The elliptical and hyperbolic outcrop patterns characteristic of periclinal folds can be used to classify structures according to different curvature attributes. Elliptical patterns indicate domal-basinal structures with synclastic curvature, that is, principal curvatures of the same sign. Hyperbolic patterns are diagnostic of anticlastic curvature (saddle-like structures). Such outcrop geometries are geological examples of Dupin’s indicatrix, the geometrical figure obtained by sectioning a curved surface on a plane parallel and almost coincident with the tangent plane. The aspect ratio of Dupin’s indicatrix is theoretically related to the ratio of the principal curvature values for the part of the structure being considered. This new method allows quantitative assessment of structures on maps and on remote sensing images. Illustrations are given from Wyoming, USA, and Yorkshire, England

Lisle, Richard John

Online Research @ Cardiff

http://journals.cambridge.org Downloaded: 24 Feb 2014 IP address: 131.251.254.13Geol. Mag. 140 (6 ), 2003, pp. 721–726. c© 2003 Cambridge University Press 721DOI: 10.1017/S0016756803008306 Printed in the United KingdomDupin’s indicatrix: a tool for quantifying periclinal folds on mapsRICHARD J. LISLE*School of Earth, Ocean and Planetary Sciences, Cardiff University, Cardiff CF10 3YE, UK(Received 16 April 2002; revised version received 26 June 2003; accepted 30 June 2003)Abstract – The elliptical and hyperbolic outcrop patterns characteristic of periclinal folds can be usedto classify structures according to different curvature attributes. Elliptical patterns indicate domal-basinal structures with synclastic curvature, that is, principal curvatures of the same sign. Hyperbolicpatterns are diagnostic of anticlastic curvature (saddle-like structures). Such outcrop geometries aregeological examples of Dupin’s indicatrix, the geometrical figure obtained by sectioning a curvedsurface on a plane parallel and almost coincident with the tangent plane. The aspect ratio of Dupin’sindicatrix is theoretically related to the ratio of the principal curvature values for the part of thestructure being considered. This new method allows quantitative assessment of structures on maps andon remote sensing images. Illustrations are given from Wyoming, USA, and Yorkshire, England.Keywords: folds, style, geometry, maps, cross-sections.1. IntroductionFleuty (1964) provided a terminological frameworkfor the description of folds that has served structuralgeologists well for four decades. Fleuty’s classic paperrepresented a significant advance because of the wayexisting basic descriptive terms such as ‘hinge’, ‘limb’and ‘interlimb angle’ were given exact definitions basedon geometry. It enabled structural geologists to applyfamiliar terms to fold structures in a more objectivefashion.Most of the definitions in that paper, includingthe fundamental terms ‘fold hinge line’, ‘fold limb’,‘axial surface’ and ‘inflection line’ are founded ona simple geometrical model of fold geometry: thecylindrical model. A cylindrical fold is one whosesurface everywhere contains a line of fixed orientation;its surface can be generated or swept out by a line whichmoves while remaining parallel to itself. Such foldsshow a curvature only in one sense. The assumption ofcylindrical geometry lies at the heart of the geometricalanalysis of folds. The methods based on this assumptionhave even been applied to structures of more complexgeometry such as refolded folds, by subdividing thesenon-cylindrical structures into smaller regions withapproximately cylindrical properties.There is a general awareness amongst structural geo-logists that all natural folds deviate from a cylindricalform to some degree. For predicting fold geometry inthe subsurface from scant information obtained at thesurface, the cylindrical model may be a satisfactory ap-proximation. However, where more complete data areavailable, as in regions investigated by 3D seismic sur-veys, the cylindrical model is seriously inadequate for* E-mail: lisle@cardiff.ac.ukcharacterizing the fold structures. This is particularly soin the analysis of folds in hydrocarbon provinces, wherestructural