Journal of Structural Geology 27 (2005) 2223–2233

www.elsevier.com/locate/jsg

Determination of fabric and strain ellipsoids from measured sectional

ellipses—implementation and applications
Patrick Launeaua, Pierre-Yves F. Robinb,*
a
Laboratoire de Planétologie et Géodynamique, Université de Nantes, Faculté des Sciences et des Techniques, 2, rue de la Houssinière, 44072 Nantes, France
b
Department of Geology, Earth Sciences Centre, University of Toronto, 22 Russell Street, Toronto, ON, Canada M5S 3B1
Received 12 October 2004; received in revised form 9 July 2005; accepted 2 August 2005
Available online 15 September 2005

Abstract
Geologists examine fabrics to constrain models of formation or of deformation of rocks, and it is often convenient to summarise the results
by a fabric ellipsoid. As fabric data are commonly collected on planar sections through the rock, estimating a fabric ellipsoid from sectional
ellipses, often with arbitrary orientations, is an important task. An algebraic method to calculate such an ellipsoid, presented in an earlier
paper, has been implemented with the program ELLIPSOID. It is used here on examples that illustrate questions and issues that arise when
collecting, selecting and processing sectional fabric data, and when assessing the results. The quality of fit of the ellipsoid to the data is
assessed in all cases. Examples include a case in which the average sizes of markers on individual sections can be used in the determination of
the ellipsoid, and other cases in which such sizes are not useful; a case in which sectional ellipses are not obtained from closed markers; and a
case in which data scatter and insufficient coverage of section orientations lead to a hyperboloid instead of an ellipsoid.
q 2005 Elsevier Ltd. All rights reserved.

Keywords: Sectional fabric ellipse; Fabric ellipsoids; Sectional ellipses; Strain analysis

1. Introduction some of the issues that arise when seeking a best-fitting

fabric ellipsoid.
Geologists often seek to obtain three-dimensional fabric Robin (2002) developed two solutions to determine a
information from rocks, although many fabric data are best-fitting ellipsoid. In ‘Case 1’, ‘with scale factor’, the
collected on two-dimensional sections. As in the type average size of markers on individual sections is deemed
example of paleostrain determination, whenever sectional significant and contributes to determination of the ellipsoid.
fabric data can be represented or summarized by an ellipse Indeed, in the field, a geologist commonly uses average
and the sought three-dimensional fabric by an ellipsoid, it is marker sizes to approximate fabric directions. One may, for
necessary to calculate the ellipsoid from measured sectional example, search sections on which average markers size is
ellipses. Robin (2002) presented an algebraic solution to the smallest as an approximation of the normal to the shape
problem of fitting an ellipsoid from three or more sectional lineation direction; or, on the contrary, one might search for
ellipses of arbitrary orientations. The method is the largest marker size to find the foliation. In Case 1, the
implemented in ELLIPSOID,1 a Visual Basic program, and data collected on each of three or more sections are typically
we present here several examples that illustrate its use and the orientation of the long axis of the sectional ellipse, and
the sizes of its long and short diameters. Sectional data,
including the actual dimensions of sectional markers, are
used to build the components of a system of six linear
* Corresponding author. Tel.: C1 416 978 5080; fax: C1 416 978 3938.
equations in the six unknown coefficients describing the
E-mail addresses: patrick.launeau@chimie.univ-nantes.fr ellipsoid.
(P. Launeau), py.robin@utoronto.ca (P.-Y.F. Robin). In ‘Case 2’, ‘without scale factor’, individual sections do
1
ELLIPSOID can be downloaded as freeware from http://www.sciences. not yield any useful size information. This may be because
univ-nantes.fr/geol/UMR6112/SPO/. the number of markers on each individual section is too
0191-8141/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. small to give a meaningful indication of average size, or
doi:10.1016/j.jsg.2005.08.003 sizes of the 3D markers are too variable, or the method to
2224 P. Launeau, P.-Y.F. Robin / Journal of Structural Geology 27 (2005) 2223–2233

