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Earth-Science Reviews 7
Fold geometry: a basis for their kinematical analysis
Fernando Bastidaa,*, Jesus Allera, Nilo C. Bobillo-Aresb, Noel C. Toimila
aDepartamento de Geologıa, Universidad de Oviedo, Jesus Arias de Velasco, 33005-Oviedo, SpainbDepartamento de Matematicas, Universidad de Oviedo, 33005-Oviedo, Spain
Received 11 December 2003; accepted 29 November 2004
Abstract
A review of geometrical methods used to describe folds is presented, with special emphasis on methods required to study
kinematical folding mechanisms. Several families of mathematical functions that approach the geometry of folded surface
profiles are considered; among these functions, conic sections are the most suitable in mathematical modelling. The Ramsay or
Hudleston classifications give a detailed functional description of the folded layer profile and are very useful in the kinematical
analysis. Nevertheless, simpler classifications of folded layers can be used to describe folds in regional studies; some of them
are simplifications of the Ramsay classification. Cleavage distribution through the folded layer, due to its relation with the finite
strain ellipsoid, is a key feature in kinematical studies; a useful graphical description of this distribution can be made
constructing the curve of cleavage dip variation as a function of layer dip.
Analysis of kinematical folding mechanisms requires extensive geometrical information to be obtained from the folds. This
analysis can be made attempting to find theoretical folds that fit natural or experimental folds by the use of the point
transformation equations for the basic mechanisms. Examples of folds theoretically modelled by tangential longitudinal strain
and flexural flow are shown, as well as an example of analysis of kinematical mechanisms in a natural fold. All these examples
make it clear how geometry can be the basis for the strain and kinematical analysis of folds.
D 2004 Elsevier B.V. All rights reserved.
Keywords: folding; geometry; kinematics; strain; cleavage
1. Introduction
Natural sciences arise from observation, from an
interaction of man with the surrounding world. Thus,
the first step of the research in this field is a description
of the objects being studied. In the study of structures in
0012-8252/$ - see front matter D 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.earscirev.2004.11.006
* Corresponding author. Fax: +34 98 5103103.
E-mail address: bastida@geol.uniovi.es (F. Bastida).
rocks, a problem is that a lot of different characteristics
can be described in a rock body, making it necessary to
choose features related to its origin and development.
Tectonic structures are the result of deformation, and
their study involves the comparison of an initial, or
undeformed, configuration with a final, or deformed,
configuration. Therefore, the geometry of geological
structures must be accurately described before con-
clusions about their formation can be drawn. In
accordance with this idea, three levels of knowledge
0 (2005) 129–164
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164130
have been traditionally considered in structural geol-
ogy: geometry, kinematics, and dynamics (e.g. Davis,
1984; Nicolas, 1984; Twiss and Moores, 1992; Ghosh,
1993). According to the usual custom in structural
geology, geometrical analysis is concerned with the
orientation, attitude, size, and morphology of the
structures; kinematical analysis is concerned with the
movements in rock bodies from the initial to the
deformed configuration, involving translations, rota-
tion, and strain; and dynamic analysis is concerned
with the stresses that produced the structures and the
mechanical factors that controlled their development. It
can be stated that the geometry is the basis for the
kinematics, just as this is the basis for the dynamics.
However, it is also true that each of these three levels of
analysis constitutes a valid goal in its own right. For
instance, constructing a geological cross section
through a thrust-and-fold belt represents an objective
for the structural geologist that can have important
implications in economic geology. Nevertheless, draw-
ing a geometrically feasible cross-section requires the
balancing and restoring of the structures, and involves
the application of kinematical concepts. Thus, geom-
etry, kinematics and dynamics are closely related
subjects.
The geometric description of folds takes different
forms depending on the aim of the work. Two main
types of descriptions can be distinguished: those made
in relation to regional geological studies and those
made in relation to detailed analyses of specific
structures. In regional studies, an assemblage of
structures is analyzed, and the use of qualitative
descriptive terms is usually dominant. The size of
folds is described using terms such as dmajorT or dminor
foldsT, indicating at most a distance that gives a
measure of the limb length. The orientation and attitude
of folds is described by the orientation of their axes and
axial surfaces, frequently using stereographic plot
diagrams, by the definition of the sense of vergence
or facing, and by the use of adjectives such as duprightT,dinclinedT, drecumbentT, dplungingT, etc. Quantitativediagrams such as that proposed by Fleuty (1964) are
rarely used. The shape of folds is usually described by
the use of terms like dantiformT, dcylindricalT,dsymmetricalT, dgentleT, dchevronT, dparallelT, dsimilarT,T, etc. Again, quantitative geometrical methods such as
those proposed by Ramsay (1967) or Hudleston
(1973a) have been rarely used.
When the aim is to make a detailed study of the
kinematical mechanisms that operated in a fold,
quantitative and meticulous geometrical information
is required, and we must take advantage of all the
information supplied by the fold morphology and that
of the associated structures. Unfortunately, the analysis
of kinematical folding mechanisms is, in general, a
difficult task, and, at present, it is still not always
obvious what type of geometrical information must be
obtained from the folds for use as a basis for the
kinematical analysis.
The aim of this paper is to describe quantitative
methods that can be used to analyse the geometry of
folds, and show how the description provided by
these methods can be used as a basis for the analysis
of kinematical folding mechanisms. Some examples
of the application of these methods to specific
mechanisms and folds will clarify the importance
of an adequate description of fold geometry in order
to gain insight into the kinematical evolution of
folding.
In general, natural folds affect rock layers bounded
by surfaces. Thus, for the geometrical study of folds,
it is convenient to analyse the folded surfaces and the
folded layers separately, since geometry of the layer
is characterized by the relationships between adjacent
folded surfaces. The geometrical methods used for
these analyses can be also applied to experimental
and theoretically modelled folds. Most of the folding
studies are made in 2-D, and they consider strain only
in the plane perpendicular to the fold axis. Thus,
folded surfaces appear as folded lines that define the
fold profile.
2. Attitude of folds
The attitude of a fold can be defined by the axial
surface and hinge orientations, which are usually
represented using stereographic projection diagrams.
Nevertheless, sometimes it is useful to ignore the
azimuths of these structural elements and to consider
only the axial surface dip and the hinge plunge, which
can be plotted in a diagram with several fields
corresponding to different fold types (Fleuty, 1964).
In 2-D analysis, the description of the fold attitude can
be made by simply specifying the plunge of the axial
trace on the fold profile plane. The attitude of folds is
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164 131
a feature that must be taken into account in the
kinematical analysis of folding, since, for instance, the
displacements and strains are very different in the
development of an upright fold (axial surface dipping
N808) than in the development of a recumbent fold
(axial surface dipping b108).
3. Analysis of single-folded surfaces
Two types of quantitative methods can be used to
describe the form of folded surfaces: (a) those based on
the use of parameters that define a non-functional
approach and permit some aspects of the fold shape to
be expressed; and (b) those based on the search for a
function that adequately fits each folded surface. The
second method also involves the use of several
parameters and is more complete in general than the
first one, since it allows us to use the folded surface as
an analytical mathematical object. Two basic types can
be distinguished in the geometrical analysis of folded
surfaces: cylindrical folds (defined by a cylindrical-
folded surface) and noncylindrical folds. The former
can be studied by 2-D analysis, whereas the latter
require 3-D analysis.
3.1. 2-D analysis
Several authors have tried to characterise fold
profile geometry using different parameters that do
not generate mathematical functions. Loudon (1964)
and Whitten (1966) suggested the use of the first
four statistical moments of the orientation distribu-
tion of the normals to the fold profile to express
respectively the attitude, tightness, asymmetry, and
shape of folds. They also proposed the skewness
and the kurtosis as alternative measures of the
asymmetry and shape of folds. The usual parameter
used to describe fold tightness is the interlimb angle
(h), which was used by Fleuty (1964) to differ-
entiate the following types of folds: gentle
(1808NhN1208) , open (1208zhN708) , c lose
(708zhN308), tight (308zhN08), isoclinal (h=08),and elasticas (ub08). To quantify the aspect of shape
due to the variation in curvature along the fold
profile, Ramsay (1967, p. 349–351) defined two
parameters: P1, which is the extent of the fold limbs
with respect to the hinge zone; and P2, which is
obtained by expressing the maximum curvature of
the fold surface as a ratio of the curvature of the
circle drawn with the distance between the inflection
points as a diameter. Twiss (1988) proposed a
classification of symmetric folded surfaces based
on the determination of three fold style parameters.
A problem with this classification is that the results
from its application cannot be fully represented in
two-dimensional diagrams. Moreover, the classifica-
tion of asymmetric folds requires six parameters,
and this makes its use difficult.
The use of families of functions to fit folded
surface profiles is the most useful descriptive method
when the aim is to study the kinematics of folding.
For the geometrical analysis of a folded surface
profile, it is necessary to select a reference system.
The chosen system is formed by the tangent to the
profile curve at the hinge point (x-axis) and its
normal through this point ( y-axis; Fig. 1a). In the
case of double-hinge folds, the point equidistant from
both hinges (closure point of Twiss, 1988) is chosen
as co-ordinate origin (Fig. 1b), whereas in the case of
a fold with an arc of constant curvature and without a
defined hinge point, the middle point of this arc is
taken as the origin (Fig. 1c). In chevron and cuspate
folds, the y-axis is the bisector line of the interlimb or
cusp angle, and the x-axis is perpendicular to the y-
axis through the vertex or cusp point, respectively
(Fig. 1d and e).
The unit considered for the analysis of folded
surfaces is the fold limb profile, defined as the portion
of the profile between the co-ordinate origin and an
adjacent inflection point (Fig. 1a; the quarter wave-
length unit of Hudleston, 1973a). This definition is
problematic in those folds in which the inflection
point is not defined, such as (1) folds with a line
segment in the limb, (2) chevron folds, and (3) arc-
and-cusp folds. In cases (1) and (2), the limit between
adjacent folds is taken in the middle point of the
straight part (Fig. 1d and f). In case (3), the arc
between the cusp point and an adjacent hinge is taken
as a limb. This limb is shared by the cuspate fold and
the adjacent arc fold. Hence, in this case, the same
limb may be analyzed twice with different reference
axes, one for each type of fold (Fig. 1e).
Since it is difficult to find a single simple family
of functions which approximates adequately all the
common fold morphologies found in rocks, it is
x
y
(a) (b) x
y
(c)
x
y
(d) (e) (f)x
y
x
y x'
y'
Ix0
y0
α
Stra
ight
par
t
A
H x
y
Fig. 1. Reference system and geometrical elements used in this study. (a) General case: H, hinge point; I, inflection point; A, area beneath the
limb profile; a, maximum dip. (b) Double hinge fold. (c) Fold with a circular arc. (d) Chevron fold. (e) Cuspate fold. (f) Fold with a straight
segment in the limb.
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164132
convenient to separate folds into two categories:
alloclinal folds (interlimb angleN08) and isoclinal
folds (interlimb angle=08). Folds with interlimb
angleb0 are uncommon in deformed rocks and
they are not considered here.
In the kinematical analysis of folding, it is useful
to consider an auxiliary reference line, named
bguidelineQ, which is a line that is usually, but not
necessarily, positioned midway between the layer
boundaries in the initial configuration, and enables
the monitoring of the layer geometry during folding.