closure implies the presence of doubly curvednon-cylindrical fold forms. Basic morphological termssuch as ‘inflection line’, ‘hinge line’ and ‘limb’ cannotbe employed in a strict sense when describing suchfolds.In summary, there is a general lack of tools forthe geometrical analysis of non-cylindrical folds. It iswell known that such structures give rise to distinctiveoutcrop patterns on geological maps or satellite images.For example, many textbooks describe the concentricpatterns that are diagnostic of domes and basins andthe oval disposition of formations that characterizesdoubly-plunging brachyanticlines and brachysynclines(e.g. Billings, 1954; Roberts, 1982; Lisle, 1995).This paper explains how more information can beextracted from such features and outlines the theoreticalbasis for a semi-quantitative interpretation of suchpatterns in terms of the geometry of the surface foldinvolved.2. Folded surfaces and their curvatureOnce the assumption that geological folds havecylindrical forms is relaxed, a more general approachis required to describe the geometry of their surfaces.The theoretical framework for a general treatment ofsurfaces is to be found in the literature on differentialgeometry.At any point P on a folded surface the curvature canbe defined with reference to a cross-section orientednormal to the surface, that is, a plane passing throughthe surface normal N at P (Fig. 1). The value of normalcurvature is given by kn = 1/r , where r is the radius ofcurvature of the curve obtained in the section plane.http://journals.cambridge.org Downloaded: 24 Feb 2014 IP address: 131.251.254.13722 R. J. LISLEFigure 1. Curvature of a folded surface at some point, P . Cross-sections perpendicular to the surface, that is, sections parallelto surface normal N , display different values of curvaturedepending on the direction of the section. The normal curvatureadopts extreme values in two perpendicular directions: theprincipal curvature directions.Unless the local surface geometry around point P isa flat plane or forms a part of a spherical surface, thecurvature obtained in the normal section will vary de-pending on the direction of the section plane. Forall points, regardless of the surface to which theybelong, the change of curvature with change in thedirection of the section is orderly and systematic;there is a direction in which the curvature is greatestand another perpendicular direction along which thenormal curvature has a minimum value. The extremevalues are called the principal curvatures at P andhave values k1 and k2, respectively. These values cantake positive or negative values depending on whetherthe corresponding centre of curvature lies below orabove the surface, that is, whether the curvature isconvex upwards or concave upwards. For normalsections through P in other directions the curvaturehas an intermediate value given by Euler’s equation(see Weatherburn, 1947, p. 73):kn = k1 cos2 θ + k2 sin2 θ, (1)where θ is the angle between the normal section and thedirection of the maximum curvature (Fig. 1). Althoughthe curvature properties of all points on all surfacesobey equation (1), three classes of points can bedistinguished on the basis of the signs of the principalcurvatures. Points where the principal curvatures havethe same sign, like a small fragment of an eggshell, arecalled synclastic points. Where the curvature is convexin one principal direction but concave in the other, like asaddle, the point is classified as anticlastic (‘oppositelycurved’). A third class of points which has one principalcurvature equal to zero, like a part of a cylindrical orconical fold, could be termed monoclastic because theFigure 2. (a) Normal section of a surface. The surface crops outat points A and B, distance 2d apart. The distance depends on theradius of curvature, r , and the distance between the tangent planeand the section plane, h. (b) In the section plane, the distancesd1 and d2 are in a ratio that relates to the ratio of the values ofthe principal curvatures.surface around such a point is curved in one directiononly.If we accept that natural folded surfaces are generallynon-cylindrical, then points of all three kinds might