Introduction

Superposition of folding during either progressive displacement or different phases of deformation result in three-dimensional refold structures that are exposed on two-dimensional sections as interference patterns (Ramsay, 1962, 1967). In order to understand the complex geometry of these structures several models have been presented: Earlier models used card decks, which were cut into the shape of the first fold and than sheared parallel to the second fold axial plane (e.g. Carey, 1962, O'Driscoll, 1962, Brown, 1967). Scaled physical models showed the influence of layer buckling, competence contrast between layers and the influence of the initial fold geometry on the formation of refold structures (e.g. Reynolds and Holmes, 1954, Gosh and Ramberg, 1968, Watkinson, 1981, Odonne and Vialon, 1987, Grujic, 1993, Johns and Mosher, 1995). Kinematic forward modelling computer programs have been successfully applied to simulate three-dimensional refolding and to study two-dimensional interference patterns on arbitrary oriented sections through the modelled structures (e.g. Thiessen, 1986, Perrin et al. 1988, Jessell and Valenta, 1996; Vacas Peña, 2000, Ramsay and Lisle, 2000; Moore and Johnson, 2001). Although most of these programs have the same limitations as the card deck models, that folds are assumed to be (cylindrical) similar shear folds with passive initial layering, these studies have significantly contributed to the understanding of the great variety of interference structures and the classification of refolds.

Computer animations of the development of geological structures during progressive deformation are a powerful tool both in the advancement of understanding of processes and geological education. With the help of a computer program for modelling three-dimensional refold structures and their two-dimensional interference patterns, this contribution provides computer animations, which effectively improve the understanding of natural fold superposition and are therefore ideally suited for teaching in electronic classrooms or for use in online courses on the Internet. Furthermore the animations suggest that the Type 0 refold, which actually produces no interference patterns, has to be divided in three different classes, which theoretically exist but which are probably difficult to distinguish in the field. Because the presented study is based on the kinematic forward modelling software Noddy (Jessell and Valenta, 1996) the mechanical influence of contrasting rheologies is ignored and discussed elsewhere (e.g. Johns and Mosher, 1995 and references cited therein). Despite these limitations the presented results have many geometric similarities with natural refold structures justifying the use of kinematic modelling in exploring the complex shapes and interference patterns of fold superposition (Ramsay and Lisle, 2000).

Mathematical background

The superposition of successive three-dimensional heterogeneous deformations can be expressed by a single Lagrangian equation triplet describing the superposed heterogeneous finite deformation field. This superposition is not commutative and the resulting finite deformation will differ if the order of superposition is reversed.

A deformation, which describes the transformation of an initial coordinate (x, y) to the another coordinate (x₁, y₁):

(1)

is superposed by another transformation:

(2)

deforming (x₁, y₁) to (x₂, y₂). The total finite displacement field combining Eq. (1 ) and (2) is given by:

(3)

By differentiating Eq. (3) it is possible to obtain the nine components of the three-dimensional displacement gradient tensor D in Lagrangian form:

(4)

Most of the kinematic forward modelling programs use for the functions ƒ in Eq. 1-3 a sinusoid function or Fourier series describing similar folds. Although these models do not consider layer competence contrasts that might influence the fold geometry by progressive amplification and deamplification of the layers, similar folds are mathematically simple to implement in kinematic models and a good approximation to study the geometry of natural interference structures. A simplest form of a similar fold with a vertical axial surface parallel to the xz coordinate plane with sinusoidal cross-sectional form can be mathematically described by a heterogeneous displacement:

(5)

where a is the shear amplitude of the fold. Therefore the displacement tensor D in Lagrangian form is:

(6)

The computer program Noddy (Jessell and Valenta, 1996) uses a more developed mathematical description of similar-type folds, where the heterogeneous displacement is defined by:

(7)

Parameter c controls the fold cylindricity and w is the fold wavelength. The displacement tensor D in Lagrangian form is obtained by differentiating Eq. 7:

(8)

Any spatial orientation of the modelled fold for subsequent superposition of another fold can be obtained by a displacement tensor R in Lagrangian form defining the rotation around a unit vector v by an angle of a.

(9)

Results can be either displayed by plotting a particular folded and refolded initial surface in three-dimensional space (structure-plot) or by visualizing fold interference patterns on arbitrary cross sections through the structure (pattern-plot).

