Quantification of the geometrical parameters of non-cylindrical folds

Journal of Structural Geology 100 (2017) 120e129

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Journal of Structural Geology journal homepage: www.elsevier.com/locate/jsg

Quantification of the geometrical parameters of non-cylindrical folds G. Zulauf*, J. Zulauf, H. Maul €t Frankfurt a.M., Altenho €ferallee 1, D-60438, Frankfurt a.M., Germany Institut für Geowissenschaften, Universita

a r t i c l e i n f o

a b s t r a c t

Article history: Received 20 February 2017 Received in revised form 23 May 2017 Accepted 1 June 2017 Available online 2 June 2017

The geometrical parameters of natural folds are used by structural geologists to estimate finite strain and rheological properties of deformed rocks. The relation between geometry and rheology is well understood in cases of cylindrical folds, but is still limited for non-cylindrical folds, although the latter are frequent in nature. The sparsity of quantitative geometrical data of non-cylindrical folds can be explained by the small number of 3D exposures and by the lack of robust methods to quantify their geometrical parameters in 3D space. We present a new workflow, which can be used to quantify geometrical parameters of non-cylindrical folds. 3D fold geometry is described using fold wavelength, l, arc-length, L, and amplitude, A. As most natural folds do not show ideal shapes, but are affected by various types of discontinuities, the new procedure is not fully automatic, but requires the manual selection of measuring profiles along which the geometrical parameters are constrained. The new workflow is tested using natural and experimentally produced non-cylindrical folds. The geometric parameters obtained can be used to improve our understanding of fold kinematics and fold mechanics and should assist the quantitative analysis of non-cylindrical folds present in gneiss and salt domes and in rocks containing reservoirs of hydrocarbons and minerals deposits. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Dome-and-basin structure Non-cylindrical folds 3D fold geometry

1. Introduction Folds are spectacular structures in deformed rocks affecting single or multiple layers on all scales. They can be used to calculate finite strain and to reconstruct the kinematics and the rheological conditions related to the deformation. As our observations are frequently limited by rock exposure, the geometric data obtained from natural folds are largely incomplete and often restricted to two dimensions. It is probably for this reason that current descriptive fold terminology is biased toward folds idealized in 2D cross sections (Lisle and Toimil, 2007). Most folds are assumed to result from plane strain of one or more stiff layer(s) embedded in a weak matrix, with the layer oriented perpendicular to the long axis (X) of the finite strain ellipsoid. These folds are cylindrical and their profile geometry does not change along the fold hinge (Hudleston and Treagus, 2010; Adamuszek et al., 2011). However, field observations and experimental work indicate that folds can never continue indefinitely along their hinge lines, even if they result from one single phase of folding. At some point they die out and reappear, or bifurcate at low angles. Thus, natural fold patterns

* Corresponding author. E-mail address: [email protected] (G. Zulauf). http://dx.doi.org/10.1016/j.jsg.2017.06.001 0191-8141/© 2017 Elsevier Ltd. All rights reserved.

depart from true cylindricity and display a shape variation in the third dimension (e.g. Hobbs et al., 1976; Williams and Chapman, 1979; Pearce et al., 2006; Lisle et al., 2010; Bretis et al., 2011). In cases where the degree of non-cylindricity is significant during folding, a 3D geometrical analysis of the deformed layer is required (Fletcher, 1991, 1995). Such non-cylindrical folds with a high degree of non-cylindricity may result from polyphase folding (e.g. Ramsay, 1962; Thiessen and Means, 1980; Grujic et al., 2002; Fig. 1a and b), from in-plane constriction (e.g. Ghosh et al., 1995; Schmid et al., 2008; Zulauf et al., 2016; Fig. 1c), or from lateral growth and linkage of fault-related folds (Bretis et al., 2011; Grasemann and Schmalholz, 2012; Frehner, 2014). Moreover, if the degree of noncoaxial strain in mylonitic shear zones is significant, fold hinges may rotate progressively towards the transport direction resulting in highly curvilinear sheath folds (e.g. Cobbold and Quinquis, 1980; Alsop and Carreras, 2007; Reber et al., 2012, 2013; Fig. 1d). Noncylindrical folds may also develop in transpressional (Tikoff and Peterson, 1998; Leever et al., 2011; Jacques et al., 2014; Frehner, 2016) and in transtensional settings (Venkat-Ramani and Tikoff, 2002; Fossen et al., 2013). The quality in assessing the shape of geological structures in three dimensions and on various scales has significantly improved during the last decades using new survey techniques. Map-scale structures can be quantified using high-resolution global

