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Simplified Fourier analysis of fold shapes
Tectonophysics - Elsevier Publishing Company, Amsterdam Printed in The Netherlands
SIMPLIFIED FOURIER ANALYSIS OF FOLD SHAPES C. L. STABLER Department of Geology, Chelsea College of Science and Technology, London (Great Britain) (Received January 1, 1968) SUMMARY
The shape of the smallest characteristic unit of a fold profile, a l’quarter” wavelength, can be matched by a curve made up from suitable combinations of the first and third harmonics of a sine series called an E-T curve. This approach has been developed by simplification of the complex Fourier series and is demonstrated in an analysed field example. Other examples have confirmed the speed and versatility of the method. The amplitudes of the two harmonics analyse, as well as describe, the components of shape. They have been used to plot a chart on which the range of shapes expected in folded rocks lie, and upon which the inter-relationships of deformation shapes can be studied. INTRODUCTION
With the increasing interest in statistical methods of handling geological data, there is a growing need for methods of describing basic geologic phenomena in numerical terms. This need in structural geology is being met in two very different ways. One group of geologists have been developing techniques for the automatic analysis of shapes of surfaces as depicted by mechanically plotted and contoured field data. Loudon (1964) analysed shapes statistically in terms of slope; Chappel (1964) developed the harmonic analysis of shape as a function of arc length; and Harbaugh and Preston (1965) applied detailed Fourier analysis to fold shapes. While there can be little doubt about the accuracy and future value of these objective methods, the computations gave shape constants too complex for every day use by the majority of‘geologists. The other basically different approach has been to choose a single geometrical curve as a model to fit fold profiles, Mertie (1959) adopted the ellipse, and upon it based an elaborate system of classification and measurement of folds. The author, in unpublished work, has attempted to fit hyperbolic curves to fold profiles. Although both are simple models to use, neither ellipses nor hyperbolae fit a wide enough range of shapes for general application. To satisfy the need for an easy means of depicting a wide variety of fold shapes in simple terms consistent with established structural concepts this brief communication demonstrates the basis of a method in which another curve is fitted to fold profiles. This curve is a very general type of curve made up by combining first and third sine harmonics and hereafter called F-T curve. Tectonophysics,
6 (4) (1968) 343-350
343
Hime Point
C 344
Tectonophysics,
6 (4) (1968)
343-350
THEF-TCURVE
One of the simplifying steps necessary in applying the following method .is to divide a fold into “quarter 11wavelength units. Because folds in general are asymmetric about their hinges and inflexions, analysis of a whole anticline or an anticline-syncline fold-pair would be complex. On the other hand, it is simple to analyse the single bend in a nquartertl wavelength, from inflexion point to hinge point. The method of dividing asymmetric folds into these units is demonstrated in Fig.1. Each “quarter” wavelength unit is defined by x and y reference axes, terms used by the author for datum lines from which to take measurements. The lty reference axis” is constructed as the line passing through the hinge point and parallel to the bisector of the angle formed between the tangents to the fold at its inflexion points, The ‘IX reference axis” is then the normal from each inflexion point to the ny reference axis”. The reference axes bear no fixed relationship to the axes of symmetry or the traces of hinge surfaces. Established theoretical and experimental work on buckling suggested a consideration of the closeness of fit of a natural fold profile to the form of a periodic wave of sinusoidal type. But because many folds did not closely correspond to single simple sine waves the application of Fourier analysis to the problem was studied. A Fourier series is the summation of an infinite number of harmonics of sine and cosine waves of decreasing amplitudes which may be used to describe any periodically repeating shape, from a series of circular arcs to a saw tooth. Represented mathematically the series is: a,+aIsin0+blcos0+a2sin2tI+b2cos2tJ. . . .+a n sin nO+b,
cos nC3
. . (1)
a, is a constant of position, and al, a2 etc., bl, b2 etc., are amplitudes, and 0 = 2 nx/h ? in which X is the wavelength and x is distance. Applied to fold shapes, for the quarter wavelength unit of a symmetrical anticline placed with its inflexion point at the origin of the curve, all cosine terms, even-numbered sine terms, and the constant, are zero. Thus the simplified series: al sine
+ a3 sin 30 . . . + azn-l sin(2n-1)8
(2)
represents a general curve which will describe any shape between a point of zero curvature (inflexion point) and a point of maximum curvature (hinge point). As such this general curve is thought to form a sound basis for the analysis of any geological shape.