A basic parameter in the kinematical analysis of
folding is the curvature of the guideline. If f(x) is the
function that describes this line, its curvature j is
given by:
j ¼ 1
q¼ f W xð Þ�
1þ f V xð Þ2�3=2 ð1Þ
where q is the curvature radius, and f V(x) and f U(x)
are the two first derivatives of f(x). Hence, it is
essential that the functions that describe folded
surfaces are differentiable at least twice. On the
other hand, it is also convenient that the functions
used to fit natural folds with a single hinge point
have a variation in curvature showing a continuous
increase towards a maximum at the hinge point.
The theoretical analysis of folding mechanisms
can involve the change, during folding, of the
parameters defining the function that describes the
guideline. Hence, it is convenient that the function
used to describe folded surface profiles is defined
by a small number of parameters; otherwise, the
analysis can be unsuitable. We will now offer a
critical review of the main types of functions that
have been used to describe folded surface profiles
or fold guidelines.
3.1.1. Fourier analysis
If f(x) is a periodic function with period 2p and
integrable between �k and k, the Fourier series of
this function is defined by:
f xð Þ ¼ a0
2þXln¼1
an cos nxþ bn sin nxð Þ; ð2Þ
where
an ¼1
p
Z p
�pf xð Þcos nx dx; n ¼ 0; 1; 2; . . .ð Þ;
ð3Þ
bn ¼1
p
Z p
�pf xð Þsin nx dx; n ¼ 1; 2; . . .ð Þ:
ð4Þ
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164 133
Several authors (Chapple, 1968; Stabler, 1968;
Hudleston, 1973a; Ramsay and Huber, 1987, pp.
314–316; Stowe, 1988) have used Fourier series
to describe single fold limbs. According to these
authors, each limb can be characterized by several
coefficients of a Fourier sine or cosine series
(with an=0 or bn=0, respectively). The best
known Fourier analysis for folds was made by
Hudleston (1973a), who used only the coefficients
b1 and b3 of a sine series to characterise the
profile of folded surfaces through a graphical
representation of b1 vs. b3 (Fig. 2). Thus, this
author distinguished six types of standard shapes,
from type A (box folds) to type F (chevron
folds), and five standard amplitudes (1 to 5). A
quick alternative to the measurements and calcu-
lations involved in this classification method is the
visual harmonic analysis (Hudleston, 1973a),
which allows the determination of b1 and b3 by
comparison of the fold profile with 30 idealised
fold forms with different shapes (A to F) and
amplitudes (1 to 5). Unfortunately, the use of two
Fourier coefficients gives only a rough approx-
imation to the common morphologies of natural
folds (Stowe, 1988), and shapes such as chevron,
elliptical, or box folds are not adequately fitted by
this approximation (Fig. 2, cf. Fig. 15.12 of
Ramsay and Huber, 1987, p. 316). Moreover,
the appearance in some cases of two extreme
values in the fit function within the interval
corresponding to a single limb (Fig. 2) makes
this type of functions unsuitable. As a conse-
quence, the analysis of Hudleston (1973a) allows
the labelling of folded surfaces with two param-
eters and their graphic classification, but it does
not allow a good fit to many folds. On the other
hand, an approximation to fold morphologies
based on a greater number of Fourier coefficients
provides a good global fit of the fold profiles or
guidelines, but is less satisfactory with respect to
important local properties of the curves, such as
the curvature and the slope. The common inflec-
tion points and extreme values found within the
fit interval in these fits make the kinematical
analysis of folding unfeasible. In addition, the use
of the multiple coefficients does not permit a
simple graphical classification of fold surface
profiles.
3.1.2. Functions of the form y=y0 f(x)n (nN0)
Several types of functions can be included in this
group. Bastida et al. (1999) have studied the case in
which f(x)=x/x0; that is:
y ¼ y0x
x0
� �n
; ð5Þ
within the interval [0, x0]. The meaning of x0 and y0 is
shown in Fig. 1a, and x0 is introduced in (5) to avoid
the effect of the scale factor in the classification. In
order to represent graphically both limbs of a fold (the
two adjacent limbs of an antiform or synform), the
function (5) can be modified to:
y ¼ y0jxjx0
� �n
ð6Þ
considered within the interval [�x0, x0].
Using these functions, a fold limb can be described
by two parameters: the aspect ratio (or normalized
amplitude, h=y0/x0), and the function exponent (n),
which describes the limb shape. Fig. 3 illustrates fold
morphologies obtained for various values of n and h.
The following values of n characterise some distinc-
tive fold shapes: (1) nb1, cuspate folds; (2) n=1,
chevron folds; (3) n=2/(p�2)c1.75, fit for sinusoidal
folds; (4) n=2, parabolic folds; (6) nN2, double-hinge
folds (for n values close to 2, this morphology is
visually imperceptible); and (7) nYl, perfect box
folds. Hence, this family of functions offers a
convenient description of alloclinal folds (interlimb
angleN0), and in a way that each fold can be
approximated by an explicit function.
The family of function (5) enables a good graphical
method for the classification of folded surface profiles
on a diagram of the aspect ratio (h) against the
function exponent (n). However, this family poses
some mathematical drawbacks when applied to the
analysis of kinematical folding mechanisms. In
particular, function (5) only have finite curvature at
x=0 when n=2 (parabolic folds); when nb2 the
curvature becomes infinite at that point, and when
nN2 the complete fold has two hinge points and the
curvature is zero at the coordinate origin. Visually,
these features become progressively imperceptible as
n approaches 2, but they make function (5) unsuitable
-1
1
2
3
1
2
3
0
A
EF (chevron fold)
1 2 3
1
2
3
01 2 3
1
2
3
0
sinusoidal fold
1 2 3
1
2
3
0
b3
b10 5 10
-1
-0.5
0
0.5
1 A B C
box
fold
s
sem
i-elli
pses
parabolasD
sine waves
E
F
chevron folds
1
5
4
3
2
Area with composite folds
Area with a marked inflection point in the limb
1 2 3
1
2
3
0
D (parabolic fold)
1
2
3
0
1
2
3
C (elliptical fold)
B(box fold)
1
2
3
0
-1
1
2
3
y
x
Fig. 2. Curves (thick lines) representing the result of superposition of the terms with Fourier coefficients b1 and b3 (curves in thin lines) in the
interval 0VxVp for different types of folds and the values of the coefficients indicated by big points in the b1–b3 diagram by Hudleston (1973a).
Observe the difference between the resulting curves and the corresponding shape of the fold type represented.
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164134
0-1 1 0-1 1 0-1 1 0-1 1
0-1 1 0-1 1 0-1 1 0-1 1
6
6
5
4
3
2
1
.5
n = 0.5 n = 1 n = 1.75(sinusoidal form fit)
n = 2
n = 5 n = 10 n = 1000n = 3
(parabolic)(chevron)
(box)
Fig. 3. Fold morphologies obtained from monomial functions with several values of n and h.
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164 135
for the analysis of strain patterns in folds since strain
is related to the radius of curvature. In addition, when
we want to analyse jointly the two limbs of a fold, we
must use function (6), which is not differentiable at
x=0. This represents another inconvenience for the
use of these functions in kinematical folding analysis.
Another interesting family of functions appears
when f(x)=1�cos(px/2x0) (Bastida et al., 1999).
Thus:
y ¼ y0 1� cosp2
x
x0
� �� �nð7Þ
within the interval [0, p/2]. In this case, for nb0.56,
we have cuspate folds; for nc0.56, chevron folds;
for n=1, sinusoidal folds; for nN1, double-hinge
folds (this morphology is visually imperceptible for
n values close to 1); and for nYl, box folds. The
family of function (7) has certain properties similar
to Eq. (5) and presents the same advantages and
drawbacks.
A family of functions used by Bastida et al. (2003)
and Bobillo-Ares et al. (2004) is found when:
f xð Þ ¼ uu
x
x0
� �ð8Þ
with
uu xð Þ¼ x3 2�xð Þþux2 3�5xþ2x2
; 0VuV2=3:
ð9Þ
Then, the complete family of functions is given by:
y ¼ y0 uu
x
x0
� �� �n; ð10Þ
where, in this case, nz1, and 0VxVx0.
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164136
When compared with functions (5) and (7),
family (10) has the advantage that it has no infinite
curvature in the intervals of u and x given above. It
has an inflection point at x=x0, and describes a
geometry ranging from shapes slightly sharper than
the sinusoidal shape to the box shape. Thus, this
family of functions can be used to analyse kinematical
folding mechanisms; nevertheless, a drawback of this
family is that, for nN1, the maximum curvature is at a
point with xp0 for any u value, and for n=1, the
maximum curvature is at a point with xp0 when
approximately ub0.4. Hence, when this family of
functions is used, many common fold shapes are fitted
by functions with the hinge point in xp0; this featuremakes kinematical analysis difficult. In addition, the
complete definition of this type of functions requires
us to know four parameters (x0, y0, u, n), which is one
parameter more than families of functions (5) and (7).
This is an important drawback of Eq. (10) when used
for the graphic classification of folded surface
profiles.
3.1.3. Subellipses and superellipses (Gardner, 1965)
This family of functions was used by Lisle (1988)
to approximate the shape of coarse clastic sediment
particles and by Bastida et al. (1999) to fit fold
profiles. Taking as the coordinate origin the lower
vertex of the curve, the family is defined by:
xp
xp0
þ ðy� y0Þp
yp0
¼ 1 ð11Þ
For p=2, this function describes an ellipse,
whereas for pb2, it defines a family of curves
named dsubellipsesT, and for pN2, it represents
curves named dsuperellipsesT. This function is
suitable for both alloclinal ( pb2) and isoclinal
folds ( pz2). For pb1, we have cuspate folds; for
p=1, chevron folds; for p=2, elliptical folds; and
for pYl, box folds. Eq. (11) has similar properties
than Eqs. (5) and (7) in general, but it is the only one
that produces isoclinal double-hinge folds and can be
useful in the analysis of this type of fold.
3.1.4. Cubic Bezier curves
Recently, Srivastava and Lisle (2004) have pro-
posed the use of cubic Bezier curves to fit and classify
fold surface profiles. Each segment of these curves
requires four points to be completely defined, involv-
ing therefore eight quantities (the coordinates of these
points). In order to reduce to two the number of
quantities required, these authors have introduced
some constrains that allow to define each curve
segment by three points (Fig. 4): P0 (0, h), P1 (L,
h), and P2 (1, 0). The distance L determines the shape
of the curve, whose simplified parametric equations,
with relation to the reference frame shown in Fig. 4,
are:
x tð Þ ¼ 3 1� tð Þ2tLþ 3 1� tð Þt2 þ t3 ð12aÞ
y tð Þ ¼ h�1� tð Þ3 þ 3 1� tð Þ2t
�ð12bÞ
where the parameter t gives the points of the curve
from the initial point P0 (t=0) to the end point P2
(t=1). The shape parameter L allows obtaining a good
fit of the most common fold shapes (L=0, chevron
folds; L=0.44, cosine curves; L=0.55, parabolic folds;
L=1, elliptical folds). These curves provide a very
simple method to classify folded surface profiles using
two quantities: the shape parameter (L) and the aspect
ratio (h).
