Strain and Competence Contrast Estimation from Fold Shape

Tectonophysics 340 (2001) 195–213 www.elsevier.com/locate/tecto

Strain and competence contrast estimation from fold shape

Stefan M. Schmalholz*, Yuri Yu. Podladchikov

Geologisches Institut, ETH Zentrum, 8092 Zu¨rich, Switzerland Received 14 February 2000; accepted 21 February 2001

Abstract

A new method to estimate strain and competence contrast from natural fold shapes is developed and verified by analogue and numerical experiments. Strain is estimated relative to the nucleation amplitude, AN, which is the fold amplitude when the amplification velocities caused by kinematic layer thickening and dynamic folding are identical. AN is defined as the initial amplitude corresponding to zero strain because folding at amplitudes smaller than AN is dominantly by kinematic layer thickening. For amplitudes larger than AN, estimates of strain and competence contrast are contoured in thickness-to-wavelength (H/l) and amplitude-to-wavelength (A/l) space. These quantities can be measured for any observed fold shape. Contour maps are constructed using existing linear theories of folding, a new nonlinear theory of folding and numerical simulations, all for single-layer folding. The method represents a significant improvement to the arc length method. The strain estimation method is applied to folds in viscous (Newtonian), power-law (non-Newtonian) and viscoelastic layers. Also, strain partitioning in fold trains is investigated. Strain partitioning refers to the difference in strain accommodated by individual folds in the fold train and by the whole fold train. Fold trains within layers exhibiting viscous and viscoelastic rheology show different characteristic strain partitioning patterns. Strain partitioning patterns of natural fold trains can be used to assess the rheological behaviour during fold initiation. D 2001 Elsevier Science B.V. All rights reserved.

Keywords: Strain estimation; Folding; Fold shape; Competence contrast; Strain partitioning; Fold trains

1. Introduction reconstructions of folded regions such as mountain belts. Although numerous studies have investigated A major geological question concerning folding is the mechanics of folding using analytical techniques how much strain (or shortening) is associated with (e.g., Biot, 1961; Johnson and Fletcher, 1994; Hunt et observed fold shapes. In this study, strain is defined as al., 1996), analogue experiments (e.g., Currie et al., the difference between the deformed and the initial 1962; Ramberg, 1963; Hudleston, 1973; Abbassi and length of a line element normalized by its initial Mancktelow, 1992) and numerical simulations (e.g., length. Information of strain is relevant to palinspastic Dieterich, 1970; Cobbold, 1977; Zhang et al., 1996; Mancktelow, 1999; Schmalholz et al., 2001), there is no satisfactory method to estimate strain from natural * Corresponding author. Now at Geomodelling Solutions fold shapes. One reason is that the existing analytical GmbH, Binzstrasse 18, 8045 Zu¨rich, Switzerland. Fax: +41-1- folding solutions are only valid for small limb dips 455-6390. E-mail address: [email protected] (10° to 15°, e.g., Fletcher and Sherwin, 1978), (S.M. Schmalholz). whereas most natural folds exhibit much larger limb