determine a sectional ellipse does not yield any size. The how individual incompatibility indices can be used to either
only data retained are then orientation and axial ratio of each track measurement errors or provide additional information
sectional ellipse. Given N sections, the data are used to build on mechanisms of fabric acquisition. (3) The opaque
the components of a system of (NC5) equations in (NC5) aggregates studied in thin sections of the Tellnes ilmenite
unknowns: N ‘scale factors’ (one for each section) and five deposit of SW Norway provide an example of determination
independent parameters that define a dimensionless of sectional ellipses that are not determined from closed
ellipsoid. markers, and one in which the ellipsoid determined from
In both Cases 1 and 2, solution of the system of equations sectional data can be compared with that obtained by
corresponds
pffiffiffiffi to minimizing a scalar ‘incompatibility index’, measuring the anisotropy of magnetic susceptibility (AMS).
F~ , which is a measure of the misfit between sectional (4) Pyroxene fabric in gabbro-norite from the Critical Zone,
ellipses and the ellipsoid sought (Robin, 2002). As the on the eastern limb of the Bushveld complex, is acquired
system always yields a solution, regardless of quality and fit from large numbers of grains on each section, for which we
of the data, understanding the meaning of the incompat- can compare the results obtained by using or not using
ibility index is important in order to determine pour measured scale factors. The last two examples also illustrate
ffiffiffiffiffi
confidence in the results. An incompatibility index, F I , how practical ‘resampling statistics’ can document our
is also calculated for each individual face I, and can confidence that the sample size from data sections is
therefore eventually be used to query and re-evaluate sufficient. For each example, we try to indicate the
individual data. geological significance of the results.
As in the method of Owens (1984), ELLIPSOID implements Conventions used for data entered into ELLIPSOID and
the possibility of assigning different weights to different presented in Tables 1 and 3 (specifically the right-hand rule
sections. Different weights can be assigned as a function of and the convention on orientation of rake of the long axis of
the different confidence in the data from each section the sectional ellipse) are given in Appendix A. The appendix
(Robin, 2002) or because several sections, e.g. because of also discusses coordinate system and the transformation
their similar orientations, are not considered to be from laboratory coordinates to geographic coordinates that
ELLIPSOID can effect whenever convenient.
sufficiently independent from each other (Owens, 1984).
In some cases, the quadratic surface that best fits the data
is a single-sheet hyperboloid rather than an ellipsoid. This
2. Owens’s (1984) reduction spots in a slate from
might arise in particular when the sectional data are
Dinorwic, N. Wales, UK
scattered or of modest quality and there is no section
parallel to the long axis of the fabric ellipsoid (i.e. parallel to
Owens (1984) measured reduction spots on eight
the ‘lineation’). It is important then to know what further
sections cut through an unoriented block of slate (Table 1,
data are needed to ‘close’ that hyperboloid.
columns 1–6). Each section only displays one spot. In an
Four examples presented here are selected to illustrate
actual field project, an ellipsoid determined on an oriented
several aspects of fabric ellipsoid determination. (1) Owens
sample could be used to assess the direction, style, and
(1984) was the first to establish a method to determine an
intensity of deformation within a slate belt.
ellipsoid from any number (R3) of arbitrarily oriented
Whereas one might expect reduction spots in a given
sections. We apply ELLIPSOID to reduction spots analysed in
rock to have similar sizes in three dimensions, size set by
one of Owens’s examples and compare incompatibility some characteristic diffusion distance, actual sections
indices. (2) Mafic enclaves near the border of the Mont- through them are in general not through their centres.
Louis provide an example where insufficient data yield a Therefore, the size of one marker per section is not likely to
hyperboloid rather than the ellipsoid sought, and also show carry useful strain information; only the calculation without
scale factor (Case 2) is justified. Owens (1984) assigned
Table 1 weights to his measurements (Table 1, column 7),
Sizes and orientation of reduction spots on sections of a sample of Dinorwic decreasing some on the basis of proximity of their directions
slate, N. Wales. Data from Owens (1984) to those of other sections. Fig. 1a shows the results using the
pffiffiffiffiffiffiffiffiffi same weights as Owens, whereas Fig. 1b is for equal
No. Strike Dip Rake Long Short Weight r0K1 I
Fmin (%)
axis axis weights assigned to all sections. With an incompatibility
(mm) (mm)
1
index of 2.0 and 2.1%, respectively, the fit is good. The
302 78 165 16.5 4.5 0.58 0.17 3.0
2 301 77 166 9.5 3.5 0.58 0.16 2.8
effect of weighting is small, a consequence of the fact that
3 302 75 166 20.5 6.8 0.58 0.06 1.4 sectional ellipses are closely compatible.
4 201 71 173 37 6 1 0.03 0.2 Table 1 (last column) lists the individual incompatibility
5 178 71 0 7.5 1.5 1 0.07 1.9 indices for each spot. The index for Spot No. 6 is 4.4%. If
6 18 79 10 16.7 3 0.58 0.26 4.4
7
that face is discarded, the total incompatibility index for the
17 78 8 22 4 0.58 0.06 0.8
8 19 78 7 18 3 0.58 0.11 2.1
new determination is reduced to 1.6%, the trend of the long
axis changes by 98 and A/C is reduced from 7.7 to 6.5
 P. Launeau, P.-Y.F. Robin / Journal of Structural Geology 27 (2005) 2223–2233 2225