3.1.5. Conic sections
These curves result from cutting off a cone by a
plane at different angles and have been used by Aller
et al. (2004) for the description of folded surface
profiles. The common property that characterises
these curves is that the distance from a general point
of the conical curve (P) to a fixed point (focus, O)
divided by the distance from P to a fixed straight line
(directrix) is a constant value e (eccentricity). If
0Veb1 the conic curve is an ellipse, if e =1 it is a
parabola, and if eN1 it is a hyperbola. Using a
Cartesian coordinate system such as that shown in
Fig. 1, the equations of the conics can be written in
terms of e as follows:
f e; xð Þ ¼1�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 1� e2ð Þx2
p1� e2
0Vep1: að Þx2
2e ¼ 1: bð Þ
8>><>>:
ð13Þ
O P2
P0
P1
x
y
L
1
h
Fig. 4. Reference frame and geometric elements of the simplified cubic Bezier curve used to fit and classify folded surface profiles (after
Srivastava and Lisle, 2004).
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164 137
The functions (13) for 0Veb1 correspond to
ellipses with the major axis on the y-axis. Never-
theless, for fold description, it can also be convenient
to consider ellipses with the major axis parallel to the
x-axis. These can be defined by the family of
functions given by:
y ¼ 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ð1þ e2Þx2
p1þ e2
with ez0 ð14Þ
The ratio between the major and minor axes isffiffiffiffiffiffiffiffiffiffiffiffi1þ e2
pfor these ellipses. Eq. (14) can be obtained
from Eq. (13) by substituting e=ei, being i the
imaginary unit. If e is allowed to also take pure
imaginary values, Eq. (13) describes both types of
ellipse.
The range of shapes available to represent folds
may increase through the use of a scale factor a,
which allows us to model a fold with only a sector of a
conical curve within a fixed interval [0, x0], where x0is the limb width (Fig. 1). The general expression for
the conics used in this paper is thus:
f e; a; xð Þ ¼ af e;x
a
� �; for 0VxVx0: ð15Þ
Fig. 5 illustrates how the scale factor makes it
possible for a conic defined by an e value to be used
to represent diverse fold morphologies. It is conven-
ient to characterise the geometry of the fold surface
profile by a shape parameter e, and the normalized
amplitude h=y0/x0. Taking into account that
y0=f(e,a;x0), we obtain the following equations for
the parameter h:
h ¼
x0
2afor parabolas e ¼ 1ð Þ; að Þ
a
x0
1
e2 � 1� 1þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ð1� e2Þ x
20
a2
r !
for ellipses and hyperbolas e p1ð Þ: bÞð
8>>>>><>>>>>:
ð16Þ
Examining Eq. (16), we find that h can have any
positive value in the parabola, but in the ellipse and
hyperbola h has a positive value conditioned by the
following inequalities:
hV1ffiffiffiffiffiffiffiffiffiffiffiffiffi
1� e2p for ellipses; ð17Þ
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
1.0
Circle arcs (e = 0) Ellipse arcs (e = 0.5)
h = 0.2, a =2.53
h = 0.2, a =2.510.4, 1.310.
6, 0
.92
0.8,
0.7
4
1, 0
.64
0.4, 1.270.
6, 0
.87
0.8,
0.6
7
1.0,
0.5
7
1.15
, 0.5
1
h = 0.4, a =1.17
Ellipse arcs(e = 0.9)
0.8,
0.5
2
1.2,
0.3
2
1.6,
0.2
3
2, 0.1
82.
29, 0
.16
h = 0.4, a =1.14
Parabola arcs (e = 1)
0.8,
0.4
71.2,
0.2
5
1.6,
0.1
6
2.0,
0.1
1
2.4,
0.0
8
10, 0
.00
Hyperbola arcs(e = 1.1)
Hyperbola arcs(e = 1.5)
h = 0.4, a =1.10
0.8,
0.4
1
1.2,
0.1
8
1.6,
0.0
7 2.
0, 0
.02
2.18
, 0.0
0 (c
hevr
on fo
ld)
h = 0.2, a =2.320.4, 0
.890.6, 0
.39
0.8, 010
0.89,
0.00 (
chev
ron f
old)
x
0.2 0.4 0.6 0.8x
0.2 0.4 0.6 0.8x
0.2 0.4 0.6 0.8x
0.2 0.4 0.6 0.8x
0.2 0.4 0.6 0.8x
y
0.2
0.4
0.6
y
y
y
0.2
0.4
0.6
0.8
y
0.2
0.4
0.6
y
Fig. 5. Several sets of conics in the interval [0, x0]. Each set corresponds to an e value and shows curves with different h and a values (numbers
on the curves).
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164138
with the equality corresponding to a quarter of an
ellipse, and
hb1ffiffiffiffiffiffiffiffiffiffiffiffiffi
e2 � 1p for hyperbolas: ð18Þ
In the latter, the upper limit of h is reached when x0goes to infinity:
limx0Yl
h ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffie2 � 1
p ; ð19Þ
and it corresponds to the chevron shape.
Finding the a value from Eq. (16), we obtain:
a ¼ x0
2hfor parabolas e ¼ 1ð Þ; ð20Þ
a ¼ x01þ h2 1� e2ð Þ
2h
for ellipses and hyperbolas ðe p1Þ:
ð21Þ
Introducing this value in the function f(e, a; x) [Eq.
(15)], the function g(e, h, x0; x) can be found. Several
graphical examples of conic sections with different
values of e and h (and a) are shown in Fig. 5,
illustrating the large range of fold profiles that can be
fitted by this type of curve.
The conic sections have some advantages over the
other families of functions described above. They
offer a good fit to most common fold shapes and have
finite curvature at all their points, a curvature extreme
value in the apex and at least one symmetry axis. An
important additional advantage of conics is that affine
transformations (homogeneous deformation) map
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164 139
ellipses to ellipses, parabolas to parabolas and hyper-
bolas to hyperbolas (Brannan et al., 1999, p. 85). This
is a characteristic property of conics that the other
functions considered here do not possess, and makes it
possible to analyse cases in which rotational homoge-
neous strain operates during the development of
asymmetric folds and modifies the geometry of the
folded surface profile. In these cases, if the curves that
describe the folded profile before and after strain
belong to different families, the change in shape is
very difficult to describe, and a thorny problem arises
in the analysis of the folding kinematics. Conics are
the simplest curves that do not pose this problem, and
this represents a powerful reason for them to be
preferred in the kinematical analysis of folding. In
addition, they can be defined by two parameters (a, or
h, and e) and can be easily used for a quantitative
description and a graphic classification of folded
surface profiles.
Since conic sections only range from chevron to
elliptical shapes, some natural folds cannot be fitted
by a conic curve. In particular, cuspate folds and
isoclinal folds, except those approaching a quarter of
an ellipse, cannot be fitted by a conic. Isoclinal folds
with a single hinge cannot be well fitted by any of
the functions described above either, but they can be
fitted by a function composed of a quarter of an
ellipse, as defined by Eq. (13) in the interval [0, x0],
and a line segment (x=x0, within bVyVy0) of length c,
which is a prolongation of the ellipse arc (Fig. 6), as
y
x0
P
O
y0
Fig. 6. Geometrical elements of a curve composed by a quarter of an ellip
proposed by Bastida et al. (1999). This function
allows us to extend the range of fitted folds to the box
shape.
3.1.6. Fitting fold profiles by the functions
Once the family of functions to be used to
describe the fold profile geometry has been chosen,
it is necessary to establish a fit method to
approximate natural folds using an equation of this
family. To make the necessary measurements to fit a
natural fold, it is convenient to use photographs of
the fold profile, from which this profile, the
coordinate system and the inflection point can be
drawn (Fig. 1, or Fig. 4 for fitting by Bezier
curves).
Fitting by Bezier curves can be made using
drawing software (e.g., Macromedia FreeHand,
CorelDraw, Adobe Illustrator) through the following
steps (Srivastava and Lisle, 2004): (a) mark the point
P2 on the inflection point of the fold; (b) mark the
point P0 on the hinge point and drag with the mouse
the appropriate control point in a direction parallel to
the x-axis until the shape of the fold profile is fitted;
(c) mark the point P1 and measure the distances OP2
(taken as unit), OP0 and P0P1; and (d) determine
L ¼ P0P1 =OP2 and h ¼ OP0 =OP2. These quanti-
ties can be used to classify the fold on a L vs. h
diagram (Srivastava and Lisle, 2004; Fig. 3).
Let us consider now the fit by the family of
functions given by Eq. (15). Since the y0/x0 value is
x
b
c
se and a line segment. This curve type is used to fit isoclinal folds.
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164140
a characteristic parameter of folds, we must search
for a fit curve with the same y0/x0 value as the
natural fold profile. The method described by Aller
et al. (2004) for fitting fold profiles by conics has
been named the dmiddle point fit methodT. It
consists of finding a conic that passes through the
point of the natural fold profile corresponding to
x=x0/2. The method requires various calculations to
obtain the parameters e and a of the conic and is
discussed in Appendix A for all the possible cases.
For isoclinal folds, in the case of functions
composed of a quarter of an ellipse and a line
segment, the fit function that passes through the point
(x0/2, yM) of the natural fold is defined by a c value
given by:
c ¼ y0 �yM
1�ffiffiffi3
p
2
ð22Þ
3.1.7. Classification of folded surface profiles
When the functions of the family used to fit natural
fold profiles can be characterized by two parameters,
it is possible to make a classification of fold profiles
by a diagram in which the normalized amplitude h ( y-
axis) is plotted against a parameter that characterises
the fold shape (x-axis). When conics are used, an h
versus e diagram is the easiest one to construct.
Nevertheless, a disadvantage of this diagram is that
the field suitable for representing fold limbs is small.
In addition, chevron or elliptical folds define loci
represented by curves in this diagram, and folds fitted
by a function composed of a quarter of an ellipse and
a vertical segment are not represented. Hence, the path
corresponding to the development of a fold can be
difficult to interpret; for example, the flattening of a
folded surface profile would follow a curve in the
diagram. In order to avoid these drawbacks, it is
convenient to use a normalized area defined as:
A ¼ 2A=x0y0; ð23Þ
where A is the area enclosed in the concave part of the
conic within the limb interval (Fig. 1a). The parameter
A depends on e and h, and in order to obtain this
parameter it is necessary to determine previously e
and a by the middle point fit method, and then h by
Eq. (16). As a function of h and e, the normalized area
for the fit conic is given by:
A ¼ 2� 2
X0 f e;X0ð Þ
Z X0
0
f e;Xð ÞdX ; ð24Þ
with X0 ¼2h
1þ h2 1� e2ð Þ ;Z X0
0
f e;Xð ÞdX ¼ 4
3for parabolas e ¼ 1ð Þ; ð25Þ
Z X0
0
f e;Xð ÞdX
¼ 1
1� e2
�X0 �
X0
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ e2 � 1ð ÞX 2
0
q
� 1
2ffiffiffiffiffiffiffiffiffiffiffiffiffie2 � 1
p sinh�1ffiffiffiffiffiffiffiffiffiffiffiffiffie2 � 1
pX0
� ��ð26Þ
for hyperbolas (eN1), and
Z X0
0
f e;Xð ÞdX
¼ 1
1� e2
�X0 �
X0
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 1� e2ð ÞX 2
0
q
� 1
2ffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2
p sin�1ffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2
pX0
� ��ð27Þ
for ellipses (eb1).
These equations make it possible to use a diagram
of h against A (Fig. 7). In this diagram, the typical
shapes of folded surface profiles (chevron, sinusoidal,
parabolic, and elliptic) are plotted on straight lines
parallel to the y-axis. The isoclinal folds fitted by a
function formed by a quarter of an ellipse and a line
segment are represented on the h�A diagram to the
right of the vertical line of the elliptical folds.