0040-1951/01/$ - see front matter D 2001 Elsevier Science B.V. All rights reserved. PII: S 0040-1951(01)00151-2 196 S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213 dips. An intuitively attractive estimation method is the is accommodated while amplitudes are smaller than so-called arc length method (e.g., Dahlstrom, 1969). AN is ignored because this strain is not recorded This method assumes that the observed fold arc length through the fold shape. corresponds to the initial fold wavelength (Fig. 1). The variation of fold shape with strain depends on Hence, strain can be estimated from the difference material properties. This was shown by analogue and between the observed arc length and fold wavelength. numerical experiments where the variation of geo- However, the arc length method results in large errors metrical parameters has been recorded as a function of if it is applied to folds that developed in settings where strain. Manifestations of material dependence are the the competence contrast between the folded layer and alteration of the ratios of (i) amplitude to initial its matrix is small. These errors are a result of the amplitude (Chapple, 1968; Hudleston and Stephans- relatively large component of homogeneous layer son, 1973; Abbassi and Mancktelow, 1992), (ii) arc thickening (Sherwin and Chapple, 1968). This study length to initial arc length (Hudleston, 1973; Johnson applies a new analytical solution for folding valid up and Fletcher, 1994), (iii) amplitude to thickness to large limb dips (Schmalholz and Podladchikov, (Hudleston, 1973), (iv) amplitude to wavelength (Cur- 2000) and numerical simulations of folding to develop rie et al., 1962) and (v) wavelength to thickness (Lan a strain estimation method. This method reliably and Hudleston, 1995a). One specific fold shape, estimates strain and has the additional advantage that corresponding to different material properties, may it enables estimation of the competence contrast. theoretically correspond to different strains. Without The unknown material properties and initial geom- knowledge of the material properties, a unique strain etry of natural folds are the major obstacles to strain estimate is not possible from a single geometrical estimation. An essentially flat layer with infinitesimal parameter that quantifies the fold shape. To avoid this fold amplitudes may accommodate several hundreds obstacle, we employ two observable geometrical of percent strain by homogeneous layer thickening parameters simultaneously to characterize folds. Thus, without developing observable fold limb dips. A layer during progressive folding, strain is contoured in the that appears unfolded in the field may have accom- thickness-to-wavelength (H/l) and the amplitude-to- modated substantial layer-parallel strain. With increas- wavelength (A/l) space. These ratios can be measured ing limb dip the strain that is accommodated by layer for any observed fold shape. Strain contours as a thickening decreases, and that accommodated by function of A/l and H/l provide a ‘‘strain contour folding increases. In this study, we define a nucleation map’’, which uniquely defines strain for a continuous amplitude, AN (Table 1), as the initial (or reference) range of competence contrasts. The idea of this study amplitude for strain estimation in order to circumvent is to use two observable geometrical ratios to con- the obstacle of unknown initial geometry. Only the strain two unknowns: the competence contrast and the strain that is accommodated while amplitudes are strain that is accommodated by the folds after they larger than AN is, therefore, estimated. The strain that exceed the nucleation amplitude.

Fig. 1. Major geometrical parameters of folds are the arc length (Larc), wavelength (l), amplitude (A) and thickness (H). In the initial folding stages, where amplitudes are very small, the arc length has approximately the same length than the wavelength. S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213 197

Table 1 relationships for the nonlinear folding stages (Schmal- List of symbols used in the text holz and Podladchikov, 2000) are used to choose the a general folding growth rate geometrical parameters for the strain estimation ad growth rate for ductile layers method and to evaluate its range of applicability. a dominant growth rate for ductile layers dd Next, numerical simulations are used to construct ae growth rate for elastic layers ade dominant growth rate for elastic layers the new method of strain and competence contrasts A fold amplitude estimates. All results are derived for ductile (viscous A0 initial fold amplitude pffiffiffiffiffiffi and power-law) as well as for viscoelastic layers to AC crossover amplitude: AC ¼ 1=ðp 2aÞ avoid a restriction of the strain estimation method to a AN nucleation amplitude: AN = 1/(2a) D flexural rigidity of the layer certain layer rheology. Otherwise, this rheology needs e engineering strain: e=(l0 À l)/l0 to be known by the geologist for the observed folds, en natural strain: en = ln(l0/l) which is unlikely in the majority of the cases. The new earc strain that is estimated by the method is then verified by numerical and analogue arc length method experiments and applied to constrain the strain parti- e˙ pure shear background strain rate G shear modulus of the layer tioning in natural and experimental fold trains. H thickness of the layer