Classification of refold structures

The complex geometries resulting from superposition of folding have been classified by characteristic fold interference patterns, which appear on two-dimensional sections through the three-dimensional refold structures (Ramsay, 1967, 1987, Thiessen and Means, 1980, Thiessen, 1986). Ramsay (1967) presented a classification of refold structures, which combines two-dimensional interference patterns and three-dimensional superposition geometries, suggesting that individual patterns are indicative for specific spatial angular relationships between the two folding events. These classes (Type 0 - 3) are distinguished by whether the initial fold axes and/or the initial fold axial planes are deformed during the superposed generation of folding (Fig. 1). The angle between the first (ƒ₁) and second fold axis (b₂) is called a whereas b is the angle between the pole to the first axial plane (c₁) and the second slip or transport direction of heterogeneous displacement (a₂).

Figure 1 Description of kinematic axes of the initial and superposing fold and their relative spatial orientation (after Ramsay, 1967 and Thiessen and Means, 1980). Initial fold: fold axes = ƒ1; normal to axial plane = c1; direction of heterogeneous shear displacement = d1. Superposing fold: fold axes = b2; normal to axial plane = c2; direction of heterogeneous shear displacement = a2. Angles between ƒ1 and b2 = a; between c1 and a2 = b; between ƒ1 and c2 = g; between c1 and c2 = d.

Using a kinematic modelling computer program Thiessen and Means (1980) introduced g, which is the angle between ƒ₁ and the pole to the second axial plane (c₂), d₁, which represent the normal to a plane containing c₁ and ƒ₁, and d, which is the angle between c₁ and c₂. Various orientations of axes of the first (ƒ₁, d₁, c₁) relative to axes of the second fold (b₂, a₂, c₂) can be represented by points in a cubic volume with a, b and g plotted along its edges. Because the angles are not independent, not all combinations of a, b and g are possible. Based on this orientation volume diagram and following the terminology of Ramsay (1967), Thiessen and Means (1980) concluded that b and g are appropriate angles for refold classification but that only few interference patterns are really diagnostic for superposition geometries. A single interference pattern can be produced by an infinite number of refold geometries. Furthermore, they concluded that Type 0 refolds do not create in a strict sense interference patterns and that two different geometries of Type 0 refolds exist.

The terminology used in the presented study follows the terminology of Ramsay (1967), which is well established in structural geology textbooks, and extends the ideas of Thiessen and Means (1980) in following points: (i) The terminology "Type 0-3" is used for end members of three-dimensional refold geometries and not for two-dimensional interference patterns, which have a larger variability (compare fig. 10 in Thiessen, 1986). (ii) Type 0 refolds are further subdivided in three geometrically individual end-members. Because these classes can be simply derived from the refolds Type 1-3 by rotation of the second fold by 90° around the b₂ axis, Type 0 is subdivided in Type 0₁, Type 0₂ and Type 0₃. Directions ƒ₁, d₁, c₁ of the initial fold and b₂, a₂, c₂ of the superposing fold are called in the following (orthogonal) kinematic axes of the folds (Fig. 1).

Figure 2 Classification of refold structures showing a) Type 1, b) Type 01, c) Type 2, d) Type 02, e) Type 3 and f) Type 03. and corresponding (i) physical orientation of the first and the second fold generation; (ii) resulting, idealized three-dimensional refold structure (iii) orientation of the kinematic axes of the initial and the superposing fold and (iv) stereographic projection of the orientation of the kinematic axes of the initial and the superposing fold (refold-stereoplot). Click to view larger version.

Figure 2 shows the suggested new classification plotting for each end-member type: (i) The orientation of the first and the second fold generation; (ii) The idealized resulting three-dimensional refold structure; (iii) The orientation of the orthogonal kinematic axes of the first and the second fold generation; (iv) A stereographic projection of the orientation of the kinematic axes of the first and the second fold generation. Concluding, the following six end-member refold types, with an angular relationship of the initial and superposing kinematic axes of either 0° or 90° can be distinguished:

Type 1 refold (a and b = 90°, g = 0°, Fig. 2a): The initial axial plane remains planar, but the fold axes of the superposed fold are deformed. This causes a strong undulation of the hinges of the initial folds resulting in culmination domes and depression basins, where each depression is surrounded by four culminations and each culmination is surrounded by four depressions resembling the shape of an egg-carton. Two-dimensional sections through the refold structures are often characterized by dome and basin interference patterns (Thiessen, 1986). If a < 90° the domes and basins are arranged en echelon (O'Driscoll, 1962).