The h�A diagram enables the introduction of
different families of functions that can be linked
through the normalized area, since this parameter
can also be determined for curves of the families of
functions defined by Eqs. (5), (7), (11), (12a,b), and
(15). For instance, the chart shown in Fig. 8
establishes a correspondence between A and the
shape parameter n of functions (5) and (7), the
parameter p of function (11), and c/y0 of the isoclinal
Normalised area ( )
0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Asp
ect r
atio
(h)
0
0.5
1
1.5
2
2.5
3
3.5
c/y0
0 0.2 0.4 0.6 0.8 1
Sin
e
Par
abol
a
Elli
pseC
hevr
on
Box
Fig. 7. Diagram of aspect ratio (h) vs. normalized area Að Þ showing the location of main fold shapes.
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164 141
folds. Hence, obtaining the normalized area from
Eq. (24) allows us to determine the corresponding
value of n, p, and c/y0 and therefore, to determine
the respective functions. Cuspate folds cannot be
fitted by conics, but can be introduced in the h�A
Chevron Sine fit Parabola fitCuspate
1.31.21.110.9 1.40.8
0.4 0.5 0.6 0.7 0.8 0.9 1n [function
0.9 1 1.1 1.2 1.3 1.4 1.5 p
n (monomi0.7 0.8 0.9 1.5 21
= 2A/x
Fig. 8. Correlation scales for the shape parameters n and p involved in fun
the correlation is the normalized area A ¼ 2A=x0y0Þð . Values of the shape
fits have been correlated by vertical lines; the points on them indicate the
diagram by fitting them using functions of the
families (5), (7), or (11) with an exponent lower
than one. In Fig. 7, several lines and fields can be
distinguished in which the following types of folds
can be represented: Ab1, cuspate folds; A ¼ 1,
10.80.60.40.2
BoxEllipse fit
c/y0
1.91.81.71.61.5 2
→
0
1.5 2 2.5 3 4 5 10 50 family (7)]
2 2.5 3 3.5 4 5 10
al functions) 3 4 5 10 20 100
→
88
8
→
0y0
ctions (5), (7) and (11), and c/y0 of the isoclinal folds. The basis for
parameters corresponding to chevron and sine, parabola, and ellipse
perfect fit of the corresponding shapes.
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164142
chevron folds; A ¼ 4=p ¼ 1:2732 . . . ; sinusoidal
folds; A ¼ 4=3, parabolic folds; A ¼ p=2 ¼ 1:5708. . . ; elliptical folds; A ¼ 2, box folds.
Assuming that conic sections are chosen to fit
natural alloclinal folds, to classify a limb profile of a
folded surface on the h�A diagram (Fig. 7), we
should follow these steps:
(1) Trace the fold profile from a photograph onto
transparent paper.
(2) Locate the hinge and inflection points on the
section and construct the reference frame.
(3) Determine x0 and y0 on the drawing; then,
obtain h=y0/x0.
(4) Use a fitting method to find the parameters of
the fit curve. If the middle point fit method is
used, we must first of all measure the value yMfor x=x0/2. Then, the method described in
Appendix A and Eq (24) for conics give the
normalized area.
(5) Plot the normalized area A and the aspect ratio
h on the diagram of Fig. 7.
If the fold is isoclinal, we can follow the same
procedure, except for point (4) where Eq. (22) must be
used to find c. Then, the chart of Fig. 8 allows A to be
determined.
As an example of the application of this method to
natural folded surfaces, Fig. 9 shows the classification
of a train of asymmetric folds developed in a quartz-
feldspatic layer embedded in schists from the Mon-
donedo nappe basal shear zone (Variscan belt, NW
Spain). The analysis involved finding the conical
function that gave the best fit for each of the fold
limbs in Fig. 9a and b. Fig. 9b and c enabled a visual
comparison between the geometry of the folded layer
profile drawn from the photographs (Fig. 9b) and the
folded layer model constructed by assembling all the
fit curves (Fig. 9c). The parameters h and A that
characterised the fit functions have been represented
in Fig. 9d, separating inner and outer arcs, and normal
and overturned limbs. The results for all the possible
locations exhibit increasing trends, although the slope
and the dispersion of the distribution increase from the
overturned limb outer arcs towards the normal limb
inner arcs, with the rest of the locations showing
intermediate values. The normalized area is slightly
greater for the outer than for the inner arcs.
Function families (5), (7), (11), (12a,b) and (15)
can be used to construct a chart similar to that based
on the use of the Fourier coefficients b1 and b3(Hudleston, 1973a; Fig. 12). A chart of this type with
a set of conic standard forms is shown in Fig. 10. It
facilitates the classification of natural fold profiles by
visual comparison with the standard forms. This
method is very simple and allows an approximate
classification without calculations.
3.1.8. Joint analysis of the two fold limbs: asymmetry
of folded surface profiles
When the two limbs of a folded surface profile
are drawn jointly, it can usually be observed that the
fold is asymmetric with respect to the y-axis. This
property of the folds is a result of the folding
mechanisms that formed them and it has to be taken
into account in the kinematical study. Ramsay (1967,
p. 351) considered that the fold asymmetry depends
on the relative length of the two limbs. Thus, a
simple measure of the symmetry or asymmetry of a
fold is the long limb length/short limb length ratio.
In field work, if the folds are viewed down their
plunge, they can be separated into M (symmetric),
and S or Z (asymmetric) folds (Ramsay, 1967, p.
351), in such a way that the shape of the letter
describes the shape of the fold.
In accordance with Tripathi and Gairola (1999),
the fold asymmetry depends on two factors: the
difference in amplitude and the difference in shape
between the two limbs. Thus, these authors defined
the degree of asymmetry (DA) of a folded surface
profile as the sum of two parameters, one depend-
ing on the difference in amplitude and other
depending on the difference in shape. A problem
with this method is that a specific DA value does
not allow us to determine to what extent the
asymmetry of a fold is due to difference in
amplitude or difference in shape.
An alternative method to evaluate the asymmetry
of a folded surface profile can be developed using two
new parameters:
Shape asymmetry : Sa ¼ AL=AS: ð28Þ
Amplitude asymmetry : Aa ¼ y0L=y0S; ð29Þ
where AL and AS are the normalized areas of the
long and the short limb, respectively, and y0L and
Fig. 9. (a) Asymmetric folds developed in a single quartz-feldspatic layer embedded in schists from the Mondonedo nappe basal shear zone
(Variscan belt, NW Spain). (b) Line drawing of the boundaries of the quartz-feldspatic layer. (c) Reconstruction of the layer above using fit
functions (conics, or ellipse plus line segment). (d) Classification of the folds of the layer above: the straight lines indicate the general
morphologic tendency for the different limb types.
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164 143
0
0.5
1
1.5
2
2.5
3
3.5
Sin
e
Par
abol
a
Elli
pse
Che
vron
Box
Asp
ect r
atio
(h)
0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Normalised area ( )
1
4
4.5
c/y00 0.2 0.4 0.6 0.8
Fig. 10. Chart of conic standard forms that can be used in the classification of natural fold profiles by visual comparison.
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164144
y0S are the y0 parameters of the long and the
short limb, respectively. The plot of these param-
eters in a diagram of Sa against Aa for all the
folds of a specific set allows us to visualize the
variation in asymmetry of these folds. An example
of the use of this diagram is shown in Fig. 11. In
this example, the variation range of Aa is wider
than that of Sa, indicating that in this case
amplitude asymmetry characterises the fold asym-
metry better than the shape asymmetry. In addi-
tion, Fig. 11 shows a general trend in which the
amplitude asymmetry increases as shape asymme-
try decreases.
Asymmetric folds resulting from a complex
superposition of kinematical mechanisms can
involve a migration of the hinge point during
folding. Thus, for the kinematical analysis of
folding, it is preferable to search for a single fit
function suitable for the joint analysis of both fold
limbs. The use of two different functions (one for
each limb) could offer a higher accuracy, but it
would give rise to a discontinuity in the hinge
point that can make kinematical analysis unsuitable
in many cases. Again, the conic section family is
the most adequate for the fit, which can be made
in this case by the least squares method. To make
the fit, the folded surface profile of the two limbs
is drawn from a photograph. In addition, the hinge
point is found and the coordinate system is drawn.
After that, several points of the profile P1, P2, . . .,PN, with respective coordinates (x1, y1), (x2, y2),
. . ., (xN, yN), are chosen.
0.6
0.8
1
1.2
1.4
0 5 10 15
Outer arc Inner arc
Sa
Aa
Fig. 11. Diagram of Sa against Aa for the folds of Fig. 10.
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164 145
If f(e, a; x) is the family of the conic sections [Eq.
(15)], the quadratic error associated with the fit of the
given points is defined by:
E2 e; að Þ ¼XNi¼1
f e; a; xið Þ � yi½ 2: ð30Þ
The method to obtain the best fit conic consists of
choosing the e and a values that minimise E2(e, a).
3.2. 3-D analysis
The three-dimensional quantitative geometrical
characterisation of folded surfaces is not well devel-
oped, and until now, the methods used belong
exclusively to the non-functional type. Williams and
Chapman (1979) proposed a three-dimensional clas-
sification based on two parameters (Fig. 12a): the
(a) (
β
θ
Hinge line surface
Profilesurface
Q
Fig. 12. Three-dimensional classification of folded surfaces after Williams
Classification triangle showing several standard forms.
interlimb angle (h) and the hinge angle (b). Assuming
bNh, these authors defined three other parameters: P
(degree of planarity), Q (degree of domicity), and R
(degree of noncylindrism), as:
P ¼ h=180 ð31Þ
Q ¼ 180� h þ 180� bð Þ½ 180
ð32Þ
R ¼ 180� bð Þ=180 ð33Þ
These parameters range between 0 and 1, and
accomplish the condition P+Q+R=1; they can there-
fore be plotted on a triangular PQR diagram for the
three dimensional classification of folds (Fig. 12b).
This classification only considers conical (or domical)
folds and folds with hinges contained in a plane
(cylindrical plane folds and non-cylindrical plane
b) P
R
and Chapman (1979). (a) Angles involved in the classification. (b)
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164146
folds from Turner and Weiss, 1963), and does not
consider folds with the hinge not contained in a plane.
Nevertheless, these folds are in general formed as a
result of a complex succession of deformations and
must be analyzed in terms of superimposed folding.
A parameter that has been used to characterise the
three-dimensional geometry of a folded surface is the
Gaussian curvature (Suppe, 1985, p. 313; Lisle et al.,
1990; Lisle, 1992, 1994, 2000; Lisle and Robinson,
1995; Ozkaya, 2002; Masaferro et al., 2003), defined
in any point of the surface as the product of the
principal curvatures. This parameter is a point
function and can be mapped in order to characterise
some important geometrical properties of folds.
4. Geometrical analysis of the folded layers
The study of the folded layer geometry is made
usually in the profile plane by analysis of the variation
in thickness throughout the layer. Using this param-
eter, Van Hise (1896) distinguished two basic fold
types, which he named dparallel foldsT and dsimilar
foldsT. The former folds have layer thicknesses that
are constant and they die out upward and downward.
The latter have constant thickness measured parallel
to the axial surface, so that all the folded surfaces
have the same shape; similar folds have the limbs
thinned with respect to the hinge and can be
geometrically propagated upward and downward
t'α=
(b)α
α
t = T0 0
Ttα
α
(a)
Fig. 13. (a) Definition of orthogonal thickness (ta) and thickness parallel to
(b) Fold types defined on the t Va�a diagram (after Ramsay and Huber, 19
indefinitely. Another classical term making reference
to the geometry of the folded layers is that of dsupratenuous foldsT (Nevin, 1931), which are folds in
which the beds are thinner in the crests of the
anticlines and thicker in the troughs of the synclines.