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Fig. 1. Examples of natural non-cylindrical folds. (a) Open D2 fold affecting older isoclinal D1 fold in the Phyllite-Quartzite Unit of eastern Crete; escarpment at football ground of Exo Mouliana, Crete; long side of photograph ¼ ca. 2.5 m. (b) Hinge of non-cylindrical fold in Cambrian limestone; Zonda, eastern Pre-Cordillera, Argentina (photograph by W. von Gosen). (c) Strongly curvilinear folds in the Styra nappe of the Cycladic Blueschist Unit; road-cut section at the Ochi Mt., south Evia, Greece (photograph by P. Xypolias). (d) Sheath fold portrayed by quartz band in phyllonitic high-strain mylonite of the El Llimac shear zone, Cap de Creus, Spain (photograph by P. Xypolias).

positioning systems (GPS) and light detection and ranging (LiDAR) (e.g. Pearce et al., 2006). Sub-surface structures can be reconstructed by geometric 3D modelling (Maxelon et al., 2009; Sala et al., 2014; von D€ aniken and Frehner, 2017) or using 3D seismic data (e.g. Bacon et al., 2007; van Gent et al., 2011; Vermeer, 2012). Three-dimensional structures on a smaller scale, observed in outcrops and mines, are reconstructed using terrestrial laser scanners (Slob and Hack, 2004; Bellian et al., 2005; Buckley et al., 2013; Deliormanli et al., 2014). Based on photogrammetry, multi-view photographs, obtained from different positions with respect to the object, can be used to produce high-resolution digital 3D models. After snapping a series of photos of the object from every angle that is possible, the photographs are uploaded into a photogrammetry software, which generates a 3D model. Macroscopic and microscopic structures of hand specimens of naturally and experimentally deformed rocks can be imaged without destruction using medical and micro-computertomography (e.g. Mees et al., 2003; Carlson, 2006; Cnudde and Boone, 2013; Zulauf et al., 2009, 2011b; Thiemeyer et al., 2015). Computer tomography has further been used for quantitative 3D fold and strain analyses of deformed rock analogues (e.g. Zulauf et al., 2011a, 2014; Pastorn et al., 2012; Adam et al., 2013). During rock analogue exGala periments, the continoues change in 3D shape of a deforming surface can be recorded and analyzed with progressive strain using €rtner et al., structured light and the double-scan technique (Ga 1996; Hsieh, 2001; Grujic et al., 2002). The digitized surfaces of non-cylindrical structures can be visualized using a variety of 3D visualization techniques. 3D visualization software based on advanced 3D surface reconstruction techniques and algorithms has been developed particularly for the medical imaging and design industry, but is also available for 3D geological modelling (e.g. Cowan et al., 2002). By using this type of software, it is possible to view objects interactively from different

positions at different directions. Various lighting and filtering techniques can be used to highlight shapes and surface characteristics, such as roughness. ln several cases, however, it is not sufficient to merely view the surface of 3D objects. Mostly the user wants to integrate the data into existing software packages, such as CAD or GIS, to produce 2D profiles, elevation contour lines etc. Apart from visualization, it is further necessary to quantify geometrical parameters of the 3D objects. New tools to rigorously classify the different types of non-cylindrical folds have been developed using differential geometry (e.g. Lisle and Toimil, 2007; Mynatt et al., 2007). An established technique for structural analysis is the construction of hinge lines to portray the major undulations of the folded surfaces. This technique has been extended and can be used also for folds of non-cylindrical shape (Lisle et al., 2010). Apart from these classifications, however, the geometrical parameters of folds, such as amplitude, arc- and wavelength, are of primary importance for structural geologists, as they can be used to estimate the finite strain and the rheological properties during deformation (e.g. Lan and Hudleston, 1995). The relation between fold geometry and rheology is well understood in cases of 2D structures, whereas 3D cases are only poorly constrained. One reason for this is the lack of data from 3D settings because of insufficient assessing tools. In the present paper we will present a new workflow, which can be used to quantify geometrical parameters of non-cylindrical folds. The new software will be applied to examples of natural and experimentally produced non-cylindrical folds. 2. Geometrical parameters of cylindrical folds The geometrical parameters used to describe three-dimensional folds are the same like those used for two-dimensional folds: fold arc-length, L, amplitude, A, and wavelength l. Before describing the

