Angelier et al. [1982] were the first to develop a technique is zero shearstresson the plane in the directionnormal to the as which allowed for error in the orientationof the fault plane, as slip, canbe expressed we have done here.
O'12 t = 0: O'1•11•21-JrO'2•12•22 -JrO'3•13•23
(la)
Armijo and Cisternas [1978] developed an alternative approachin which they assumedthat the data were perfectbut where o12' is the shearstressin the directionX2' on the plane thatthe stresstensorvariedin the regionof study.They then with normalXl'; Ol, 02, and 03 are the principalstressmagnifound the orientationof the principalstresses that minimizedthe variationsrequiredin the relative sizesof the stressesneededto fit the data perfectly. We find this approachunsatisfactory becausethereclearly are errorsin the observations, and nonuniformity in the stresstensormay involve variationin principal stressdirections,as well asvariationin stressmagnitudes. To date, all these inversion techniques, when applied to earthquakefocal mechanismdata, suffer from uncertaintyas to
tudes;and•]qis an anglecosinebetween theprimed(firstsubscript)and unprimed(secondsubscript)coordinates.Combining (la) with an expressionfor orthogonalityof the Xl' and x2' directions,
(•b) it follows that
(2a)
whichnodalplaneis the truefault plane.Thesemethodsrequire
that the investigatorselectthe preferrednodal plane from each fault plane solution. Of course, normally this is done on the Except for differencesin notation, this is equivalentto (14) of basisof knowledgeof the local geologyand tectonics.Often, McKenzie [1969]. The expressionon the left is the parameter however, there is no objective meansby which to make this identifiedby Etchecoparet al. [1981] and assignedthe label R, selection. The method we describe here enables an objective hence selectionof the preferrednodal plane from each focal mecha-
R = •2•
nism.
•12•22
THEORY
(2b)
Given the observedfault plane geometryand assumingthe Our objectiveis to gain as muchinformationas we can about principalstressdirectionsare known, (2) describes the magnithe best-fittingregionalstresstensorfrom observedearthquake tudesof the principalstresses that are consistent with slip occurfocal mechanisms,which tell us the orientationof fault planes ring in the x3' direction. Assumingthat (Jl•(J2•(J3, the value and the directionof slip on theseplanes(for now we will ignore of R is limitedby 0_