(Fig. 1c). Considering how small the data set is, these Table 2
changes may be deemed small. Owens’s (1984) best-fit Results of Owens (1984) for symmetry axes of ellipsoid calculated from
sections of reduction spots in a slate sample from Dinorwic, N. Wales, UK;
solution, shown in Table 2, is essentially identical to the to be compared with results in Fig. 2a
solution found with ELLIPSOID after elimination of Spot
No. 6. A B C
Owens (1984) calculates the equivalent of an Normalized 2.34 1.20 0.36
incompatibility index for each section by applying to length
Trend (8) 29 122 265
each sectional ellipse a virtual ‘retrodeformation’ defined Plunge (8) 10 14 73
by the ellipsoid found. This retrodeformed axial ratio, rI0 ,
can be compared with that of a circle, i.e. to 1, or to the
average value of these ratios for all sections. Owens’s 3. Mafic enclaves in the hercynian Mont-Louis granite,
values of rI0 K1 are given in Table 1 (column 8). Plot of eastern central Pyrenees
one incompatibility index vs. the other shows an
approximate linear correlation between the two. The Geologists concerned with granite emplacement gener-
discussion by Owens (1984) on the use of such an index ally interpret shapes of enclaves found in granitic rocks as
to identify ‘rogue data’ remains entirely appropriate for providing some record of deformation of these enclaves.
determinations with ELLIPSOID and will be reviewed again However, that deformation does not relate simply to any
in the next example. well-defined strain in the host granite, and it is also expected

Fig. 1. Two outputs of ELLIPSOID for sectional data on reduction spots in slates from Dinorwic, N. Wales, UK. Data, from Owens (1984), are shown in Table 1.
Each window shows the number of faces, N, used in the calculation, the coefficients of the inverse shape matrix calculated by the method of Robin (2002),
eigenvalues of that matrix, direction cosines of the corresponding eigenvectors, the corresponding diameters of the best-fit ellipsoid normalized to ABCZ1,
their directions given by their trends and plunges, axial ratios, strike and dip of the ‘foliation’ found, and rake of the ‘lineation’ within the foliation plane.
‘Flinn’ is the shape parameter, ðA=BK1Þ=ðB=CK1Þ; P 0 and T are, respectively, the intensity and shape parameters defined by Jelinek (1981) for the AMS tensor
and applied here to the shape ellipsoid. The equal-area spherical projection shows the poles to the planar sections used (C), the directions of the long axis of the
ellipsoid found (,), of its intermediate axis (6), and of its short axis (B), the plane of ‘foliation’ (i.e. plane A). (a) Ellipsoid found for sectional data weighted
as by Owens (1984). See Table 1. (b) Ellipsoid found when all sections are weighted equally.
2226 P. Launeau, P.-Y.F. Robin / Journal of Structural Geology 27 (2005) 2223–2233

that different populations of enclaves may show different close to the southern contact of the Mont-Louis granite, near
combinations of deformation and of rigid rotation, as Site 10 of Gleizes et al. (1993, fig. 3).
recently emphasized by Paterson et al. (2004). Still, Each oriented section measured is a rock face on which
enclaves often exhibit a well-defined anisotropic distri- one enclave section is seen and, as with reduction spots,
bution that must record some common history. We describe sectional areas of these enclaves are not expected to carry
a sequence of measurements and results obtained in the field any useful size information: only sectional axial ratios and

Fig. 2. Ellipsoid determinations from mafic enclaves in the Mont-Louis granite. (a) The first 14 enclaves measured yield a hyperboloid instead of an ellipsoid.
(b) Ellipsoid obtained with 21 additional measurements. (c) Elimination of seven measurements with individual compatibility indices above 30% does not
change the result much but decreases the overall incompatibility index.
 P. Launeau, P.-Y.F. Robin / Journal of Structural Geology 27 (2005) 2223–2233 2227

