1. Introduction
Structural geologists have long used fold shapes as indicators of the kinematic, dynamic and rheological behaviour of rocks during deformation (Biot, Reference Biot1961; Hudleston & Lan, Reference Hudleston and Lan1993; Lan & Hudleston, Reference Lan and Hudleston1996). Fold shape studies are also crucial in exploration of ore and petroleum deposits that are commonly concentrated in zones of high curvature, that is, the hinge zones of folds. The great significance of folds, as well as their prolific occurrence in layered rock sequences, has been the source of motivation for development of a number of methods for fold shape analysis during the last four decades (De Sitter, Reference De Sitter1958; Ramsay, Reference Ramsay1967; Twiss, Reference Twiss1988; Lisle, Reference Lisle1994; Pearce et al. Reference Pearce, Jones, Smith, McCaffrey and Clegg2006; Lisle et al. Reference Lisle, Martínez, Bobillo-Ares, Menéndez, Aller and Bastida2006).
Identification of the hinge point, that is, the point of maximum curvature on a folded surface, is an essential requirement in several methods of fold shape analysis. Some of these methods also require selection of the inflection point, the point at which the curvature changes its sign and crosses zero. In practice, however, these points are routinely selected by a visual estimation. This article addresses the issue of errors in visual estimation of the hinge point and the inflection point, and proposes a user-friendly computer-based method for precise selection of these points on a fold profile.
2. Methods of fold shape analysis
Accuracy in selection of the hinge points and/or inflection points is important in the analysis of fold shapes on profile sections by two types of methods: (1) methods that analyse a folded layer that is bounded by the outer and inner arcs, and (2) those methods that analyse a single folded surface, that is, the trace of a folded surface on the profile section.
Methods of the first type use the difference between curvatures of the outer and inner arcs to distinguish several standard shapes of the fold layers, such as Class 1A, Class 1B, Class 1C, Class 2 or Class 3 folds, and in some instances, to estimate the strain that concentrates during flattening stages of folding (Ramsay, Reference Ramsay1967, pp. 411–14). Several versions of these methods exist: the tα´/α method of Ramsay (Reference Ramsay1967, p. 413), the ϕα/α method of Hudleston (Reference Hudleston1973), the ‘inverse thickness’ method of Lisle (Reference Lisle1992, Reference Lisle and Sengupta1997), the ‘retrodeformational method’ of Srivastava & Shah (Reference Srivastava and Shah2006), the ‘Wellman’ and the ‘Mohr circle methods’ of Shah & Srivastava (Reference Shah and Srivastava2006), and the ‘isogon rosette’ method of Srivastava & Shah (Reference Srivastava and Shah2008). Of these, the tα´/α method is used most commonly and application of this method requires the selection of hinge points on the inner and outer arcs.
Methods of the second type define the shape of a single folded surface using quantitative shape parameters, such as the ratio of two harmonic coefficients (Mertie, Reference Mertie1959; Ramsay & Huber, Reference Ramsay and Huber1987, pp. 314–17). These methods analyse the shape of a single folded surface by using Fourier analysis (Stabler, Reference Stabler1968; Hudleston, Reference Hudleston1973; Stowe, Reference Stowe1988), conic section analysis (Bastida, Aller & Bobillo-Ares, Reference Bastida, Aller and Bobillo-Ares1999; Aller et al. Reference Aller, Bastida, Toimil and Bobillo-Ares2004) or Bézier curve analysis (Bézier, Reference Bézier1966, Reference Bézier1967; De Paor, Reference De Paor and De Paor1996; Wojtal & Hughes, Reference Wojtal and Hughes2001; Srivastava & Lisle, Reference Srivastava and Lisle2004; Coelho, Passchier & Grasemann, Reference Coelho, Passchier and Grasemann2005; Lisle et al. Reference Lisle, Martínez, Bobillo-Ares, Menéndez, Aller and Bastida2006). Of these, the Bézier curve method is the most rapid and easy to use (Srivastava & Lisle, Reference Srivastava and Lisle2004; Lisle et al. Reference Lisle, Martínez, Bobillo-Ares, Menéndez, Aller and Bastida2006).
3. Errors in visual estimation
3.a. Hinge points
We address the issue of errors in visual estimation of the hinge point with the help of two examples, one of a single layer fold and the other of a single folded surface. The fold in each example is analysed by considering two alternative positions of the hinge point.
3.a.1. Single layer fold
We give a natural example of a folded layer to demonstrate that uncertainty in the selection of hinge point can lead to significantly different strain estimates (Fig. 1a). As the inner arc is relatively sharp-crested, its hinge point can be marked fairly easily by visual estimation (point 3 in Fig. 1a). In contrast, the curvature distribution on the broad hinge zone of the outer arc is such that there is an uncertainty in the selection of a precise hinge point by visual estimation. Two possible and alternative positions of the hinge point, 1 and 2 are marked on the outer arc by visual estimation (Fig. 1a).
