Strain Patterns in Rocks. A Selection of Papers Presented at the International Workshop, Rennes, 13–14 May 1982

231 Pages·1983·12.581 MB·English

by  P. R. Cobbold and W. M. Schwerdtner (Eds.)

STRAIN PATTERNS IN ROCKS
A Selection of Papers Presented at the International Workshop,
Rennes, 13-14 May 1982
Special Editors: P. R. Cobbold, W. M. Schwerdtner
General Editor: S. H. Treagus
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Printed in Great Britain by A. Wheaton & Co. Ltd., Exeter
Journal of Structural Geology, Vol. 5, No. 3/4, p. i, 1983
Pergamon Press Ltd. Printed in Great Britain
PREFACE
WITHIN the last decade or so, geologists have made many measurements of permanent tectonic strain in rocks, using
deformed objects of known initial shape or orientation. The basis for such measurements has been the theory of
homogeneous strain, it being argued that strain is usually homogeneous on the scale of a rock sample. Studies on
a regional scale have shown that the state of strain is often heterogeneous, sometimes markedly so, the strain
ellipsoids being distributed in patterns which are characteristically associated with certain tectonic structures, or are
diagnostic of certain processes. Thus there is a current interest in studying natural strain patterns and their
occurrence. Furthermore, each pattern results from a unique displacement field which can, in principle, be calculated
mathematically. For this and other reasons, much is to be learned from studying the theory of heterogenous strain.
The first international meeting on the subject of strain patterns was a workshop (Table Ronde) organized by P.
Cobbold at the University of Rennes (France) from 13 to 14 May 1982. The meeting was sponsored by the Centre
National de la Recherche Scientifique, the Tectonic Studies Group (U.K.) and the University of Rennes. It attracted
82 participants from 10 countries. The programme included 25 informal lectures, 19 posters and 4 practical
demonstrations. The audience at lectures was seated in a U-shaped pattern, facing the speaker. All these
arrangements led to a relaxed informal atmosphere and a free interchange of ideas. The meeting proper was followed
by two day-excursions, one to the South Armorican Shear Zone (led by P. Choukroune), the other to the He de
Groix (led by P. Cobbold and C. Audren).
The Centre National de la Recherche Scientifique contributed 40,000 FF towards travelling expenses of
contributors. The University of Rennes provided lecture facilities and 5000 FF towards secretarial expenses and
refreshments. The contributors wish to thank P. Cobbold and all the staff and students at Rennes for organising
such an enjoyable and memorable Table Ronde. The following are some conclusions drawn from the meeting:
(1) There is a large body of available theoretical work on deformation and strain, although much of it needs
reprocessing before it becomes useful to the structural geologist.
(2) There is no general consensus on the best terminology and nomenclature, nor is there ever likely to be one,
although a few names and symbols appear to be in the process of becoming adopted by a majority of
specialists. A notable example is the use of capital letters and lower-case letters for quantities in the
undeformed and deformed states, respectively. Another example is the use of "c" for the reciprocal quadratic
stretch (Cauchy's) tensor, represented by the strain ellipsoid.
(3) Much is to be gained by constructing theoretically valid models of strain fields and comparing these with
natural patterns.
(4) The Mohr circle is a very practical tool for illustrating and analysing strains and rigid rotations in a
deformation field.
(5) Compatibility equations are powerful, but as yet rather obscure, tools for analysing variations in strain and
rotation.
(6) The use of finite elements is a powerful and practical method of removing strains, although further work needs
to be done on examples where there are volume changes.
(7) Natural volume changes continue to be difficult to detect and even more difficult to measure.
(8) There are as yet few regional studies in which strain has been measured at frequent intervals. Thus there are
few examples of well-defined natural strain patterns. This is mainly due to a lack of adequate strain gauges.
(9) Discontinuities, especially faults, are difficult to deal with, especially if the throws are unknown, as they often
are in nature.
This Special Issue Strain Patterns in Rocks contains 21 contributions from the Rennes meeting, and one in
addition. Papers have been arranged on two broad themes, homogeneous to heterogeneous strain, and strain theory
to geological strain patterns. Many contributions include both theory and natural examples so the issue has not
been subdivided formally into chapters.