directions are used to determine an ellipsoid (Case 2, eigenvalue, shown as a square in Fig. 2a, is often close
‘without scale factor’). to the long direction of the ellipsoid that is found when
more data are obtained.
3.1. The ‘hyperboloid problem’
3.2. Incompatibility indices and elimination of outlying data
The first 14 measurements, from a road-cut outcrop, were
An additional 21 enclaves measured within 60 m of the
on rock faces with a relatively narrow range of orientations,
original site yield the result shown in Fig. 2b, in which the
shown by ELLIPSOID as poles (with a C symbol) in a lower
long axis of the ellipsoid is indeed close to the ‘complex
hemisphere equal-area projection (Fig. 2). They yield one
axis’ originally found and shown in Fig. 2a. Paradoxically,
negative eigenvalue for the inverse-shape matrix describing
the incompatibility index, 21.6%, is larger than before
the ellipsoid (Robin, 2002), indicating that the conic surface
(11.6%), no doubt a reflection of the complexity of fabric
found is a single-sheet hyperboloid rather than an ellipsoid.
development in enclaves. Such an increase does not
The corresponding axial ‘length’, rather than shown as a
necessarily mean that our confidence
pffiffiffiffi in our estimate has
complex number, is shown by ELLIPSOID in red and preceded decreased. This is because F~ can be thought of as an
by an asterisk (*). As explained below, finding a estimate of the standard deviation of the population of
hyperboloid is not due to an algebraic failure: some data sectional ellipses. It describes that population itself. In
may fit a hyperboloid best. analogy with the confidence in the estimate of a mean, and
Fig. 3a shows a model of three measured sectional without any claim to statistical rigour, we may p take
ellipses with an insufficient coverage in their orientations. ffiffiffiffiffiffiffiffiour
interval of confidence in the result as varying as F=N ~ ,N
Sizes of the measured sections are assumed not to be a being the number of sections measured. As it turns out, this
useful parameter, but they have been drawn here so that ratio does increase slightly here, from 3.10 for the 14
each area matches the area of their respective elliptical measurements to 3.65 for the 35 measurements. In other
section of the conic surface found. It is intuitively easy words, our confidence has indeed decreased somewhat,
to see that some distributions could fit a hyperboloid, and supporting the suggestion that the population of enclaves is
even do so with a low incompatibility index. Yet a complex and somewhat heterogeneous.
hyperboloid, even though it might be a solution to the Table 3 shows all 35 ffi data and the calculated
pffiffiffiffiffiffiffiffi
geometrical problem, is not an acceptable geological incompatibility index, Fmin I
, for each face I. We note
solution. If an additional, better-oriented face, such as that the index for Face 18 is 82.3%. This may be treated
Face 4 (Fig. 3b) is measured, it serves to ‘close’ the as an exceptional outlier, and we investigate the effects
conic surface, and the calculation now yields an of eliminating it (by assigning it a weight of 0). All
ellipsoid. In the authors’ experience, the symmetry axis individual incompatibility indices change somewhat; we
of the hyperboloid corresponding to the complex can now eliminate the enclave with the next highest
index. Repeating this procedure until all data with an
index O30% are eliminated yields the results shown in
Fig. 3d. After eliminating seven enclaves in this manner,
the overall incompatibility index has decreased from 21.6
to 10.4%, the foliation and lineation directions have not
changed significantly, but the Flinn parameter,
ðA=BK1Þ=ðB=CK1Þ, has changed from 2.30 to 0.56.
With this procedure, ellipsoid determination thus pro-
vides a tool and the data to investigate the competing
mechanisms leading to a final enclave fabric as well as
potential measurement errors (Owens, 1984), outliers can
be examined, e.g. for petrographic characteristics or for
shapes that are more or less likely to rotate than to
deform passively, etc.

3.3. Mont-Louis marginal enclaves

As mentioned earlier, the site studied is near the southern

margin of the Mont-Louis granite, where we would expect
the latest magmatic strain to be dominantly shear parallel to
Fig. 3. (a) In some cases, the quadratic surface that best fits measured
ellipses is a hyperboloid rather than an ellipsoid. (b) Additional data, the local direction of magma movement with respect to the
particularly from planar sections close to the direction of the axis [A] of the wall. For this site, with BZ0.91–1.08, i.e. close to 1, data
hyperboloid, should ‘close’ the ellipsoid. are compatible with such magmatic strain (at least that part
2228 P. Launeau, P.-Y.F. Robin / Journal of Structural Geology 27 (2005) 2223–2233

