2003 Delvaux-Sperner Tensor Program

8/10/2019 2003 Delvaux-Sperner Tensor Program

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ew aspects tectonic stress inversion withreference to the T SOR program

D. DELVAUX 1,2 B. SPERNER 3

Royal Museum r Central Africa, Department Geology-Mineralogy, B-3080 Tervuren,Belgium e-mail: [email protected]@skynet.be)

2Present address: Vrije University, Amsterdam, The Netherlands3Geophysical Institute, Karlsruhe University, Hertzstrasse 16, 76187 Karlsruhe, Germany

bstract Analysis of tectonic stress from the inversion of fault kinematic and earthquake focalmechanism data is routinely done using a wide variety of direct inversion, i terative and gridsearch methods. This paper discusses important aspects and new developments of the stressinversion methodology as the critical evaluation and interpretation of the results. The problemsof data selection and separation into subsets, choice of optimization function, and the use ofnon-fault structural elements in stress inversion (tension, shear and compression fractures) areexamined. The classical Right Dihedron method is developed in order to est imate the stressratio R widen its applicability to compression and tension fractures, and provide a compatibilitytest for data selection and separation. A new Rotational Optimization procedure for interactivekinematic data separation of fault-slip and focal mechanism data and progressive stress tensoroptimization is presented. The quality assessment procedure defined for the World Stress Mapproject is extended in order to take into account the diversity of orientations of structural dataused in the inversion. The range of stress regimes is expressed by a stress regime index R ,useful for regional comparisons and mapping, All these aspects have been implemented in acomputer program TENSOR, which is introduced briefly. The procedures for determination ofstress tensor using these new aspects are described using natural sets of fault-slip and focalmechanism data from the Baikal Rift Zone.

Analysis of fault kinematic and earthquake focalmechanism data for the reconstruction of past andpresent tectonic stresses are now routinely done inneotectonic and seismotectonic investigations.Geological stress data for the Quaternary period areincreasingly incorporated in the World Stress Map(WSM) (Muller Sperner 2000; Muller et at.2000; Sperner et at. 2003 .

Standard procedures for brittle fault-slip dataanalysis and stress tensor determination are nowwell established (Angelier 1994; Dunne Hancock 1994). They commonly use fault-slip data toinfer the orientations and relative magnitude of theprincipal stresses.

A wide variety of methods and computer programs exist for stress tensor reconstruction. Theyare either direct inversion methods using leastsquare minimization (Carey-Gailhardis Mercier1987; Angelier 1991; Sperner et at. 1993) or iterative algorithms that test a wide range of possibletensors (Etchecopar et at. 1981) or grid searchmethods (Gephart 1990b; Hardcastle Hills 1991;Unruh et at. 1996). The direct inversion methodsare faster but necessitate more complex mathemat-

ical developments and do not allow the use of complex minimization functions. The iterative methodsare more robust, use simple algorithms and are alsomore computer time intensive, but the increasingcomputer power reduces this inconvenience.

This paper presents a discussion on the methodology of stress inversion with, in particular, the useof different types of brittle fractures in addition tothe commonly used fault-slip data, the problem ofdata selection and the optimization functions. Twomethodologies for stress inversion are presented:new developments of the classical Right Dihedronmethod and the new iterative Rotational Optimization method. Both methods use of the full rangeof brittle data available and have been adapted forthe inversion of earthquake focal mechanisms. Theinterpretation of the results is also discussed fortwo important aspects: the quality assessment inview of the World Stress Map standards and theexpression of the stress regime numerically as aStress Regime Index for regional comparisonsand mapping.

All aspects discussed have been implemented inthe TENSOR program (Delvaux 1993a which can

From: NIEUWL ND D. A (ed.) New Insights into Structural Interpretation and Modelling, Geological Society, London, Special Publications, 212, 75-100. 0305-8719/03/ 15 The Geological Society of London 2003.


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76 D. DELVAUX B. SPERNER

be obtained by contacting the first author. A program guideline is also provided with the programpackage.

Stress inv rsion m tho ologi s

Stress analysis considers a certain volume of rocks,

large enough to sample a sufficiently large data setof slips along a variety of different shear surfaces.The size of the volume sampled should be muchlarger than the dimensions of the individual brittlestructures. For geological indicators, relativelysmall volumes or rock 100-1000 m ) are necessary to sample enough fault-slip data, while forearthquake focal mechanisms, volumes in the orderof 1000-10 000 krrr are needed.

Stress inversion procedures rely on Bott s (1959)assumption that slip on a plane occurs in the direction of the maximum resolved shear stress.Inversely, the stress state that produced the brittle

microstructures can be partly reconstructed knowing the direction and sense of slip on variably oriented fault planes. The slip direct ion on the faultplane is inferred from frictional grooves or slickenlines. The data used for the inversion are the strikeand dip of the fault plane, the orientation of the slipline and the shear sense on the fault plane. Theyare collectively referred to as fault-slip data. Focalmechanisms of earthquakes are also used in stressinversion. The inversion of fault-slip data gives thefour parameters of the reduced stress tensor: theprincipal stress axes (maximum compression),a2 (intermediate compression) and a3 (minimum

compression) and the Stress Ratio R = a2 -a3 / al - a3 . The two addit ional parameters ofthe full stress tensor are the ratio of extreme principal stress magnitudes a3/al and the li thostaticload, but these two cannot be determined from faultdata only. We refer to Angelier (1989, 1991, 1994)for a detailed description of the principles and procedures of fault-slip analysis and palaeostressreconstruction.

Weare aware of the inherent limitations of anystress inversion procedures that apply also to thediscussion proposed in this paper (Dupin et al.1993; Pollard et al. 1993; Nieto-Samaniego

Alaniz-Alvarez 1996; Maerten 2000; Roberts Ganas 2000).

The question was raised as to whether fault-slipinversion solutions constrain the principal stressesor the principal strain rates (Gephart 1990a). Wewill not discuss this question here, and leave readers to form their own opinions on how to interpretthe inversion results. The britt le microstructures(faults and fractures) are used in palaeostressreconstructions as kinematic indicators. The stressinversion scepticals (e.g. Twiss Unruh 1998)argue that kinematic indicators are strain markers

and consequently they cannot give access to stress.Without entering in such debate, we consider herethat the stress tensor obta ined by the inversion ofkinematic indicators is a function that models thedistribution of slip on every fault plane. For this,there is one ideal stress tensor, but this one is onlycertainly active during fault initiation. After faults

have been initiated, a large variety of stress tensorscan induce fault-slip by reactivation.

Stress and strain relations

In fault-slip analysis and palaeostress inversion, weconsider generally the activation of pre-existingweakness planes as faults. Weakness planes can beinherited from a sedimentary fabric such as bedding planes, or from a previous tectonic event. Aweakness plane can be produced also during thesame tectonic event, just before accumulating slip

on it, as when a fault is neoformed in a previouslyintact rock mass. The activated weakness plane can be described by a unit vector n normal to (bold is used to indicate vectors). The stress vectora acting on the weakness plane has two components: the normal stress v in the direction of nandthe shear stress T, parallel to These two stresscomponents are perpendicular to each other andrelated by the vectorial relation a= v + T.

The stress vector a represents the state of stressin the rock and has rrl , a2 and a3 as principalstress axes, defining a stress ellipsoid. The normalstress v induces a component of shortening oropening on the weakness plane in function of thissign. The slip direction d on a plane is generallyassumed to be parallel to the shear stress component T of the stress vector a acting on the plane. is possible to demonstra te that the direction ofslip d on depends on the orientations of the threeprincipal stress axes, the s tress ratio R = a2 -a3 / al - a3 and the orientation of the weaknessplane n (Angelier 1989, 1994).

The ability of a plane to be (re)activated dependson the relation between the normal stress and shearstress components on the plane, expressed by thefriction coefficient:

the characteristic friction angle c of the weakness plane with the stress vector a acting on itovercomes the line of initial friction, the weaknessplane will be activated as a fault. Otherwise, nomovement will occur on it. This line is defined bythe cohesion factor and the initialfriction angle c o


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TECTONIC STRESS INVERSION AND THE TENSOR PROGRAM 77

Data types and their meaning in stressinversion

Determination of palaeostress can be done usingtwo basic types of brittle structures: 1) faults withslip lines slickensides) and 2) other fractureplanes Angelier 1994; Dunne Hancock 1994).

In the following discussion as in program TENSOR, under the category faul t , we understandonly fault planes with measurable slip lines slickensides), while we refer to all other types ofbrittle planes that did not show explicitly traces ofslip on them as fractures .

Brittle structures other than the commonly usedslickensides can also be used as stress indicators Dunne Hancock 1994). They are generallyknown as joints or fractures . The term jointformally refers to planes with no detectable movements on them, both parallel and perpendicular tothe plane Hancock 1985). Others use the term

joint to refer to large tensional fractures and in thiscase, tension fractures are mechanically similar tojoints Pollard Aydin 1988). To avoid confusion,we use here the general term fracture for allplanar surfaces of mechanical origin, which do notbear slip lines on the fracture surface. Brittle fractures as understood here bear some information onthe stress stage from which they are derived Dunne Hancock 1994). In order to assess therelationship between fracture planes and stressaxes, it is important to determine their geneticclasses. They may form as tension fractures tension joints) or shear fractures that may appear

as conjugate pairs. Hybrid tension fractures with ashear component cannot be considered for stressanalysis, as the mechanical conditions for their formation are intermediate between those responsiblefor tension fractures and for shear fractures.