H0 initial thickness of the layer l fold wavelength 2. The nucleation amplitude for ductile and visco- l0 initial fold wavelength elastic layers ldd dominant wavelength for ductile layers lde dominant wavelength for elastic layers leff effective dominant wavelength A layer subjected to layer-parallel shortening Larc fold arc length accommodates strain by both layer thickening and Larc0 initial fold arc length rotation. The deformation process, during which con- ml viscosity of the layer siderable rotation of fold limbs takes place, is termed mm viscosity of the matrix n power-law exponent of the layer folding or buckling (e.g., Ramsay and Huber, 1987; P layer-parallel stress Price and Cosgrove, 1990). Layer thickening and R dominant wavelength ratio: folding can occur simultaneously, but it is possible R = ldd/lde or alternatively R = ade/add to discriminate between the velocity fields of the two Reff effective dominant wavelength ratio: mechanisms (e.g., Biot, 1965; Johnson and Fletcher, R = n max(1,R) eff 1994). If the layer and the matrix exhibit the same vceff effective viscosity contrast Vdyn dynamic folding velocity material properties, the amplification velocity of the Vkin kinematic folding velocity top layer boundary is controlled exclusively by the kinematic velocity, which arises due to layer thickening (e.g., Biot, 1965; Johnson and Fletcher, 1994): The aim of this paper is to provide a practical tool for field geologists, which enables strain and compe- H V ¼ e_ : ð1Þ tence contrast estimation from fold shape. We also kin 2 show all steps of the mathematical derivation of our strain estimation method to present the assumptions and limitations of our method. The study starts from The kinematic velocity is a linear function of the ˙ deriving the nucleation amplitude (the reference pure shear background strain rate e_ and the layer amplitude for strain estimation). In the next section thickness H (Fig. 1, for symbols see Table 1). In pure we derive an amplitude, designated the crossover shear, the vertical ( y)distanceofthetoplayer amplitude, at which the existing theories of folding boundary from the origin of the coordinate system break down. Both sections, quantifying the initial (point of zero velocity) is half of the layer thickness H stages of the folding instability, are potentially useful plus the amplitude A. The contribution of the ampli- for strain corrections if the initial fold amplitude is tude to that distance can be neglected to calculate the known. In the succeeding section, new analytical kinematic velocity, because in the initial stages (i.e., 198 S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213 very small limb dips) the amplitude is much smaller with respect to l0 to zero and solving for l0. This than the thickness of the layer (Fig. 1). maximum corresponds to the dominant wavelength If the material properties of the layer are different (Fletcher, 1974; Smith, 1977): than those of the matrix, then the dynamic velocity is l ¼ 2pH ð6nÞÀ1=3ðm =m Þ1=3: ð4Þ non-zero because of the instability of the layer. This dd 0 l m component of the velocity is (e.g., Biot, 1965; John- The second subscript ‘‘d’’ indicates that the son and Fletcher, 1994): wavelength is the dominant one. It is assumed that the dominant wavelength is selected and preserved Vdyn ¼ Aae_: ð2Þ during the initial stages of folding because perturbations with this wavelength grow exponentially faster The dynamic velocity is a linear function of the than all other perturbations. Substituting Eq. (4) into fold amplitude A, a dimensionless growth rate a and Eq. (3) gives the dominant growth rate for ductile the strain rate. layers: For ductile (viscous and power-law) layers embed- pffiffiffi 2=3 ded in a viscous matrix the thin-plate theory (e.g., 4 n ml Timoshenko and Woinowsky-Krieger, 1959) provides add ¼ : ð5Þ 3 mm the dimensionless growth rate (e.g., Fletcher, 1974): ! 2 In Eq. (5), the growth rate is explicitly dependent 1 pH0 n l0 mm ad ¼ n= þ : ð3Þ on material properties. Alternatively, add can be 3 l0 2p H0 ml expressed through ldd and H0 using Eq. (4). This makes the dominant growth rate explicitly dependent Here ml, mm, H0, l0 and n are the effective viscosity of the layer, the viscosity of the matrix, the initial on geometric parameters: layer thickness, the initial wavelength of a sinusoidal 2 perturbation and the power-law exponent of the layer, n ldd add ¼ 2 : ð6Þ respectively. The subscript ‘‘d’’ indicates that the p H0 growth rate corresponds to a ductile layer. The growth rate has a maximum as a function of wavelength, In Fig. 2, the kinematic velocity and several which is obtained by setting the derivative of Eq. (3) dynamic velocities, for different material properties

Fig. 2. Growth of kinematic and dynamic velocities during shortening. (A) The dynamic velocities (solid lines) increase with increasing viscosity contrast (vc). The natural strain, en, at which the dynamic velocity equals the kinematic velocity depends on the initial amplitude to thickness ratio (here A0/H0 = 0.02). (B) For smaller A0/H0 ratios (0.01) than in (A), more strain is necessary for the dynamic velocities to exceed the kinematic velocities. The intersection between dynamic and kinematic velocities defines the nucleation amplitude. S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213 199