Table 3 production in the world. It is interpreted as a late dike

Measurements of enclaves near the south wall of the Mont-Louis Massif, (w920 Ma) within an anorthosite host. The ilmenite,
East Central Pyrenees, by Patrick Launeau and Gérard Gleizes
actually a hemo-ilmenite (Duchesne, 1999), is interstitial
pffiffiffiffiffiffiffiffiffi
I Strike Dip Rake Long Short I
Fmin (%) to the sub- to euhedral crystals of plagioclase, orthopyrox-
(8) (8) (8) axis axis ene and olivine (see Fig. 4a). Its texture has been interpreted
(cm) (cm)
as a result of coalescence and recrystallization between the
1 99 49 170 11 2.5 24.4 silicate minerals of earlier isometric-shaped ilmenite grains
2 93 63 155 13.5 2.5 19.5
3 93 63 157 5.5 2 10.7
(Diot et al., 2003). In a study of the norite, Diot et al. (2003)
4 151 74 133 7 2.5 11.1 compared the fabric of its opaque mineral phases, as
5 159 88 127 18.5 6 21.9 obtained from sectional image analysis, with measured
6 157 71 122 5.5 2.5 11.2 anisotropy of magnetic susceptibility (AMS). Sample TS 06
7 166 62 140 6 1.2 35.3
is reported here to illustrate the use of a sectional elliptic
8 148 83 134 4 0.9 19.8
9 150 43 145 6 1.7 31.7
fabric that is based on interconnected rather than closed
10 95 74 139 5 1.5 28.3 markers.
11 105 80 154 3.5 1.6 5.5 The determination of an ellipsoid from such a texture
12 144 83 154 8.5 1.6 37.9 requires some discussion. If a section characterized by
13 127 48 152 7 1.8 20.7
only two ‘phases’—meaning here all opaque minerals
14 117 29 152 3.7 1.6 5.5
15 321 74 50 10 5.1 13.6 vs. all non-opaque minerals—is isotropic, orientations of
16 320 73 51 8 2 28.2 sectional boundaries are equally distributed over all
17 325 76 49 6 2.8 15.4 orientations within the plane of the section. This means
18 146 78 20 11 1.3 82.3 that a ‘characteristic shape’ for the image—which can
19 0 87 60 40 18 24.9
20 317 78 53 6.5 2.1 24.8
be obtained in principle by sorting all boundaries
21 311 77 44 13.5 4 19.2 according to their orientation and drawing them in
22 83 35 0 14 5 18.5 sequence (e.g. Launeau and Robin, 1996, Section 4.5)—
23 72 30 5 5.5 2.2 16.7 is a circle. In three dimensions, the characteristic shape
24 66 38 10 2.8 1.7 4.7
would be a sphere and sectional characteristic shapes
25 60 78 25 19.5 8.7 36.3
26 96 49 170 7.5 2.2 20.1
are sections of that sphere. If such a rock undergoes a
27 200 41 47 7 2.8 44.0 homogeneous strain, the sphere becomes an ellipsoid,
28 146 67 129 6.5 2.7 7.2 and sectional characteristic shapes become elliptical
29 61 12 0 6.2 2.8 30.4 sections through that ellipsoid. Ellipsoid determination
30 156 39 115 34 9.5 15.0
from sectional ellipses would therefore be no different
31 72 90 7 6.5 2.5 33.0
32 77 42 0 12 4.1 19.3
here from that done where fabric elements are closed
33 110 44 140 4.3 1.2 26.4 markers. More generally, any fabric element that can be
34 95 22 159 15 6 18.2 used as a strain gauge can in principle give rise to
35 185 87 56 13.7 4 74.1 sectional strain ellipses and to a strain ellipsoid
determination. In the case of the Tellnes norite, the
of the strain responsible for the shape of the enclaves), with texture is not interpreted to be the result of an actual or
magma coming up from a west-north-west direction. The of some virtual strain, and therefore the existence of an
same orientation is found at a site with 48 enclaves located ellipsoidal characteristic shape in three dimensions is an
11 km to the east along the same margin. This relatively approximation. But it is the same approximation as that
simple and robust pattern suggests that late emplacement for many closed markers that are not interpreted as
strain may still have played a dominant role in enclave strain markers.
shape fabric, in spite of the expected variety of enclave
rigidities and deformation histories. This may be because 4.1. Methodology
the host magma in that final stage, close to the margin, was
sufficiently crystallized for the viscosity contrast to be low, Samples were first analysed by AMS, using standard
and was thus able to impose a significant strain on most orientation procedures. Principal directions of the magnetic
enclaves. susceptibility tensor are shown on an equal-area projection
in Fig. 4b. Large thin sections (3.25 cm!5 cm) were cut
normal to these principal directions and labelled (xy), (yz)
4. Image analysis in the Tellnes Ilmenite deposit, SW and (xz), according to the convention discussed in Appendix
Norway A. Although automated, the analysis of a site remains
labour-intensive and three approximately orthogonal sec-
The Tellnes ilmenite norite ore body of SW Norway is tions are a good choice that provides sufficient fabric
the second most important magmatic titanium deposit in information for the determination of a fabric ellipsoid
 P. Launeau, P.-Y.F. Robin / Journal of Structural Geology 27 (2005) 2223–2233 2229

