Natural fault patterns formed in response to a single tectonic event often display significant variation in their orientation distribution. The cause of this variation is the subject of some debate: it could be “noise” on underlying conjugate (or bimodal) fault patterns or it could be intrinsic “signal” from an underlying polymodal (e.g. quadrimodal) pattern. In this contribution, we present new statistical tests to assess the probability of a fault pattern having two (bimodal, or conjugate) or four (quadrimodal) underlying modes and orthorhombic symmetry. We use the eigenvalues of the second- and fourth-rank orientation tensors, derived from the direction cosines of the poles to the fault planes, as the basis for our tests. Using a combination of the existing fabric eigenvalue (or modified Flinn) plot and our new tests, we can discriminate reliably between bimodal (conjugate) and quadrimodal fault patterns. We validate our tests using synthetic fault orientation datasets constructed from multimodal Watson distributions and then assess six natural fault datasets from outcrops and earthquake focal plane solutions. We show that five out of six of these natural datasets are probably quadrimodal and orthorhombic. The tests have been implemented in the R language and a link is given to the authors' source code.

Faults are common structures in the Earth's crust, and they rarely occur in
isolation. Patterns of faults, and other fractures such as joints and veins,
control the bulk transport and mechanical properties of the crust. For
example, arrays of low-permeability (or “sealing”) faults in a rock matrix
of higher permeability can produce anisotropy of permeability and preferred
directions of fluid flow. Arrays of weak faults can similarly produce
anisotropy, i.e. directional variations, of bulk strength. It is
important to understand fault patterns, and quantifying the geometrical
attributes of any pattern is an important first step. Faults, taken as a
class of brittle shear fractures, are often assumed to form in conjugate
arrays, with fault planes more or less evenly distributed about the largest
principal compressive stress,

Schematic diagrams to compare conjugate fault patterns displaying
bimodal symmetry with quadrimodal and polymodal fault patterns displaying
orthorhombic symmetry.

Fault patterns are most often visualised through maps of their traces and equal-angle (stereographic) or equal-area projections of poles to fault planes or great circles. Azimuthal projection methods (hereafter “stereograms”) provide a measure of the orientation distribution, including the attitude and the shape of the overall pattern. However, these plots can be unsatisfactory when they contain many data points or the data are quite widely dispersed. Woodcock (1977) developed the idea of the fabric shape, based on the fabric or orientation tensor of Scheidegger (1965). The eigenvalues of this second-rank tensor can be used in a modified Flinn plot (Flinn, 1962; Ramsay, 1967) to discriminate between clusters and girdles of poles. These plots can be useful for three of the five possible fabric symmetry classes – spherical, axial and orthorhombic – because the three fabric eigenvectors coincide with the three symmetry axes. However, there are issues with the interpretation of distributions that are not uniaxial (Woodcock, 1977). We address these issues in this paper. Reches (Reches, 1978, 1983; Aydin and Reches, 1982; Reches and Dieterich, 1983) has exploited the orthorhombic symmetry of measured quadrimodal fault patterns to explore the relationship between their geometric–kinematic attributes and tectonic stress. More recently, Yielding (2016) measured the branch lines of intersecting normal faults from seismic reflection data and found they aligned with the bulk extension direction – a feature consistent with their formation as polymodal patterns. Bimodal (i.e. conjugate) fault arrays have branch lines aligned perpendicular to the bulk extension direction.

Stereographic projections (equal area, lower hemisphere) showing
two natural fault datasets.

The fundamental underlying differences in the symmetries of the two kinds of
fault pattern – (i) bimodal and bilateral or (ii) and polymodal and
orthorhombic – suggest that we should test for this symmetry using the
orientation distributions of measured fault planes. The results of such
tests may provide further insight into the kinematics and/or dynamics of the
fault-forming process. This paper describes new tests for fault pattern
orientation data and includes the programme code for each test written in the
R language (R Core Team, 2017). The paper is organised as follows:
Sect. 2 reviews the kinematic and mechanical issues raised by conjugate
and polymodal fault patterns, in particular the implications for their
orientation distributions. Section 3 describes the datasets used in this
study, including synthetic and natural fault orientation distributions.
Section 4 presents tests for assessing whether an orientation distribution
has orthorhombic symmetry, including a description of the mathematics and
the R code. The examples used include synthetic orientation datasets of
known attributes (with and without added “noise”) and natural datasets of
fault patterns measured in a range of rock types. A discussion of the issues
raised is provided in Sect. 5 and is followed by a short summary. The R
code is available from

Stereographic projections (equal area, lower hemisphere) showing
the eight synthetic datasets designed to model conjugate (bimodal) fault
patterns in this study.

Stereographic projections (equal area, lower hemisphere) showing
the eight synthetic datasets designed to model quadrimodal fault patterns in
this study.

Conjugate fault patterns should display bimodal or bilateral symmetry in
their orientation distributions on a stereogram and ideally show evidence
of central tendency about these two clusters (Fig. 1d; Healy et al.,
2015). Quadrimodal fault patterns should show orthorhombic symmetry and,
ideally, evidence of central tendency about the four clusters of poles on
stereograms (Fig. 1e). More general polymodal patterns should show
orthorhombic symmetry with an even distribution of poles in two arcs (Fig. 1f). For data collected from natural fault planes some degree of intrinsic
variation, or “noise”, is to be expected. Two natural example datasets are
shown in Fig. 2. The Gruinard dataset is from a small area
(

Stereographic projections (equal area, lower hemisphere) showing
the six natural datasets used in this study. All plots show poles to faults,
the majority of which are inferred to be normal.