The first quantitative study of the folded layer
geometry was made by Ramsay (1962), who defined
the orthogonal thickness and the thickness parallel to
the axial surface (Fig. 13a), in addition to measuring
the variation of orthogonal thickness with the distance
along the bedding plane, and the variation of thickness
parallel to the axial surface with the distance to the
axial plane. He deduced that pure parallel and similar
folds are rare in deformed rocks, and that a wide range
of intermediate possibilities between them exists. He
also observed that the existence of maximum orthog-
onal thickness in the axial traces is very common and
interpreted this feature, when it occurs in competent
layers, as a result of a flattening superposed on parallel
folds (homogeneous strain with shortening perpendic-
ular to the axial plane). Moreover, Ramsay (1962)
showed the characteristic variation of orthogonal
thickness with layer dip graphically for several values
of the flattening amount. Subsequently, Ramsay
(1967, pp. 359–369) devised a fold classification
based on three alternative parameters: orthogonal
thickness (ta), thickness parallel to the axial plane
(Ta), and dip isogons (lines joining top and bottom
points of equal dip). The measure of the two thickness
must be made taking the thickness at the fold hinge
tαt0
1,0
0,5
0,00 30 60 90
Class 1A
Class 1B Parallel
Class 1C
Class
2Sim
ilar
Class 3
Dip α
axial surface (Ta) of a folded surface (after Ramsay, 1967, p. 361).
87, p. 349).
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164 147
(t0=T0) as a unit; that is to say, t Va=ta/t0, and T Va=Ta/T0.
Among the three parameters above, t Va is the most
commonly used, and to make the classification it must
be plotted against the dip a (Fig. 13b), so that each
fold limb is represented on the diagram by a curve
whose position and geometry determine the fold class.
Ramsay differentiated three fold classes, class 1, class
2 (similar folds), and class 3, with three subclasses of
class 1 folds: 1A, 1B (parallel folds), and 1C. As can
be seen in Fig. 13b, similar and parallel folds are
represented by specific lines, and between them a
wide range of intermediate geometries are represented
in the field of class 1C. In supratenuous folds,
anticlines are examples of class 1A folds, and
synclines examples of class 3 folds. In order to make
the classification more precise, Ramsay (1967, p. 370)
also suggested plotting the first two derivatives of the
thickness against the dip, since the fold class is
controlled by the slope variation of the tangent to the
curve t Va, rather than by its position inside the fields in
Fig. 13b. In order to simplify the curves representative
of the folded layers, Hudleston (1973a) used a
diagram of t Va2 against cos2a, in order that the curve
representative of the similar fold (t Va=cosa) became a
straight line.
A problem with Ramsay’s classification is the
difficulty to obtain the best location of the reference
thickness (t0) to determine t Va. In order to solve this
problem, Hudleston (1973a) modified Ramsay’s
(b)
9
-9
φα
o
iA
A
φ
A
Aφ
o
i
φ Positive φ Negative
(a)
Fig. 14. (a) Definition of angle Ua for classification of folded layers and
Hudleston, 1973a,b). Classification diagram of the folded layers (after Hu
classification by the application of a new parameter
(Ua), defined as the angle between the isogon and the
normal to the tangents drawn to either fold surface at
angle of dip a. According to the convention shown in
Fig. 14a, Ua can be positive or negative. Hudleston
(1973a) constructed a diagram of Ua to classify folded
layers (Fig. 14b). In this diagram, the straight line
Ua=0 represents parallel folds, and the straight line
Ua=a represents similar folds.
A criticismmade to the Ramsay classification is that
it differentiates very few classes of folds (Hatcher,
1995; Lisle, 1997). Zagorcev (1993) subdivided the
subclass 1A folds in three types, using the straight line
Ua=�a, symmetrical to Ua=a in the Hudleston
diagram (Fig. 14b). The line Ua=�a (equivalent to
t Va=sec a in the Ramsay diagram) defines the fold type
1A2 and separates two fields: one for folds of the type
1A1 (�UaNa), and another for folds of the type 1A3
(�Uaba). Zagorcev (1993) also subdivided class 3
folds in three subclasses: 3A, 3B, and 3C. To do this,
he defined the complementary shape of the parallel
fold in the way illustrated in Fig. 15. This can be
defined as the shape that we must add to a parallel-
folded layer to obtain a similar fold. This complemen-
tary shape defines the subclass 3B, and is represented
by the curve given by tanU=tana/(2cosa�1) on the
Ua–a diagram and by t Va=2cosa�1 on the t Va–adiagram. Subclass 3B allows us to separate two fields:
one for subclass 3A (cosaNt VaN2cosa�1), and another
Angle of dip α
0
0
-90 900
0
Class 1A1
Class 1
A2
Class 1B
Class 1C
Class 1A3
Class 2
Class 3
32
1C
1B
1A3
1A11A
2
sign convention; i=inner arc, o=outer arc; AAV=dip isogon (after
dleston, 1973a,b, with the modification by Zagorcev, 1993).
0,00,0 0,5 1,0
0,5
1,0
β2
β1
t 22
t 12
A
B
O
sin2α
sin2α
t α2
Class 1A
1C
3
1B
2
22m sin2α m
Fig. 16. Diagram of t Va2 against sin2a, and definition of angles b1 and
b2, and t V 21 and t V 22 from a curve representative of a fold.
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164148
for subclass 3C (2cosa�1Nt Va). The folds of subclasses3A and 3C can be defined as complementary shapes of
the folds of subclasses 1C and 1A, respectively.
Nevertheless, according to Zagorcev (1993), folds of
types 1A1 and 3C are rare in rocks deformed under
natural conditions. A different subdivision of Ram-
say’s fold classes was made by Lisle (1996), who
divided class 3 folds into new classes 3A, 3B and 3C,
defined in a different way. Nevertheless, Lisle’s
classification scheme involves kinematical concepts
and will be considered below.
The use of Ramsay’s classification is laborious but it
provides an excellent tool for analysing the geometry of
single folds, and it is indispensable for making studies
of kinematical folding mechanisms in specific folds,
since it provides an accurate functional description of
folded layer geometry. Nevertheless, when Ramsay’s
classification is applied to the analysis of large data
sets, it gives rise to swarms of t Va curves from which it is
hard to draw conclusions based on a truly quantitative
analysis. This makes this classification unsuitable in
these cases, and new methods must be developed for
the statistical analysis of fold populations in regional
studies or in the analysis of multilayers.
The need to synthesise the results obtained from
Ramsay’s classification was considered by Hudleston
(1973b), who analyzed a large number of minor folds
developed in the Moine rocks of Monar (Scotland). In
order to transform a t Va curve into a single parameter,
Hudleston projected t Va2 against cos2a and obtained
the intercept value of the best-fit straight line. This
method implies a loss of information regarding
Subclass 3BT'α = 2 - sec α t' α = 2cos α−1
Parallel (subclass 1B)T'α = sec α , t' α = 1
Similar (class 2)T'α = 1t' α = cos α
Fig. 15. Complementary shape of the parallel fold (after Zagorcev,
1993).
individual fold geometry, and is laborious, but it is
useful and can complement the use of t Va curves.
Bastida (1993), taking into account the simplicity
of the t Va curves, defined two parameters, p V1 and p V2, todescribe the geometry of any t Va curve and to classify
folded layers. These parameters are closely related to
the slope of the curve. Nevertheless, this analysis can
be considerably simplified if we consider the Hudles-
ton diagram of t Va2 vs. cos2a, modified to represent
t Va2 vs. sin 2a, and define two parameters, ds1T and
ds2T (similar to p V1 and p V2) as follows.
Let us consider a curve of t Va2 vs. sin2a for a fold
limb (Fig. 16), and two points, A and B, on this curve.
A is the point of the curve where the abscissa equals
(sin2am)/2, and B is the final point of the curve with
abscissa sin2am (am is the maximum dip of the folded
layer). After drawing the line segments, OA and AB,
we define s1 and s2 as (Fig. 16):
s1 ¼ tanb1 ¼2 1� t V21 sin2am
ð34Þ
s2 ¼ tanb2 ¼2 t V21 � t V22 sin2am
ð35Þ
1
1,5
0,5
0
-0'5
1A - 3
1A - 2
1B-
3
s2
1C - 3
1C - 2
2-
3 3
3 - 22
Fold limb
1C1A - 1C
2-
1C 3 - 1C
1A - 1B
1B-
1C
1B
-0,5
1A
1C - 1B
1B-
1A
0,5
3 - 1B
1 1,5
1C - 1A
s1
2-
1A
3 - 1 A
Fig. 17. Fields and lines defined on the s1–s2 graph, and their corresponding fold classes (after Bastida, 1993).
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164 149
Thus, a simple s1 vs. s2 graph (Fig. 17) offers a
means of classifying folds, since each fold limb can
by represented by a single point. The t Va2 vs. sin2a
line corresponding to parallel folds (Ramsay’s
subclass 1B) in Fig. 13b will be represented by
the origin of the s1–s2 diagram, and the curve
corresponding to similar folds (Ramsay’s class 2) by
point 2, with coordinates (1, 1) (Fig. 17). The graph
of this figure contains several fields and lines on
which points that represent fold classes can be
plotted. 1A, 1C, and 3 fold classes lie in three of
these fields. The other fields and all the lines
represent folds composed of two different classes
in Ramsay’s classification, one class corresponding
to parameter s1 and the other to s2. In addition, the
straight line s1=s2 contains the points that represent
fold limbs whose graphs in the t Va2 vs. sin2a diagram
are straight lines. The general equation that describes
the orthogonal thickness of these folds is:
taV ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� msin2a
p; ð36Þ
where mb0, for class 1A folds, m=0 for class 1B
folds, 1NmN0 for class 1C folds, m=1 for class 2
folds, and mN1 for class 3 folds.
Fig. 18 shows representative t Va2 vs. sin2a curves
for different fields and lines of the s1–s2 diagram. The
fold class is determined by the slope of the curve and
not strictly by the field in which the curves are
situated in the t Va2 vs. sin2a diagram. In this sense, the
s1–s2 diagram sometimes represents the fold class
better than the t Va2 vs. sin2a curve itself, or than the
Ramsay’s curve of t Va vs. a, since s1 and s2 define the
mean slope of the corresponding parts of the curve. In
addition, the method allows a natural subdivision of
the Ramsay classes in the composite types shown in
Figs. 17 and 18.
The s1–s2 method only requires the measurement
of three values of the orthogonal thickness (t0, t V1and t V2) for each limb, from which s1 and s2 can be
obtained by Eqs. (34) and (35), and plotted on the
diagram as a point that characterises the geometry of
the folded layer. The dip a1 corresponding to the x-
1
1,5
0,5
0
-0'5
1A - 3
1A - 2
1B-
3
s2
1C - 3
1C - 2
2-
3
3 - 22
1A - 1C
2-
1C
3 - 1C
1A - 1B
1A - 1B
1B-
1C
1B
-0,5
1A
1C - 1B
1B-
1A
1B - 1A
0,5
3 - 1B3 - 1B
1 1,5
1C - 1A
s1
2-
1A
2 - 1A
3 - 1 A
1B - 3
3
2 - 3
1A - 2 3 - 2
1C
1C
1C
Fig. 18. t V2a curves representative of the folds corresponding to different fields and lines of the s1–s2 diagram (after Bastida, 1993).