Compression fractures form a particular typethat can also be used in palaeostress analysis. Thepressure-solution type of cleavage plane can beconsidered as a compression fracture if it can bedemonstrated that no shear occurred along theplane. Stylolites seams are considered separatelyfrom compression fractures for which the stylolitecolumns tend to form parallel to the direction of

lTl.For faults, one can measure directly the orien

tations of the fault plane and the slip line, anddetermine the slip sense normal, inverse, dextralor sinistral) using the morphology of the fault surface and secondary structures associated as in Petit 1987). In vectorial notation, this corresponds to n unit vector normal to the fault plane) and d slipdirection, defined by the orientation of the slip lineand the sense of slip on the plane). These form theso-called fault-slip data.

Data on slip surface and slip direction can also

be obtained indirectly by combining several brittlestructures, as suggested by Ragan 1973). Conjugate sets of shear fractures 1 and can be usedfor reconstructing the potential slip directions onthe fracture planes, and to infer the orientations ofprincipal stress axes. In conjugate fracture systems,lTl bisects the acute angle o/\oz lT is determined

by the intersection between 0 1 and 0 z and lT3bisects the obtuse angle. The slip directions d andd z respectively on shear planes defined by 1 and0 z are perpendicular to lT in a way that the twoacute wedges tend to converge.

Similarly, the slip direction on a shear planewithout observable slip line can be inferred if tension fractures are associated at an acute angle tothe shear plane Ragan 1973). For shear plane n,and associated tension fracture n., lTl is parallel tothe tension fracture, lT is determined by the intersection between n, and n., and l is perpendicularto the tension fracture parallel to n.). The slip

direction d, on the shear plane defined by n, is perpendicular to lT so the block defined by the acuteangle n, \ t tend to move towards the block definedby the obtuse angle.

In these two cases the slip direction d on theshear planes n is reconstructed and these can beused in the inversion as additional fault-slip data.

For the stress inversion purpose, we distinguishbetween three types of brittle fractures.

Tension fractures plume joints without fringezone, tension gashes, mineralized veins, magmatic dykes), which tend to develop perpendicular lT3 and parallel to IT1. The unit normalvector nt represents an input of the directionlTl.

Shear fractures conjugate sets of shear fractures, slip planes displacing a marker), whichform when the shear stress on the plane overcomes the fault friction. It corresponds to theinput of a fault plane OS but without the slipdirection d.

Compression fractures cleavage planes),which tend to develop perpendicular to lTl andparallel to lT3 The vector n, represents aninput of the direction o l.

For the stylolites, it is more accurate to use theorientation of the stylolite columns as a kinematicindicator for the direction of maximum compression IT 1) instead of the plane tangent to thestylolite seam i.e. input of direction lTl).

Earthquake focal mechanisms are determinedgeometrically by the orientations of the p- and tkinematic axes bisecting the angles between thefault plane and the auxiliary plane. They can bedetermined also by the orientation of one of thetwo nodal plane 0 1 or oz) and the associated slipvector d 1 or d z or by the orientation of the two


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78 D. DELVAUX B. SP R R

nodal planes 1 and ) and the determination ofthe regions of compression and tension (i.e. inputof either and or P and t .

In addition to the orientation data, it is alsoimportant to record qualitative information for allfault-slip data: the accuracy of slip sense or fracturetype determination (slip sense confidence level), a

weighting factor (between 1 and 9, as a functionof the surface of the exposed plane), whether thefault is neoformed or has been reactivated, the typeand intensity of slip striae, the morphology andcomposi tion of the fault or frac ture surface, estimations of the relative timing of faulting for individual faults, based on cross-cutting relationshipsor fault type.

Data selection and separation into subsets

After measuring the fault and fracture data in thefield or compiling a catalogue of earthquake focal

mechanisms, the data on brittle structures are introduced in a database. This raw assemblage of geometrical data forms a data set (or pattern if wefollow the terminology of Angelier 1994). The rawdata set is used as a starting point in stress inversion. For rock masses that have been affected bymultiple tectonic events, the raw data set consistsof several subsets (or systems of brittle data. Asubset is defined as a group of faults and fractures(in fault-slip analysis) or of focal mechanisms (inseismotectonic analysis) that moved during or havebeen generated by a distinct tectonic event. Moreover, the movement on all the weakness planes of

the subset can be fully described in a mechanicalpoint of view by the stress tensor characteristic ofthe tectonic event.

For an appropriate constraint on a stress tensor,data subsets should be composed of more than twofamilies of data. Using the definit ion of Angelier(1994), a family is a group of brittle data of thesame type and with common geometr ica l characteristics. In stress inversion, movement on a systemof weakness planes is modelled by adjusting thefour unknowns of the reduced stress tensor. Therefore, the stress tensor will be better constrained fordata subsets with the largest amount of families of

data of different type and orientation. This conceptis used in the diversity criteria for the quality ranking procedure, described later.

In raw data sets, it is frequently observed thatall the br ittle data do not belong to a single subsetas defined above. This is often due to the action ofseveral tectonic events during the geological history of a rock mass. But the observed misfits canalso be the consequence of other factors such asmeasurement errors, the presence of reactivatedinherited faults, fault interact ion, non-uniformstress field and non-coaxial deformation with

internal block rotation (Dupin et al. 1993; Pollardet al. 1993; Angelier 1994; Nieto-SamaniegoAlaniz-Alvarez 1996; Twiss Unruh 1998;Maerten 2000; Robert Ganas 2000).

The frequent occurrence of multiple-event datasets and the numerous possible sources of misfitshave important implications in fault-slip analysis

and palaeostress reconstruction. necessitates theseparation of the raw data sets into subsets, eachcharacterized by a different stress tensor. Thischeck is necessary even in the case of a singleevent data set, as there are several possibilities tocreate outliers. Errors might occur during fieldwork (e.g. uneven or bent fault planes, incorrectreading), during data input (incorrect transmission),or during data interpretation (e.g. not all events aredetected due to the lack of data). A certain percentage of misfit ting data c 10-15 ) is normal, butthey have to be eliminated from the data set forbetter accuracy of the calculated results.

In the iterative approach for stress tensor determination, data are excluded on the basis of a misfitparameter that is calculated for each fault or fracture as a function of the model parameters 1, o I,u3 and the stress ratio R that best fit the entire setof data. A first stress model is determined on theraw data set, and then the data with the largest misfit are separa ted from the raw data set. After a firstseparation, this process is repeated and the originaldata set is progressively separated (split) into a subset containing data more or less compatib le withthe stress model calculated, and non-compatib ledata which remain in the raw data set. After the

separation of a first subset from the raw data set,this process is repeated again on the remaining dataof the raw data set, to eventually separate a secondsubset. We will discuss this procedure in moredetail later when presenting the Right Dihedronand the Rotational Optimization methods.

In summary, data separation is performed duringstress inversion as a function of misfits determinedwith reference to the stress model calculated on thedata set. This is done in an interactive and iterativeway and the two processes (data separation intosubset and stress tensor optimization for thatsubset) are intimately related.

The first step of the selection procedure starts inthe field. Faults or fractures of the same type, withthe same morphology of fault surface, the sametype of surface coat ing or fault gauge are likely tohave been formed under the same geological andtectonic conditions. Already in the field, they canbe tentatively classified into different families, andfamilies associated into subsets.

Cross-cutting relations might help to differentiate between different families of faults, but it isnot always easy to interpret these relations in termsof successive deformation stages. relations


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between two types of structures are consistent andsystematic at the outcrop scale (i.e. a family of normal faults systematically younger than a family ofstrike-slip faults), they can be used to differentiatefault families in the field and to establish their relative chronology. But this is not enough to differentiate brittle systems of different generation, as data

subsets should be composed of several families ofstructures in order to provide good constraints during stress inversion. As much qualitative fieldinformation and as many observations as possibleof the relations between pairs of fault or fracturesare needed. But they are often insufficient alone toidentify and differentiate homogeneous families offault and fractures related to a single stress event.

The starting point for fault separation and stressinversion can be the first separation performed inthe field. I f this is not possible, a rapid analysis ofthe p and t axes associated with the faults and fractures can help differentiate between different famil

ies of faults and fractures, based on their kinematicstyle and orientation (e.g. normal, strike-slip andreverse faulting, tension and compressional joints).This might be helpful if the measured faults andfractures belong to deformation events of markedlydifferent kinematic styles. The separation done inthe field has to be checked and refined during theinversion. In most cases, however, the selection of

data relies mostly on an interactive separation during the inversion procedure.

However, we want to warn of pure automateddata separation, because this might result in completely useless subsets. For better clarity we illustrate this risk with a 2D data set (depending on twoparameters) instead of a 4D example (depending

on four parameters), as it would be necessary forfault-slip data (three principal stress axes plusstress ratio). Figure I shows how the automatedseparation of a data set with two clusters results inthree subsets, all of them representing only parts ofthe two clusters of the original data set. The samehappens if multi-event fault-slip data sets are separated automatically. This problem can be handledby first making a rough manual separation andthen doing the first calculation. In the 2D exampleof Figure I this can easily be done by separatingthe data of the two clusters into different subsets.The manual separation of fault-slip data is not so

straightforward because it is a 4D problem. Thefirst requirement for a successful separation arefield observations which indicate the existence ofmore than one event and can be used to discriminate the character of the different events. The bestindicators for a multi-event deformation are differently oriented slip lines on one and the same faultplane which, in the best case, show different min-

Original dataset

o

... :

..Data which are compatible

(a)

Determination ofmean value (M,) (c)

\\I

I

\\

~o 0

o 0o

Rest : Two datasets withSeparation . e: : ~ .. \ :

~ : ~ :

(b) :I\

(d) . . . . .