(viscous rheology, n = 1) and initial amplitudes, are nant wavelength for elastic layers (e.g., Biot, 1961; plotted versus the natural strain, which is defined by: Turcotte and Schubert, 1982): pffiffiffiffiffiffiffiffiffi lde ¼ 2pH0 G=P: ð12Þ en ¼ lnðl0=lÞ; ð7Þ The corresponding dominant growth rate for elastic where l is the current fold wavelength. The dynamic layers is then: velocity grows faster than the kinematic velocity with increasing natural strain, and exceeds the kinematic ve- p P H locity after a certain amount of natural strain. This a ¼ 0 : ð13Þ de 3 m e_ l amount of natural strain depends on the material pro- m de perties and the initial amplitude. The amplitude, at which the dynamic velocity is equal to the kinematic To obtain ‘‘effective’’ parameters valid for ductile velocity, is termed nucleation amplitude AN. and viscoelastic layers, we now employ linear viscoe- AN is derived by equating Eqs. (1) and (2) and lastic folding theory (Schmalholz and Podladchikov, solving the result for A/H: 1999, 2001) to express the elastic dominant wave- A 1 length and growth rate. To this end, it is assumed that A ¼ ¼ ; ð8Þ finite amplitude folding occurs in lithospheric regions N H 2a dominated˙ by ductile behaviour (i.e., Deborah num- which is dimensionless (the amplitude is measured in bers mle_/G < 1 (e.g., Poliakov et al., 1993; Schmalholz units of the layer thickness). Characterizing the gro- and Podladchikov, 1999)). Consequently, the mem- wth rate by the value obtained for the dominant wave- brane stress P is expected to become˙ limited by the length yields: viscous membrane stress 4mle_ (e.g., Turcotte and Schubert, 1982) at˙ very small strain. Therefore, in _ 2 2 this study P =4mle is assumed. Folded layers that p 1 H0 AN ¼ : ð9Þ exhibit growth rates derived for elastic layers are, in 2 n ldd this study, viscoelastic because the membrane stress is assumed to be viscous (Schmalholz and Podladchi- For elastic (subscript ‘‘e’’) layers embedded in a kov, 2000). The dominant wavelength ratio R intro- viscous matrix, the thin-plate theory provides the duced by Schmalholz and Podladchikov (1999) is dimensionless growth rate (Turcotte and Schubert, defined as: 1982): 1=3 1=2 ldd 1 ml P ! R ¼ ¼ ; l 6n m G 1 2p 2p 2 de m ae ¼À D ÀPH0 ; ð10Þ ldd 4m e_ l l lde ¼ : 14 m R ð Þ

The dominant wavelength ratio R relates the dom- where D and P are the flexural rigidity of the layer and inant growth rates through the proportionality: the layer-parallel membrane stress (averaged over layer thickness), respectively. For incompressible ma- ade ¼ addR: ð15Þ terials, the flexural rigidity is: For layers exhibiting a viscoelastic Maxwell rheo- 1 logy (i.e., elastic and viscous element connected in D ¼ GH 3; ð11Þ series), the folding mode (ductile or elastic) that 3 0 exhibits the largest growth rate will tend to dominate in nature. Therefore, the R parameter is controlling if where G is the shear modulus of the layer. As in the a compressed viscoelastic (Maxwell model) layer ductile case, the growth rate function yields a domi- folds ductilely (R < 1) or elastically (R >1) (Schmal- 200 S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213 holz and Podladchikov, 1999, 2001). Comparison of and substituting add and ade yields then an ‘‘effec- this criterion with the relationship between the dom- tive’’ viscosity contrast: inant wavelengths (Eq. (14)) shows that the faster ()ffiffiffi 3=2 p mode of folding is characterized by shorter dominant 3 a ¼ n ml ; R < 1 4 dd mm vceff ¼ pffiffiffiffiffiffiffiffi wavelength. 3 3=2 3 3=2 3 ml ade ¼ ðaddRÞ ¼ nR ; R > 1 Eq. (5) establishes a relation between the viscosity 4 4 mm pffiffiffi contrast and add. To preserve this relationship in the 3=2 ml ¼ nðmaxð1; RÞÞ : (16) context of viscoelastic folding, an effective viscosity mm contrast is defined by the condition that the dominant growth rates are given by expressions identical to Eq. In a similar way an effective dominant wavelength (5). Solving Eq. (5) (n = 1) for the viscosity contrast can be introduced that preserves the relation between