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procedures used in the present study to determine these parameters in 3D space, we will first consider the 2D case. Principal locations of a 2D fold are the inflection points, the hinge and the extremity (Fig. 2). The hinge is where the maximum degree of curvature is attained. The inflection point marks the place where the curvature of the interface changes its sign. In cases of upright folds, inflection points are separating synforms from antiforms. In horizontal fold trains like that shown in Fig. 2, the extremities are the upper- or lowermost points of the fold. In cases where the fold train is inclined, the extremities are the points furthest remote from the line connecting the inflection points. In regular fold trains, the fold arc-length, L, is the interface length between two inflection points, the latter with the same change in sign. Thus, L implies components of both the syn- and the antiform of a fold (Fig. 2). In cases of regular periodic fold trains, the arclength components of syn- and antiforms are almost the same (Folds 1, 2 and 4 in Fig. 2). In cases where the fold is not regular and periodic, the arc-length components of syn- and antiform may differ where the fold train changes its shape (Fold 3 in Fig. 2). There are at least nine possibilities how fold amplitude, A, can be defined (Adamuszek et al., 2011, and references therein). However, due to large uncertainties in the description of the traces of median, axial, and enveloping surfaces, and discarding those, which are not invariant with respect to rotation, only three definitions of A are regarded to be robust (Adamuszek et al., 2011). In the present study we are adopting one of these definitions, where A is the distance between the line joining two inflection points and the extremity of the fold (Ramsay and Huber, 1987). In cases of upright symmetric folds (Fold 1 and 2 in Fig. 2), this distance is identical with the distance from the hinge to the line joining two inflection points (definition suggested by Park, 1997). For the definition of the fold wavelength, l, at least eight possibilities have been found by the review of Adamuszek et al. (2011). Because of the restrictions mentioned above, only four of these definitions turned out to be robust. In the present study, l is defined as twice the distance between adjacent inflection points (Ramsay and Huber, 1987; Price and Cosgrove, 1990). In cases of upright symmetric folds (Fold 1 and 2 in Fig. 2), this distance is identical with the distance between alternating inflection points (Price and Cosgrove, 1990) and with the distance between two alternating hinges (definition suggested by van der Pluijm and Marshak, 2004). It should be emphasized that the layer thickness, h, is not considered in the present study, and the above definitions of fold parameters only refer to a single interface of a folded layer. The reason for this approach is that in most cases of natural and experimentally produced folds, it is hardly possible to ensure that corresponding inflection and hinge points exist on the two interfaces of a deformed layer. Moreover, 3D folds are frequently analyzed in plan-views from which the thickness of the layer and other fold elements, such as axial surface, cannot be determined.

3. Geometrical parameters of non-cylindrical folds There are several reasons why the assessment of geometrical

Fold 1

inflection point arc-length (L)

Fold 4

Fold 3

Fold 2

extremity wavelength (λ)

hinge amplitude (A)

Fig. 2. Geometrical parameters of 2D cylindrical folds.