Fig. 1. Separation of an artificial data set into subsets using an automated procedure. The original data set shows twoclusters (a), but the calculated mean M, lies in between them (b). Separation according to the deviation from M,results in a data set compatible to M, (c) and two other data sets with own means M 2 and M 3 (d). Thus, the automatedseparation leads to three subsets (instead of two) with three different means M 1 M 3 . The mean of the largest subsetM, (c) has nothing in common with the means of the two clusters in the original data set (a).


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eralization, so that the mineralization can then beused as a separation criterion. Additionally to fieldobservations, obviously incompatible data can beseparated through careful inspection of the dataplots. As shown in Figure 2 some of the faultplanes might have similar orientations, but different slip lines (e.g. down-dip versus strike-slip

movements; see encircled data in Fig. 2). Separating these data into different subsets can serve as astarting point for the assignment of the other data,so that consistent subsets emerge (Fig. 2). Thesesubsets are then checked for compatibility by feeding them into a computer method like the RightDihedron and the Rotational Optimizationimplemented in the program TENSOR.

In this paper, we strongly advise making aninitial separation in function of field criteria, a careful observation of the data plots and p-t analysis.The raw data set or the preliminary subsets shouldthen be further separated into subsets while optim

izing the stress tensor using successively theimproved Right Dihedron method and theRotational Optimization method in an interactiveway.

Improved Right Dihedron method

The well-known Right Dihedron method was originally developed by Angelier Mechler (1977) asa graphical method for the determination of therange of possible orientations of o Land u stressaxes in fault analysis. The original method wastranslated in a numeric form and implemented indifferent computer programs. We discuss here aseries of improvements that we developed to widenthe applicability of this method in palaeostressanalysis. These new developments concern (1) the

estimation of the stress ratio (2) the complementary use of tension and compression fractures, and(3) the application of a compatibility test for dataselection and subset determination using a Counting Deviation.

Although the Improved Right Dihedron methodwill still remain downstream in the process of

palaeostress reconstruction, it now provides a preliminary estimation of the stress ratio R and dataselection. It is typically designed for buildinginitial data subsets from the raw data set, and formaking a first estimation of the four parameters ofthe reduced stress tensor. The Improved RightDihedron method forms a separate module in theTENSOR program. The original method isdescribed first briefly, before focusing on theimprovements.

General principle The Right Dihedron method isbased on a reference grid of orientations (384 here)

pre-determined in such a way that they appear asa rectangular grid on the stereonet in lower hemisphere Schmidt projection. For all fault-slip data,compressional and extensional quadrants aredetermined according to the orientation of the faultplane and the slip line (Fig. 3a-c) , and the senseof movement. These quadrants are plotted on thereference grid and all orientations of the grid falling in the extensional quadrants are given a coun-ting value of 100 while those falling in the compressional quadrants are assigned 0 . Thisprocedure is repeated for all fault -slip data (Fig.3f). The counting values are summed up and divided by the number of faults analysed. The grid ofcounting values for a single fault defines its characteristic counting net The resulting grid of averagecounting values for a data subset forms the average

\\

0 3

0 ,

very unrel iable (4) ,

Reliability of slip sense(in brackets: confidence level):

b157b 7a

bl 7 t certain 1)t reliable (2)1 unreliable (3)

Fig. 2. Separation of a fault-slip data set (bI57) into two subsets (b157a and bI57b). Obviously incompatible slickensides that have similar fault orientations, but different slip directions (for examples see encircled data) are separatedinto different data sets. The calculated palaeostress axes for the subsets are plotted.


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Normal fault B/ Dextral strike slip fault C/ Reverse fault

N N N

-+ -

+D/ Tension fracture E/ Compression fracture

\

N

/ Resulting ount ing net n d stressaxis determin tion

weighedSLOWER

n = 5

t

@ S l: 02/181: 9 x 0 SZ: 88/016 S3 : 02/271: 9 x 100R, CD : .7 5 28 .1 8

xt nsiv STRIKE SLIP91-1010 ZO ...::

20 30 30-10 10-50

i 80-9090-99100Fig. 3. Principle of the Right Dihedron method Schmidt projections, lower hemisphere). a-e different kinds ofsimple fault-slip data used with their characteristic counting net and projection of the fault or fracture plane, and theircombination to produce the resulting average counting net f). The orientation of tTl and t axes are computed asthe mean orientat ion of the points on the counting grid that has respec tively values of 0 and 100. The two axes areset perpendicular to each other and the orientation of the intermediate tT2 axis is obtained, orthogonal to both tTl andtT3. The count ing value of the point on the counting net which is the closest to tT2 serves for the est imation of the ratio: 0= 100 - S2vaI)/100, where S2val is the count ing value of the nearest point on the count ing grid.

ounting net for this subset. The possible orientations of tTl and t are defined by the orientationsin the average counting net that have values of 0and 100 , respectively.

This method is particularly suitable for the stressanalysis of earthquake focal mechanisms. For faultslip data, it gives only a preliminary result, as itdoes not verify the Coulomb criteria. Problems


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occur for data sets with only one orientation offault planes (and with identical slip direction). Inthis case, 1 and 0 3 deviate by 15 from the correct position, if they are placed in the middle ofthe compressional/extensional quadrants. For conjugate faults the position in the middle of the0 /100 area corresponds with 1 and 0 3 (Fig.4). Moreover, it can be applied only when the senseof movement is given; otherwise the compressionaland extensional quadrants are undetermined.

The graphical method gives a range of possibleorientations of 1 and 0 3. These correspond to thegrid orientations on the resulting counting net withrespectively values of 0 and 100 . The meanorientations of these reference points give respectively the most probable orientations for 1 and 0 3.Because 1 and 0 3 are determined independently,they are not always perpendicular to each other.

A

1

They can be set orthogonally by choosing either0 1 or 0 3 fixed and rotating the other axis arounda rotation axis defined as the normal to the planecontaining o Iand 0 3. The orientation of the intermediate 0 2 axis can then easily be deduced.

One problem with this method is that it does notdetermine the stress ratio R R = 0 2 - 0 3)/ 0 10 3 and that 1 and 0 3 are undefined when theextreme values on the counting net do not reach0 and 100 .

Estimation stress ratio R The stress ratio Rdefined as equivalent to 0 2 - 0 3)/ 0 1 - 0 3) isone of the four parameters determined in the stressinversion, with the three principal stress axes 0 1,0 2 and 0 3. Until now, the Right Dihedron methodhas given only an estimate of the orientations of0 1, 0 2 and 0 3, but not of the stress ratio R. Acareful observation of the Right Dihedron countingnets, however, shows that their patterns differ as afunction of the type of stress tensor (extensional,strike-slip or compressional). We therefore investigated a way to express this pattern as a functionof a parameter that would be an estimation of thestress ratio R.

A way to do this is to compare the orientationof the previously determined 0 2 axis with the distribution of counting values on the average counting net. Using the position of the 0 2 axis on thecounting grid, we found that a good estimation ofthe stress ratio R can be obtained with theempiric relation:

R (100 - S2val)/100

Fig. 4. Uncertainty in determination of stress axes withthe Right Dihedron method. (a) For a single normal faultthe middle of the compression andof the extension quad-rant crosses) is not identical with ul and u (15 off).(b) Two conjugate normal faults and their compressionand extension quadrants: if ul and u are placed in themiddle of the 0 and 100 area respectively, they arecorrectly located.

B 1

3

where S2val is the counting value of the point onthe reference grid nearest to the orientation of 0 2.This formula is only valid for large fault populations with a wide variety of fault plane orientations.

The accuracy of this method for the estimationof the R ratio has been validated using models withsynthetic sets of faults obtained by applying different stress tensors on a set of pre-existing weaknessplanes of different orientation and computing the

shear stress component7'

on the plane and the friction angle p The differents sets are then submittedto stress analysis using the Improved Right Dihedron method. The values of the stress ratio Robtained are in general close to the ones used toproduce the models, within a range of R 0.1(Fig. 5). Similarly, the orientation of the stress axesgenerally match within a few degrees the ones usedto generate the synthetic sets. Experience gainedusing this method in conjunction with othermethods of direct inversion (like the RotationalOptimization method described later) on a largenumber of sites shows that the stress ratio R esti-


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Correlation foro Extensional regimeA Strike slip regimeo Compressional regime

ce 0 0.8Q) c + -

c 0.60>

c+ -

0.4 0Q)C

0.20

o 0.2 0.4 0.6 0.8R use to pro u e mxlels

Fig. 5. Relation between stress ratios R used to producemodels of synthetic fault sets by applying different stresstensors on a set of 152 pre-existing weakness planes ofdifferent orientation, and values obtained by analysingthese sets with the Improved Right Dihedron method.Models were produced with = 0, 0.1, 0.3, 0.5, 0.7, 0.9and 1.0. for extensional, strike-slip and compressionalstress regimes.

mated using the Improved Right Dihedron method

is generally close to the one obtained by theRotational Optimization method (see hereafter).We conclude that for all types of stress tensors

(extensional, strike-slip and compressional) witha l > a2 > a3 and 0.25 < R < 0.75, the ImprovedRight Dihedron method successfully estimates thefour parameters of the stress tensor. In the extremecase of flattening a2 = a3 R = 0) or constriction l = a2 = 1) only one of the extreme valueswill be well defined (0 fo r flattening, 100 forconstriction); the other one will have mediumvalues and a circular distribution. Therefore onlya l is well defined when = 0 and a3 when = 1 .