Fig.1. Fold terms and “quarter” wave length units: A. descriptive fold terms on a symmetrical fold-pair; B. “quarterl1 wave length units on upright asymmetrical fold; C. “quarter”wave length units on inclined asymmetrical fold. Each unit is bounded by x and y reference axes. The y reference axis is constructed parallel to the bisector, b, of the two tangents, c and d, to the fold at its inflexion point I. The x reference axis is drawn as the normal from the inflexion points to the y reference axis. Tectonophysics, 6 (4)(1968)343-350
345
cm
-
Fig.2 Analysis of a fold - a random section through folds in Moinian, at Camas-Luinie, Killilan, Kintail, Ross-shire. Three measurements, y1 , 3’2 and A, of one of the fold profiles are used in the text to form simultaneous equations of the odd sine harmonic series. Their solutions give the F-T curve: 2.55 sin 0 + 0.015 sin 30. This curve is a very close fit and is constructed beside the profile for comparison. The size of the quarter-wavelen~h profile, L, is 1.7 inches.
When applied to examples of natural folds this general curve was found to be capable of further simplification since high harmonics became numerically insignificant. The application of the curve to the analysis of a fold is now presented in detail, Fig.2, while the general conclusions about this and other examples follow. The “x reference axis”, constructed as demonstrated in Fig. 1, was used as a datum from which to make measurements. In this example, three distances, yl, y 2 and A were measured at 30° intervals and used to form three simultaneous equations from the first three terms of the series in eq.2. These simultaneous equations are: Yl = alsin300 + a3sin900 + a5sin1500
y2 = aIsin600 + a3sin1800 + a5sin3000 A
= alsin900
+ a3sin2’700 + assin90O
Their solutions expressing measurements are: 346
values of amplitudes in terms of the three
Tectonophysics, 6 (4)(1968)343-350
Anomalovi show (dual infkxionr)
Fig.3. F-T curve chart - showing the range of shapes of F-T curves plotted with hinges at their Fand T coordinates. F is the scaled amplitude of the first harmonic al/L and T is the scaled amplitude of the third harmonic as/L, L being the length of the “quarter” wave length.
Tectonophysics, 6 (4) (1968) 343-350
347
al = l/3
(A + yl) + [y2/(2sin60°)]
aa = l/3
(2~1 - A) [1/(2sin60°)
a5 = l/3
(A + yl)*[(-ya)/(2sin60°)]
= 0.5761
The taking of Only three measurements on the fold is justified by the fact that it has been found by more detailed analysis that the resultant value of the fifth harmonic is so small that the first and third harmonics of the general curve represent and analyse the shape of the fold profile well within the errors of measurement. Several outcropping sections of folds of different sizes from various parts of the British Isles have been analysed in a similar way. The results confirm that only the first and third terms of the curve are needed for a close match (to within l!%d. This curve has here been called an F-T curve, P for first and T for third harmonic. However, in the rarer cases of folds resembling circular arcs and folds with very sharp hinges, P. Hudleston (personal communication, 1967) has shown that a curve with afew more harmonics would be needed for a good fit. Such examples could be analysed by taking more measurements over closer spacings, and from simultaneous equations derived from eq.2, evaluating the higher harmonics. In order to be able to compare the true shapes of folds of different sizes it is essential to eliminate the factor of size. This has been achieved in this method by reducing all tlquartern wavelengths to unity, whereupon the harmonics are reduced by dividing them by the tlquarteru wavelength, L. These scaled values of the harmonics F = al/L, T = as/L are dimensionless parameters which can be used alone in an analysis to describe shape in terms of its harmonic components. (The ‘Iquarter wavelength measurement is considered as the unique size parameter). The range of curve shapes which can be formed by various combinations of F and T in F-T curves is demonstrated on the F-T curve chart, Fig.3. These curves have unit nquarteru wavelength and are constructed with hinges at their F and T co-ordinates. Two interesting points emerge. Firstly, T