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164150
value (sin2am)/2 for which t V1 must be measured is
given by:
a1 ¼ sin�1 sinamffiffiffi2
p� �
: ð37Þ
An advantage of this method is that by averaging
the slope of the t Va2 vs. sin2a curve it reduces the effect
of the error in selecting the reference points where the
dip is taken as zero. An example of the application of
the s1–s2 diagram to natural folds is shown in Fig. 19.
The use of the s1–s2 diagram involves a loss of in-
formation with respect to the maximum a value of the
fold limbs. This problem can be mitigated using dif-
ferent symbols for different intervals of a. Anotherpossibility is to construct a graph of the mean value of
s1 and s2 [s=(s1+s2)/2)] against the maximum a value
of the fold limbs. This diagram is complementary to the
s1–s2 diagram, and the value of s indicates the fold class
to which a specific fold limb dominantly belongs. An
example of the application of this diagram is shown in
Fig. 20.
Srivastava and Gairola (1999) extended the classi-
fication of folded layers to multilayers. They drew
several dip-isogons and for each of them determined
-1
-2 -1 1
Navia unitMondoñedo nappe unit
1C-3 3
3-1C
3-1A1C-1A1A
1A-1C
1A-3
1C
s2
s1
1
0
Fig. 19. s1–s2 diagram for natural folds developed along the Cantabrian coast in two units of theWestasturian–Leonese zone (Variscan belt of NW
Spain). Most of the folds belong to class 1C, although some of them lie on other fields. It can be also seen that the points representing the folds of
Mondonedo nappe unit (with a more internal location in the orogenic belt than the Navia unit) present a lesser dispersion than the others units.
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164 151
the orthogonal thickness (or the thickness parallel to
the axial surface) through the different layers. They
then calculated the standard deviation (rn) for the
thickness data of each dip-isogon. The rn values
obtained were plotted as a function of the dip and the
points of the different isogons were joined by a smooth
curve. The authors divided the rn–a coordinate space
in several fields bounded by curves representative of
standard fold types. These fields characterise different
-2
-1,5
-1
-0,5
0
0,5
1
1,5
30
(s +
s )
/21
2
3
1C
1A
1B
2
Class
Fig. 20. s-am diagram for the natural folds of Fig. 19. Higher values of t
internal Mondonedo nappe unit.
types of folded multilayers and allow their classifica-
tion. Hence, the field or fields where a rn–a curve lies
determines the corresponding type of fold.
Nevertheless, in the cases studied, these curves are
very complicated and commonly cross all the fields
(Srivastava and Gairola, 1999; Figs. 7 and 9), with the
result that the classification is not easily interpreted.
In addition, although the classification is based on the
parameters involved in the Ramsay classification, the
60
Navia unitMondoñedo nappe unit
Maximum dip 90
he maximum dip and a lesser range of s are observed in the more
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164152
fold classes distinguished can hardly be related to the
classes defined by Ramsay. Alternatively, a plot of
the different folded layers on a s1–s2 diagram, where
each folded layer is represented by a point, is suitable
for the description of the geometrical differences bet-
ween the folded layers of a multilayer in terms of the
classes established by Ramsay.
5. Cleavage distribution in folds
It is generally accepted that slaty cleavage, and in
general the tectonic foliations formed directly over a
sedimentary fabric, define planes very close to the XY
principal plane of the finite strain ellipsoid (XNYNZ).
Thus, knowing the detailed geometrical pattern of the
cleavage associated with a fold is essential in order to
obtain the orientation of the strain ellipse through the
profile and to unravel the folding kinematics.
To quantitatively describe the cleavage distribution
through a fold, Treagus (1982) proposed an angular
measure, angle b, defined as the angle between the
cleavage trace and the normal to the bedding. This
parameter is similar to Hudleston’s angle Ua (1973b),
and uses the same sign convention. b values may be
plotted against normalized limb dip a (a=0 in the
hinge points), and this plot may be made together with
the Hudleston graph of Ua against a. Thus, angle ballows a fold classification according to the cleavage
pattern, which is complementary to the Hudleston
classification. The fold classes obtained by the
Treagus method are the same as those obtained by
the Hudleston or Ramsay methods, but they are given
in Roman numerals (classes I, II, and III). Bastida et
al. (2003) used a graph of bedding dip a against
cleavage dip / (taking dip zero in the hinge points) to
analyse kinematical folding mechanisms by compar-
ing curves obtained from measurements in natural
fold profiles with curves of the plunge of the strain
ellipse major axis versus bedding dip obtained from
theoretical models of fold profiles.
6. Application of the fold geometry to the study of
kinematical folding mechanisms
2-D numerical modelling of folds by combination of
kinematical mechanisms can be done theoretically by
addition of successive folding steps from an initial
configuration of the layer profile. The program
dFoldModelerT, developed in the MATHEMATICAkenvironment (Bastida et al., 2003; Bobillo-Ares et al.,
2004), allows us to carry out this analysis. The first
step in the modelling process is to define the initial
configuration of the layer profile that will be folded to
give a fold limb. To do this, the profile is divided into
a grid of quadrilaterals that are small enough to as-
sume nearly homogeneous strain within them. The
nodes of the grid define the points that will be trans-
formed by the different kinematical folding mecha-
nisms, and allow the strain after folding to be analysed.
The initial configuration of the layer profile is defined
by the parameters of the guideline (if represented by a
conic, maximum x-value, eccentricity and scale factor),
the thickness of the layer above and below the
guideline, the number of quadrilaterals in every row,
and the number of rows above and below the guideline.
Once the initial grid has been defined, we can
deform it applying folding steps. Every step of a
specific mechanism implies the application of its trans-
formation equations to the nodes of the given config-
uration in order to obtain the corresponding images in
the new configuration. Thus, to define a folding step, it
is necessary to specify the folding mechanism that will
be applied and the changes that the step has to produce
in the parameters of the guideline (aspect ratio
variation and shape change). As a result, we achieve
theoretical folds formed by the combination of several
folding mechanisms. Of special note among the
outputs that we can obtain from the modelled folds are:
– The drawing of the folded layer, showing the
deformed grid, the distribution of strain ellipses
and their axes, and a variation in the grey level
depending on the ellipticity of the strain ellipse.
– Curves showing the variation in plunge of the
major axis of the strain ellipse as a function of the
layer dip for the inner and outer arcs of the folded
layer (/–a curves).
– Curves of the strain ellipse axes ratio against the
layer dip for the inner and outer arcs of the folded
layer (Rs–a curves).
– Ramsay’s classification of the fold limb.
– The bulk shortening associated with folding.
– The parameters defining the final function that
describes the guideline (e, a).
O
n(X )
Q(X )
q(X )
NL
l
Fig. 21. Transformation by folding of an initial point Q(X) in a
deformed point q(X). L and l, original and deformed guidelines; N y
n, initial and final unit normal vectors to the guideline.
OQ(X0)
q(X )
NL
l
n(X )0
0
P
p
Fig. 22. Images q(X0) and p of two points Q(X0) and P folded by
tangential longitudinal strain, and located on the guideline and
outside of it, respectively.
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164 153
– The value of the final amplitude and width of the
fold limb (measured on the guideline), as well as
the ratio between these two parameters, which
represents the final normalized amplitude ( y0/x0).
– The ratio between the layer thickness at the point
with dip zero and the amplitude of the outer arc of
the folded layer (t0/yoa).
Modelling by this method allows predictions to be
made about the geometrical properties of folds pro-
duced by a combination of several types of kinematical
mechanisms, and of the structures that can appear
associated with these folds. In addition, if adequate
geometrical information is available from a natural
folded layer, it is possible to shed light on the combi-
nation of kinematical folding mechanisms that could
give rise to such a fold and obtain some conclusions
about the strain distribution in the folded layer profiles.
In order to show the use of the results of the geo-
metrical analysis of folds in the kinematical analysis,
we will now describe the transformation equations of
some basic mechanisms, and some modelling exam-
ples of theoretical folds, as well as an example of a
natural fold fitted using this method.
6.1. Tangential longitudinal strain and flexural flow
In these basic mechanisms, the guideline, consid-
ered originally horizontal and midway between the
layer boundaries, does not undergo length changes
during folding (Fig. 21). After deformation, this
guideline (L) becomes the arc l, which will be
described mathematically by a conic with parameters
(e, a). LetQ(X) be a point on the original guideline L, at
a distance X from O. The deformed image q(X) of this
point is fully determined, since it corresponds to the
point of the conic l at a distance X from O measured
along the curve. N and n(X) are the unit normal vectors
to the guideline at the points Q(X) of L and q(X) of l.
We will now describe the kinematical folding
mechanism using the geometric elements above. We
will use the common notation in affine geometry
according to which, given a point A and a vector v, the
bsumQ A+v gives the point B, so that v ¼ ABY.
Tangential longitudinal strain is a kinematical
folding mechanism defined by the following con-
ditions (Bobillo-Ares et al., 2000):
1. Existence in the layer of a surface, named dneutralsurfaceT, defined by points without finite strain.
This surface is located inside the folded layer and
separates an outer zone with tangential stretching
from an inner zone with tangential shortening. The
neutral surface is referred as dneutral lineT on the
fold profile. A corollary of this condition is that
lines originally normal to the neutral line remain
normal to it as folding progresses. According to this
condition, the image q(X0) of point Q(X0), at a
distance X0 from O, is found assuming that the
length of the arc Oq is also X0 (Fig. 22).
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164154
2. Straight lines originally normal to the neutral line
remain straight lines after folding. Hence, points
not located on the guideline remain always in the
same line normal to the guideline. Thus, the
images of points of the straight line PQ, normal
to L at Q, are on the straight line pq, normal to l at
q (Fig. 22).
3. The area remains constant during folding in all the
parts of the folded layer profile.
Let P be a point to a distance Y from Q; then:
P ¼ Qþ YN; ð38Þ
The image p of P is at a distance h from point q.
That is to say:
p ¼ qþ hn Xð Þ; ð39Þ
(b)
(c)
(a)
DLT
Fig. 23. Folding of a grid (a) by tangencial longitudinal strain (b and c)
the folded guideline are conics with an aspect ratio of 0.8, and eccen
(hyperbola arc) for panels (c) and (e). In the initial grid, circular m
configuration.
Assuming the condition of area preservation, the h
value is given by (Bobillo-Ares et al., 2000):
h ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2Yj Xð Þ
p� 1
j Xð Þ ð40Þ
where j is the curvature of the conic section l at
point q.
The modelling of folds formed by tangential
longitudinal strain can be easily carried out with
FoldModeler superposing a single step of this
mechanism to the initial configuration. The incre-
ments in the normalized amplitude of the limb, and
the eccentricity of the conic that describes the
guideline, must be also indicated. Two examples of
folds produced by this mechanism with different
forms of the guideline are shown in Fig. 23b and c.
(d)
(e)
FF
and flexural flow (d and e). The functions describing the form of
tricity values of 0.5 (ellipse arc) for panels (b) and (d), and 1.3
arkers have been drawn to visualise the strain in the folded
OQ(X )
q(X )
NL
l
O'
0
n(X )0
P
p
L'
l'
h
rq(X )1
Fig. 24. Images q(X0) and p of two points Q(X0) and P folded by
flexural flow, and located on the guideline and outside of it
respectively. L and l, original and deformed guidelines; LV and lV,lines parallel to the guideline in the original and deformed
configuration.