Fig. 4. IlmeniteCmagnetite fabric in the Tellnes deposit, SW Norway. (a) Digital image of thin section, showing only opaque and non-opaque minerals. (b)
Lower hemisphere equal-area spherical projection of the result of anisotropic magnetic susceptibility (AMS) measurements. Below it is the plot of Jelinek’s
parameters T vs P 0 . (c) Spherical projection showing orientation of the fabric ellipsoids calculated, with, below, Jelinek’s parameters for the fabric ellipsoid
found.

without undue thin section preparation and associated found with AMS and image analysis of opaque minerals are
petrography. in reasonable agreement, with all ellipsoid directions falling
In thin section, as shown in Fig. 4a, individual grains of in the respective 2s cones around the AMS mean principal
interstitial ilmenite cannot be easily separated. Instead, directions. Even axial ratios are close, perhaps fortuitously:
boundaries between opaque minerals—with no distinction k1 =k2 =k3 Z 1:08=0:98=0:94, while A=B=CZ 1:11=0:99=0:91.
between ilmenite and magnetite—and non-opaque minerals The incompatibility index for the ellipsoid is 1.6%, better
were analysed with the method of intercepts (Launeau and than that for the reduction spots. This lower index is related
Robin, 1996), using the latest version of the program not only to the large sampling provided by the intercept
INTERCEPT downloadable from the same page as ELLIPSOID. As method, but is also a function of the low anisotropy of the
discussed above, this method provides a record of the fabric. A justifiable normalization of the incompatibility
orientation distribution of the boundaries separating opaque index to the anisotropy of the rock is yet to be devised.
from non-opaque phases and a sectional ellipse can be Diot et al. (2003) argue that the directions found with
extracted from that record. The images do carry size AMS and with image analysis of opaque/non-opaque
information, but only the treatment without scale factor boundaries record the direction of the dike walls along
(Case 2) is reported here. which the noritic crystal mush was emplaced and the
direction of magma flow within that fracture.

4.2. Scatter analysis and results

Fig. 4b shows the principal directions of AMS, with the 5. Pyroxene fabric in a gabbro-norite from the critical
traditional scatter of directions obtained from processing zone, eastern limb of the Bushveld complex
several samples. A comparable plot can be obtained from
image analysis. The three large thin sections were each The ca. 2 Ga Bushveld complex is one of the world’s
divided into halves, all halves were analysed separately, and largest deposits of base metals and platinum group
the resulting sectional ellipses were combined in the eight elements. Yet the mechanism of emplacement and
different ways possible, thus yielding eight different formation of its magmatic layers is still enigmatic.
ellipsoids. Fig. 4c shows the symmetry directions of these Fabric studies have been undertaken on a number of
eight ellipsoids. We note that the scatter in the directions units within the complex in order to document a possible
found is quite small, smaller than that of AMS; it indicates anisotropy within the plane of layering and therefore a
that the fabric is homogeneous over the scale of the large possible magmatic flow direction. Using image analysis,
thin sections and that the sample size is sufficient. Directions Auréjac (2004) has analysed the pyroxene fabrics of
2230 P. Launeau, P.-Y.F. Robin / Journal of Structural Geology 27 (2005) 2223–2233

Fig. 5. Pyroxene fabric in a gabbro-norite from the critical zone, eastern limb of the Bushveld Complex. (a), (b) and (c) Number of pyroxene grains, n, used to
calculate (by the inertia tensor method) the three orthogonal sectional ellipses shown, and parameters of the ellipses found. r, axial ratio of ellipse, f, rake of
long axis of ellipse, as defined by the convention shown in Fig. 7a. (e), (f) and (g) Digital images of the thin sections; the orientation of the long axis of each
pyroxene grain is shown by its colour as per the rose in (d). The rectangle (1) of (g) is one example of nine quarter-size counting windows used to calculate 729
(Z93) ‘subset ellipsoids’. (h) Ellipsoid parameters calculated. (i) Lower hemisphere equal-area projection of the results. Solid symbols are for the ‘full
ellipsoid’, i.e. that found by using all the pyroxene grains in each section. Open symbols are mean parameters of 729 ‘subset ellipsoids’ (see text). Surrounding
cone projections are at 2s from each mean. For the two ellipsoids: squares indicate the long axes, A; triangles, B; circles, C. (j) T vs. P 0 for the same ellipsoids
with previous mean and s convention.
 P. Launeau, P.-Y.F. Robin / Journal of Structural Geology 27 (2005) 2223–2233 2231