fold parameters of the 3D folds described above is more complicated than assessing these parameters from 2D folds: (1) the arcand wavelength of one and the same structure may vary in different directions, (2) hinges or extremities of three adjacent structures may not be situated along a straight line, (3) apart from real axialsymmetric dome-and-basin structures, the hinge may show different degrees of curvature in various directions; in cases of general domes and basins (symmetrical pericline folds), these normal curvatures change systematically as the normal section plane is turned, and reach extreme values in two orthogonal directions of the section plane (Lisle et al., 2010). Because of these problems, the assessment of 3D fold parameters requires particular procedures, which are described in the following sections. 3.1. Construction of a 3D model The spatial data obtained from the methods described above are analyzed using the following software packages (Fig. 3): ImageJ/FiJi (http://imagej.net/Fiji), Meshlab (Visual Computing Lab e ISTI e CNR, http://meshlab.sourceforge.net), and Smoooth (rt-mp Software Development, [email protected]). In a first step, a volume data set is obtained by filtering and data conversion using ImageJ. This volume data set is used to create a polygon model using SmoothCreateModel. This software supports production, visualization, and analyses of volumetric data and can be used for 3D reconstructions based on histological sections even if the latter are free from external landmarks. The presentation of data by polygons and voxels allows the combination of polygon models with arbitrary number of sections, the latter with any orientation when cutting through the volumetric data set. In most cases the initial polygon models are very complex and thus have to be simplified and smoothed using various filter techniques. Smoothing of the polygon models is attained with a Laplace filter. An example of such a model, based on computer-tomography, is shown in Fig. 4a. This model was produced experimentally under bulk pure constriction using non-linear viscous plasticine as rock analogues (Zulauf et al., 2016). A 2.0 mm thick competent layer, situated in an incompetent matrix, was shortened parallel to the layer until a finite shortening strain, eY¼Z, of 35% was attained. At the applied strain rate, e_ ¼ 5 * 105 s1, the viscosity contrast, m, between layer and matrix was 5, and the stress exponent, n, of both materials was 7. In-plane shortening led to significant layer thickening (ca. 80%) and to the dome and basin geometry shown in Fig. 4a. Note, that only the competent layer of the deformed model is shown in Fig. 4, whereas the matrix is neglected (for further details of the model production and computer-tomographic analysis, see Zulauf et al., 2016). The size of the individual polygons depends on the degree of curvature. In cases of larger curvature (small radius), the polygons are small, whereas small curvature (large radius) results in large polygons (Fig. 4b). Apart from central projection, there is a possibility to present the 3D models in parallel projection when using the software Smoooth. 3.2. Classification of non-cylindrical folds A classification of non-cylindrical folds using a triangular plot and based on measurements of interlimb angle and hinge angle has been suggested by Williams and Chapman (1979). The apices of the triangular diagram are labelled P, Q and R, which represent the degree of fold planarity, domicity, and noncylindrism, respectively. The end-members are planes, cylindrical isoclines, and isoclinal domes (Williams and Chapman, 1979). This PQR diagram can be used to classify individual three-dimensional shapes of cylindrical and noncylindrical folds, and varying degrees of fold tightness and

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CT

μCT

DEM

Opt. Scan

123

Simulation

Data import Filtering Data conversion

ImageJ

Volume dataset Polygon mesh production

Smoooth-CreateModel

Polygon model Cropping Filtering Mesh simplification

MeshLab

Curvature calculation Dome/Basin classification

Smoooth-ClassifyCurvature

Dome/Basin editing

MeshLab

Dome/Basin model Extremity calculation Selection of section lines Determination of reference plane Determination of basal plane Inflection point calculation Piercing point calculation

Smoooth-AnalyzeModel

Fold parameter calculation Fold geometry λ, L, A, A‘ Statistics Normalization

MS-Excel

Normalized fold parameters Fig. 3. Workflow described and applied in the present study to quantify geometrical parameters of non-cylindrical folds. Blue boxes show data sets. Red boxes show individual operations. The software packages used are listed italicized on the right-hand side. A ¼ orthogonal amplitude,A' ¼ oblique amplitude, L ¼ arc-length, l ¼ wavelength, CT ¼ computer-tomography, mCT ¼ microcomputer-tomography, DEM ¼ digitial elevation model. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 4. 3D model of stiff plasticine layer shortened experimentally under bulk constriction in a weaker plasticine matrix resulting in non-cylindrical folds (dome-and-basin structures). The model is based on a computer-tomographic analysis (for details of the experiments, see text and Zulauf et al., 2016). (a) Oblique view of upper surface of the model; black frame indicates extent of detail shown in (b), (b) detail of (a) showing the polygons (triangles) constituting the surface of the model. Different grey levels result from different degree of shading (inclination of the surface with respect to the light source). Width of the quadratic model ¼ 6 cm.