beexpected to occur upon them. The question that nowarises is how such points may be recognized from thetwo-dimensional outcrop patterns observed on outcropsurfaces or geological maps.3. Planar sections through non-cylindrical foldsIt is well known that a cross-section of a fold revealsgeometrical features of the fold. For a cylindrical fold,the most logical and informative cross-section is oneoriented perpendicular to the fold axis. For surfacesthat are more complex, the equivalent natural cross-section does not exist because the curvature attributesvary from point to point in such a way that cannot bedisplayed in a single planar slice. Nevertheless, a cross-section almost coincident with one of the tangent planesof the surface (Fig. 2) is most revealing about the localcurvature characteristics. It is demonstrated below thatsuch unorthodox cross-sections correspond to certainoutcrop patterns observed on geological maps.Figure 2a illustrates how a planar cut through asurface on a section plane AB just below the plane oftangency at point P will intersect the surface twice,at points A and B. The folded surface with radiusof curvature r will crop out at these two points withhttp://journals.cambridge.org Downloaded: 24 Feb 2014 IP address: 131.251.254.13Dupin’s indicatrix 723Figure 3. Air photograph of a periclinal fold in Wyoming (U.S.Geological Survey photo). The elliptical outcrop has an aspectratio of 4.2 indicating the principal curvatures are in the ratio of4.22 = 18.0.distance 2d apart. Referring to Figure 2a, by Pythagoraswe haver 2 = (r − h)2 + d2. (2)Provided the section plane AB is close to the tangentplane, that is, when h is very small and therefore h2 isof negligible magnitude, the curvature is approximatelygiven bykn = 1r= 2hd2. (3)If this relationship is applied in the two principaldirections at a synclastic point (Fig. 2b), the distancesof separation of the two outcrop positions 2d1 and 2d2will be in the ratioRd = 2d22d1. (4)From (3) it can be deduced that the ratio of the aspectratio of the oval outcrop, Rd , relates to the ratio of theprincipal curvatures in absolute values:R2d =(d2d1)2= max |k|min |k| . (5)This is an important result because it permits an in-terpretation of the geometry of the oval outcrop shape interms of the principal curvatures at the synclastic pointon the folded surface. This is illustrated in Figure 3,where the elliptical outcrop has an aspect ratio of 4.2,indicating a curvature ratio of 18.0:1.4. Dupin’s indicatrixThe result above can be generalized for all threetypes of points. By combining (1) with (3) weobtain the equation in polar co-ordinates of the curvecorresponding to the ‘glancing slice’ of a fold, that is,the polar distance d for each direction θ :d =√2hk1 cos2 θ + k2 sin2 θ. (6)The curve (d, θ), illustrated in Figure 4, is a conic calledDupin’s indicatrix after Charles Dupin, the Frenchmathematician who investigated its properties in theearly nineteenth century. It is the figure obtained bytaking a very thin slice off a folded surface, parallel tothe tangent plane. The radius of the indicatrix in anydirection is proportional to the square root of the radiusof normal curvature in this direction (Hilbert & Cohn-Vossen, 1952). If the point is synclastic (k1 and k2 havethe same sign) the curve is an ellipse (Fig. 4a). If theprincipal curvatures differ in sign, as they do in the caseof an anticlastic point, the real values of d define a pairof hyperbolas (Fig. 4b). Finally if the local curvature ismonoclastic (k1 or k2 are zero), the figure consists of apair of straight parallel lines (Fig. 4c).Where Dupin’s indicatrix of elliptical, hyperbolic orstraight-line types can be recognized on a geologicalmap or cross-section, we have the opportunity toinvestigate principal curvatures and their directions forthat small part of the folded structure. The practicalitiesof the method are explained in the next section.5. MethodThe first stage in the analysis of local curvaturefrom maps or sections is to correctly identify Dupin’sindicatrix, that is, elliptical, hyperbolic or straight-line outcrop patterns. In the case of observationsfrom maps, care must be taken not to confuse thepatterns