gabbronorite samples from 78 different sites spread When each parameter of the ‘full’ ellipsoid (i.e. that
northward from the vicinity of the Tweefontein mine calculated with all the data) falls within the 2s intervals
over a north–south distance of ca. 60 km. No lineation is around the mean for the corresponding parameter of subset
detected in the field. The analysis of Sample 50B is ellipsoids, we conclude that the size of the counting window
presented here to illustrate the use of scale factors in is sufficient; the 2s interval can then be seen as a confidence
calculating the ellipsoid and to continue a reflection on interval. On the contrary, when some or all the parameters
statistical testing of the results. of the full ellipsoid fall outside the 2s range defined by
subset ellipsoids, we conclude that the counting window is
5.1. Methodology of image analysis too small for the heterogeneity of the data. Such an ellipsoid
usually displays a high global incompatibility index. The
The main fissility of the rock is subparallel to the sensitivity of this technique increases with the number of
layering. Each sample was therefore oriented in the field by subsets by the use of smaller counting windows and
marking and recording the orientation of its face parallel to narrower scan steps, but a number of grains greater than
that layering. The sample was then sectioned along three 100 per subset is often necessary.
orthogonal cuts, one (xy) parallel to the layering and two, In the current example, the perfect match between the
(yz) and (xz), normal to the layering and to each other full and the subset ellipsoids (Fig. 5h–j) shows that even
(Fig. 5). though the preferred orientation of the markers is weak, the
Oriented thin sections were then analysed on the rotating directions of the axes of symmetry can be measured with
polarizer microscope developed by Fueten and Goodchild high confidence. Auréjac (2004) finds that the lineation
(2001). Orthopyroxene grains were identified and isolated direction within the gabbro-norite layers is consistently
by the same technique as that used by Fueten and Goodchild WNW–ESE over the field area, whereas similar pyroxene
(2001) for quartz grains. The resulting images of individual grain lineation in the pyroxenite that underlies the gabbro-
grains are shown in Fig. 5e–g. Shape orientations of norite within each cycle is consistently NE–SW. Auréjac
individual grains were determined by the inertia tensor (2004) attributes the latter to primary emplacement
method using program SPO (Launeau and Cruden, 1998; direction of the magma in each cycle, and the former to
Launeau, 2004) downloadable from the same page as secondary magma migration within the chamber.
ELLIPSOID.

5.2. Results 6. Discussion and conclusion

Sectional results, without scale factor (Case 2), are As shown with these examples, calculation of ellipsoids
shown in Fig. 5a–c, and the ellipsoid found, rotated back from sectional ellipses is a tool that can provide valuable
into geographic coordinates, is given in table form in geological information such as direction of magma
Fig. 5h. The low value of the incompatibility index, 1.7%, is transport, of emplacement mechanism, or of deformation.
again a consequence of the low axial ratios of the faces and Ellipsoid calculations should be equally interesting in
of the resulting ellipsoid as well as of the good fit. If the studies of current direction, of paleoslope direction, or of
average sizes of the grains are used to provide a scale factor diagenesis and compaction.
for each face and these scale factors are used in the Individual incompatibility indices for each section are
calculation (Case 1), the results differ by less than 1% and essential tools to assess the data. As already pointed out by
by a fraction of a degree from the results shown here, while Owens (1984), they can be used to search for errors in
the incompatibility index increases to 2.2%. Thus, there is measurement or data entry. But also, as suggested with the
no significant difference between the results obtained by the Mont-Louis enclaves, identification of outliers may help
two methods here. document competing mechanisms pffiffiffiffi of fabric acquisition. A
The ratio A/B found within the plane of the layering is global incompatibility index, F~ , is also essential, all the
only 1.08. This is quite weak, which accounts for why no more necessary as the method always yields a result,
lineation could be seen in the field. In order to assess our regardless of whether the data are very compatible or very
confidence in the data, grains in each of the three sections incompatible. But that index must not be interpreted too
were split into nine subsets by scanning a quarter size literally when comparing several data sets or comparing
counting window in three staggered, evenly spaced calculation methods. Modelling still needs to be done to
positions along both the x- and y-directions; each subset evaluate how the incompatibility index varies as a function
contains around 230 grains. As was done for the Tellnes of the axial ratio.
norite, subset ellipsoids are calculated from all possible There is at present no theory to translate the confidence
combinations of sectional subsets, i.e. 729 (Z93) ellipsoids, intervals for the measurements on individual faces, and the
with each section weighted by its number of grains. Similar distribution of face orientations into a confidence interval on
analysis with subsets is commonly used in mathematical the parameters describing the ellipsoid found. But partial
morphology (Serra, 1982; Coster and Chermant, 1989). sampling tests along the line of what was done for the
2232 P. Launeau, P.-Y.F. Robin / Journal of Structural Geology 27 (2005) 2223–2233