Use of compression and tension fractures aspalaeostress indicator The Right Dihedron methodalso allows the use other types of britt le data suchas compression and tension fractures as definedabove for estimating the four parameters of thestress tensors (Fig. 3d, e). For tension fractures, a3is considered as oriented within a cone angle of degrees around the normal to the plane and a l islocated at an acute angle : Sf3) of the tension plane.The opposite is true for compression fractures, with

a l located within the cone angle and a3 at anacute angle (: S 3) to the tension plane. The orientations between the cone angle and the surfacegenerated by the revolution of a line inclined atan angle from the fracture plane are consideredas intermediate.

is possible to define on the counting nets, areas

in compression (value 0), in extension (value 100),and intermediate areas (value 50). The individualcounting nets are summed up and averaged as inthe case of faults, to obtain the average countingnet. This allows the combined analysis of the threedifferent types of data (slickensides with knownsense of movement, tension and compression fractures, Fig. 3).

For the value of the angle we use the commoninitial friction angle of 16.7 given by Byerlee(1978). The use of this value is justified by theshear stress/normal stress relations in the initialfriction law (Jaeger 1969). If the angle between the

tension fracture or the normal to the compressionfracture and l is larger than 16.7, the resolvedshear stress is theoretically high enough to causeslip on that plane, which is in contradiction to thepostulated nature of that plane. Other values forthe angle can be used, without modifying thegeneral principle.

When working with a database composed onlyof fracture data, this method provides a way to estimate the stress ratio while this could not bedetermined by formal inversion.

Counting Deviation Another important improve

ment of the original Right Dihedron method is thedevelopment of a parameter for estimating thedegree of compatibility of the individual countingnets with the average counting net of the subset. relies on the calculation of a Counting DeviationCD (expressed in ) for each datum by comparingits counting net with the average counting net.Fault or fracture data with a low CD value contribute in a positive way to the average counting net(reinforce the extreme counting values) and thedata with higher CD values contribute in a negativeway (weaken the extrema).

The principle of Counting Deviation and its use

in compatibility testing are presented hereafter withreference to Figure 6. The first (and forward) stepis to compute the average counting net by summingthe values of each point on the reference grid forall the individual counting nets (Fig. 6a-c), takinginto account the weighting factor associated witheach datum (Fig. 6d). After this operation, thesecond step is performed in a reverse way. Eachindividual counting net is compared with the average counting net. As a result, a differential counting net is obtained for each datum by subtractingfor each orientation of the reference grid the coun-


10/26


Normal fault B/ Dextral strike-slip fault C/ Reverse fault

N N N

X

\X X

0 Resulting average counting ne t and histogram of counting deviations

N 1-167.

lG Z6 /.

2 e ~ 3 O / 30 1 El: : 19 50;;;

j ~ ~ '-9'J10e:.:

E/ Differential counting netsN N

co =17.2 CO=31.2 CD =49 9 Fig. 6. Use of the Counting Deviation CD to filter the data. As in Figure 3, data from three different types of brittlestructure a-c are used to produce the average counting net (d), with consideration of the weighting factor. For alldata, each counting value of the individual counting nets is multiplied by the weighting factor (4 for the normal faultand 1 for the other data) . The stronger influence of the normal fault on the resulting average counting net is clearlyvisible (d). The differential counting nets (e) are obtained by subtracting respect ively the counting values of theindividual counting nets (a, b or c) from the ones o f the average counting net (d), without application of the weightingfactor. The Counting Deviation (CD) is the average counting value for the 384 points on the differential countingnets, expressed as a percentage (e). On the average counting net diagram (d), the CD values are displayed in ahistogram, weighted according to the weighting factor associated with the individual data. The average countingdeviation for the whole data set is 25 12.3 Lrr). In this case, the reverse fault has a CD value above the averageCD + a and can be eliminated from the data set. the process is repeated on the two remaining data, the dextralstrike-slip fault will in tum be eliminated.


11/26


ting values of the individual nets from the those ofthe average counting net Fig. 6e . The CountingDeviation CD is the average of all the 384 valueson the differential counting net. As the countingvalues in the counting net are expressed as percentages, the unit for the Counting Deviat ion is alsothe per cent.

The compatibility of individual data with theentire data subset is estimated by the dispersion ofthe CD values away from the arithmetic mean ofall the CD values. The homogeneity of the data setis expressed by the standard deviation of the CDvalues: smaller standard deviations suggest thatmost data of the subset are compatible with thefinal counting net and hence possibly belong to thesame kinematic event. In the TENSOR program,the fault planes can be rejected if their CD valuesare larger than the arithmetic mean -t-Irr or 2lT,depending on the choice of the user. This is usedto control the compatibility of each data with thefinal result, and to reject it from the subset if necessary.

Using this system, a first solution is computedon the initial data set single-event deformation oron the manually separated subsets multi-eventdeformation , and the incompatible faults arerejected. The procedure is repeated several timesuntil well-defined areas of 0 and 100 values areobtained in the counting net. An example of progressive separation is given in Fig. 8a and Table 1for fault-slip data set and in Fig. 9a and Table 3for a focal mechanism data set. The final resultand the corresponding subset will serve as a starting point for the rotational optimization.

In summary, the Improved Right Dihedronmethod allows a first estimation of the orientationsof the principal stress axes and of the stress ratioR and a first filtering of compatible fault-slip data.The selected fault-slip population and the preliminary tensor can be used as a starting point in theiterative inversion procedures like the RotationalOptimization method described hereafter.

Rotational Optimization method

The Rotational Optimization method presentedhere is a new iterative inversion procedure. As forall iterative inversion methods, it is based on thetesting of a great number of different stress tensors,with the aim of minimizing a misfit function. Inprinciple the whole range of orientations for thethree stress axes and the stress ratio R has to betested to find the minimum value of the misfit function grid search . Considering these four parameters, the problem is close to a four-dimensionalone, with an additional constraint that the threeprincipal stress axes have to be orthogonal. This

leads to a large number of different configurationsof the stress tensor to be tested with the data set.

To find the solution in a more direct way, wedeveloped the Rotational Optimization method andpropose to initiate the search procedure using thestress tensor estimated with the Right Dihedronmethod. It allows restriction of the search area during the inversion, so that the whole grid does nothave to be searched.

In the following, we introduce a few classictheoretical notions, discuss in detail the misfit functions to minimize, and present the Rotational Optimization procedure with the help of a naturalexample from the Baikal Rift Zone in Siberia.

Shear stress construction and misfit parametersThe basic idea in the direct inversion is to test anumber of stress tensor on all faults and fracturesof the data set by computing the direction, sense

and magnitude of the shear stress 7 acting on theplane, the magnitude of the related normal stress lTn the characteristic dihedral angle 28 and thefriction angle c

In TENSOR, this is done using the method ofMeans 1989 . In the inversion of fault-slip data,the isotropic part of the stress tensor is missing.Thus the reduced stress tensor is identical with thedeviatoric part of the stress tensor. The magnitudesare expressed in a relative way because the absolute values cannot be determined using geologicaldata only see Angelier 1989 . By convention, themagnitude of lTl is fixed at 100 and the magnitude

of l at 0 in abitrary units . The relative magnitudes are therefore in the range 0 = l lT2 lTl = 100. The magnitude of the lT2 axis is fixedby the stress ratio R, as a function of the magnitudes lTl and lT3.

The occurrence of slip on a pre-existing rock discontinuity is governed by friction laws. On a Mohrdiagram, the corresponding point is enclosedbetween the initial friction curve and the maximumfriction line failure curve . The relation betweenthe minimum shear stress 7 and the normal stress v is approximately linear Angelier 1989 . Hence,the sliding criteria can be simplified by assuming

that the friction angle c must be greater than theinitial friction angle and smaller than the maximumfriction angle. The friction criteria of Byerlee 1978 are used here as default values initial friction angle c oof 16.7 and maximum angle c m x of40.4 . If c < c o no reactivation of pre-existingdiscontinuities will occur; if c > c m x failure willoccur with the development of a new fracture.

inimization functions A great advantage of theiterative approach for stress tensor inversion is thatthe complexity of the function to minimize is not


12/26

86 D. L VAUX B. SP RN R

fl i = a i = function Fl in TENSOR program

fz i = sin 2 a i / 2

= function F2 in TENSOR program

a limiting factor and the function can be easilychanged without changing the algorithm.

