Fold geometry: a basis for their kinematical analysis

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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: [email protected] (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.


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


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.


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.


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.


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).


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).


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


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.


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.


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.


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.


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.


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.


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)


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).


(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.


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).


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).


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.


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


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.


– 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).


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


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


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


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


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


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.


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


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.


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


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.

Fold geometry: a basis for their kinematical analysis

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