can be positive or negative, modifying the basic shape of the first harmonic by flattening it when added or sharpening it when subtracted, as explained in Fig. 4. Secondly, there is a central area on the chart in which common fold shapes lie, extending to high values of F. Areas on either side, for values of T greater than about one tenth of P, contain anomalous curves. Such E-T charts can be used to plot calculated F and T harmonic values for folds, each fold having a unique plot. This was done for .a variety of natural fold profiles representing the whole range of continuous deformation phenomena from buckling folds in limestones to ductile flows in gneiss. As was expected, gentle folds lay at low values of F, tight folds at high values of 1”; blunt-ended folds had high positive T values and sharp folds had high negative T.values. All fold types can be described in this wav without having to take into account any theories of their mode of origin. This chart was also designed for quick shape determinations in the field by holding a transparent print of it up to the eye, at a suitable distance to scale the size and match exposed folds through it. The analysis of folds in terms of their component harmonics, could be achieved by compilation and interpretationeither graphically by the field worker using F-T charts, or statistically by the experimenter using data storage 348
Tectonophysics, 6 (4) (1968) 34S350
------sharp
F-T
curve
=
Fig.4. Variations in the shape of an F-T traction (B) of first and third harmonics.
curve by addition (A) sub-
methods. Analysis by geologists working on the many different aspects of rock folding and flowage will uncloubtably lead to an increased understanding of formation and a meaningful representation of geometrical variability.
CONCLUSIONS
This study illustrates how the formidable task of analysing geological features by complex mathematical functions can be overcome once the features have been broken down into simpler units. In this case the Fourier series has been reduced to a curve made up of only two terms, the first and third sine harmonics, for matching the shape of llquarteru wavelength units of fold profiles. This F-T curve, tested on a wide range of folds, has immediate application. It is quick and easy to use and needs only four measurements on the folded surface. The values of the amplitudes of the two constituent harmonics may be used to compare related folds, either by graphical representation on an F-T chart or by statistical analysis. It is hoped that geologists working with similar problems will be able to test, confirm and develop the applications of these curves. It is intended to extend the method to three dimensional analysis of folds, the relationship between successive folded surfaces, and thence to the prediction of structure at depth.
Tectonophysics, 6 (4) (1968) 343-350
349
ACKNOWLEDGEMENTS
The author wishes to thank the following people whose discussions most helpful in evolving the theme of this communication: H.A. Taylor, A.J. Barber, L.V. Illing, P. Hudleston, J.G. Ramsay and G.D. Hobson.
were
REFERENCES Benny,
L.B., 1958. Mathematics for Students of Engineering and Applied Science. Oxford Univ. Press, London, 783 pp. Chappel, W.M., 1964. A mathematical study of finite - Amplitude rock folding. Trans. Am. Geophys. Union, 45(l): 104. Harbaugh, J.W. and Preston, F.W., 1965. Fourier series analysis in geology. Symp. Computer Appl. Mineral Exploration, 6th, Tucson, Ariz., Rl-R46. Loudon, T.V., 1964. Computer analysis of orientation data in structural geology. Office Naval Res., Geograph, Branch, Tech. Rept., 13: 130 pp. Matthews, D.H., 1958. Dimensions of asymmetric folds. Geol. Mag., 95(6): 511-513. Mertie, J.B., Jr., 1959. Classification, delineation and measurement of non-parallel folds. U.S. Geol. Surv., Profess. Papers, 314-E: 91-124. Ramsay, J.G., 1967. Folding and Fracturing of Rocks. McGraw-Hill, New York, N.Y., 568 pp. Stone, R.O. and Dugundji, J., 1965. A study of microrelief - its mapping, classification, and quantification by means of a Fourier analysis. Eng. Geol., l(2): 8s-187. Whitten, E.H.T., 1966. Sequential multivariate regressive methods and scalars in the study of fold-geometry variability. J. Geol., ‘74(5,2): 744-763.