The issue begins with techniques of strain measurement (Lloyd, Ribeiro et al., Lacassin & van den Driessche),
an orthographic analysis of deformation (De Paor) and two applications of the Mohr circle to inhomogeneous
deformation (Means, Cutler & Elliott). It continues with methods of strain removal (Cobbold & Percevault), a
general transformation to simulate heterogeneous strain states (Hirsinger & Hobbs), the significance of isotropic
points (Brun) and detection of volume changes (Gratier). Analyses of strain discontinuity at interfaces (Cobbold)
and strain refraction through contrasting layers (Treagus) are presented, followed by strain patterns in ductile shear
zones (Inglιs) and at the tips to shear and thrust zones (Coward & Potts). The remaining contributions describe
natural strain patterns: in mylonite zones (Mawer), in granites (Choukroune & Gapais, Schwerdtner et al.), in Alpine
nappes (Harris et al., Siddans), in linearly anisotropic rocks (Watkinson), in an ice cap (Hudleston) and in a boudin
model (Hildebrand-Mittlefehldt).
P. R. Cobbold
S. H. Treagus
Journal of Structural Geology, Vol. 5, No. 3/4, pp. 225 to 231,1983 0191-8141/83 $3.00 + 0.00
Printed in Great Britain © 1983 Pergamon Press Ltd.
Strain analysis using the shape of expected and observed continuous
frequency distributions
GEOFFREY E. LLOYD
Centre for Materials Science, University of Birmingham, Birmingham B15 2TT, U.K.
(Received 12 July 1982; accepted in revised form 10 December 1982)
Abstract—An initial attempt to develop a method of strain analysis based on shape modifications to continuous
frequency distributions of structural data during deformation is described. The method uses the dimensionless
coefficients of skewness (/3, = skewness^/variance^) and kurtosis {β2 = kurtosis/variance^) and although the
justification for adopting these parameters is complex, the actual calculation of and fij is relatively simple.
Different frequency distributions can be accurately distinguished by plotting graphs of against β2. Since the
effect of strain on a frequency distribution is to modify its shape, theoretically determined shape modifications
can be followed on β i \$β2 graphs for increasing strain and hence the graphs are automatically contoured in terms
of strain. The strains involved in natural deformations can then be estimated by plotting on the graphs the βι, β2
values for observed continuous frequency distributions. Various examples of the application of this technique are
discussed using data from the literature.
INTRODUCTION equations y = i{x), y = φ(χ) is required in which
dy/dx = 0 for specific values of jc (e.g. at the maximum
THIS brief contribution outlines an initial attempt to and at the end of the curve where there is contact with
develop a method of strain analysis using the shape of the X axis). This suggests that
continuous frequency distributions. Distribution shape
is preferred to other more commonly employed statistics (2)
(e.g. mean, variance, median and mode) since it is
usually characteristic for a particular distribution and since dy/dx = 0 if y = 0 or jc = JCQ; the former corres­
can be represented simply by a pair of dimensionless ponds to the contact with the χ axis at one end of the
coefficients. The philosophy behind this approach is that curve and the latter represents the curve maximum if XQ
any initial continuous frequency distribution of struc­ is defined as the distance between the origin and the
tural elements (e.g. orientations, spacings, lengths, etc.) mode. Rearranging equation (2) and expanding ¥{x) by
is transformed by progressive deformation into a strain- Maclaurin's theorem yields
modified distribution, the shape of which is a function of
the type and magnitude of the deformation. In most
ibo + b,x^b2x'^ . . . )^ = y{x-^xo), (3)
cases distribution shape can be described in terms of
the dimensionless coefficients of skewness and kurtosis in which bo,bi,b2,ttc. are constants.
(Elderton & Johnson 1969, Harr 1977), The majority of continuous frequency distributions
can be generated from equation (3) firstly by multiplying
βι = skewness^/variance^
(1) by jc" and then integrating with respect to x.
β2 = kurtosis/variance^.
