Numerical paleostress analysis – the limits of automation

Abstract: The purpose of this paper is to highlight the main problematic questions limiting theautomation of numerical paleostress analysis of heterogeneous fault-slip data sets. It focusses on prob-lems concerning the creation of spurious solutions, the separation of data corresponding to several tec-tonic phases, and the precision of measurement. It offers some ways of dealing with these problems andtrying to control their influence on the accuracy of solutions.

Keywords: numerical paleostress analysis, fault-slip data, principal stresses, reduce stress tensor.

1Department of Geological Sciences, Faculty of Science, Masaryk University, Kotlarska 2, 611 37 Brno, Czech Republic.

*e-mail: mujstock@email.cz

M. KERNSTOCKOVA1* AND R. MELICHAR1

The stress field reconstruction of deformed rocks is akey to understanding tectonic events in studied areas.It is usually based on an analysis of fault-slip data sets.Fault-slip data (e.g. orientation of faults, orientationof striation, sense of slip) code information about thestress state which causes these brittle structures.Changes of stress field can cause new fault formationor the reactivation of older ones. Multiple striationson some fault surfaces indicate this reactivation.Numerical searching for appropriate stress states act-ing on studied areas is the focus of many studies (e.g.Fry, 1999; Yamaji, 2000; Shan et al., 2004a).Automation of this process is complicated by manyfactors such as the variability of the stress field in geo-logical time, the precision of measurements, the orig-ination of spurious solutions, etc.

Homogeneous vs. heterogeneous data sets

There are several approaches to determine possiblepaleostress conditions. The simple inverse method(Carey and Brunier, 1974; Nemcok and Lisle, 1995;Yamaji, 2000; Shan et al., 2004a; among others) canbe applied to fault-slip data, which represent a singletectonic phase – it means that all faults were activat-ed under the same stress state. In this case the fault-slip data set is called homogeneous (Angelier et al.,1982). In fact, we use a direct solution to calculate

the reduced stress tensor (Angelier et al., 1982) for afour-element fault-slip group. The reduced stresstensor is represented by three directions of principalstresses σ1, σ2, σ3 (σ1 ≥ σ2 ≥ σ3) and its relative valueis expressed by a shape parameter (e.g. Lode’s ratioμL). Otherwise, the stress ellipsoid represents theexisting stress state constituted by the nine-dimen-sional stress tensor vector Tσ (Melichar andKernstockova, 2009).

If we process a homogeneous data set with morethan four faults, the optimal stress tensor is distin-guished simply, just like the best-fit solution basedon the least square method. Let N be the numberof all faults with striation. Each fault-slip datumcan be expressed as a nine-dimensional unit vectorC (Shan et al., 2004b; Li et al., 2005, Melicharand Kernstockova, 2009). Let Ci be the i-th vectorand δi the angle between vector Ci (for i = 1...N)and any other 9D vector (e.g. the normalized stressvector Tσ we are looking for, which must be per-pendicular to all Ci vectors). Let us choose cosδi asthe representative of deviation from the perpendi-cular position. This choice enables us to find thedirection with minimal deviation mathematically.Afterwards, for a homogeneous data set, the opti-mal stress vector Tσ can be calculated just like abest-fit solution by minimizing the function:

Trabajos de Geología, Universidad de Oviedo, 29 : 399-403 (2009)

(1),

in which cosδi can be replaced with the scalar productof unit vectors Ci and normalized stress vector Tσ:

(2),

where vector Tσ is unknown but constant. Thus equa-tion (2) can be rearranged as:

(3).

Symbol [M]:

(4),

denotes the orientation matrix of a fault-slip data set.

This orientation matrix is symmetric; the eigenvectorcorresponding to the second smallest eigenvalue ofthis matrix designates a stress vector with a minimalsum of deviations between vectors Ci and Tσ, i.e. theoptimal stress vector which is as perpendicular as pos-sible to each of the Ci vectors. This is the completely9D parallel to Fry’s (1999) procedure in 6D.

However, the stress conditions vary in geological timeand thus field data are usually heterogeneous. In thiscase a multiple inverse method (Yamaji, 2000) is usedto process data and estimate possible stress states.

The heterogeneity of a field data set is the basic prob-lem of fault-slip data paleostress analysis. There aresome indicators to distinguish the minimal number ofstress phases (e.g. multiple striations on a single faultsurface) or the relative ages of different stress states,but it is not so easy to assign an individual fault to cer-tain tectonic phases.

A field data set is a mixture of polyphase stress staterecords on fault surfaces which we must processtogether. The fault-slip data are combined just intofour-element groups and the reduced stress tensor(shape parameter μL and orientation of stress ellip-soid) is calculated for each group. Any of the fourfault-slip data is a vector C in 9D space and the stresssolution is represented by vector Tσ perpendicular toeach of them. It means C.Tσ = 0, where symbol “dot”marks the scalar (“dot”) product of two vectors.