Angle
Ort
ho
go
nal
th
ickn
ess
t'α
0.00 10 20 30 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Fig. 25. Curves t Va�a for flattened parallel folds for sever
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164 155
In flexural flow the displacements of points are
produced along lines parallel to the boundaries of the
layer. This displacement involves two basic proper-
ties: (a) the orthogonal thickness is constant along the
folded layer (folds of class 1B) and equals the
original thickness; and (b) the arc length measured
along the layer boundaries or along lines parallel to
them is maintained constant during folding. A
corollary of these properties is that the area through
the folded layer profile remains constant during
folding. The strain in the layer is a heterogeneous
simple shear with variable shear direction parallel to
the layer boundaries, so that the shear sense is
different in both fold limbs. Locally, the deformation
can be solved in a homogeneous simple shear plus a
rotation.
The above conditions are sufficient to determine
the image of any point of the initial configuration.
Consider point P of Fig. 24; its image p is obtained
of dip α
50 60 70 80 90
λ2
λ1= 0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
al values offfiffiffiffiffiffiffiffiffiffiffiffik2=k1
p(after Ramsay, 1967, p. 413).
(a)
(b)
(c)
(d)
(e)
Class 1A
Class 1B (parallel)
Class 1C
Class 2 (similar)
Plotting scheme
t 1/t
t1/t
t1/t
t 1/t
1/t
α
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164156
by finding the point q(X1) for which the arc length
O Vp along the curve l V is X0. That is to say:
p ¼ q X1ð Þ þ hn X1ð Þ: ð41Þ
Details of the calculation of X1 can be found in
Bastida et al. (2003).
Modelling of folds formed by flexural flow is
similar to that of the tangential longitudinal strain
folds described above, and requires the definition of
the variations in shape involved in the corresponding
folding step. Two examples of folds produced by this
mechanism with different guideline forms are shown
in Fig. 23d and e.
6.2. Pure deformation (layer shortening and fold
flattening)
This mechanism represents a homogeneous
strain superposed to the fold. According to the
orientation of this strain and its timing in relation
to folding, three types of pure deformation deserve
our attention:
– Layer shortening; this occurs in the early stages of
folding and its maximum shortening direction is
perpendicular to the axial surface.
– Fold flattening; this occurs in the late stages of
folding and its maximum shortening direction is
perpendicular to the axial surface.
– Shortening with a maximum shortening direction
contained in the axial surface (or perpendicular to
the bedding when this shortening is previous to the
folding). This is not generally associated with
progressive folding, and is a response to a different
process. Diagenetic compaction is a particular type
of this shortening.
Class 3 (f)
t1/t
Fig. 26. Plot in polar coordinates of the inverse orthogonal thickness
(1/ta) versus the dip a (after Lisle, 1997): (a) plot method and (b–f)
result of the plot for different classes of folds.
Several methods have been developed to determine
the geometry of flattened folds and the amount of
flattening. In the case of flattened parallel folds,
Ramsay (1967, pp. 411–415) deduced that the
orthogonal thickness is given by:
taV ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos2a þ k2
k1sin2a
s; ð42Þ
where k1 and k2 are the principal quadratic elonga-
tions of the homogeneous strain associated with
flattening. Function (42) was represented graphically
by Ramsay (1967) for several values offfiffiffiffiffiffiffiffiffiffiffiffik2=k1
p(Fig.
25). These results show that parallel folds become
class 1C folds when they are flattened, and come
closer to the similar fold geometry as the ratio
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164 157
ffiffiffiffiffiffiffiffiffiffiffiffik2=k1
pdecreases, that is, as the ellipticity of the
flattening strain ellipse increases. Curves of Fig. 25
become straight lines when they are plotted in the
diagram of t Va2 against sin2a. Hence Eq. (42) is
equivalent to Eq. (36) when m=1�(k2/k1). By
changing k1 by k2 and vice versa in Eq. (42), the
equation that describes the orthogonal thickness for
compacted parallel folds is obtained, and this is
equivalent to the Eq. (36) when m=1�(k1/k2). Their
representative curves in the diagram of t Va2 against
sin2a are straight lines inside the field of class 1A
folds.
An ingenious method to measure the amount of
flattening and classify folds was developed by Lisle
(1997), who for this purpose used the polar plot of the
coordinates of the inverse orthogonal thickness (1/taor 1/t Va) versus the dip a (Fig. 26). By this method,
class 1B (parallel) folds are represented by circles,
class 1A folds obtained by compaction of parallel
folds are represented by ellipses with the major axis
1A
1C
3A
3C
3
3
2.8
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
F = 1
0.8
0.6
0.4
56798-9-6-4
-3
-2
F = -1
-0.5
Fig. 27. Curves resulting from the polar plot of inverse normalized thickne
index F (after Lisle, 1997).
parallel to the polar axis, class 1C folds obtained by
flattening of parallel folds are represented by ellipses
with the major axis perpendicular to the polar axis,
and class 2 (similar) folds are represented by a pair of
straight lines perpendicular to the polar axis. In
addition, Lisle defined pure class 3 folds (or class
3B folds) as those represented by an equilateral
hyperbola. Flattening of these folds yields folds
represented by a hyperbola whose asymptotes have
a slope with absolute value greater than 1 (class 3A
folds), and compaction of these folds yields folds
represented by a hyperbola whose asymptotes have a
slope with absolute value lower than 1 (class 3C
folds). This division of the class 3 folds is different to
that made by Zagorcev (1993). Lisle (1997) deduced
an expression for the orthogonal thickness of all these
folds given by:
t V2 ¼ sin2a1
FffiffiffiffiffiffiF2
p � 1
� �þ 1; ð43Þ
3A
3C
-0.5
F = -1
-2
-3
-4-6-9897654.5
4
.5
ss (1/t Va) versus dip a for folds with different values of the flattening
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164158
where F is a flattening index defined as the ratio
between the conic axes perpendicular and parallel to
the polar axis. In the case of the hyperbola, it is
(a) (b
(f)
(c)
(h)
φ
α
(d)
20406080
100120140160180
φ 11111
10 20 30 40 50 60 70 80 90
p1 =
(g)
(i)
iniG
blockblockblock
Asp
20 40 60 80
0.2
0.4
0.6
0.8
1
1.2
1.4
α
t'α
ec
1
2
3
4
Rs
a10 20 30 40 50 60 70 80 90
1.52
2.53
3.54
4.55
Rs
assumed that their axes have opposite signs, so that F
is negative. The different fold classes are characte-
rized by the following F values: class 1A, 1NFN0;
)
(e)
204060800020406080
α10 20 30 40 50 60 70 80 90
{0, 1, 1, 100000}
Theoretical Bulk Shortening = 55.98%
rid = [p1, 1/18, 1/18, 18, 1, 1]
1 = {1, {3, 0.86, 1}}2 = {1, {1, 0.73, 0.3}, {2, 0.04, 0}}3 = {1, {3, 0.66, 1}}
ect RatioNatural FoldTheoretical Fold
1.768 1.766
0.269 0.270
(y /x )f f
t /y0 oa
centricity (e) 1.063 1.090
α10 20 30 40 50 60 70 80 90
.52
.53
.54
.55
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164 159
class 1B, F=1; class 1C, lNFN1; class 2, F=l;
class 3A, �1NFN�l; class 3B, F=�1; class 3C,
0NFN�1. Curves with different F values are repre-
sented in Fig. 27. Eq. (43) is equivalent to Eq. (36)
with m ¼ 1� 1=FffiffiffiffiffiffiF2
p� �.
Modelling of folds involving pure deformation
requires, besides the step with this mechanism, the
application of at least one step producing a folded
form. In a step of pure deformation, the change in the
guideline does not need a specific description, since it
is defined by the general parameters that describe the
folding step: the stretch in the x-axis direction
(generallyffiffiffiffiffik2
p), and the area change with strain
(final area/initial area).
6.3. Folds produced by a combination of kinematical
mechanisms
Folding by a complex superposition of kinematical
mechanisms can be modelled by an adequate
sequence of folding steps. To organize the informa-
tion, the folding steps are grouped in blocks; each
block consists of a sequence of folding steps and a
natural number indicating the times that the sequence
has to be executed; a complete program to produce a
theoretical fold is defined by a sequence of blocks,
which will be executed in the order indicated.
Models with a combination of mechanisms can be
used to fit natural folds and to analyse their kine-
matics. This is done by searching, by a trial and error
method, for a theoretical fold with the same geo-
metrical characteristics of a given natural fold. To
conduct this analysis, it is essential to obtain from the
natural fold the normalized amplitude ( y0/x0), the
ratio between the layer thickness at the point with dip
zero and the amplitude of the outer arc (t0/yoa), the
parameters of the conic that will describe the guide-
line (e, a), the orientation of cleavage vs. dip (/–acurves), Ramsay’s classification, and if possible, some
strain measurements along the folded layer (Rs–a
Fig. 28. Example of a good fit of a natural fold with a theoretical one form
dFoldModelerT. (a) Antiform developed in a lower Cambrian sandstone ne
characterising the theoretical fold (p1, iniGrid, block1, block2, block3),
corresponding parameters of the natural fold (t0/yoa is the ratio between th
classification of the theoretical folded layer (line) and the natural fold
configuration of the theoretical layer showing the strain ellipses and shadin
diagrams for the outer and inner arcs, respectively, showing the data obtain
(h and i), Rs–a diagrams for the outer and inner arcs, respectively, showi
curves). Since the location of the guideline is not
known in the natural fold, the line located in the
middle of the layer is taken as guideline.
An example of a good fit of a natural fold by a
theoretical fold is shown in Fig. 28. Before carrying out
the fit, apart from obtaining all the possible geometrical
information from the natural fold, it is necessary to give
an approximate value to the initial thickness of the
layer. This value can be modified during the modelling
by a trial and error method in order to improve the fits.
In the case analyzed, the initial thickness/length ratio is
0.111, which is the thickest layer for which we can
achieve good fits. Good fits other than this are also
possible for any initial thickness/length ratio of V0.111.Fig. 28 shows a sequence of three simple blocks that
fits the natural fold. This sequence is formed by an
initial pure deformation withffiffiffiffiffikx
p¼
ffiffiffiffiffik2
p¼ 0:86 with-
out area change, that is,ffiffiffiffiffiffiffiffiffiffiffiffik1=k2
p¼ 1:35; a folding step
of tangential longitudinal strain that gives an y0/x0 of
0.73; a folding step of flexural flow giving rise to an y0/
x0 increase of 0.04; and finally a step of pure
deformation withffiffiffiffiffikx
p¼
ffiffiffiffiffik2
p¼ 0:66 without area
changeffiffiffiffiffiffiffiffiffiffiffiffik1=k2
p¼ 2:30
� �. The final shape of the
folded layer is fitted introducing variations in eccen-
tricity e and normalized amplitude h in the fit conic
inside the blocks with tangential longitudinal strain or
flexural flow, until the shape of the natural folded layer
is obtained. Nevertheless, it must be remembered that
apart from these mechanisms, pure deformation also
produces variation in e and h. For the initial thickness/
length ratio of 0.111, several good fits can be achieved,
apart from the one shown in Fig. 28, and all of them
have similar amounts of the mechanisms involved in
the fits. If a thinner layer is considered, the amounts of
the mechanisms are only slightly different, except for
the initial layer shortening. Strain data, including area
change, are needed to reduce the field of possible fits
and establish the correct thickness/length ratio of the
initial layer. As regards the timing distribution of the
different mechanisms, in all the fits, an initial step with
ed by the superposition of strain patterns modelled by the program
ar Cudillero (Spain) whose right limb has been fitted; (b) Input data
and comparison of some outputs of the theoretical fold with the
e hinge thickness and the amplitude of the outer arc); (c) Ramsay’s
(dots); (d) initial configuration of the theoretical layer; (e) folded
g proportional to the aspect ratio of the strain ellipses; (f and g) /–aed from the theoretical folded layer (line) and the natural fold (dots);
ng the strain pattern predicted for the theoretical fold.