In general, the minimization function has the following form:

These two functions are still insufficient in the caseof newly formed conjugate fault systems. In thiscase 1 and oJ might lie anywhere in the planeperpendicular to the two conjugate fault sets,because the slip deviation will be minimum for anyconfiguration of stress axes. As long as a2 is parallel to the intersection line of the two fault systems,the orientation of a l and a3 will not influence theslip deviation. Thus additional constraints arenecessary, taking into account the ability of thefault to slip Angelier 1991 . This can be either the

The first term is for the minimization of the deviation angle a and corresponds to the f2 i definedabove, mult iplied by 360 to ensure that this termremains dominant with regards to the second term.The second term allows increasing the tendency ofthe fault to slip, by maximizing while minimizing Ivl. Tinv i maximizes when minimizingf3 i : Tinv i = a /2 - Hi 1 with a corresponding to the maximum possible value of Ivei lminimizes the normal stress component on theslip surface.

With magnitudes of a l and a3 fixed respectivelyat 0 and 100, the minimum possible value for theexpression Tinv i + Nnorm i is 29.7 on a Mohrcircle construction. To allow this term to convergeto zero in the most favourable cases, 29.7 is substracted from the result of this expression. Th e proportionality factor p also allows the first term to bekept dominant with regard to the second term. InTENSOR, it is set by default to 5, but can be modified.

The TENSOR program can also use fracture datato constrain the stress tensors, and is not restrictedto the analysis of slickenside data only. Toimplement this, other minimization functions havebeen developed.

For shear fractures slip surface with observedmovement but without slip lines , the second termof function fii can be used as defined above:

f3 i = lf2 i X 360 + Tinv i + Ivei l- 29.7 /p

friction angle f or the shear stress magnitude on the fault plane, which both should be maximized. As the mechanical properties of the faultedrocks are generally unknown in standard palaeostress investigation, it is more appropriate to use to express the tendency of the fault to slip seealso Morris et al. 1996 . Similarly, the normalstress component on the slip surface v also influences the ability of the fault to slip. should beminimum in order to lower the fault friction andhence to favour slip. These three parameters canbe taken into account simultaneously by combiningin the same function the slip deviation a and thenormal stress magnitude I vi that should be minimized, with the shear stress magnitude that shouldbe maximized. This is done in function f3 i :

For tensional fractures plume joints, mineralizedveins, magmatic dykes , the normal stress Iv i 1applied to the fracture surface should be minimal tofavour fracture opening. Simultaneously the shearstress magnitude 17 01 should also be minimal toprevent slip on the plane:

f4 i = Tinv i + lveol - 29.7 /p

1

L jj i X weiF ~ ~ ~ ~

J n X L wei

For slickensides with known sense of movement,the slip vector is uniquely defined and a rangesbetween 0 and 180. When the sense of movementis not known, only the orientation of the slip vectoris defined, and two opposite directions are possible.In this case, the maximum slip deviation is at 90from the slip direction. is generally assumed thatfor a slip deviation of a :::; 30 the observed faultmovement is compatible with the theoreticalshear vector.

is also common to use a least-square functionfor the minimization. This implies a Gaussian-typedistribution of individual misfits. A good exampleis function S4 Angelier 1991 , already proposedby the same author in 1975:

where wei is the weight of the individual data andjj i is the function that has to be minimized.

In the following, by fault-slip data we understand fault planes defined by the unit vector n normal to the slip plane and associated slip vectors don the slip planes, either observed or reconstructedas explained above. Faults with associated slipstriae are also known as slickenside. In this category, we include also fault planes with slip direction only direction of movement unknown . For

those fault-slip data, the most classic misfit parameter is the angular deviation between the slip vector d and the shear stress t slip deviation a , computed in function of the stress tensor and theorientation of the slip plane n:


13/26


j5 i = lv


14/26


Rotation around o l axisRotation angle : +34

Function F5 value: 20.515

3.

Rotation around r axisRotation angle: 0


- 5 2 2 5 22 5

Rotation angle 0)

Rotation around cr3 axisRotation angle : -16


15 - 5 2 2 5 ) 22 SRotation angle C)

Optimisation of R RatioR value: 0.75


28

5 2 2 S 22 S

Rotation angle C)..5

.3

.15

Z5 se 75R values 52-53)/ 51-53)

108

Fig. 7. Principle of the 4D Rotational Optimization. The stress tensor is rotated successively around the o l , u andu3 axes by 12.5 and 45, a polynomial regression curve is computed by least square minimization using values ofthe optimization function F5 here). The minimum of the function is found then additional rotations of 5 from theangle corresponding to the minimum value of the function are again performed. A new regression curved is computedand the minimum is taken to define the rotation angle to apply to the initial tensor +34 for the rotation around rrl) .The equivalent rotation is performed and the procedure is repeated for the next two axes. A similar procedure is donefor optimizing the stress ratio by checking a range of possible values between 0 and 1. The range of rotation anglesand values of R is progressively decreased to provide finer constraints during the next runs.

procedure of tensor optimization and fault separation is executed successively with a decreasingvalue of the maximum slip deviation e.g. from 4to 30; Fig. 8 and Table 1). Because a stress tensoris adjusted on a particular fault population, a newstress tensor has to be recalculated after each modification of the fault population. Using this procedure, the first tensor obtained corresponds to thesubset, which is represented in the original population by the greatest number of faults. The rejectedfaults are then submitted to the same procedure, toextract the next subset.

Figure 8 and Table 1 present an example of progressive stress tensor optimization and fault separ-

ation using successively the Right Dihedron andthe Rotational Optimization methods on a naturalfault-slip data set measured along a border fault ofthe Central Baikal basin in the area of Zama Delvaux et al 1997b, 1999). This fault data setcontains a minority of reverse or thrust faultsrelated to an older brittle event, and a majority ofnormal or oblique-slip faults related to Late Cenozoic extension.

We want to point out again that an uncontrolledautomated separation presents a great risk becauseit might lead to useless results see remarks above).Thus, we would recommend first doing a roughseparation using field observations, careful inspec-


15/26

TECTONIC STRESS INVERSION AND TH E TENSOR PROGRAM

(a)

89

1/ Initial data base: 54 fault-slip data 2/ Initial data base: tangent-lineation diagramlint .. . ~ l h c de ....... s ww

N -51 N n t ~ 5 1

~ t y

i -1\ \, \, . hode. 4 ,w + ~. I I Sf All 51 1\ ~ ~ i\,

3/ Right Dihedron solution on initial database: 43 valid data

@ S I : -.. . .r.3: z IS&. sz: IiI.434l l 33: fVl31 : Z II: IZI OK: .li'J 16.4 es

5/ Right Dihedron solution after newelimination of 5 data: 34 valid data

@ S J ; ~ l l f j R: ..... ZJ5l l $3: W...:nlI: 1 x '3ro C IY.: . 95 48 .8 B

4/ Right Dihedron solution after firstelimination of 4 data: 39 valid data

6/ Right Dihedron solution after lastelimination of? data: 27 valid data

- ~ - -@ : 6StZ3Z: 1 I l l&. SZ: z s . .Mll l 33: , , 133 : weII. Olol: .e 3l.Z 6. 3

Fig. 8. Example of progressive kinematic separation and stress tensor optimization on a natural polyphase fault-slipdata set measured along a border fault of the Central Baikal basin in the area of Zama Delvaux t al 1997b, 1999).The ini tial data base contains 54 fault -s lip data, some related to an older compression stage but the major ity of themrelated to Late Cenozoic extension. All stereograms have Schmidt lower hemisphere projections. The different parameters used for est imat ing the quality resul ts are reported in Table 1. a, 1 and 2) Initial data base; (3-6) progressivefault separation using the Right Dihedron, leading to a first subset and starting tensor for subsequent rotational optimization. b, 7) After successive optimizations and elimination of incompatible data 8-12), the final solution is obtainedfor the first separated set, represented by the greates t number of faul t-data 13-14). For the remaining 25 faul t-sl ipdata that were exc luded from the initial data base c, 15 and 16), a new series of separat ion was done using RightDihedron 17), leading to a second subset which was again progressively opt imized and separated 18 and 19) untilthe final solut ion is reached for the second set 20). Cross-cut ting relat ions and fault plane morphology observed inthe field show that the second set corresponds to an older stress state, compatible with the Early or Mid-Palaeozoiclocal stress field Delvaux t al 1995b and that the first set is compatible with the Late Cenozoic stress field Delvaux t al 1997b

tion of data plots, and then only starting the optimization procedure.

At the end of the procedure, when two or moresubsets are separated from the original fault population, it is necessary to test the stress tensors of

each subset on the total fault population, to checkif the fault separation has been done in an optimalway. Effectively, the faults-slip data are consideredcompatible with a stress tensor as soon as the deviation angle ex is less than 30. In the process of


16/26

90 D. DELV AUX & B. SPERNER

(b)

7/ Shear stress solution using final RightDihedron result: 38 valid data

8/ Solution after optimisation with 45 range

:Enxw 13

: ~nxw 52

UN

(i)SI:?St2:1B. 32: 1ZI l6@~ S ;1V.MElII 6 7 4 8

N

@SI,6s.-23Z.i.SZ:ZSt'EI11~ S : VITIft : i lS.S1

:L nxw 52

9/ Solution after elimination of 4 datawith misfit a> 40: 34 data

10/ Solution after new optimisation with20 range: 34 data

\)

~ ~II nxw SS

@SI:7'},; ' I 'J7&. sa. t361 9 l l[ jS3:1e..-3i8 I : .6113.12:

( i ) S l : 7 5 / Z l l J, i ,SZ:l2t 'e68~ S :18/346B -. .111 re.ez

II I Solution after new elimination of 5 datawith misfit a > 30: 29 data

~@ S t : 7 V l o n,i,.sz:EIEw(In~ 3 3 ; 1 Q / 3 mB , , , : .61 ' J . t 7

: ~e nx w 55

12/ Solution after new optimisation with10 range: 29 data

3 1 : 8 1 / 1 3

~ S :e1.lzw( JS3:e ') .I35G1 ' . i ' 9 7 . 6 8

:Lnxw 6813/ Final solution after last optimisation with

5 range: 29 data14/ Final solution, tangent lineation diagram

U U

I'J

;\~ S t : I I 1 / 1 6 3 i S l : S V 1 b 3 1

SZ'IIVZ63 S2: 8Vz 3( J 53: M 5 [ j S 3 : 6 ' : V l 5 3

. , : .Z l 6.18 B -: .2:16.18

; O{ +(;:0 a. @

:L :L~

\ ~ ~\. ' jnx w 8 nx w Fig. 8. Continued.

stress tensor inversion and fault separation, a givenfault-slip datum is attributed to the first populationfor which it is compatible with the characteristicstress tensor. But it can also be compatible withthe stress tensors of the other subsets, sometimeswith an even smaller deviation angle Q' . During thefinal cross-check not shown in the example of Fig.