Fig. 3. (A) The alteration of the nucleation amplitude AN versus leff (Eq. (17)). AN is the amplitude at which the kinematic velocity equals the dynamic velocity. AN increases strongly for decreasing leff. (B) The alteration of the crossover amplitude AC versus leff. AC is the amplitude at which the linear theories break down. AC increases strongly for decreasing leff. (C) The continuous folding process can be separated in three stages: (i) kinematic layer thickening (A < AN), (ii) exponential growth of the amplitude (AN < A < AC), and (iii) layer length controlled growth of the amplitude (AC < A). (D) The maximum strain that can be accommodated between AN and AC is 40% for a viscosity contrast of 2. For a viscosity contrast of 100, the maximum strain reduces to 13%. The strain decreases strongly if viscosity contrasts or Reff are increased, where Reff = n max(1,R). S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213 201 the dominant wavelength and the growth rate given in Podladchikov, 2000). The linear theory cannot pro- Eq. (6). The effective dominant wavelength can be vide its own limits and, therefore, a new nonlinear written: finite amplitude theory (Schmalholz and Podladchi- kov, 2000) is used to establish the limits of the linear () pffiffiffi theory. The nonlinear theory estimates the crossover ldd n; R < 1ðductile foldingÞ H0 ffiffiffi leff ¼ pffiffiffip amplitude at which the linear theory breaks down as ldd n R; R > 1ðelastic bucklingÞ H0 (cf., Schmalholz and Podladchikov, 2000): ffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ldd p ¼ n maxð1; RÞ: (17) A 1 1ffiffiffiffiffiffi ffiffiffi1 H0 AC ¼ p ¼ p : ð20Þ l p 2a 2leff

The effective dominant wavelength yields a gen- The crossover amplitude is a dimensionless num- eral growth rate valid for ductile and elastic layers: ber that measures the amplitude in units of the current wavelength. Amplitudes smaller than AC grow expo- 2 nentially (as predicted by the classical, linear solu- leff a ¼ : ð18Þ tions), whereas amplitudes larger than AC grow slower p than exponential, being controlled by the layer’s resistance to stretching (Fig. 3B). This growth is This growth rate can be substituted into Eq. (8) to designated as layer length controlled growth (Schmal- provide a general nucleation amplitude for ductile and holz and Podladchikov, 2000). To view nucleation viscoelastic layers: (Eq. (19)) and crossover amplitudes on a single plot, 2 AN is multiplied by the initial dominant thickness to 1 1 p wavelength ratio. Thus, A is also normalized by the AN ¼ ¼ : ð19Þ N 2a 2 leff wavelength. In Fig. 3C, the nucleation and crossover amplitudes are plotted versus the dominant wave- Decreasing values of leff cause larger nucleation length to thickness ratio for viscous rheologies. This amplitudes because the growth rates decrease with phase diagram distinguishes three stages of viscous decreasing values of leff and this requires larger folding, which are kinematic thickening, exponential amplitudes to maintain the same dynamic velocity growth and layer length controlled growth (Fig. 3B). (Eq. (2), Fig. 3A). The formation of large amplitude The strain that is accommodated between AN and folds having large arc length to thickness ratios AC during folding can be calculated using the linear requires fast initial growth rates, i.e., a>>1 and theory. Classical, linear analytical theories assume an leff>>1, which result in small nucleation amplitude exponential growth of the fold amplitude such that values (Eq. (19), Fig. 3A). Therefore, the nucleation (e.g., Biot, 1965; Johnson and Fletcher, 1994): amplitude is likely to be smaller than the real initial amplitude for observable folds of practical interest, a A ¼ A expðae Þ; ð21Þ conclusion of primary importance for the following 0 n discussion. where A0 is the initial amplitude of the fold. To calculate the strain accommodated between AN and 3. The crossover amplitude for ductile and visco- AC, A in Eq. (21) is replaced by AC and A0 is replaced elastic layers by AN. The strain is then: The classical linear theory of folding described 1 AC ldd en ¼ ln above becomes invalid at a certain amplitude, because a AN H0 ffiffiffi the assumptions that allow linearization are strictly 2 p 2 valid only for infinitesimal amplitudes (or very small p ffiffiffiffiffiffiffi2 leff ¼ ln p 2 : ð22Þ limb dips) (e.g., Chapple, 1968; Schmalholz and leff Reff p 202 S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213