noncylindrism are discernible. However, no information is portrayed concerning orientations of structural elements, and fold structures, which imply both syn- and antiformal geometry, are not considered. For this reason we do not use the PQR diagram, but a classification based on differential geometry. By applying this technique to calculate surface normal and tangent vectors, mathematically rigorous geometric descriptions of geologic structures can be obtained, including various measurements of surface curvature (e.g. Pollard and Fletcher, 2005; Lisle and Toimil, 2007; Mynatt et al., 2007, and references therein). In the present study, the classification is carried out using the software MeshLab and Smoooth-ClassifyCurvature (Fig. 3). The first and second derivatives are calculated and the principal curvatures (kmin and kmax) are found. Convex upward surfaces are arbitrarily assigned as positive curvature, whereas concave upwards surfaces are assigned as negative values. The mean curvature (kmean ¼ (kmin þ kmax)/2) and the Gaussian curvature (kGauss ¼ kmin kmax) are computed (Fig. 5a and b) and used for fold classification (Fig. 6). Some fold types

shown in Fig. 6 do hardly exist in nature, as they require at least one principal curvature to be zero. This holds for the plane, saddle and cylindrical antiform and synform. Measurement error and the inherent irregularity of geologic surfaces preclude these possibilities, although geologists often approximate folds as cylindrical to simplify description and analysis, knowing such perfect shapes do not exist (Mynatt et al., 2007). Fig. 7 shows domes, basins, antifomal and synformal saddles, which have been separated from the model shown in Fig. 4 using the method described above. 3.3. Measuring procedure to assess geometrical parameters of domes and basins In this section we will show how the geometry of domes and basins of a 3D fold model can be quantified. First of all the domes and basins have to be separated from adjacent structures using the software Meshlab (Fig. 3). The next step is to assess the position of the extremities of domes and basins. As in the hinge area the degree

Fig. 5. (a) Mean curvature and (b) Gaussian curvature of the model depicted in Fig. 4a. Histograms at the bottom show the surface area of the model with respect to the variation in mean and Gaussian curvature, respectively.

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Fig. 6. Geologic curvature classification (modified from Mynatt et al., 2007, and references therein). The geologic curvature of a point on a surface can be determined from the Gaussian (kGauss) and mean (kmean) curvatures at the point. The color code is used throughout the paper. Note that the classes without color (plane, synform, antiform, saddle) do not or do hardly occur in the deformed stiff layers and thus can be neglected. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

of curvature may change (see above), we do not use the hinges but the extremities to constrain the geometrical parameters. In most cases the geometry of domes and basins is not related to the real vertical and horizontal directions of a model. This holds for both natural and experimentally produced non-cylindrical folds. For this reason the determination of extremities of domes and basins requires a local reference direction, which is comparable to the axialplane of a cylindrical fold. In the digital models described above, the surface of domes and basins is defined by numerous polygons (triangles), which are different in area and orientation (Fig. 4b). Both the area and orientation of the polygons are quantified. The orientation is defined by the normal to the plane (red lines with arrow in Fig. 8b). The directions of all surface normals, each weighted (multiplied) by the area of the corresponding polygon,

Fig. 7. Different fold types of the model depicted in Fig. 4a.

are used to determine the reference direction (reference line in Fig. 8c). The extremity (E) of a dome or of a basin is then assessed by shifting a plane, which is normal to the reference direction, until it is tangent to the extremity of the dome or of the basin (Fig. 8c). Shifting the reference line through the extremity, results in the extremity line (Fig. 8d). Extremities of domes and basins of the model shown in Figs. 3, 4 and 6 are depicted in Fig. 9. The surface area and the geometrical parameters of domes and basins are determined from the digital 3D models using the software Smoooth-AnalyzeModel (Fig. 3). The surface area is determined as absolute values, but is commonly presented as relative values in area per cent of the entire structure considered. Based on the position of the extremities, profiles are selected along which the inflection points are determined. Inflection points are shown in green color in Fig. 9. The distance from an extremity of a dome to an extremity of a basin is called segment. For one measuring profile, along which the fold parameters are quantified, at least three segments with inflexion points are required (yellow lines in Fig. 9). The following aspects have to be considered when selecting the measuring profiles: (1) an extremity of a dome is connected with an extremity of a basin and vice versa; (2) the profile should not cross another dome or basin that is not part of the profile and measurement, respectively; (3) in most cases the profiles are not straight, but show a kink at the extremity of the dome (or basin) situated between two basins (or between two domes); (4) it is not required to consider all domes and basins of the model; (5) one dome or one basin can serve for more than one profiles; (6) profiles shouldn't be measured twice. Subsequent to the selection of the measuring profiles, the fold parameters, L, l, A as well as the area of the domes and basins are quantified and listed in a table. The orthogonal amplitude, A, is measured as the distance between a basal plane and the extremity (E) of the structure (Fig. 10). The basal plane contains both inflection points (I1 and I2), and is perpendicular to the reference plane (shown in blue in Fig. 10). The reference plane is parallel to the line joining both inflection points but is also parallel to the reference direction defined above. Apart from the orthogonal amplitude, A, it is also possible to assess the oblique amplitude, A’, which is the