350
Tectonophysics,
6 (4) (1968) 343-350
SIMPLIFIED FOURIER ANALYSIS OF FOLD SHAPES C. L. STABLER Department of Geology, Chelsea College of Science and Technology, London (Great Britain) (Received January 1, 1968) SUMMARY
The shape of the smallest characteristic unit of a fold profile, a l’quarter” wavelength, can be matched by a curve made up from suitable combinations of the first and third harmonics of a sine series called an E-T curve. This approach has been developed by simplification of the complex Fourier series and is demonstrated in an analysed field example. Other examples have confirmed the speed and versatility of the method. The amplitudes of the two harmonics analyse, as well as describe, the components of shape. They have been used to plot a chart on which the range of shapes expected in folded rocks lie, and upon which the inter-relationships of deformation shapes can be studied. INTRODUCTION
With the increasing interest in statistical methods of handling geological data, there is a growing need for methods of describing basic geologic phenomena in numerical terms. This need in structural geology is being met in two very different ways. One group of geologists have been developing techniques for the automatic analysis of shapes of surfaces as depicted by mechanically plotted and contoured field data. Loudon (1964) analysed shapes statistically in terms of slope; Chappel (1964) developed the harmonic analysis of shape as a function of arc length; and Harbaugh and Preston (1965) applied detailed Fourier analysis to fold shapes. While there can be little doubt about the accuracy and future value of these objective methods, the computations gave shape constants too complex for every day use by the majority of‘geologists. The other basically different approach has been to choose a single geometrical curve as a model to fit fold profiles, Mertie (1959) adopted the ellipse, and upon it based an elaborate system of classification and measurement of folds. The author, in unpublished work, has attempted to fit hyperbolic curves to fold profiles. Although both are simple models to use, neither ellipses nor hyperbolae fit a wide enough range of shapes for general application. To satisfy the need for an easy means of depicting a wide variety of fold shapes in simple terms consistent with established structural concepts this brief communication demonstrates the basis of a method in which another curve is fitted to fold profiles. This curve is a very general type of curve made up by combining first and third sine harmonics and hereafter called F-T curve. Tectonophysics,
6 (4) (1968) 343-350
343
Hime Point
C 344
Tectonophysics,
6 (4) (1968)
343-350
THEF-TCURVE
One of the simplifying steps necessary in applying the following method .is to divide a fold into “quarter 11wavelength units. Because folds in general are asymmetric about their hinges and inflexions, analysis of a whole anticline or an anticline-syncline fold-pair would be complex. On the other hand, it is simple to analyse the single bend in a nquartertl wavelength, from inflexion point to hinge point. The method of dividing asymmetric folds into these units is demonstrated in Fig.1. Each “quarter” wavelength unit is defined by x and y reference axes, terms used by the author for datum lines from which to take measurements. The lty reference axis” is constructed as the line passing through the hinge point and parallel to the bisector of the angle formed between the tangents to the fold at its inflexion points, The ‘IX reference axis” is then the normal from each inflexion point to the ny reference axis”. The reference axes bear no fixed relationship to the axes of symmetry or the traces of hinge surfaces. Established theoretical and experimental work on buckling suggested a consideration of the closeness of fit of a natural fold profile to the form of a periodic wave of sinusoidal type. But because many folds did not closely correspond to single simple sine waves the application of Fourier analysis to the problem was studied. A Fourier series is the summation of an infinite number of harmonics of sine and cosine waves of decreasing amplitudes which may be used to describe any periodically repeating shape, from a series of circular arcs to a saw tooth. Represented mathematically the series is: a,+aIsin0+blcos0+a2sin2tI+b2cos2tJ. . . .+a n sin nO+b,
cos nC3
. . (1)
a, is a constant of position, and al, a2 etc., bl, b2 etc., are amplitudes, and 0 = 2 nx/h ? in which X is the wavelength and x is distance. Applied to fold shapes, for the quarter wavelength unit of a symmetrical anticline placed with its inflexion point at the origin of the curve, all cosine terms, even-numbered sine terms, and the constant, are zero. Thus the simplified series: al sine
+ a3 sin 30 . . . + azn-l sin(2n-1)8
(2)
represents a general curve which will describe any shape between a point of zero curvature (inflexion point) and a point of maximum curvature (hinge point). As such this general curve is thought to form a sound basis for the analysis of any geological shape.