The former coefficient defines the symmetry of the A:"(¿o + bix -h ¿>2JC^ + -Oy
distribution and the latter its peakedness. The justifica
tion for using these two parameters is presented in the
[nbox"-' + (n + l)6iJc" + (n + 2)b2X^^' + .. .]ydx
next section.
yx^'-^dx -h yxox'^dx. (4)
THEORETICAL JUSTIFICATION:
DISTRIBUTION CLASSIFICATION
The expression λ:"(6ο 4- b^x +62^^ 4- .. .)y vanishes at the
The following theoretical discussion is based on the ends of a frequency curve. Thus by writing μ„ = \ yx^'dx
classification of continuous frequency distributions and rearranging, equation (4) becomes
suggested by Karl Pearson (for a complete review of his
work see Elderton & Johnson 1969). Χομη + nboμn-^ + (π + 1)6ιμ„
In general, continuous frequency distributions can be + (« -f 2)¿>2Mn+l + . . . = -μη^ν (5)
considered to begin at zero, rise to a maximum and then
fall away (often at a different rate) such that there is For successive positive integer values of η from zero to 5,
usually high contact at the ends of the distribution. To 5+1 equations are formed which allow the constants to
represent this behaviour mathematically a series of be determined:
225
226 G. Ε. LLOYD
Χφο + 0/?o + l¿>i^o + 2^2^! = -μι [4(2/32 - ^β\ - 6)(4j82 - 3βι)] which is usually denoted
by the symbol K.
Χομι + Ifeo^o + 2&ιμι + 3í?2M2 = "^2
(6) Pearson called Κ the criterion and used it to classify
Χομ2 + 2&ομι + 3ί?ιμ2 + 462μ3 = -^3
the different types of continuous frequency distributions
Χομ3 + 3Ζ7ομ2 + ^b^^ + 5&2μ4 = -μ4. (although he found it useful to employ other 'criteria' as
well; see Elderton & Johnson 1969). However, since all
It is usually convenient to make the mean the origin of
the 'criteria' are functions of the parameters βι and β2 it
the distribution, in which case XQ becomes the distance
is possible to distinguish between different distributions
between the mean and the mode and it is necessary to
using these terms; Harr (1977) suggested they should be
amend the other terms accordingly. Assuming grouped
plotted against each other (Fig. 1). It is proposed that
data, these become
plots of βι vs β2 are capable of distinguishing between
1 initial and strain-modified distributions and hence can
= sample mean
be used to give estimates of the magnitude of finite
/=1
strains and also the types of strains involved. The next
m
Σ {fiixi - μχΫ) = sample variance section discusses this approach.
i=\
(7)
m EXPONENTIAL
- μχΫ) = sample skewness
Σ {fiiXi
Ν
m
Σ {fÁXi - μι)'*} = sample kurtosis NO DISTRIBUTIONS
in which m is the number of data groups, and/¡ are the
midpoint and frequency of the /th group and Ν is the
sample size. It is also convenient to treat μο as unity and
together these simplifications lead to a set of simultane­
ous equations which can be solved and substituted into
equation (3) to give
1 φ; ^ X + (Μ1/Λ/2)
CLUSTERED
y ax (M3 + M,x + M^x^)IM2
increasing peakedness
with: M, = μψβΤ{β. + 3) 9 10
(8)
BETA TWO
M2 = 2(5β2 - 6j8, - 9)
Fig. 1. Graph of β, vs ^2 distinguishing regions of different distribution
Μ, = M2(4j82 - 3/3,) types (modified from Harr 1977).
M4 = 2β2 - 3j8, - 6
where βχ = μ^/μΐ, β2 = μ^Ιμι [in agreement with equa­
tion (1)] and xo = μΗ^β\\β2 + 3)/2(5)32 " (^βχ " 9);
STRAIN ANALYSIS USING vs ^2
note that μψ is the standard deviation.
GRAPHS
By inserting the values of the sample moments into
equation (8) it is possible to obtain a formula representa­
Although graphs of /3i vs β2 are capable of distinguish­
tive of the distribution data. However, this formula is
ing between different continuous frequency distribu­
not of exactly the same form as the original data. A truly
tions, in this simple form they do not reveal any informa­
representative formula can be obtained if {x + JCQ)/
tion on strain. The use of such graphs in strain analysis
(i>o + bix + b2X^ . . .) is integrated. This is possible by
requires that the effects of different types and amounts
recognising equation (8) as a general expression for
of strain on initial frequency distributions are known.
integration and observing that the form the integral
The values of β χ and β2 for the various theoretical
takes depends on the particular values of the coefficients
strain-modified distributions would then be calculated
of X in the denominator for
and hence the graphs contoured in terms of strain mag­
nitudes, with different graphs constructed for different
b^x + bnx^ = b: X - types of strain. It would then be a relatively simple
2b2
matter to calculate the βχ and β2 values for natural data
and hence to determine the type and magnitude of the
-b, - {b\ - Abφ^) iir
X — (9) natural finite strains.