Effect of uncertainty in visual selection of the hinge point on a single layer fold. (a) Profile section of a fold in the Almora crystalline, Kumaun Lesser Himalaya. 1 and 2 – two alternative positions of the hinge point on the outer arc, 3 – hinge point on the inner arc. (b, c) Isogon patterns with respect to axial traces 1–3 and 2–3, respectively. The isogons on left limb are more convergent in (b) as compared to those in (c). (d, e) t ´α /α plots with respect to the axial traces 1–3 and 2–3, respectively. Curves in (d) and (e) indicate different shapes and estimates of flattening.
To demonstrate the importance of precision in selecting the hinge point, we analyse the fold in Figure 1a by the tα′/α method, first by considering line 1–3 as the axial trace and then by considering line 2–3 as the axial trace (Fig. 1b, c). At low angles of limb dip (≤ 20°), the left limb shows Class 1A geometry if line 1–3 is the axial trace (Fig. 1d). If, however, line 2–3 is the axial trace, then the same limb shows Class 1C geometry irrespective of the angle of limb dip (Fig. 1e). Although the right limb shows Class 1C geometry with respect to both the axial traces, the tα´/α plots for this limb imply a significantly higher estimate of flattening strain if line 2–3, rather line 1–3, represents the axial trace (Fig. 1d, e). This example demonstrates that the results of shape analysis and estimates of flattening strain could be significantly different depending upon the choice of the hinge point or the axial trace.
The above example shows only one of several possible effects of the error in selection of the hinge point. For the sake of simplicity and application of the t α´/α method, it assumes that tangents at the inner and outer arcs are parallel irrespective of whether the axial trace is 1–3 or 2–3. In a strict sense, however, the tangents at the hinge points of the outer and inner arcs could parallel each other in only one of these two situations. A slight error in the selection of the hinge point on a sharp-crested arc results in a significant difference in the slope of the tangent at the hinge point, and no estimates of strain can be obtained if the tangents at the outer and inner arcs are non-parallel (Hudleston, Reference Hudleston1973).
3.a.2. Single folded surface
Srivastava & Lisle (Reference Srivastava and Lisle2004) and Aller et al. (Reference Aller, Bastida, Toimil and Bobillo-Ares2004) point out that subjectivity in the selection of the hinge point and the inflection point can be a potential source of error in fold shape analysis. In this article, we present an example to illustrate the issue of error in the visual estimation of the hinge point on a single folded surface (Fig. 2). Let us consider the profile section of a single folded surface and mark two possible hinge points, 1 and 2, by visual estimation (inset in Fig. 2). The analysis of this fold by the Bézier curve method reveals that a small difference in position of the hinge point leads to significant differences in the shape parameter as well as the aspect ratio of the fold (Fig. 2).
Effect of uncertainty in visual selection of the hinge point on a single folded surface. Inset shows a folded surface with two alternative positions, 1 and 2, of the hinge point. l and r – left and right limbs. The graph shows results of shape analysis by the Bézier curve method. l1, r1 and l2, r2 – shapes of left and right limbs with respect to hinge points 1 and 2, respectively. Aspect ratio (R) – amplitude/quarter wavelength; Shape parameter (L) – length of Bézier handle/quarter wavelength (for details see Srivastava & Lisle, Reference Srivastava and Lisle2004).
3.b. Inflection points
The uncertainty in the visual estimation of inflection points is also a potential source of error in the shape analysis of single folded surfaces. The inset in Figure 3a is an example of a folded surface on which two possible positions of the inflection points, i1 and i2 are marked by visual estimation. The analysis of this fold using the Bézier curve method yields significantly different values of the shape parameter and the aspect ratio depending upon the choice of the inflection point (Fig. 3b).
Effect of uncertainty in visual selection of the inflection point. (a) h – hinge point, i1 & i2 – two alternative positions of the inflection point on a fold limb. (b) Results of shape analysis of the fold limb by the Bézier curve method. I1 and I2 – shapes of the fold limb with respect to the inflection points i1 and i2, respectively. Aspect ratio (R) and Shape parameter (L) same as in Figure 2.
4. Precise selection of the hinge and the inflection points
4.a. The algorithm
The curvature C(x) at any point (x, y) on the profile section of a folded surface, y = f(x), is given by (Ramsay, Reference Ramsay1967, p. 347):
The algorithm calculates the derivatives dy/dx and d 2y/dx 2 and substitutes these values in Eq. 1 to determine the curvature C(x) at different points (x, y) on the fold. The method produces a curvature distribution curve for selection of the point (x max, y max) where curvature C(x) attains the maximum value, and the point (x inf, y inf) where it changes its sign and crosses zero.