Graphs showing the ratios of eigenvalues of the orientation
matrices for the synthetic datasets (Flinn, 1962; Ramsay, 1967; Woodcock,
1977).

We use two sets of synthetic data to test our new statistical methods, both
based on the Watson orientation distribution (Fisher et al., 1987 Sect. 4.4.4; Mardia and Jupp, 2000 Sect. 9.4.2). This is the simplest
non-uniform distribution for describing undirected lines, and has
probability density

We use six natural datasets of fault plane orientations from regions that
have undergone or are currently undergoing extension; i.e. we believe the
majority of these faults display normal kinematics (Fig. 5). The Gruinard
dataset (Fig. 5a) is from Gruinard Bay in NW Scotland (UK) and featured
in previous publications (Healy et al., 2006a, b). The most important thing
about this dataset is that the fault planes were all measured from a small
area (

The

The

The

We calculated the second-rank orientation tensor (Woodcock, 1977) for each
of the synthetic datasets shown in Figs. 3 and 4 (bimodal and quadrimodal,
respectively). The eigenvalues of this tensor (

To get a suitable general setting for our tests, we formalise the
construction of the bimodal and quadrimodal datasets considered in Sect. 3.1. Whereas the datasets considered in Sect. 3.1 necessarily have equal
numbers of points around each mode, for datasets arising from the
distributions here, this is true only

Suppose that axes

The bimodal equal mixture model can be thought of intuitively as obtained by
“pulling apart” a unimodal distribution into two equally strong modes, angle

The quadrimodal equal mixture model can be thought of intuitively as
obtained by “pulling apart” a bimodal equal mixture distribution into two
bimodal equal mixture distributions with planes angle

The problem of interest is to decide whether the parent distribution is Eq. (1) or Eq. (2).

Given axes

Some algebra shows that

Table 1 gives the

Eigenvalue ratio plot for the natural datasets shown in Fig. 5.
All but one dataset (central Italy) lie above the line for

Eigenvalue ratio plots of synthetic data to illustrate the impact
of dispersion on the ability of this plot to discriminate between conjugate
(bimodal) and quadrimodal fault data.

Eigenvalue ratio plots comparing the relative performance of the
two tests proposed in this paper. The red lines denote

Table 3 gives the

In the analysis described above and the tests we performed with synthetic datasets, we assumed that bimodal and quadrimodal Watson orientation distributions provide a reasonable approximation to the distributions of poles to natural fault planes. In terms of the underlying statistics this is unproven, but we know of no compelling evidence in support of alternative distributions. New data from carefully controlled laboratory experiments on rock or analogous materials might provide important constraints for the underlying statistics of shear fracture plane orientations.

We have tested our new methods on synthetic and natural datasets. Arguably,
six natural datasets are insufficient to establish firmly the primacy of
polymodal orthorhombic fault patterns in nature (Fig. 7). However, we
reiterate the key recommendation from Healy et al. (2015): to be useful for
this task, fault orientation datasets need to show clear evidence of
contemporaneity among all fault sets through tools such as matrices of
cross-cutting relationships (Potts and Reddy, 2000). In addition, as shown
above, larger datasets (

The Chimney Rock dataset is probably not orthorhombic according to the two
tests and lies close to the line for

A final point concerns dispersion (noise) in the data. Synthetic datasets of
bimodal (conjugate) and quadrimodal patterns with low values of

To assess the relative performance of the two tests presented in this paper,
we generated synthetic bimodal and quadrimodal distributions and compared
the resulting

Bimodal (conjugate) fault patterns form in response to a bulk plane strain with no extension in the direction parallel to the mutual intersection of the two fault sets. Quadrimodal and polymodal faults form in response to bulk triaxial strains and constitute the more general case for brittle deformation on a curved Earth (Healy et al., 2015). In this contribution, we show that distinguishing bimodal from quadrimodal fault patterns based on the orientation distribution of their poles can be achieved through the eigenvalues of the second- and fourth-rank orientation tensors. We present new methods and new open source software written in R to test for these patterns. Tests on synthetic datasets in which we controlled the underlying distribution to be either bimodal (i.e. conjugate) or quadrimodal (i.e. polymodal, orthorhombic) demonstrate that a combination of fabric eigenvalue (modified Flinn) plots and our new randomisation tests can succeed. Applying the methods to natural datasets from a variety of extensional normal-fault settings shows that five out of the six fault patterns considered here are probably polymodal. The most tightly constrained natural dataset (Gruinard) displays clear orthorhombic symmetry and is unequivocally polymodal. Most map-scale natural faults evolve and grow through interaction, splaying and coalescence, and in some cases through reactivation under stress rotation. Variation within fault orientation datasets is therefore inherent. Statistical tests can help to discern this variation and guide the interpretation of any underlying pattern. We encourage other workers to apply these tests to their own data, assess the symmetry in the brittle fault pattern and to consider what this means for the causative deformation.

Datasets used in this study are available from the first author on request (d.healy@abdn.ac.uk).

DH devised the study, and PJ formulated the statistical tests and wrote the R code. DH collated the data and wrote the paper with input from PJ.

The authors declare that they have no conflict of interest.

David Healy gratefully acknowledges receipt of NERC grant NE/N003063/1 and thanks the School of Geosciences at the University of Aberdeen for accommodating a period of research study leave, during which time this paper was written. We thank two anonymous reviewers, plus Atilla Aydin (Stanford) and Nigel Woodcock (Cambridge), for comments which helped us improve the paper. Edited by: Federico Rossetti Reviewed by: Nigel Woodcock, Atilla Aydin, and two anonymous referees