opaque grains in the Tellnes deposit and for the pyroxene

grains in the Bushveld gabbronorite suggest a possible
practical approach to this statistical problem.
In a recent paper, Gee et al. (2004) repeatedly assert
that ellipsoid determinations from sectional ellipses
require that sections be parallel to the symmetry
directions of the fabric ellipsoid sought and also that
absolute dimensions of the sectional ellipses must be
known. These authors thus directly contradict several of
the papers that they cite, including that of Robin (2002);
as demonstrated here, both assertions are in fact
incorrect. To find a remedy to what are in fact non-
existent problems, Gee et al. (2004) propose to
determine fabric ellipsoids by using the traces of planar
crystallographic features (cleavage and exsolution
lamellae in pyroxene grains, albite twin planes in
plagioclase grains) on three orthogonal thin sections.
They consider each such trace to approximate the Fig. A1. Convention for the description of planes by strike (a) and dip (q),
and that of a direction within that plane by its rake (f). A convention such
projection on the section plane of the long direction of
as the one illustrated here is necessary for efficient and unambiguous entry
the grain; this is of course only true if the planar feature of orientation data into a computer program. Although structural geologists
used happens to be perpendicular to that section, i.e. would normally choose a between 0 and 3608, q between 0 and 908 and f
rarely, and never for all three sections. Still, it remains between 0 and 1808, ELLIPSOID imposes no restriction on the signs and values
true that in an isotropic rock, such traces would be of these angles that can be entered for them.
isotropically oriented in all sections and also true that if
such directions are then assumed to rotate passively
during a deformation, their distribution can be used to azimuthal direction. The strike is between 0 and 3608,
calculate a sectional strain. However, in this case, counted clockwise from north. The dip q can in principle be
calculating a sectional ellipse from these data and from
greater than 908 (e.g. as one may want to measure on an
the resulting ‘characteristic shape’ (e.g. Launeau and
overhung face); but in Tables 1 and 3, dips are always
Robin, 1996) for each section and combining sections
smaller than 908, and strike is chosen accordingly. The rake
with ELLIPSOID is a much simpler process than the multi-
(/pitch) of the long axis of the sectional ellipse, f, is the
step and iterative method proposed by Gee et al. (2004).
angle where the axis makes with the azimuthal line selected
as per the above right-hand rule. Rake can vary between
0 and 1808. All calculations in ELLIPSOID assume a
Acknowledgements
right-handed geographic coordinate system with [x]Znorth,
We are grateful to Gérard Gleizes for his contribution [y]Zeast and [z]Zdown.
to the collection of enclave data in the Mont-Louis When making measurements on three orthogonal faces
granite, and to Jean-Baptiste Auréjac for sharing of a block in the laboratory, it is sometimes convenient
pyroxene shape data from the Bushveld and letting us instead to refer these measurements to an equally right-
refer to some of his conclusions. Reviews of versions of handed laboratory coordinate system [x], [y] and [z] (Fig.
the manuscript by W.M. Schwerdtner, by Toshiko A2a). The transformation from the laboratory coordinate
Shimamoto, and, particularly, by W.H. Owens lead to system to the geographic coordinate system is completely
significant improvements of the paper. Jacques Girardeau defined by a set of Euler angles corresponding to the strike a
helped make P.-Y.R.’s visits to Nantes, and consequently of the top face (xy) of the block, the dip q of that face, and
the project, possible. Part of the research was supported the rake f of the laboratory [x] axis within that top face (Fig.
by the Canadian National Science and Engineering A2b). If measurements on faces of the rectangular block are
Research Council. made in accordance with the reference directions shown in
Fig. A2a and the above Euler angles are entered, ELLIPSOID
can transform the orientation of the ellipsoid from lab
Appendix A coordinates to that in geographic coordinates. All Euler
angles can have positive or negative values.
The orientation of each section is given by a strike, a, and Alternatively, users can convert their sectional data to
a dip q. In the right-hand rule used by ELLIPSOID, illustrated in geographic coordinates before entering them into ELLIPSOID.
Fig. A1, the strike is the azimuth of the strike line such that A simple and practical way, particularly useful when the
the dip of the plane is measured down from the right of that faces of the block are not orthogonal, is to set the block in a
 P. Launeau, P.-Y.F. Robin / Journal of Structural Geology 27 (2005) 2223–2233 2233

sand box, re-orient it to its field orientation, using the

preserved measured field face, and then measure the
orientations of the faces and of the rakes with a compass
and clinometer.

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