A group with four homogeneous fault-slip data (i.e.four fault-slip data corresponding the same tectonicphase) provides the true results which characterize thereal paleostress conditions activating the movementalong these faults. The stress tensor calculated for aheterogeneous four-fault group is not reliable.Projections of directions calculated from heteroge-neous data sets – false results – are dispersed, whereasthe true results obtained from homogeneous data aregrouped in clusters.

The density maximum indicates some of the possibledirections of considered principal stress. The numberof such clusters indicates a minimal number of pale-ostress phases. The best way to recognize the densitymaxima of the correct solution is the direction densi-ty analysis in (9 – 4 =) 5-dimensional space (Melicharand Kernstockova, 2009).

For the density distribution calculation we use theWatson distribution usually used for contour plots, gen-eralized on multidimensional space. Searching for theoptimal stress vector Tσ can be graphically expressed bythe density distribution h(a) in direction a:

(5),

where W(a, Ci) is an effect of Cith measurement of

density in direction a (Fig. 1):

(6),

where k is a constant determining that the probabili-ty in all directions is equal to one (to compute onlythe relative concentrations is correct by simply apply-ing k = 1); κ is the shape parameter termed the con-centration parameter, because the larger the value of κ,the more the distribution is concentrated arounddirection a (Fisher et al., 1987).

Paleostress analysis – limits of automation

The rate of automation is an important criterionwhen calling the software “user friendly”. However, inmany cases it is difficult to transform the problemat-ic task in a numerical way. Usually it is possible tofind an algorithm to solve the problem automatically(e.g. Shan, et al., 2004a; Yamaji et al., 2006) but atthe expense of the accuracy of solution. In paleostressanalysis of heterogeneous fault-slip data sets there arealso some difficulties of this kind. To make the prob-

M. KERNSTOCKOVA AND R. MELICHAR

∑n

icos2 δi

∑N

i(TσT . Ci)2=∑

N

iTσT .Ci .CiT . Tσ

∑i

W(a,Ci)h(a) =

W(a,Ci) = .exp[κ.(a.Ci)2]

∑N

iTσT .Tσ =TσT . [M].TσCi .CiT

∑N

i[M]= Ci .CiT

1k

400

lem more graphic we can use the parallelism betweenpaleostress analysis of fault-slip data in 9D and foldaxes analysis of bedding planes taken from several dif-ferent folds in 3D space.

Spurious solutions

In the paleostress analysis of a heterogeneous fault-slipdata set there is a problem of some analysis outputsbeing spurious and not reflecting real stress condi-tions in rock. These solutions are products of stresstensor numerical calculation from heterogeneous fourfault-slip data groups.

The analogous situation is well known in fold-axis analy-sis, where we are looking for the fold β-axis just as a direc-tion perpendicular to the bedding planes unit normal vec-tors n1, n2. Fold β-axis is calculated like a vector product:

(7).

In the studied area we commonly measure severalbedding planes of every fold. In a similar way as twobedding planes (n1, n2) of two different folds producea β-axis of a nonexistent fold, four fault-slip data (CI,CII, CIII, CIV as symmetrical components in 5D sub-space, see Melichar and Kernstockova, 2009) corre-sponding to different tectonic phases produce spuri-ous solutions of the stress state Tσ:

(8).

These spurious solutions are dispersed whereas thecorrect ones are clustered, so they can be simply dis-tinguished by the density distribution function h(a)mentioned above.

The situation is more complicated because of the pre-ferred orientation of field data. Preferred orientationof faults brings spurious density maxima of solutions.Let us demonstrate the problem using the fold-axisanalysis parallel.

The analysis of bedding orientation usually leads toseveral clusters of bedding normals instead of whole“great circles”. Consequently, in the case of a morethan one-fold system, the determination of the cor-rect “great circles” is not explicit: some β-axis calculat-ed as a pole to the “great-circle” defined by pairs ofclusters may not represent real fold axes (Fig. 2a).Analogously, due to the preferred orientation offaults, only segments of “great hypercircles” are gener-ated. The determination of an appropriate stress stateTσ as a pole to this “great-hypercicle” is at risk ofresulting in the same error.

How can this problem be addressed? Spurious maxi-ma could be recognized from the character of faultsurfaces (e.g. material of accretion steps, alterations).Faults with different characters separated into onegroup probably represent a spurious tectonic phasecreated by the preferred orientation of input data,whereas faults with the same characters separated intoone group were probably activated by common realstress.

Numerical separation of fault-slip data corresponding toseveral tectonic phases

Another problem is related to the fact that some fault-slip data correspond to several phases, so it is difficult toseparate them correctly and assign them their corre-sponding stress state. Analogously, in fold-axis analysisthis sorting problem is well known as well. Data fromtwo or more folds bring two or more “great circles” ofappropriate bedding poles. Bedding poles surroundingthe intersection point of these two or more “great cir-cles” are disputative in terms of their assignment to acertain β-axis represented by the pole of “great circle”(Fig. 2b). Analogously, heterogeneous fault-slip datacreate points of several “great hypercircles”. The poles ofthese “great hypercicles” indicate possible stress states, apossible tectonic phase. Consequently, fault-slip datasurrounding the intersection of two “great hypercircles”are similarly ambiguous with respect to their assignmentto a certain tectonic phase. In practice, this separationcan be performed numerically, but this is only an artifi-cial separation. If there are common characters of sometectonic phases, the accuracy of this separation could beverified. Fault-slip data with the same characters of faultsurfaces should belong to the same group.