2b2 An important pre-requisite for this technique is that
the theory exists for the modification of an initial distri­
The basis for fixing the particular form is the same as bution by a particular type of strain. To date, only two
that for the nature of the roots of the equation such theories are known in the literature and concern the
5o + bxx + b2X^ = 0; that is ¿?i/(46o^2)- Βγ substitut­ effects of homogeneous irrotational strains on initially
ing from equation (8) this gives [βι{β2 + 3)^]/ Gaussian and uniformly distributed data.
Strain analysis using continuous frequency distributions 227
Strain-modified Gaussian distributions
Sanderson (1973) studied the effect of homogeneous
LU
irrotational strain on orientation data with an initial
Gaussian distribution. He showed that the Gaussian
CO
distribution (maintaining the present notation) CO
<
_l
o
Ν / 1 \ *o
F = (10a)
(T
{2πμ2) \ 2μ2 UQi.
>-
is modified to O
ζ
LU
Ν _]_
F' = pjexp (90 - ef a
'2μ; LU
{xrñ ^^^^^
ORIENTATION
in which is the sample size, μψ is the standard Fig. 2. Homogeneous irrotational strain-modified Gaussian distribu­
deviation of the initial distribution, θ is the angle tions (modified from Sanderson 1973); X/Y is the strain-ratio, μ^'^ is
the standard deviation of the initial distribution and φ is the angle
between the observed direction and the extension direc­
between the mean direction of the initial distribution and the normal
tion {X) of the finite strain ellipsoid (A' ^ 7 ^ Z), φ is to the extension direction (Λ').
the angle between the initial mean and the normal to the
extension direction and XIY is the strain ratio. For
φ = 0 the initial distribution is perpendicular to X but as tion towards smaller values of /32. For φ # 0 the distribu­
the strain increases the distribution spreads and then tions are not symmetrical and plot further away from the
splits into two maxima symmetrical about X (Sanderson β2 axis with increasing φ, lying on distinctive curves for
1973; see Fig. 2a). For φ 0 the initial distribution is each value of φ. Note that while φ remains small the
obUque to X and as the strain increases the peak rotates strain-modified distributions eventually become bi-
towards X with a subsidiary peak developing after a modal, but for larger values of φ they remain unimodal.
certain value of strain (Sanderson 1973; see Fig. 2b). Having drawn the distinctive curves for each value of
The βι and β2 values of strain-modified Gaussian φ it is possible to join together the points of equal strain
distributions have been determined and plotted on a (Fig. 3). For small strains these are slightly curved but
graph of βχ vs β2 for different strains and values of φ but for larger strains {X/Y >^3) they may be considered as
constant standard deviation (Fig. 3). For φ = 0 all the Unear. This operation therefore contours the βχ vs β2
distributions are symmetrical and therefore plot along graphs in terms of strain and makes them extremely
the β2 axis but since increasing strain causes a change useful in the estimation of finite homogeneous irrota­
from unimodality to bimodahty there is a gradual migra- tional strains as the following examples show.
15 Η
1 Η
05 Η
1-
3 35
BETA TWO
Fig. 3. Strain contoured β, vs β2 graph for homogeneous irrotational strain-modified Gaussian orientation distributions with
= 20°.