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Fig. 8. 2D section through a dome, to show the determination of the extremity (E) and of the extremity line. (a) The polygons, which define the surface of the dome, are shown by individual sections. (b) The length of the normals to the polygons (red lines with arrow) depends on the area of the polygons. (c) The directions of all surface normals are used to determine the reference direction (reference line); the location of the extremity (E) of the dome (yellow filled circle) is assessed by shifting a plane, which is normal to the reference line, until it is tangent to the extremity of the dome. (d) Shifting the reference line through the extremity results in the extremity line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

distance between the extremity (E) and the piercing point (PP) measured along the extremity line (Fig. 10). The half values of arcand wavelength, respectively, are measured from one to the other inflexion point of a selected profile. The half arc-length results from

Fig. 9. Separated domes (in blue) and basins (in pink) of the model shown in Fig. 4a. The extremities of domes and basins are shown by blue and red dots, respectively. The extremity lines of domes and basins are shown by blue and red lines, respectively (see Figs. 8 and 10 for explanation of extremity lines). The yellow curved lines indicate selected measuring profiles, which consist of at least three sections. Inflection points are shown by green dots (for further explanation, see text). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

the curved black lines (I1-E-I2) shown in black in Fig. 10. The half wavelength is measured along the basal plane from inflection point to inflection point via the piercing point of the extremity line (I1PP-I2 in Fig. 10). To compare structures with different absolute size, it is recommended to use normalized values, such as amplitude divided by

Fig. 10. Sketch showing the geometric elements required for the determination of arclength, L, wavelength, l, orthogonal amplitude, A, and oblique amplitude, A0 , of a dome, which is not axial-symmetric. The orthogonal amplitude, A, is the distance between the extremity (E) and the orange basal plane. The basal plane is parallel to the line joining the inflection points (I1 and I2), and is perpendicular to the blue reference plane. The reference plane is also parallel to the line joining the inflection points and is still parallel to the extremity line. The extremity between the inflection points forms a kink, which is situated in front of the reference plane. The extremity line and the arcs joining I1 and I2 with E are also situated in front of the reference plane. The distance I1PP-I2 is equivalent to l/2. The curved distance I1-E-I2 is equivalent to L/2. The oblique amplitude is measured along the extremity line as the distance between PP and E. The situation would be analogue for a basin, but with the extremity (E) being located below the basal plane. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

a

arc-length

27

40

22

18

b

wavelength

15 18 15

20

U1

U2

strK

slK

slC

strC

measuring procedure

amplitude, A [mm]

arclength, L, wavelength, λ [mm]

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127 amplitude orthogonal

27 18

amplitude oblique 15

22 18 15

2

U1

U2

strK

slK

slC

strC

measuring procedure

Fig. 11. Geometric data of the experimentally produced domes and basins shown in Fig. 4a. Number of measurements, N, used to calculate mean values and standard deviation of the geometric data are depicted. (a) Arc-length and wavelength; (b) Orthogonal and oblique amplitude. The different data shown in each diagram result from: measurements by different users (U1, U2); preferred selection of strongly kinked (strK) and slightly kinked (slK) measuring profiles; and preferred selection of strongly curved (strC) and slightly curved (slC) domes and basins. User 1 (U1) has considered profiles defined by distinct and relatively large domes and basins. User 2 (U2) has considered almost all possible profiles.