Figure 1. Probability density function of the Watson distributionfor κ = 50 demonstrating one measurement effect on density.

NUMERICAL PALEOSTRESS ANALYSIS – THE LIMITS OF AUTOMATION

[n1 ×n2]=β

CI × CII × CIII × CIV =Tσ

401

Moreover, in the partitioning method there is also thesingularity problem. The main condition for separation– the fault-slip data vector C is perpendicular to thestress tensor vector Tσ (i.e. C · Tσ = 0) – is always validfor faults perpendicular to any principal stress σ1, σ2,σ3. But in this case, the shearing stress reacting on thefault surface is zero in any direction; thus it is zero indirection l and m too and the fault could not be everreactivated (vector n is the normal to the fault surfaceand is down directed, vector l is a vector oriented in thedirection of hanging-wall movement (i.e. parallel tostriation) and the vector m lies on the fault surface at aright angle to the striation, which means it is perpendi-cular to n and l).

To evaluate the possibility that the fault will be reac-tivated, the relative Mohr criterion Δτ’ based on theestimated angle of internal friction φ of the deformedrock, the normalized shear stress τ and the normalizednormal stress σ was proposed:

(9),

where:

(10),

and:

(11),

Symbol τ denotes tangential and σn normal stresses onthe fault surface. The higher the value of Δτ’, thehigher the probability of reactivation (Fig. 3a).

Precision of measurement

The precision of measurement is an issue for all kindsof analysis. Precision of measurement in fault-slip dataanalyses controls the precision of solutions. Input datarounded to thousands of degrees provide distinct solu-tions; solutions of analysis from data rounded to wholedegrees are still acceptable but the increasing inaccura-cy of input data rapidly diminishes the precision ofsolutions. The more the fault-slip data are analyzed, themore precise the calculated solutions of the stress statesbecome. Especially if a small amount of fault-slip datais analyzed, the position of measured striation must betested in accordance with the orientation of the meas-ured fault surface. If striation does not lie on the faultsurface and the deviation is small, data should beorthogonalized; if the deviation is large it would be bet-ter to exclude the data from the analysis.

Moreover, a stress tensor calculated from nearly paral-lel data (it means nearly parallel faults with nearly par-allel striation) is numerically correct but practicallyerror prone and this solution should be excluded. Thescale of parallelism is indicated by the Gram determi-nant G of vectors CI, CII, CIII, CIV, Axyz, Axy, Axz, Ayz(Melichar and Kernstockova, 2009). The magnitudeof the square root of the Gram determinant √G nearto zero indicates that two or more of the pieces ofinput data are nearly parallel and this solution should

Figure 2. A problem with separa-tion of heterogeneous fault-slip dataschematically illustrated on analo-gous situation with β-axis analysis.Full circles represent bedding nor-mals of fold 1, empty circles denotebedding normals of fold 2. Symbolsβ1, β2 indicate β-axes of fold 1 andfold 2. a) Problem of data preferredorientation: the dashed line and β‘symbol represent one possibility ofthe false β-axis determination,b) problem of numerical separationin the case of data corresponding tomore tectonic phases: short inter-secting lines illustrate one possibleartificial separation producingmatching of some bedding normalsto incorrect β-axis.

M. KERNSTOCKOVA AND R. MELICHAR

Δτ´= σ.tanφ–τtan2φ+1

τ= 2τσ1– σ3

σ=2σn–σ1–σ3

σ1–σ3

402

be excluded. Otherwise, the more the magnitude ofthe square root of the Gram determinant approxi-mates to superior limit 1/8 (Melichar andKernstockova, 2009), the more perpendicular are thevectors CI, CII, CIII, CIV and the more stable is thesolution. Consequently, the magnitude of the Gramdeterminant is a good criterion for controlling the sta-bility of solutions (Fig. 3b).

Conclusions

The automation of the numerical paleostress analysisof heterogeneous fault-slip data has some limits which

emerge especially in the effort to separate or matchfault-slip data to certain tectonic phases. To dealwith these limits some control parameters are pro-posed (κ, Δτ’, √G). Therefore, numerical automationof paleostress analysis is practicable but in the chargeof a geologist, particularly when the analysis of theresults are interpreted.

Acknowledgements

Thanks to Matthew Nicholls for his proofreading. The study wassupported in part by the grant project MSM 0021622412 and byGA AVCR, the grant IAAA3013406.

Figure 3. Distribution of thereactivity and stability of solu-tions of real fault-slip data fromthe Brno Massif granitoids.a) The best point to limit the areaof reactivation is the point wherethe distribution of reactivitysteeply increases, b) visualizationof stability distribution enables usto exclude the unstable solutionsfrom the analysis.

NUMERICAL PALEOSTRESS ANALYSIS – THE LIMITS OF AUTOMATION

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