228 G. Ε. LLOYD
Table 1. Comparison between strain magnitudes obtained using Sanderson's (1973) technique for homogeneous irrotational strain-modified
Gaussian orientation distributions and those using β, vs β2 graphs; μ]/^ = standard deviation, X/Y = strain ratio; φ = angle between initial
mean and normal to final extension direction (i.e. Y-axis)
Sanderson 1973 Roberts & Sanderson 1974
OObbsseerrvveedd ffoolldd aaxxiiss OOrriiggiinnaall ffoolldd aaxxiiss Strain modified fold axis distribution
ddiissttrriibbuuttiioonn ddiissttrriibbuuttiioonn
BBoossccaassttllee CCrraaiiggnniisshh Knapdale steep belt Loch Tay inversion Aberfoyle Anticline
((FFiigg.. 44)) ((FFiigg.. 55aa)) (Fig. 5b) (Fig. 5c) (Fig.5d)
μμψψ == 2200°° μμψψ == 2211°° μψ = 20° μψ = 20° μψ = 25°
Sanderson
X/Y 4-5 very low (~1) 3.5 4 4
5° 11.6° 5° 15° 3°
Φ
This paper
X/Y 6.5 1.2 4.5 5.75 6.75
Φ 8° 26.5° 4.5° 17° 6°
β^ 0.765 0.112 0.171 3.922 0.298
βι 2.012 3.416 1.589 5.748 1.484
Examples of strain-modified Gaussian distributions inversion to the Aberfoyle Anticline (Fig. 5). There is
also some variation in the values of μ2'^ and φ, especially
Sanderson (1973) applied his model to fold axes which near Loch Tay where the initial mean fold axis direction
occur oblique to the regional trend. He assumed that the was somewhat oblique to the stretching direction (Table
fold axes originally had a Gaussian distribution {μ2^ = 1). Roberts & Sanderson (1974) showed that there must
20°) about the Y axis of the strain ellipsoid but were have been a progressive increase in strain southeast-
subsequently rotated towards the Ζ direction by further wards from Craignish (X/Y ~ 1) towards the Loch Tay
(stretching) deformation within the axial planes of the inversion and Aberfoyle Anticline {X/Y — 4). The
folds. The initial distribution was consequently modified technique described here has been applied to the data
by passive rotation and relative elongation of fold shown in Fig. 5, using strain-contoured βι vs β2 graphs
axes within the XY plane of the deformation to pro­ for the different values of μ2^, and shows a similar trend,
duce a slightly asymmetric final distribution with a sub­ although the strain continues to increase into the
sidiary maximum (Fig. 4), suggesting that φ > 0°. Aberfoyle Anticline (Table 1). The present technique
Sanderson therefore used φ = 5° and thus determined also recognises the more oblique initial fold-axis
that 4 < X/Y < 5. The method described here, using a distribution in the Loch Tay inversion and suggests that
strain-contoured β\ β2 graph for μ2^ = 20°, gives the distribution at Craignish is not the initial Gaussian
φ = 8° andZ/y = 6.5 (Table 1). distribution but is the result of very slight stretching
In a subsequent use of Sanderson's model Roberts & {X/Y = 1.2) of an obhque {φ = 26.5°) fold axis
Sanderson (1974) have analysed the variation in patterns distribution.
of fold axis distributions across the Scottish SW High­ In general, the strain estimates obtained using the
lands, which they attribute to varying amounts of defor­ method described here are somewhat larger than those
mation and to the initial attitude of the mean fold axis obtained using Sanderson's original approach. The most
relative to the finite strain axes. They argued that the likely reason for the discrepancy lies with the method of
folds formed with a mean fold axis nearly perpendicular comparing the observed and expected distributions.
to the stretching direction and that this original Gaussian Sanderson uses coincidence of modes but this does not
distribution can still be recognised in areas where the reflect variations in other regions, especially where fre­
subsequent deformation was low (e.g. at Craignish, see quencies are low. In contrast, the theoretical derivation
Fig. 5a). There is a progressive modification of the initial of Pearson's classification and βι and β2 contains
distribution southeastwards across the region from implicitly a very rigorous comparison test which consid­
Craignish through the Knapdale steep belt and Loch Tay ers equally the whole range of the different distributions.
Strain-modified uniform distributions
Sanderson (1977) considered also the effect of
homogeneous irrotational strain on orientation data
LU
a with an initial uniform distribution. He showed that the
uniform frequency distribution in a sector subtended by
an angle a,
F=Na/27r (11a)
90 0 90
ORIENTATION is modified to
Fig. 4. Orientation frequency distribution histogram of angles between
stretching lineation (X) and fold axes, Boscastle, Cornwall (modified F = ^ [tan-^ {R, tan θ^) - tan"^ {R, tan θ[)] (lib)
from Sanderson 1973). 2π
Strain analysis using continuous frequency distributions 229
30n (a) -, (c)
204
10^
O
Ζ
LU
a
-PL
LU
20i(b) (d)
10
90 90 90 90
ORIENTATION
Fig. 5. Orientation frequency distribution histograms of angles between stretching direction (X) and fold axes from four
localities in the Scottish SW Highlands (modified from Roberts & Sanderson 1974).