Fig. 12. Natural example of non-cylindrical fold analyzed in the frame of the present study. The non-cylindrical fold results from at least two different Variscan folding phases  Me sto nad Metují, Czech Republic. The sample is deposited at the Institute of Geology and Paleontology, affecting chloriteesericite phyllite from the Metuje River valley near Nove Charles University, Prague, Czech Republic. Long side of sample is ca. 1.0 m. (a) Photograph of the folded sample. (b) Digital 3D model. (c) 3D model showing the different fold types; color coding of domes, basins, antiformal and synformal saddles is the same as in Fig. 6; black frame shows the outline of the analyzed area depicted in (d). (d) Close-up view of the area depicted in (c). The model shows only domes in blue and basins in pink. The extremities of domes and basins are shown by small dark blue and red diamonds, respectively. The depicted measuring profiles have been used to study the geometrical parameters of folds with axes trending from left to the right. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

wavelength. If the thickness of the folded layer is known, the arclength can be divided by layer thickness. 4. Application to an experimental example of dome-andbasin folds The experimental model shown in Fig. 4 has been used to test the workflow described above. As the measuring profiles have to be selected manually, there is a certain impact on the results depending on the user. For this reason, the first sets of data were produced by two different users to evaluate the impact of subjectiveness. The results are shown in Fig. 11a and b. A first user (U1)

considered only those profiles, which include distinct and large domes and basins. For this reason the number of data is relatively low (18). The second user (U2) considered almost all profiles available. It is obvious from Fig. 11a and b, that the impact of subjectiveness is moderate. The mean values of each set of measurements obtained by one user are within the standard deviation of the values obtained by the other user. This holds for all parameters measured. According to our expectations, the profiles selected more critically by U1 yielded lower standard deviations than those selected by U2. Moreover, the arc-length, L, is in all cases larger than the wavelength, l, and the oblique amplitude, A’, is in all cases slightly larger than the orthogonal amplitude, A (Fig. 11b).

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We further checked the impact of the degree of straightness of a measuring profile. We measured one series with strongly kinked profiles (strK in Fig. 11) and another series with slightly kinked profiles (slK in Fig. 11). It is obvious that straight measuring profiles yield arc- and wavelengths, which display a lower standard deviation than strongly kinked profiles. The mean values, however, are similar. The amplitude is slightly lower in cases of kinked profiles, whereas the standard deviation is similar. In a third set of measurements, we tried to check the impact of the degree of hinge curvature. A first series of measurements was carried out considering strongly curved hinges (strC in Fig. 11). A second series was carried out focusing on slightly curved hinges (slC in Fig. 11). The mean values of wavelengths are almost the same in both cases (Fig. 11a). However, the standard deviation is much lower in cases of slightly curved hinges. The arc-length of strongly curved hinges is higher than the arc-length of slightly curved hinges. Moreover, the difference between arc- and wavelength is larger in cases of strongly curved hinges. If the arc-length increases, while the wavelength remains almost constant, the amplitude has to grow. This is actually the case. The amplitude of the strongly curved hinges is significantly larger than the amplitude of the slightly curved hinges (Fig. 11b). 5. Application to a natural example of a refolded fold The workflow described above was further used to quantify geometrical parameters of a natural example of non-cylindrical folding. The studied sample of chlorite-sericite phyllite is derived  Me sto nad Metuji, situated from the Metuje River valley near Nove  hory of the Czech Sudetes (European in the foreland of the Orlicke Variscides). The non-cylindrical fold patterns result from at least two different Variscan folding phases. A top-view photograph is shown in Fig. 12a. The fold axes of the dominant folds (F2) are trending from the left to the right in the photograph, whereas another set of fold axes are trending from the bottom to the top. The latter folds (F1) are probably older and thus less distinct. They are hardly visible in both the photograph and in the 3D model shown in Fig. 12 b. However, they are clearly visible in the 3D model, in which the different fold types are distinguished by different colors (Fig. 12c). Based on a sequence of multi-view photographs, the 3D model was produced using the software Agisoft PhotoScan. There is a clear interference pattern of the F1-and F2-folds shown in Fig. 12c. However, as the F2-folds are dominant, a distinct dome-and-basin pattern like that described above from the experimental model does not exist. The fold pattern is characterized by (1) antiformal saddles, trending from the left to the right, which are interrupted by elongated domes, trending from the bottom to the top, and (2) synformal saddles, trending from the left to the right, which are interrupted by elongated basins, trending from the bottom to the top. We measured the geometrical parameters of domes and basins of the area bounded by the black broken frame in Fig. 12c. The distribution of domes and basins allows selection of measuring profiles only in the vertical direction. These profiles are shown in yellow in the close-up view of Fig. 12d. In this figure, domes and basins as well as the corresponding extremities are depicted. The values for orthogonal and oblique amplitude are the same (1.8 ± 1.2 mm). Because of the very low amplitude, the arc- and wavelength is also almost the same. The arc-length is 49.8 ± 17.8 mm, and the wavelength is 49.0 ± 17.0 mm. 6. Discussion and conclusions Fold analysis in the past was largely based on stereographic projections of poles to bedding and poles to cleavage, if present. The construction of a real surface and the reconstruction of the