Four potential sources of error are implicit in our approach. First, any irregularity on the fold limb may appear as a spurious hinge point due to an anomalously high curvature. Our method obviates appearance of such erroneous hinge points by fitting a smooth n th degree polynomial curve, y = anxn + a n−1x n−1 + ⋅⋅⋅ + ax + a 0, on the given fold.
Second, the accuracy in selection of the hinge point critically depends upon the smoothness of the curve that connects the successive points on the fold profile (Fig. 4a). The nature of curvature distribution with respect to the manually digitized points is, however, commonly uneven in natural folds (Fig. 4b). By best-fitting a polynomial curve through points on the fold, we obtain a curvature distribution curve that is smooth enough for precise selection of the hinge and inflection points (Fig. 4c, d).
(a) Digitized points on an image of the fold profile. (b) Irregular nature of curvature distribution. No polynomial curve is fit to the digitized points. (c) Smooth curvature distribution obtained by fitting a polynomial of degree 14 to the digitized data points. C(x) – curvature, C max – the point of maximum curvature, C inf – the point where the curvature changes sign and crosses zero. (d) Hinge point H and inflection point I on the fold.
Third, the precision in the results also depends upon the accuracy in curve-fitting. One simple way to improve the accuracy is to use a large number of points on the fold profile. Extraction of more than 100 points on a mesoscopic-scale folded surface is, however, quite time-intensive. Our method overcomes this limitation by using the interpolation technique that generates several intermediate points between the digitized data points. Typically, the interpolation increases the number of points from 100 to 500–1000.
Finally, the curvature analysis in some folds may report false hinge and inflection points even after best-fitting the smooth polynomial curve and increasing the points by interpolation. The false points correspond to unavoidable sharp turns that occur towards the end points of the high degree polynomial fitted to the fold. Our method excludes the sharp turns by delimiting the search domain to a small segment of the polynomial curve that is likely to contain the hinge point. Similarly, another segment of the curve may be selected for searching the inflection point. The flexibility of choosing a segment of the fold facilitates piecewise analysis of the fold.
Figure 4c represents an example where the polynomial fitting results in a sharp turn towards the edge of left limb of the fold. We obviate the sharp turn by selecting a potential segment that is likely to contain the hinge point. Similarly, we select a potential segment containing a single zero crossing for searching the inflection point on the right limb. This selection of a potential segment also obviates the zero crossing towards the edge of right limb, possibly an artifact of polynomial fitting and numerical differentiation. The code can mark multiple inflection points within the chosen segment and this feature can be utilized to analyse the folds with multiple hinge points or the parasitic folds.
4.b. ‘HingeInflex’
The Matlab code, namely, ‘HingeInflex’ (online Appendix at http://journals.cambridge.org/geo), imports the image of a fold profile, digitizes different points on the image, increases the number of points by interpolation and fits a smooth polynomial curve through the points. It calculates the first derivative and the second derivative at each point on the polynomial, substitutes these values in Eq. (1) and determines the values of curvature at different points.
We have used the cubic spline smoothing method for interpolating the digitized data points, least-squares fitting procedure for polynomial fitting and numerical differentiation for determining derivatives at different points on the fold. The degree of best-fitted polynomial has been determined by minimizing the residuals.
The step-by-step procedure for selection of the hinge point and the inflection point is as follows:
(1) Import the image of the given fold profile into Matlab and digitize it by using any commonly available software, such as the free program ‘GRABIT’ (J. Doke, unpub. data, 2005; http://www.mathworks.com/matlabcentral/fileexchange/7173).
(2) Extract and store the co-ordinates (x, y) of all the points in a Matlab data file. The code prompts for the file selection menu. Use the menu to load the data file and plot the digitized data points (Fig. 5a).
(a) Digitized data points and the best-fitted polynomial on an asymmetric fold. (b) Residuals at different points on the fold. (c) Curvature distribution curve showing the maximum curvature point C max and the zero crossing point C inf. (d) Precise positions of hinge point H and inflection point I on the fold.
(3) Obtain the best-fit curve through the points by using the provisions, given in the code, for fitting a polynomial curve and for changing the degree of polynomial. This operation produces a single smooth curve through the extracted data points.
The code prompts for polynomial fitting and asks for the degree of polynomial. It fits the polynomial of chosen degree and displays the fitted curve through the data points (Fig. 5a). It also calculates and displays the residuals at different points. By changing the degree of the polynomial, the user obtains the best-fit that corresponds to the minimum value of the residuals (Fig. 5b).
The code calculates curvature C(x) at a large number of points on the fitted polynomial curve and gives the curvature distribution curve that shows variation in C(x) with respect to x (Fig. 5c).