in an arc subtended by θ[ and Θ2 measured from the 7). The displacement along the β2 axis for each 0.1
maximum principal strain axis (X). In these equa­ increment of strain is considerable but approximately
tions, Ν is the sample size and is the strain ratio (Rs = constant, even up to large values of strain. Thus, the
(λι/λ2)^^^). Initially the frequency is independent of graphs may be used to give sensitive estimates of strain
orientation but as the strain increases a preferred orien­ over a wide range of values.
tation develops which is symmetrical about the extension
direction (X) of the finite strain ellipsoid {X ^ Y ^ Z);
Example of strain-modified uniform distributions
the dispersion about X also decreases with increasing
strain (Fig. 6).
Beach (1980) studied the orientation of belemnites
Since the strain-modified uniform distributions are all
from Britain and France. He argued that in the unde-
symmetrical they plot along the βι (symmetry) axis of
formed state the orientation distribution of belemnites is
the βι vs βι graph, migrating away from the position of
approximately uniform (Fig. 8a) but, where the rock
the theoretical uniform distribution {βι = 0, β2= 1.8)
suffered an homogeneous irrotational strain, the belem­
towards larger values of ^2 with increasing strain (Fig.
nites show a preferred orientation symmetrical about
the principal extension direction (Fig. 8c). Beach did not
attempt any strain estimates from his examples but by
using the technique described here all the distributions
are shown to be symmetrical (Fig. 9) and hence probably
due to homogeneous irrotational deformation. The
undeformed example plots exactly at the position of the
uniform distribution while an example thought by Beach
to be only very sUghtly deformed (Fig. 8b) is found to
have a strain ratio of 1.1. The example, said to be typical
of the belemnite orientation distributions found in the
.ORIENTATION
French Maritime Alps (Fig. 8c), gives a strain ratio of
Fig. 6. Homogeneous irrotational strain-modified uniform orientation
distributions (modified from Sanderson 1977). 3.07.
01
X/Y = 1 15
uniform gaussian
0 1„
10
8
BETA TWO
Fig. 7. Strain-contoured /3, vs β2 graph for homogeneous irrotational strain-modified uniform orientation distributions.
230 G. Ε. LLOYD
10 η (α) In this respect it is worth noting that Sanderson (1977)
did not really restrict his analysis of the strain modifica­
tion of initially uniform distributions to irrotational
5Η strain, but considered the rotational components to play
no part in determining the shape of the strain modified
distribution, at least for passive markers. This was sub­
sequently justified by Sanderson & Meneilly (1981,
"go ' ' 180
pp. 109-111) using Owens' (1973) development of the
theory of strained angular density distributions (March
1932). In general, the form of the strain-modified
frequency distribution is determined by the initial
distribution (shape and orientation) and the deforma­
tion gradient tensor (relative to some defined reference
frame). However, for the particular case of the initially
uniform distribution, since its frequency is constant, the
deformation gradient tensor can be factorized into a
stretch tensor and a rotation tensor. The operation of
the rotation tensor on a uniform distribution leaves the
frequency unchanged and so irrotational and rotational
deformations should produce similar strain-modified
distributions (e.g. pure shear and simple shear result in
the same strain-modified frequency distributions, D.J.
Sanderson, personal communication 1982). Thus, distri­
butions of belemnite orientations which are asymmetric
ORIENTATION
with respect to the principal extension direction (Beach
Fig. 8. Belemnite orientation frequency distribution histograms (after 1980, see Fig. 10 for examples), and therefore plot off
Beach 1980); (a) undeformed; (b) very sHghtly deformed; (c)
the β2 axis (Fig. 9), are more Hkely to be due to non-uni­
deformed. See text for discussion.
form initial distributions rather than rotational strains. It
DISCUSSION is possible that original sedimentary influences (e.g.
palaeoslopes and/or current activity) were responsible
(a) Generality of the technique for inducing slight preferred orientations such that the
initial distributions responded as (platykurtic) Gaussian
The technique described here is not itself a method of distributions.