Fig.1. Fold terms and “quarter” wave length units: A. descriptive fold terms on a symmetrical fold-pair; B. “quarterl1 wave length units on upright asymmetrical fold; C. “quarter”wave length units on inclined asymmetrical fold. Each unit is bounded by x and y reference axes. The y reference axis is constructed parallel to the bisector, b, of the two tangents, c and d, to the fold at its inflexion point I. The x reference axis is drawn as the normal from the inflexion points to the y reference axis. Tectonophysics, 6 (4)(1968)343-350
345
cm
-
Fig.2 Analysis of a fold - a random section through folds in Moinian, at Camas-Luinie, Killilan, Kintail, Ross-shire. Three measurements, y1 , 3’2 and A, of one of the fold profiles are used in the text to form simultaneous equations of the odd sine harmonic series. Their solutions give the F-T curve: 2.55 sin 0 + 0.015 sin 30. This curve is a very close fit and is constructed beside the profile for comparison. The size of the quarter-wavelen~h profile, L, is 1.7 inches.
When applied to examples of natural folds this general curve was found to be capable of further simplification since high harmonics became numerically insignificant. The application of the curve to the analysis of a fold is now presented in detail, Fig.2, while the general conclusions about this and other examples follow. The “x reference axis”, constructed as demonstrated in Fig. 1, was used as a datum from which to make measurements. In this example, three distances, yl, y 2 and A were measured at 30° intervals and used to form three simultaneous equations from the first three terms of the series in eq.2. These simultaneous equations are: Yl = alsin300 + a3sin900 + a5sin1500
y2 = aIsin600 + a3sin1800 + a5sin3000 A
= alsin900
+ a3sin2’700 + assin90O
Their solutions expressing measurements are: 346
values of amplitudes in terms of the three
Tectonophysics, 6 (4)(1968)343-350
Anomalovi show (dual infkxionr)
Fig.3. F-T curve chart - showing the range of shapes of F-T curves plotted with hinges at their Fand T coordinates. F is the scaled amplitude of the first harmonic al/L and T is the scaled amplitude of the third harmonic as/L, L being the length of the “quarter” wave length.
Tectonophysics, 6 (4) (1968) 343-350
347
al = l/3
(A + yl) + [y2/(2sin60°)]
aa = l/3
(2~1 - A) [1/(2sin60°)
a5 = l/3
(A + yl)*[(-ya)/(2sin60°)]
= 0.5761
The taking of Only three measurements on the fold is justified by the fact that it has been found by more detailed analysis that the resultant value of the fifth harmonic is so small that the first and third harmonics of the general curve represent and analyse the shape of the fold profile well within the errors of measurement. Several outcropping sections of folds of different sizes from various parts of the British Isles have been analysed in a similar way. The results confirm that only the first and third terms of the curve are needed for a close match (to within l!%d. This curve has here been called an F-T curve, P for first and T for third harmonic. However, in the rarer cases of folds resembling circular arcs and folds with very sharp hinges, P. Hudleston (personal communication, 1967) has shown that a curve with afew more harmonics would be needed for a good fit. Such examples could be analysed by taking more measurements over closer spacings, and from simultaneous equations derived from eq.2, evaluating the higher harmonics. In order to be able to compare the true shapes of folds of different sizes it is essential to eliminate the factor of size. This has been achieved in this method by reducing all tlquartern wavelengths to unity, whereupon the harmonics are reduced by dividing them by the tlquarteru wavelength, L. These scaled values of the harmonics F = al/L, T = as/L are dimensionless parameters which can be used alone in an analysis to describe shape in terms of its harmonic components. (The ‘Iquarter wavelength measurement is considered as the unique size parameter). The range of curve shapes which can be formed by various combinations of F and T in F-T curves is demonstrated on the F-T curve chart, Fig.3. These curves have unit nquarteru wavelength and are constructed with hinges at their F and T co-ordinates. Two interesting points emerge. Firstly, T can be positive or