8), each fault datum is attributed to the populationfor which the composite function is minimum.Because each subset was modified, a new optimization run has to be performed on each subset.

As explained above, the deviation angle Q aloneis not enough to unambiguously discriminatebetween faults of different data subsets or between


17/26

TECTONIC STRESS INVERSION AND THE TENSOR PROGRAM

(c)

91

15/ Remaining 25 fault-slip data

N

17/ Final Right Dihedron solution afterelimination of 5 data

16/ Remaining, tangent lineation diagram

N

~ ~ T

\ \ \ '}'> ;

+

/

\

18/ Solution after successive optimisationsand data eliminations with respectively45 and 20 0 range and a> 40 and 30 0

@ S l : ~ 8 : ? : l x 9

&. 82: ZSI 2?S[ ] S3; S:V149: s lC leef1,CD ';;;.12 31.1 f>.Z

CD%

E=n x w

(j)@ S l ; z e . t 3 1sz:9Z1 976[BS3 :76 /172R a : . 1 1 1 . 8 7

19/ Final solution after additionaloptimisations with 10 and 50 range

u

6 6 ~ae

ne x Z

Fig. 8. Continued.

focal planes of the same focal mechanism, whenthey all fit the basic requirement that a < 30 ando> 00 . The criteria on the slip deviation a and thefriction angle 0 are used to eliminate fault-slip dataor focal planes from the data set, while the composite function is generally used as a minimizationfunction during the inversion and for the finer discrimination between data subsets or focal planes.

Interpretation results and qualityassessment

The quality of the results obtained by stress inversion is dependent on a number of factors such asthe number of data per subset, the slip deviation,

20/ Final solution, tangent lineation diagram

(\) , N@~ S l : l f i ; o 3

S2:ae, . ,913

I \ \ '}OS3; 1/171R ; . n 8.59 +a:

: ~ / ,1\ \ i

nxw the type of data, and even on the experience of theuser. Thus, for a rating of the quality and a bettercomparability of the results it is necessary to introduce a quality ranking. A first quality rankingscheme was proposed by Delvaux et t l995b .I t was later improved by Delvaux et at. l997b .In the first release of the World Stress Map (WSM;Zoback 1992), the quality ranking proposed forgeological indicators was defined in a way that allstress tensors obtained from the inversion ofQuaternary-age fault-slip data obtained the bestquality. In co-operation with the WSM project(Sperner et at. 2003) and on the basis of these earlier approaches, the quality ranking schemeincluded in the TENSOR program was further


18/26


Tab le Value of the parameters used fo r estimating the quality o f the results of progressive stress tensor optimizationand fault slip separation using successively the Right Dihedron and Rotational Optimization for the example in Fig-ureFirst subset, Right Dihedron

Step Difference between Average counting deviationextreme counting values ( , ::t::10

3 67 46.4 t4 75 44.5 6.45 87 40.8 8.06 100 31.2 6.3

First subset, Rotational Optimization

Step n nlnt CL w a F5 DT w Plen Slen QRw QRt QI

7 38 0.7 0.73 35.5 42.0 1.0 0.82 0.68 E E 6.18 38 0.7 0.73 17.5 19.2 1.0 0.82 0.68 D D 12.59 34 0.63 0.75 14.0 11.6 1.0 0.83 0.72 C C 12.2

10 34 0.63 0.75 13.1 11.3 1.0 0.83 0.72 C C 13.0 29 0.54 0.76 9.2 6.4 1.0 0.83 0.75 B C 13.212 29 0.54 0.76 7.7 5.7 1.0 0.83 0.75 B C 15.713 28 0.52 0.76 6.2 5.0 1.0 0.83 0.75 B C 17.5

Second subset, Right Dihedron

Step Difference between Average counting deviationextreme counting values ( , ::t::10

17a 86 39.5 7.617 100 31.4 6.2

Second subset, Rotational Optimization

Step n nlnt CL w a w F5 w Plen Slen QRw QRt QI

18a 20 0.37 0.9 28.8 30.1 1.0 0.60 0.54 E E 13.118 13 0.24 1.0 11.9 7.4 1.0 0.77 0.59 D D 9.919a 13 0.24 1.0 9.0 5.4 1.0 0.77 0.59 D D 12.919b 13 0.24 1.0 8.2 5.1 1.0 0.77 0.59 D D 14.219 13 0.24 1.0 8.6 5.0 1.0 0.77 0.59 D D 13.6

This data set contains fault-slip data belonging to two different stress stages. The subset represented by the largestnumber of data will be extracted first. The CLw parameter progressively increases due to the elimination of data with

less constrained slip sense determination. For the first subset, the values Slen

increases and limit the final qualityof the tensor QRt to C quality, while the WSM quality QRw remains to B. This is justified by the fact that theretained fault population consists mainly in two conjugated sets of fault-slip data. As a result, the stress ratio isnot well constrained.

developed to meet the more strict requirements ofthe new release of the WSM project.

In accordance with the new ranking scheme forthe WSM, the quality ranges from A (best) to E(worst), and is determined as a function of threshold values of a series of criteria. The threshold

values have been chosen more or less arbitrarilyand then tested with field data and checked to findwhether they gave a good range of quality according to our feeling of the results. A given qualityis assigned if the threshold values corresponding tothat particular rank are met for all the criteria.


19/26


For the WSM quality ranking scheme the following parameters are used, with threshold valuesas defined in Table 2:

n, number of fault-slip data used in the inversionn/nt, ratio of fault-slip data used relative to the totalnumber measured m deviation between observed and theoreticalslip directionsCi w , slip sense confidence level for individualfaults, applied for the TENSOR program:

Certain 1.00Probable 0.75Supposed 0.50Unknown 0.25

In the case of fractures, Cl.; corresponds to the confidence level in type determination shear, tensional, compressional) and for focal mechanisms,to the quality of focal mechanism determination

DT w , fault-slip data type:

Slickenside 1 00Focal mechanism 0.75Tension/compression fracture 0.50Two conjugate planes 0.50Stylolite column 0.50Movement plane tension fracture 0.50Shear fracture 0.25

As discussed in Spemer et al. 2003), the diversity of orientation of fault planes and of slip lineations is also an important parameter to consider. t is clear that the stress tensors are better constrained if the fault planes and the slip lines havea large variety of orientations rather than if theyare all parallel to each other. The distribution oforientation data can be expressed by the normalized length of a vector obtained by addition ofunit vectors representing the poles of the faultplanes p or the slip direction s:

~ x + p y2 + p z2Plen = -

n

~ S x sy2 sz2Slen = -

n

where p; p P, and s., y s, are respectively theCartesian co-ordinates of the unit vector p and s expressed as direction cosines). The normalizedlengths range between 1 for unimodal populations all poles/slip lines parallel) to :: 6 for homogeneously distributed populations.

Unfortunately the computation of such distribution has to be done on the original fault-slip data,which are generally not given in published results.For this reason, it was decided not to use these twocriteria in the definition of the WSM quality rankfor geological indicators here named QRw,

Table 2).In the TENSOR program, the diversity criteria

using Plen and Slen were also implemented andcombined with the five first parameters, to determine the TENSOR quality rank QRt Table 2). Inthis way, Plen and Slen are additional criteriawhich lower the quality of the result if the diversityof orientation of data is insufficient regarding tothe WSM quality rank QRw.

A quality index QI) expressed numerically hasbeen previously proposed by Delvaux et al. 1995b, 1997b , based on the following formula:

QI = n 2 X n/nt /a w

with threshold values of QI 2 : 1.5 for A quality,1.5 > QI 2 : 0.5 for B quality, 0.5 > QI 2 : 0.3 forC quality and QI < 0.3 for D quality. However,this system proved to be unsatisfactory for areliable quality assessment and we prefer to abandon it.

Table 2. Threshold values as defined in Sperner et al. 2003 fo r the individual criteria used in the quality rankingscheme for the WSM Quality Rank QRw n to DTw

WSM quality n n/nt CL w a w DT w Plen Slen Tensor qualityrank QRw rank QRt

A 22 5 2 6 2 7 :s 9 2 9 :s0.80 :S0.80 AB 21 5 2 4 2 55 :S12 2 75 :S0.85 :S0.85 BC 2 1 2 3 2 4 :S15 2 5 :S0.92 :s0.92 CD 2 6 2 15 2 25 :S18 2 25 :s0.95 :s0.95 DE < 6 < 15 < 25 >1 8 95 > 95 E

Proposed threshold values for the Plen and Slen are added for the Tensor quality rank QRt.