strain analysis. However, it does represent a very sensi­
tive way of evaluating data from analytical methods (b) Comparison of different data distributions
which involve the modification of continuous frequency
distributions by deformation. Although the examples The βι vs β2 graphs (e.g. Figs. 3 and 7) show that if the
considered concern only orientation data, homogeneous type and magnitude of strain is constant then the shape
irrotational strain and Gaussian and uniform initial dis­ of the strain-modified distribution is a function only of
tributions, the technique is nevertheless applicable to the initial distribution. Thus, if the fold axes and belem-
any type of continuous data distribution and mode of nites considered previously had occurred in the same
deformation. The only requirement is that a model region and had suffered similar amounts of homogene­
exists for the effects of a particular type of strain on a ous irrotational strain they would nevertheless show
particular initial distribution. different frequency distributions. Furthermore,
gaussian 3
BETA TWO
Fig. 9. Analysis of belemnite orientation frequency distributions using strain contoured β, vs J82 graph for homogeneous
irrotational strain-modified uniform orientation distribution. Examples plotted on the J82 axis are symmetrical and therefore
due to homogeneous irrotational strain of initially uniform orientation distributions. The other examples are probably due
to deformation of initially nonuniform distributions.
Strain analysis using continuous frequency distributions 231
of the deformational parameters a range of values is
obtained which may vary considerably depending on the
imprecision in defining the initial standard deviation.
Unfortunately, the definition of the pre-deformation
data distribution may not be easy and in any case will be
subject to similar data sampHng problems as those dis­
cussed above.
Finally, although the examples discussed in this con­
tribution involve orientation data, they have, neverthe­
less, been analysed using linear rather than circular
sample statistics and distributions. It is therefore neces­
sary to construct the histograms of the data about some
external reference direction defined as zero orientation.
ORIENTATION
The range of the histograms can then be considered to be
Fig. 10. Asymmetric beiemnite orientation frequency distribution
±90° provided data < -90° or > -h90° are transferred to
histograms (after Beach 1980).
the opposite regions of the histogram (e.g. 95° is plotted
as -85°). This approach is convenient but ultimately
although sUght variations in homogeneous irrotational should be replaced by a more rigorous technique based
strain magnitudes have little effect on the modified on the βι, β2 analysis of circular distributions.
shape of initially uniform distributions (see Fig. 7),
initially Gaussian distributions can vary markedly. For
CONCLUSIONS
example, using Fig. 3, a strain variation of 1 < XIY < 3
acting on an initial Gaussian distribution with μ^'^^ = 20°
(1) A method of strain analysis is described based on
and φ = 10° can produce distributions ranging from
the modification of continuous frequency distributions
unimodal bell-shapes, through J or re verse-J shapes to
by progressive deformation.
bimodal shapes. In spite of this, it is still possible to
(2) Such distributions can be accurately described in
determine accurate estimates of deformational para­
terms of their shape via the dimensionless coefficients of
meters using the techniques described here.
skewness {βι) and kurtosis (JS2); graphs of βι vs ^2 can
therefore be used to distinguish different distributions.
(c) Data collection (3) Theoretical studies of the effects of deformation
on initial frequency distributions (i.e. strain-induced
While the use of βχ and J82 to classify continuous shape modifications) can be used to contour βι vs β2
frequency distributions is theoretically soundly based, in graphs in terms of strain.
practice it does involve several assumptions concerning (4) The positions of natural data distributions on
the collection of the actual data. In particular, it is these graphs consequently give the natural strain mag­
assumed that the data are a representative sample of the nitudes.
total population whereas in reality it is more likely to be
Acknowledgements—The technique described here forms part of a
either random or biased. The data distribution is there­
statistical model of rock deformation currently under development. I
fore only an estimate of the total population distribution. should like to express my gratitude to Colin Ferguson for his interest,
Thus, the values of βχ and β2 calculated from the data are advice and encouragement during the course of this work. I should also
like to thank Dave Sanderson for his comments on the application of
only estimates of the real values and hence do not
this technique to his data. Joyce Heathcote typed the manuscript and
necessarily define the best-fit theoretical distribution, Josie Wilkinson drew the figures.
which in turn means that the derived deformational
parameters (i.e. XIY and φ) may not be the true values.
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