(4) By using the provision in the code, select two potential segments of the curvature distribution curve, one containing the hinge point and the other containing the inflection point(s). The code searches for the point C max where C(x) attains the maximum and for the point(s) C inf where it crosses zero, and marks these points on the curvature distribution curve (Fig. 5c). Finally, it marks the points corresponding to C max and C inf as the hinge point H and the inflection point(s) I on the fold (Fig. 5d).
Fitting a polynomial curve in step (3) is difficult on overturned folds that have two or more points with different values of y for a single value of x (Fig. 6a). Such folds require an additional step between step (2) and step (3) given above. The additional step rotates the digitized image through an arbitrary angle such that every point on the fold has a unique value of y (Fig. 6b). A polynomial curve can now be easily fitted to the digitized points on the rotated fold image (Fig. 6c). The rotation also facilitates fitting of a polynomial curve to those folds in which the tangent at the hinge point is not horizontal.
(a) Digitized data points on the image of an overturned fold. Several points on the right limb have same value of x but different values of y. (b) Rotated image of fold in (a). (c) Best-fitted polynomial through digitized points on fold in (b). (d) Distribution of residuals along the fold. (e) Curvature distribution with the maximum curvature point at C max and the zero crossing point at C inf. (f) Hinge point H and inflection point I on the fold.
5. Examples
We have tested our code on a large number of natural and graphically-simulated folds. Here we list three examples: (1) an open asymmetric fold that has a broad hinge zone (Fig. 5a), (2) an overturned asymmetric fold with relatively narrow hinge zone (Fig. 6a) and (3) a natural cuspate fold (Fig. 7a, traced from a part of figure 17.4 in Ramsay & Huber, Reference Ramsay and Huber1987, p. 350).
(a) Natural example of a cuspate fold (part of figure 17.4 in Ramsay & Huber, Reference Ramsay and Huber1987, p. 350). (b) Digitized data points on the fold. (c) Rotation of data points in (b) by 10° and the best-fit polynomial through the data points. (d) Residual distribution along the fold. (e) Curvature distribution showing the maximum curvature point C max and the zero crossing points C inf. Inset shows the magnified distribution around zero axis. (f) Inflection points I and the hinge point H are marked on the fold.
Visual estimation of the hinge point of the fold in Figure 5a involves uncertainty due to the nature of curvature distribution on the broad hinge zone. The proposed method generates 500 points by cubic smoothing spline interpolation to the digitized data points and fits a polynomial of degree 9. These operations facilitate fitting a single smooth curve through the points (Fig. 5a). The quality of fitting is ensured by monitoring and minimizing the residuals (Fig. 5b). The distinct peak C max and the zero-crossing C inf in the curvature distribution plot (Fig. 5c) correspond to the hinge point H and the inflection point I on the fold profile (Fig. 5d).
The overturned asymmetric fold in Figure 6a is an example of a curve on which a few points have a common value of x but different values of y. In order to fit a polynomial curve to such a fold, it is necessary to rotate the fold image prior to the fitting (Fig. 6b). Figure 6c shows the digitized data points on the image, rotated through an angle of 33°, along with the smooth fitted polynomial of degree 17. The small values of residuals as shown in Figure 6d indicate the satisfactory curve fitting. The curvature C(x) can now be calculated (Fig. 6e) to search for precise positions of the hinge and inflection points (H and I in Fig. 6f).
Figure 7a is an example of a natural cuspate fold and Figure 7b shows the digitized data points on this fold. We have rotated the fold by 10° to facilitate polynomial curve fitting (Fig. 7c). Small values of residuals at different points on the curve indicate satisfactory curve fitting (Fig. 7d). The curvature distribution curve shows the peak C max and the zero crossings C inf corresponding to the hinge point H and the inflection points I, respectively (Fig. 7e). These points are also marked on the fold (Fig. 7f).
6. Summary and conclusions
The shape analysis of folds is sensitive to a lack of precision in the selection of the hinge and inflection points. ‘HingeInflex’ is a user-friendly method for the mathematical determination of these points on those folds that can be fitted with a smooth polynomial curve. We give a Matlab code that best-fits a smooth polynomial curve and selects the hinge and the inflection points by calculating and comparing the curvature at different points of the curve.
The method has an in-built flexibility for changing the degree of polynomial curve and ensuring the best-fit by monitoring and minimizing the residuals. It also provides an option to delimit the potential search domain for obviating the appearance of spurious hinge or inflection points and obtaining the rapid results. Tests on several natural and computer-simulated folds reveal that ‘HingeInflex’ is a robust method that accurately identifies positions of hinge and inflection points.
Acknowledgements
Jyoti Shah, Dipanwita Nandi, Alisha Mehra and Rajit Ghosh tested our method on several examples of computer simulated and natural folds. We are grateful to an anonymous referee, Richard Lisle and Richard Jones for erudite reviews and to the Earth Science System Division of Department of Science and Technology, Government of India, for the funding.