20/26

94 D. DE L V AUX B. SPERNER

Stress regime index

The stress regime can be expressed numerically bythe stress regime index R defined in Delvaux et t I997b). The main stress regime is a functionof the orientation of the principal stress axes andthe shape of the stress ellipsoid: extensional when 1 is vertical, strike-slip when 1I2 is vertical andcompressional when I is vertical. For each ofthese three regimes, the value of the stress ratio Ris fluctuating between 0 and I. When the value isclose to 0.5 plane stress), the stress regimes aresaid pure extensional/strike-slip/compressional.The transition between the three regimes isexpressed by opposed values of R. An extensionalregimes with R = I is equivalent to a strike-slipregime with R = 1. Similarly, a strike-slip regimewith R = 0 is equivalent to a compressional regimewith R =

To facilitate the representation of the range of

stress regimes, Delvaux et at 1997 b) defined astress regime index which expresses numericallythe stress regime as follows:

= R when III is vertical extensionalstress regime)

R = 2 - R when 1I2 is vertical strike-slipstress regime)

R = 2 R when I is vertical compressionalstress regime).

In the WSM, the naming of the stress regimes iscorrespondingly: normal faulting/strike-slip faulting / thrust faulting regime. The index forms acontinuous progression from 0 radial extension =flattening) to 3 radial compression = constriction),while R is successively evolving from 0 to 1 in theextensional field, 1 to 0 in the strike-slip field andagain 0 to 1 in the compressional field. R hasvalues of 0.5 for pure extension, 1.0 for extensionalstrike-slip, 1.5 for pure strike-slip, 2.0 for strikeslip compressional and 2.5 for pure compression.The intermediate stress regimes are sometimes alsonamed transtension for the transition betweenextension and strike-slip and transpression for thetransition between strike-slip and compression.

The stress tensors are displayed in map view bysymbols representing the orientation and relativemagnitude of the horizontal stress axes, assug-gested by Guiraud et t 1989) and furtherdeveloped by Delvaux et t 1997b).

In regional studies it is sometimes necessary toobtain the mean regional stress directions of a series of closely related sites. In Delvaux et at1997b), we introduced the concept of a weighed

mean tensor , without defining it clearly. This wasbased on the calculation of mean SHmax, Shminand stress axis by vectorial addition, taking intoaccount the number of fault-slip data used for the

stress inversion of each tensor. Similarly, the meanstress ratio was computed as the average stressratio index R defined above. In this procedure, theorientations of I, 1I2 and I axes are assessed toSHmax, Shmin and Sv as a function of the stressregime extensional, strike-slip or compressional),expressed as azimuth and plunge. Now, we prefer

to use simply the average SHmax azimuth asdefined in the World Stress Map) and the averagestress regime index as defined above. These twoparameters describe fully the orientations of thestress axes in terms of SHmax and Shrnin,assuming Sv vertical), and the stress regime R =0-1 for extensional regime, 1 2 for strike-slipregimes and 2-3 for compressional regimes).

pplic tions

Inversion of earthquake focal mechanism

dataEarthquake focal mechanisms are defined by twoorthogonal nodal planes, one of them being theplane that accommodated the slip during seismicactivation fault plane) and the other being theauxiliary plane. In the absence of seismological orgeological criteria, both nodal planes are potentialslip planes and they cannot be discriminated. TheRight Dihedron method is particularly well adaptedfor the stress inversion of focal mechanisms, as italso uses two orthogonal planes to define compressional and extensional quadrants. TheImproved Right Dihedron method is useful for afirst estimation of the stress tensor, and also for aninitial separation of mechanisms from the database.The preliminary stress tensor and filtered focalmechanism data set are used as a starting point inthe Rotational Optimization procedure Fig. 9 andTable 3).

In the Right Dihedron method, both nodal planesof an incompatible focal mechanism are eliminatedsimultaneously as they both have the same Counting Deviation values. In the Rotational Optimization procedure, all the nodal planes, which haveslip deviations greater than the threshold value of,for example, 40 at the beginning of the analysisprocedure or 30 at the end, are eliminated. Thetwo nodal planes have generally different valuesfor the slip deviation 0 . If one of the planes has 0greater than the threshold value, it is considered asan auxiliary plane and is eliminated. both 0values are higher than the threshold value, theentire focal mechanism is eliminated. both 0values are lower than the threshold value, the twonodal planes are kept for further processing. Thisresults in the progressive selection of the probablefault plane for each focal mechanism.

The final choice of the presumed fault plane for


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(a) Initial data base : 24 focal mechanisms

48 focal planes).2/ Initial data base: Tangent-lineation

diagram

N N

3/ Right Dihedron solution on initial database: 24 mechanisms 48 focal planes)

N

4/ Right Dihedron solution after eliminationof 2 mechanisms: 22 mechanisms

44 focal planes)

V

o81 : 84..-239: 59 x 9

:. 82: E16 .. .wbl lS3 : 811 12l>: 4 : ' Jx lgea , C D 7 .: . 3 6 2 4 .4 3 .' 1

CD%

n x w 4.8

N~ y~@ 81: 8'ldll9: 51 x 9 82: OO..1JJll l 53; 91 121; 22 x 190R, CD '-:.31 2 f 7 ~ 3

CD

nxw 44

N

6/ Shear stress solution using final RightDihedron result: 16 mechanisms

32 focal planes)

oVi )S l :82? l< r3 82: 8 AMl l 33: 83/317J l a : 5 6 11.61

C

nx w 16

N

@ S I : 82.


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(b)

7/ Solution after optimisation with 45 range 8/ Solution after selection of one focal planefor each mechanism

a

~nxw zz

a

:L) nx w 16

N

@Sl:8l1,. ' l11 sz: e6.- l:1[ l S 3 : 6 7 r J l 1R, .. : . 1 1 7 . 1 1

N

@ S I : o o l 7 1~ l Z efve11[ jSJ:Q?- 3tiR , I l - : .11 11.S3

9/ Solution after new optimisation with 20range

N

10/ Solution after checking previouslyrejected mechanisms

(one focal plane for each mechanism)

V51 : 71/172

. . 52: Ul.I9'll[ ] 33: 12/lO9R , a - : . 4 1 5 . 5

N

Va

:Lnxw 12 : Final solution, tangent-lineation diagram1/ Final solution, after last 2 optimisations

with 5 range

N

@ 51: 78/175~ S l : 8 J . - e J 7f .]33:871 396B , a - : .4 3.85

a

:Lnx w 30

V

i) 51: 713/175

sa. 9'3.1 37[ ] 33: 67 .. .. Of>B. . . . - . . 11 3 .8 5

a

:L) nx w 3El

N

Fig. 9. Continued.

fault plane can be influenced by pre-existing planesand might be more unfavourably oriented than anew plane.

A special option has been implemented in theRotational Optimization module of TENSORthatrejects automatically the non-compatible mechanisms and discriminates between the fault planeand the auxiliary plane of a focal mechanism, forany given stress tensor.

A similar (but less automated) procedure wasused in Petit et al (1996) for stress inversion offocal mechanisms from the Baikal Rift Zone. Theycompared their results with those obtained usingthe Carey-Gailardis Mercier (1987) method onthe same data set. It proved that, in most cases, the

two methods give similar results. The differencesresult mainly from the different selections of faultand auxiliary planes, not from the inversion itself.

ests with synthetic fault sets

To validate the results obtained with the RightDihedron and the Rotational Optimizationmethods, we performed a series of tests using asynthetic set of fault and fracture data representinga strike-slip regime with north-south compression,vertical intermediate axis and = 0.4. This wasproduced by applying the corresponding tensor ona set of 152 pre-defined weakness planes of variousorientations distributed homogeneously on a ster-


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Table 3. Value o f the parameters used fo r estimating the quality o f the results of progressive stress tensor optimisationand focal mechanism data separation using successively the Right Dihedron and Rotat ional Optimisation for theexample in Figure 8Right Dihedron

Step Difference between Average counting deviationextreme counting values ( , u

3 100 26.7 4.34 100 24.4 3.45 100 20.0 3.2

Rotational Optimization

Step nt N n nt CL w a F5 w Plen Slen QRw QRt QI

6 48 32 0.67 1.0 11.7 6.2 0.75 0.68 0.68 B B 31.37 48 32 0.67 1.0 11.5 5.5 0.75 0.68 0.68 B B 31.78 24 16 0.67 1.0 7.1 2.7 0.75 0.70 0.71 B B 34.39 24 16 0.67 1.0 4.0 1.9 0.75 0.70 0.71 B B 61.6

10 24 22 0.92 1.0 4.9 3.0 0.75 0.69 0.71 B B 82 24 22 0.92 1.0 3.9 2.3 0.75 0.69 0.71 B B 104.9

For the Rotational Optimization, the procedure started on all focal planes nt = 48) then one focal plane is selectedfor each mechanism at step 8 nt =24). In step 10, mechanisms previously excluded by the Right Dihedron procedureare reincorporated n = 16-22). The highest possible quality for focal mechanisms is quality B because of theuncertainty in the differentiation between actual focal plane and auxiliary plane for each mechanism. The valuesof Plen and Slen increase with increasing separation, reflecting a decreasing in diversity of focal planes and associated slip directions. In this example the quality index (QI) regularly increases

eonet (Fig. 10-1). To simulate the different typesof brittle structures, we used an initial friction

angle of 10, a cohesion factor of 10 (arbitraryunits) and a ratio u3/a-I of 0.2. On a Mohr diagram(Fig. 10-2), the individual data are plotted in different areas, corresponding to the types of structures generated: 88 faults, 9 tension fractures, 24compression fractures and 31 shear fractures.When working separately with all the faults, tension fractures and compression fractures, the stresstensors obtained with both methods are alwaysclose to that used for generating the data set (Fig.10-3 to 10-6) . The orientations of the stress axesalways fit closely to those used for generating thedata set. The R value obtained for the set of 88

faults is 0.51 with the Right Dihedron method and0.4 for the Rotational Optimization method(exactly the original value). For the tension andcompression fracture sets, the R value is estimatedonly with the Right Dihedron method and the samevalue is used in the Rotational Optimization. TheR values obtained (0.75 for the set of 9 tensionfractures and 0.35 for the set of 24 compressionfractures) reflect the weaker constraint on R usingfracture instead of fault data. The relatively smallnumber of data in the tension fracture set mightalso influence the accuracy of the result.