with similar shapes arising from the effectsof topography, in particular where there are isolatedhills (elliptical shapes) and mountain cols (hyperbolicforms). In general, the accuracy of the method willbe detrimentally affected by any non-planarity ofthe cross-section surface. Inspection of topographiccontours in the vicinity of the shape should help checkwhether the pattern is indeed a valid Dupin’s indicatrix,that is, whether the pattern is an expression of folding.Where geological boundaries exhibit a nested ar-rangement of ellipses or hyperbolas, the shape of theinnermost boundary should be taken as the Dupinindicatrix (DI) (Fig. 4). These are shapes produced bysurfaces whose tangent plane is closest to the sectionplane, a condition that produces an outcrop patternclosest in form to a DI. To obtain a true DI, the sectionplane must be close to the tangent plane (h is small; seeFig. 2a). In this respect, it is not simply the distance hbut this value as a proportion of the radius of curvaturethat influences the degree to which the outcrop shapedeparts from the ideal DI. Where h/r is large theoutcrop pattern may deviate significantly from the idealshape of a DI. This is illustrated in the fold interferencehttp://journals.cambridge.org Downloaded: 24 Feb 2014 IP address: 131.251.254.13724 R. J. LISLEFigure 4. Dupin’s indicatrix. (a) Elliptical type indicating synclastic curvature. (b) Hyperbolic type indicating anticlastic curvature.(c) Pair of straight lines indicating a fold with a zero principal curvature value (‘monoclastic’).pattern in Figure 5 by the crescentic shapes of the tracesof outermost surfaces.5.a. Elliptical patternsOnce an elliptical DI is identified, we can concludethat the local curvature of fold is synclastic in nature.Furthermore, the directions of the minor and majoraxes of the ellipse indicate the principal curvaturedirections, k1 and k2. The ratio of the lengths ofminor and major axes Rd can be used to calculatethe ratio of the principal curvatures using equation (4).The elliptical pattern on its own does not allowthe sign of the curvatures to be determined, thoughthe short axis direction indicates the direction ofgreatest curvature (ignoring the sign). Therefore,without additional information, the DI will not indicatewhether the structure is convex (antiformal) or concave(synformal).5.b. Hyperbolic patternsDupin indicatrices of hyperbolic type indicate that thefolded layering in the vicinity has anticlastic curvature,that is, they are associated with points on saddle-Figure 5. Interference pattern due to refolding (based onRoberts, 1989, fig. 241). Innermost closed traces of bandingdefine an elliptical Dupin’s indicatrix, whereas outer traces showmore complex forms.http://journals.cambridge.org Downloaded: 24 Feb 2014 IP address: 131.251.254.13Dupin’s indicatrix 725Figure 6. The analysis of hyperbolic patterns. See text forexplanation.shaped structures, or their upside-down equivalents,called ‘shoe horns’ by Lisle (1995, p. 35). The principalcurvature directions are determined from the symmetryof the hyperbolic pattern, but from the DI geometryalone it is not possible to distinguish k1 from k2.The ratio of the absolute maximum and minimumcurvatures is found as follows (Fig. 6):(1) Locate the points of maximum curvature on eacharm of the hyperbola (points A and B in Fig. 6). One ofthe principal curvature directions is parallel to the lineAB, whilst the other principal direction is perpendicularto AB. We refer to these principal curvatures as k p andkq respectively.(2) From the mid-point of line AB, sketch the linesthat are asymptotic to the hyperbola. These representthe directions of lines of zero normal curvature (kn = 0)within the folded surface.