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164160
pure deformation is necessary, and this may take place
simultaneously with some tangential longitudinal
strain. Then comes a second stage of tangential
longitudinal strain, with or without a small amount of
flexural flow, and finally fold flattening.
7. Conclusions
The geometrical description of folds is mainly used
in regional geological studies and in the detailed study
of specific structures. In the first case, the use of
qualitative terms is common, whereas in the second
case quantitative information is required. Neverthe-
less, both approaches are closely related. The methods
used to describe folds quantitatively can be divided
into two types: those based on parameters defining a
non-functional approach to fold geometry, and those
based on the use of a function that describes the
variation of the fold geometry. The kinematical study
of folding requires methods of the latter type because
they allow the use of folds as mathematical tools to
analyse strain through the folded layers. In the
geometrical description of folds, two aspects must
be considered separately: the analysis of folded
surfaces, and the analysis of folded layers.
Two types of folded surfaces can be distinguished
in the geometrical analysis: cylindrical and non-
cylindrical. The study of the former only requires 2-
D analysis, whereas the analysis of the latter requires
3-D analysis. 2-D analysis is made on a section
perpendicular to the fold axis (folded surface profile).
The curvature of the folded surface profile and the
reference line, or bguidelineQ, are essential parameters
in the kinematical analysis of folding. Thus, these
parameters and their variation along the fold must be
taken into account in order to define functions that
describe the geometry of the fold profile. These
functions must be derivable at least twice, and in
order to make the geometrical and kinematical
analysis of folds feasible, they must be defined by a
small number of parameters.
Several families of functions can be used to
describe alloclinal fold forms: functions defined by
Fourier series, functions of the form y=y0 f(x)n
(nN0), subellipses and superellipses, cubic Bezier
curves, and conic sections, among others. Conic
sections accomplish the conditions mentioned above
and are defined by two parameters. In addition,
conics are transformed into conics of the same type
by affine transformations (homogeneous strain), and
therefore proved to be the most useful functions for
the kinematical analysis of folding. Nevertheless, all
the functions mentioned that can be defined by two
parameters are suitable for the classification of folded
surface profiles. One of these parameters represents
the normalized fold amplitude (h=height/width ratio
of the fold, represented in the y-axis), whereas the
other is more concerned with the fold shape (x-axis).
The different shape parameters can be linked by the
normalized area Að Þ enclosed by the concave part of
the limb profile curve. Hence, this area becomes a
parameter of general value for the classification of
folded surface profiles. The geometry of the profile
of isoclinal folds can be approached using functions
made up of a quarter of an ellipse and a line segment,
or using superellipses when isoclinal double-hinge
folds are considered. These functions allow a good fit
of isoclinal folds up to the box shape.
Asymmetry of the folded surface profile can be
quantified using two parameters: shape asymmetry
(Sa), which measures the y0 ratio between the long
and the short limb, and amplitude asymmetry (Aa),
which measures the normalized area ratio between the
long and the short limb. The plot of these parameters
in a diagram of Sa against Aa for all the folds of a
specific set allows the asymmetry variation of these
folds to be visualized.
The functional 3-D description of folded surfaces is
difficult and it is not well-developed at present.
Several parameters have been used for classification
or description of folded surfaces, these being mainly
the hinge angle and the degree of planarity and non-
cylindrism (Williams and Chapman, 1979), or the
Gaussian curvature (Suppe, 1985).
The most detailed and useful description of folded
layer geometry for kinematical folding analysis is
based on the Ramsay classification, which involves a
functional relationship between orthogonal thickness
and dip, or alternatively on Hudleston classification,
which uses the variation of the angle between the
isogon and the normal to the tangents drawn to either
fold surface at a specific dip, as a function of the layer
dip. Simplifications of Ramsay classification to
represent a fold limb by a point on a diagram instead
of by a curve, or to describe a fold by a single
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164 161
parameter, have been made by several authors.
Hudleston (1973b) used the intercept value of the
curve of the square of the orthogonal thickness versus
the square of the dip angle cosine to characterize
folds. Bastida (1993) defined two parameters based on
the slope variation of the curve of orthogonal thick-
ness versus dip. Lisle (1997) defined a flattening
index from a plot in polar coordinates of the inverse
orthogonal thickness versus the dip. These simplifi-
cation methods involve an information loss with
respect to the classic Ramsay or Hudleston classi-
fications, but they are useful in regional descriptions
involving a large number of folds.
The cleavage distribution on a folded layer profile
bears a close relationship to the associated strain
distribution. Hence, knowledge of the cleavage dis-
tribution is essential in the kinematical analysis of
folding. A simple method to describe this pattern is to
obtain the functional relationship between cleavage
dip and bedding dip through the fold limb profile, by
measurements in the field or from photographs.
The analysis of kinematical folding mechanisms by
comparison of natural folds with mathematical models
requires the application of the geometrical methods
detailed above, and in particular of those involving a
functional description of folds. Hence, measurements
made in natural folds must be used as inputs in
theoretical models or as data for comparison with the
geometrical outputs from the models. The main input
data are the parameters that describe the geometry of
the folded guideline, which must be preferably fitted
by conic sections from photographs of the natural fold
profile. The eccentricity and scale factor values
obtained for the conic must coincide with those of
the guideline of the modelled fold. The modelled fold
offers complete geometrical information that must be
checked in detail with data from the natural fold, such
as the cleavage and strain distributions, and the
Ramsay classification of the folded layer. These data
must be fitted by the theoretical models to reasonably
infer the kinematical mechanisms that operated in the
natural fold.
Acknowledgements
The present paper has been supported by the
BTE2002-00187 project funded by the bMinisterio
Espanol de Ciencia y TecnologıaQ and the bFondoEuropeo de Desarrollo Regional (FEDER)Q. We are
grateful to J. Poblet and an anonymous referee for
many valuable suggestions that notably improved the
paper. We thank Robin Walker for his revision of the
English of the manuscript.
Appendix A. Middle point method to obtain a
conic section that fits a fold limb profile
We will describe here how to obtain the parameters
e and a of the conic of the family defined by Eq. (15)
that passes through the points (x0/2, yM) and (x0, y0).
The algorithm to calculate these parameters is
synthesized in the flow chart of Fig. A1. Parameters
e and a can be obtained solving the system:
yM ¼ f e; a; x0=2ð Þy0 ¼ f e; a; x0ð Þ
�: ðA1Þ
Making K=y0/yM, we obtain the equivalent system:
K ¼ f e; a; x0ð Þf e; a; x0=2ð Þ
y0 ¼ f e; a; x0ð Þg: ðA2Þ
For the parabola (e=1), the system (A2) takes the
simple form:
K ¼ 4; y0 ¼x202a
; ðA3Þ
which leads to the solution:
e ¼ 1; a ¼ x202y0
: ðA4Þ
For e p1, the system (A2) takes the form:
K ¼ g Að Þ; ðA5Þ
y0 ¼1
E1�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4A
p� �; ðA6Þ
with
g Að Þ ¼ 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4A
p
1�ffiffiffiffiffiffiffiffiffiffiffiffi1� A
p ; ðA7Þ
A ¼ 1
41� e2 x0
a
� �2; ðA8Þ
START
Data entry
k = 4?
No
NoNo conic
END
Yes
Calculate
Yes
Calculate
Calculate
Output
D>1?
END
Recumbent ellipseε
Yes
No
Vertical ellipseHyperbola
e
END
=x0, y0, ky0yM
END
2
1
1- 3
A =4k(4-5k+k2)16-8k2+k 4
E =1- 1- 4A
y0
a =E x0
2
4A
y0D = ( ) (1- 1- 4A)
4Ax0
22
ε = D-1
e = 1-D
Parabola
e = 1
a =x0
2y0
?2<k<
2
Fig. A1. Chart of the algorithm used to calculate the parameters e
and a of the conic that gives the best fit for a fold limb profile by the
middle point method.
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164162
E ¼ 1� e2
a: ðA9Þ
g(A) is an increasing function in its definition domain,
�lbAV1/4. Taking into account that
limAY�l
g Að Þ ¼ 2 and g1
4
� �¼ 1
1�ffiffiffi3
p
2
; ðA10Þ
the range of values permitted for K=y0/yM is
2VKV1
1�ffiffiffi3
p
2
: ðA11Þ
In this range of K values, it is easy obtained that
A ¼ 4K 4� 5K þ K2ð Þ16� 8K2 þ K4
: ðA12Þ
With this A value, the E value is obtained from Eq.
(A6):
E ¼ 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4A
p
y0: ðA13Þ
From the E and A values [Eqs. (A12) and (A13)], e
and a are obtained solving the system formed by Eqs.
(A8) and (A9). To do this, by making:
D ¼ x0
y0
� �2 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4A
p 24A
; ðA14Þ
the e values are obtained:
e ¼ffiffiffiffiffiffiffiffiffiffiffiffi1� D
pfor DV1;
e ¼ffiffiffiffiffiffiffiffiffiffiffiffiD� 1
pfor DN1 e ¼ eið Þ: ðA15Þ
In both cases,
a ¼ Ex204A
: ðA16Þ
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Fernando Bastida received his degree in
geology in 1973 from the University of
Oviedo (Spain), where he also completed
an MSc in 1974 and a PhD in 1980. He is
currently a permanent professor in the
Department of Geology at the University
of Oviedo. His main research interests are
the structure and tectonics of the Variscan
belt in Spain, where he works on the
tectono-thermal evolution of the foreland
thrust and fold belt, and the study of several
structure types, mainly folds, tectonic foliations, and microstruc-
tures. He is currently especially interested in fold modelling by
superposition of strain patterns, and the analysis of kinematical
folding mechanisms in natural folds by their fit and comparison
with folds produced theoretically.
Noel Canto Toimil was born in Langreo
(Spain), and after obtaining his degree in
geology in 1999, he received his MSc
from University of Oviedo, where he is
currently working as assistant lecturer.
Since his graduation, he has been research-
ing into folding geometry by analysing
numerical and experimental models and
natural folds. He also was at ETH of
Zurich for several months in order to
collect data of natural folds in the Alps.
interested in the mat
nisms in natural fold
F. Bastida et al. / Earth-Science Reviews 70 (2005) 129–164164
Jesus Aller is a permanent professor of
Geology at the University of Oviedo, Spain,
where he earned a degree in geology in
1978 and a PhD in structural geology in
1984. His early work was in structural
geology and tectonics of the Iberian pen-
insula. At present, he is especially inter-
ested in the analysis of kinematical
deformation mechanisms in natural folds,
using models of theoretical folds produced
by superposition of strain patterns.
At present, he is finishing his PhD on kinematical mechanisms
in competent layers.
Nilo Bobillo-Ares is a permanent profes-
sor of Mathematics at the University of
Oviedo (Spain). He earned a degree in
Telecommunication Engineering in 1978
from the Polytechnic University of Madrid
(Spain), where he also completed an MSc
in 1979 and a PhD in 1990. His early
work was in mechanics of continuous
media (free boundary problems in Nav-
ier–Strokes equations using Lagrangian
coordinates). At present, he is especially
hematical analysis of deformation mecha-
s.