Regional applications

The program TENSOR is now in use in more than30 different laboratories worldwide and has beenused in a large spectrum of geotectonic settings,for the late Palaeozoic to the Neotectonic period.These include the East African rift (Delvaux et at.1992, 1997a; Delvaux 1993b, , the Baikal RiftZone (Delvaux et at. 1995b, 1997b; Petit et at.1996; San kov et at. 1997), the Dead Sea Rift (ZainEldeen et at. 2002), the Oslo graben (Heeremanset at. 1996), the Apennines (Cello et at. 1997;Ottria Molli 2000), the Neogene evolution ofSpain (Stapel et at. 1996; Huibergtse et at. 1998and the Altai belt in Siberia (Delvaux et at. 1995a,

1995c; Novikov et at. 1998). Inversion of focalmechanisms has also been performed using thisprogram (Petit et at. 1996 .

on lusions

New aspects of the tectonic stress inversion havebeen discussed. The separation of data into subsetsduring the inversion is an integral part of the stressanalysis process. has to be performed with muchcare and has to rely first on an initial manual separation on the basis of careful field observations.


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Tensorappliedto 152 weaknessplanes 2/ EquivalentMohrdiagram

conesfactcr.

Areaaffaul ts

3/ Right Dihedron, 88 faults 4/ RotationalOptimisation on 88 faults

i

i s l : e e ....oo. i . S Z : J 6 / l O O~ S : A l 9 9f t : 4 1

@ S l : 8 Q I ( t 8 1.i. 32: IlEr.'Z7e[ ] S3: iH.o MlK. , , : .75

6/ Rotational Optimisation, 9 tension fractures

@Sl:llEVUl9 S Z ; 1 8 / : 1 6 6[ ]33:99/Z7QR : . 5 1

@Sl:Ele.-18e.SZ: l SI Z781 JS3:851 8 t6

: . 1 5

5/ RightDihedron, 9 tension fractures

7/ Right Dihedron, 24 compression fractures 8/ Rotational Optimisation,24 compr. fractures

Fig. 10. Example of stress analysis of a synthetic data set produced by applying a str ike-sl ip stress tensor north

south, rrl, vertical a R = 0.40) on a reference set of 152 pre-existing weakness planes. (1) Tangent-lineation diagram,lower hemisphere projection of all 152 planes, with arrow showing the directions of slip for activated faults. 2) Mohrdiagram illustrating the areas of faults and different types of fracture. 3 and 4) Right Dihedron and Rotational Optimization tangent-lineation diagram) solutions for the set of 88 faults. 5 and 6) Right Dihedron and Rotational Optimization great circle projection) solutions for the set of nine tension fractures. 7 and 8) Right Dihedron and RotationalOptimization great circle projection) solutions for the set of 24 compression fractures.

estimatation of the stress ratio R 2) the use oftension and compression fractures in combinationwith slickenside data, and 3) an initial separationof the data to be performed on the basis of theCounting Deviation. The stress tensor and prelimi-

Not only faults with slip lines slickensides) canbe used in the stress inversion, but also fracturestension, shear and compressional) and focal mech

anisms.The improved Right Dihedron method allows 1)


25/26


nary data set obtained after the use of the RightDihedron are used as a starting point for theRotational Optimization. This method is based ona controlled grid search with rotational optimization of a range of misfit functions. Of the differentmisfit functions proposed, the composite functionallows simultaneous minimization of the slip deviation a for slickensides, maximization of shearstress T for fault planes and shear fractures, minimization of normal stress v for tension fractures andmaximization of normal stress v for compressionfractures and stylolites.

A quality ranking scheme was developed in collaboration with the World Stress Map to estimatethe quality of the results obtained. It is based on aseries of parameters, for which threshold values arepre-defined. The type of stress regime is alsoexpressed numerically by a stress regime index RThe application of the Right Dihedron andRotational Optimization is illustrated by a syntheticdata set and natural data sets of fault-slip data andfocal mechanism data from the Baikal Rift Zone.

All the aspects discussed here were implementedin the TENSOR program developed by the firstauthor available on request . This program is atool for controlled interactive separation of faultslip or focal mechanism data and progressive stresstensor optimization using successively theImproved Right Dihedron method and the iterativeRotational Optimization method.

The development of this method for s tress tensor inver

sion and the related Quick-Basic computer program wasperformed by D. Delvaux during a series of projectsrelated to the East African and Baikal rift systems, fundedby the Belgian federal state former FRFC-IM , nowSSTC and hosted at the Royal Museum for CentralAfrica. The Quali ty ranking procedure was set up joint lywith the World Stress Map Project of Karlsruhe University. This paper was written during an ISES VisitingResearch Fellowship of D. Delvaux at the Vrije Universiteit Amsterdam. I thank S. Cloe tingh for his invi ta tionand support and D. Nieuwland for his edi torial effor ts .

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ANGELlER J. 1994. Fault slip analysis and paleostressreconstruction. In: HANCOCK P. L. ed. ContinentalDeformation. Pergamon, Oxford, 101-120.

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BOTT M. H. P. 1959. The me chani sm of oblique-slipfaulting. Geological Magazine 96, 109-117.

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DELVAUX D. 1993a. The TENSOR program for paleostress reconstruct ion: examples from the east Africanand the Baikal rift zones. In: Terra Abstracts. Abstractsupplement No.1 to Terra Nova 5, 216.

DELVAUX D. 1993b. Quaternary stress evolution in EastAfrica from da ta of the western branch of the Eas tAfrican rift. In: THORWEIHE U. SCHANDELMEIERH eds Geoscientific Research in Northern Africa. Balkema, Rotterdam, 315 318.

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DELVAUX D., FERNANDEZ ALONSO M., KLERKX J. etal. 1995a. Evidences for active tectonics in Lake Teletskoye Gorny-Altai, South-Siberia . Russian Geologyand Geophysics 36 10 , 100 112.

DELVAUX D., MOEYs R., STAPEL G., MELNIKOV A. ERMIKOV V. 1995b. Paleostress reconstructions andgeodynamics of the Baikal region, Central Asia. Par tI: Paleozoic and Mesozoic pre-rift evolution. Tectonophysics 5 61 101.

DELVAUX D., THEUNISSEN K., VAN DER MEER R. BERZIN N. 1995c. Formation dynamics of the GornoAltaian Chuya-Kurai depression in Southern Siberia:paleost res s, t ectonic and c lima ti c cont rol . RussianGeology and Geophysics 36 10 , 26 45.

DELVAUX D., KERVYN R., VITTORI E., KAJARA R. S.A. KILEMBE E. 1997 a. Late Quaternary tectonicactivity and lake level fluctuation in the Rukw a riftbasin, East Africa. Journal of African Earth Sciences26 3 , 397 421.

DELVAUX D., MOEYS R., STAPEL G. et al. 1997b.

Paleostress reconstruct ions and geodynamics of theBaikal region, Central Asia. Part II: Cenozoic tectonicstress and fault kinematics. Tectonophysics 282 1-4 ,1-38.

DELVAUX D., FRONHOFFS A., Hus R. POORT J. 1999.Normal fault splays, relay ramps and transfer zones inthe central par t of the Baikal Rift basin: ins ight fromdigital topography and bathymetry. Bulletin desCentres de Recherches Exploration-Production El fAquitaine 22 2 , 341 358 2000 issue .

DUNNE W. M. HANCOCK P. L. 1994. Paleostressanalysis of small-scale brittle structures. In: HANCOCKP. L. ed. Continental Deformation. Pergamon,Oxford, 101 120.


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DUPIN, J. -M., SASSI W. ANGELIER J. 1993. Homogeneous stre ss hypothesis and actual fault slip: a distinct element analysis. Journal o Structural Geology,15, 1033-1043.

ETCHECOPAR A. VASSEUR G. DAIGNIERES M. 1981.An inverse problem in microtectonics for the determination of s tress tensors from fault s tr ia tion analysis.Journal o Structural Geology, 3, 51-56.

GEPHART 1. W. 199 a St re ss and the d ir ec tion of slipon fault planes. Tectonics, 9 4), 845-858.

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