(3) Measure the angle, θ , between the asymptote andthe line AB (see Fig. 6).The angle of inclination of the asymptotic lines, θ ,relates to the ratio of the principal curvatures. By settingkn to zero in equation (1), and rearranging, we obtain∣∣∣∣kpkq∣∣∣∣ = tan2 θ. (7)The maximum curvature (ignoring sign) is therefore inthe q direction when θ is less than 45◦ and in the pdirection when θ exceeds 45◦.5.c. Straight line patternsWhere the outcrop of a geological boundary consistsof two closely spaced parallel lines, the structuralinterpretation is straightforward. This figure impliesa monoclastic geometry for the folded surface, that is,the case where one of the principal curvatures equalszero and ratio of the principal curvatures is infinite. Thedirection of the bisector of the straight lines indicatesthe zero principal curvature direction.Figure 7. Curvature analysis from Dupin’s patterns in theChesterfield area, England. Geological boundaries taken fromSheet 112, 1 inch scale map, Geological Survey of Great Britain.The ratio of the principal curvatures are shown, together withthe directions of the principal curvature axes.6. ExamplesDupin patterns are common. This is because thesheet dip of folded horizons is frequently closeto horizontal so that the topographic surface oftenintersects such horizons tangentially. Examples aboundon geological maps of the Variscan fold belt of theArdennes, the Cantabrian Mountains and South Walesand on Landsat images of the Appalachian fold beltin Pennsylvania. Figure 7 is a geological map ofa region 8 km southeast of Chesterfield, England,where Dupin patterns are defined by the outcroppattern of contacts within the Upper CarboniferousCoal Measures. Elliptical and hyperbolic outcropshapes both indicate that direction of greatest absolutecurvature varies from 019◦ and 049◦N. The ratio ofthe principal curvatures varies between 2.0 and 2.7 andrelates to the relative intensities of Variscan foldingin two directions analysed by other means (Lisle,1999).Dupin patterns are also abundant on structurecontour maps because each structure contour portraysthe shape of a horizontal slice of a geologicalsurface. Figure 8 illustrates examples of elliptical andhyperbolic types on the structure contour map (topof El Abra Limestone) of the Atn field, Veracruz,Mexico. These can be interpreted in terms of principalcurvatures and their directions.http://journals.cambridge.org Downloaded: 24 Feb 2014 IP address: 131.251.254.13726 Dupin’s indicatrixFigure 8. Examples of elliptical (E) and hyperbolic (H) Dupinindicatrices on structure contour maps. Atún oilfield, Veracruz,Mexico (from Viniegra & Castillo-Tejero, 1970).7. ConclusionsThe qualitative significance of the various patterns re-ferred to in this paper is already well known. However,the new method outlined here delivers quantitativeinformation of three-dimensional geometry from suchtwo-dimensional structural images from outcrop maps,structure contours, seismic time slices and satelliteimages. It is stressed that information relates to localstructural characteristics and its accuracy is adverselyaffected by irregularities of the cross-section surface(that is, the flatness of the land surface) and the distanceseparating the section surface and the tangent plane tothe folded surface.Acknowledgements. Alex Maltman and a second refereemade useful suggestions for improving the manuscript.ReferencesBILLINGS, M. P. 1954. Structural Geology. Englewood Cliffs,N.J.: Prentice-Hall.FLEUTY, M. J. 1964. The description of folds. Proceedings,Geologists Association 75, 461–91.HILBERT, D. & COHN-VOSSEN, S. 1952. Geometry and theimagination. New York: Chelsea Publishing Co.LISLE, R. J. 1995. Geological Structures and Maps. 2ndedition. Oxford: Butterworth-Heinemann.LISLE, R. J. 1999. A new Variscan deformation map ofEngland and Wales. Tectonophysics 309, 27–39.ROBERTS, J. L. 1982. Introduction to geological maps andstructures. Oxford: Pergamon Press.ROBERTS, J. L. 1989. A field guide to geological structures.London: Macmillan.VINIEGRA, F. O. & CASTILLO-TEJERO, C. 1970. GoldenLane Fields, Veracruz, Mexico. In Geology of GiantPetroleum Fields (ed. M.T. Halbouty), pp. 309–25.American Association of Petroleum Geologists, Memoirno. 14.WEATHERBURN, C. E. 1947. Differential geometry ofthree dimensions. Volume 1. Cambridge: CambridgeUniversity Press.

Dupin's indicatrix: a tool for quantifying periclinal folds on maps

https://orca.cardiff.ac.uk/id/eprint/10411/1/Lisle%202003.pdf

Dupin's indicatrix: a tool for quantifying periclinal folds on maps

Abstract

Similar works

Full text

Available Versions

Online Research @ Cardiff