complete shape from this data set are in most cases not possible resulting in subjective interpretations of the fold geometry. This holds particularly for non-cylindrical folds, the 3D surfaces of which may differ in length scale, amplitude, closure direction etc. In cases where the entire surface of the fold has been assessed using modern 3D survey equipment, the resulting digital 3D model can be used to classify the geometry of the folds completely and objectively using differential geometry (Lisle and Toimil, 2007; Mynatt et al., 2007). The classification of the folds by differential geometry, however, does not inevitably imply that geometrical parameters, such as amplitude, wavelength and arc-length are known. Moreover, natural folds do not show idealized shapes because of different types of irregularities (e.g. strain inhomogeneity because of polyphase folding or compositional inhomogeneity). For these reasons we used a semi-automatic method to constrain the geometric parameters of non-cylindrical folds. The workflow and related software were introduced and tested in the previous sections. It has to be emphasized that the procedure described does not consider the whole range of 3D fold geometries as revealed by differential geometry. The main focus of the present study is the dome-and-basin geometry, which, however, is present in most types of non-cylindrical folds. The preliminary tests of the workflow have shown that the selection of measuring profiles plays an important role for the quality of the obtained geometrical data. Fully automatic assessment of geometrical parameters of domes and basins is recommended only in cases of idealized axialsymmetric structures resulting in an egg-carton shape. Such a structure, however, does not exist in nature. As soon as the shape of the domes and basins becomes asymmetric in plan-view, the arcand wavelengths vary in different directions, which require a manual selection of measuring profiles. Moreover, given that smallscale structures are present, which result from irregularities mentioned above, these structures have to be deleted and/or ignored during the measuring procedure. Based on the new framework, a novel practical method is devised for the analysis of amplitude, arc- and wavelength of domes and basins. The geometric parameters so obtained should assist the quantitative analysis of structures present in gneiss and salt domes and structures containing reservoirs of hydrocarbons and mineral deposits. Moreover, in cases where the layer thickness is known, the geometric parameters obtained can be used to check theories developed for 3D fold kinematics and 3D fold mechanics (e.g. Ghosh, 1970; Fletcher, 1991, 1995; Frehner, 2014). Acknowledgements We would like to thank M. Peinl, the author of the program Smoooth, for providing additional information of the internally used methods of Smoooth-AnalyzeModel. We further acknowledge the constructive reviews by M. Dabrowski, M. Frehner and an anonymous referee, which helped to improve the quality of our paper. The studies were supported by a grant of Deutsche Forschungsgemeinschaft (DFG, grant Zu 73/26-1), which is kindly acknowledged. References Adam, J., Klinkmüller, M., Schreurs, G., Wienecke, B., 2013. Quantitative 3D strain analysis in analogue experiments: integration of X-ray computed tomography and digital volume correlation techniques. J. Struct. Geol. 55, 127e149. Adamuszek, M., Schmid, D.W., Dabrowski, M., 2011. Fold geometry toolbox automated determination of fold shape, shortening, and material properties. J. Struct. Geol. 33, 1406e1416. http://dx.doi.org/10.1016/j.jsg.2011.06.003. Alsop, G.I., Carreras, J., 2007. The structural evolution of sheath folds: a case study from Cap de Creus. J. Struct. Geol. 29, 1915e1930. Bacon, M., Simm, R., Redshaw, T., 2007. 3-D Seismic Interpretation. Cambridge University Press.

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