Type 2 refold (a = 90°, b and g = 0°, Fig. 2c): The initial axial plane and the initial fold axis are deformed. If the refold structure is progressively unroofed perpendicular to b₂,interference patterns with circular forms, rounded triangulars, crescent shapes and typical dome-crescent-mushroom patterns characterize the sections (Ramsay and Huber, 1987). However, oblique sections, especially if the refold structures deviate from the end-member orientation, show a great variability of complex interference patterns (Thiessen, 1986).

Type 3 refold (g = 90°, a, b = 0°, Fig. 2e): The initial axial plane is deformed but the initial fold hinges are not bent by superposing folding. Cross sections parallel to the fold axes (ƒ₁ and b₂) will not develop a complex interference pattern but show parallel, straight lines. However, cross sections normal to the fold axes will show complex convergent-divergent or hook shaped interference patterns (Thiessen, 1986).

Type 0 refold structures always develop when b and g = 90°. On two-dimensional sections through the refold structures no characteristic interference pattern is developed resembling a cross section through a single-phase fold. Although two different end-member geometries of Type 0 refolds have already been noted by Thiessen and Means (1980), these type of refold structures have attracted little attention of structural geologists mainly because their differences would be difficult to observe in the field. However, if the initial fold generation is cut by a marker plane (e.g. a dyke or vein) at a high angle toƒ₁, differences become obvious and three classes have to be distinguished. Note that additionally d, the angle between the normals to the axial planes (c₁ and c₂), is needed for this discrimination (Table 1):

	ƒ1| b2	c1|a2	ƒ1|c2	c1|c2
Type 1	90°	90°	0°	90°
Type 01	90°	90°	90°	90°
Type2	90°	0°	0°	90°
Type 02	90°	90°	90°	0°
Type 3	0°	0°	90°	90°
Type 03	0°	90°	90°	0°

Table 1 Angles between the kinematic axis of the initial and the superposing fold and the corresponding end-members.

Type 0₁ refold (a, b, g, d = 90°, Fig. 2b): The shearing direction of the superposing fold is parallel to ƒ₁ but axial planes c₁ and c₂ are perpendicular to each other. The resulting refold structure is identical to the shape of the initial fold. However, a planar passive marker normal to ƒ₁ clearly demonstrates the superposition of heterogeneous deformation. Although cross sections normal to ƒ₁ shows a simple section through a cylindrical fold the deformation is markedly non-plane strain. By rotation of the superposing fold around b₁, end-member structures Type 1 and Type 0₁ can be continuously transformed into each other (Fig. 4a and b).

Type 0₂ refold (a, b, g = 90°, d = 0°, Fig. 2d): The shearing direction of the superposing fold is parallel to ƒ₁ and the axial planes c₁ and c₂ are parallel to each other. The resulting refold structure is again identical to the shape of the initial fold but a planar passive marker normal to ƒ₁ reveals the superposition of heterogeneous deformation. Again the deformation in a two-dimensional section normal to ƒ₁ is markedly non-plane strain. By rotation of the superposing fold around b₂, end-member structures Type 2 and Type 0₂ can be continuously transformed into each other (Fig. 5a and b).

Type 0₃ refold (b, g = 90°, a, d = 0°, Fig. 2f): Axial planes and fold axes of the initial and the superposing fold are parallel to each other. However, the resulting refold structure is not identical in shape of the first fold but may be amplified, overprinted or theoretically cancelled. In case of out-of-phase relationship of the waveform of the superposing fold, generation of second order folds on the first fold generation may occur (polyharmonic folds). However, the deformation in a section normal to ƒ₁ is plane strain. This Type 0₃ refold structure correspond to the Type 0 redundant superposition (Ramsay, 1967; Thiessen and Means, 1980; Ramsay and Huber, 1987). By rotation of the superposing fold around b₂, end-member structures Type 3 and Type 0₃ can be continuously transformed into each other (Fig. 6a and b).