negative, modifying the basic shape of the first harmonic by flattening it when added or sharpening it when subtracted, as explained in Fig. 4. Secondly, there is a central area on the chart in which common fold shapes lie, extending to high values of F. Areas on either side, for values of T greater than about one tenth of P, contain anomalous curves. Such E-T charts can be used to plot calculated F and T harmonic values for folds, each fold having a unique plot. This was done for .a variety of natural fold profiles representing the whole range of continuous deformation phenomena from buckling folds in limestones to ductile flows in gneiss. As was expected, gentle folds lay at low values of F, tight folds at high values of 1”; blunt-ended folds had high positive T values and sharp folds had high negative T.values. All fold types can be described in this wav without having to take into account any theories of their mode of origin. This chart was also designed for quick shape determinations in the field by holding a transparent print of it up to the eye, at a suitable distance to scale the size and match exposed folds through it. The analysis of folds in terms of their component harmonics, could be achieved by compilation and interpretationeither graphically by the field worker using F-T charts, or statistically by the experimenter using data storage 348
Tectonophysics, 6 (4) (1968) 34S350
------sharp
F-T
curve
=
Fig.4. Variations in the shape of an F-T traction (B) of first and third harmonics.
curve by addition (A) sub-
methods. Analysis by geologists working on the many different aspects of rock folding and flowage will uncloubtably lead to an increased understanding of formation and a meaningful representation of geometrical variability.
CONCLUSIONS
This study illustrates how the formidable task of analysing geological features by complex mathematical functions can be overcome once the features have been broken down into simpler units. In this case the Fourier series has been reduced to a curve made up of only two terms, the first and third sine harmonics, for matching the shape of llquarteru wavelength units of fold profiles. This F-T curve, tested on a wide range of folds, has immediate application. It is quick and easy to use and needs only four measurements on the folded surface. The values of the amplitudes of the two constituent harmonics may be used to compare related folds, either by graphical representation on an F-T chart or by statistical analysis. It is hoped that geologists working with similar problems will be able to test, confirm and develop the applications of these curves. It is intended to extend the method to three dimensional analysis of folds, the relationship between successive folded surfaces, and thence to the prediction of structure at depth.
Tectonophysics, 6 (4) (1968) 343-350
349
ACKNOWLEDGEMENTS
The author wishes to thank the following people whose discussions most helpful in evolving the theme of this communication: H.A. Taylor, A.J. Barber, L.V. Illing, P. Hudleston, J.G. Ramsay and G.D. Hobson.
were
REFERENCES Benny,
L.B., 1958. Mathematics for Students of Engineering and Applied Science. Oxford Univ. Press, London, 783 pp. Chappel, W.M., 1964. A mathematical study of finite - Amplitude rock folding. Trans. Am. Geophys. Union, 45(l): 104. Harbaugh, J.W. and Preston, F.W., 1965. Fourier series analysis in geology. Symp. Computer Appl. Mineral Exploration, 6th, Tucson, Ariz., Rl-R46. Loudon, T.V., 1964. Computer analysis of orientation data in structural geology. Office Naval Res., Geograph, Branch, Tech. Rept., 13: 130 pp. Matthews, D.H., 1958. Dimensions of asymmetric folds. Geol. Mag., 95(6): 511-513. Mertie, J.B., Jr., 1959. Classification, delineation and measurement of non-parallel folds. U.S. Geol. Surv., Profess. Papers, 314-E: 91-124. Ramsay, J.G., 1967. Folding and Fracturing of Rocks. McGraw-Hill, New York, N.Y., 568 pp. Stone, R.O. and Dugundji, J., 1965. A study of microrelief - its mapping, classification, and quantification by means of a Fourier analysis. Eng. Geol., l(2): 8s-187. Whitten, E.H.T., 1966. Sequential multivariate regressive methods and scalars in the study of fold-geometry variability. J. Geol., ‘74(5,2): 744-763.
350
Tectonophysics,
6 (4) (1968) 343-350