This strain is small for folding parameters of from fold shapes without knowledge of material practical interest (Fig. 3D), and from now on we properties and initial geometries. The strains are express strain by the so-called engineering strain, e, estimated with respect to the nucleation amplitude AN. which is related to the natural strain en (Eq. (7)) by: Numerical simulations of viscous and viscoelastic single-layer folding were performed to provide a basis for the analytical treatment using a combined spectral/ l0 À l l 1 e ¼ ¼ 1 À ¼ 1 À : ð23Þ finite-difference method (Schmalholz et al., 2001) l l expðe Þ 0 0 n (Fig. 4). These simulations establish known or real strains that can be compared to estimated strains. The overall strain accommodated by most natural Applying the arc length method assumes that folds is dominated by the layer length controlled observed fold arc lengths (Larc, cf. Fig. 1) correspond growth stage of folding because the strain accommo- to initial fold wavelengths. Therefore, the ‘‘arc length dated between AN and AC is small (Fig. 3C and D). strain’’ can be calculated by: The linear analytical solutions for fold amplification (e.g., Biot, 1961; Johnson and Fletcher, 1994) are, Larc À l therefore, unsuitable to derive a strain estimation earc ¼ : ð24Þ method and we have to use a new nonlinear solution Larc valid for finite amplitude folding (Schmalholz and Podladchikov, 2000). As expected, the arc length strains estimated for viscous folds that developed in settings where viscosity contrasts are small exhibit the largest deviations 4. The strain contour map from the real strain (Fig. 5). This deviation is caused by the large component of homogeneous layer thick- In this section, a strain contour map is constructed ening (Sherwin and Chapple, 1968). Layers with large that allows strain and competence contrast estimation growth rates (either large viscosity contrast or R>1)

Fig. 4. Numerically simulated single-layer fold shapes for three different material properties. The three folds initially exhibited a sinusoidal fold shape corresponding to the theoretical dominant wavelength to thickness ratios. Increasing viscosity contrasts cause larger wavelength to thickness ratios and faster amplitude growth. Numbers above folds indicate engineering strains in percent. S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213 203

in Eq. (26) is identical to that for the arc length method (Eq. (24)). The finite amplitude solution provides the arc length method in the limit of very large growth rates. Schmalholz and Podladchikov (2000) showed that the arc length of a folded layer with initial sinusoidal shape can be approximated by:

L p2ðA=lÞ2 arc ¼ 1 þ : ð27Þ l 1 þ 3ðA=lÞ2

Substituting Eq. (27) into Eq. (26), replacing the natural strain by the engineering strain (Eq. (23)), assuming that l0/Larc0 1 and solving the resulting equation for A/l yields: Fig. 5. Accuracy of the arc length method. The arc length method is applied to numerically simulated fold shapes and strain estimates pffiffiffi A e obtained by the arc length method are compared with real strains pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð28Þ known from numerical simulations. The dotted line for ‘‘chevron l p2 Àðp2 þ 3Þe type’’ folding corresponds to folding with pure limb rotation where the initial arc length is unaltered during folding. The arc length method gets more inaccurate with decreasing viscosity contrast Eq. (28) gives the finite amplitude evolution with (vc). The arc length method is accurate for viscoelastic layers with increasing strain for folded layers exhibiting large R = 2 and different ratios of membrane stress to shear modulus growth rates. The amplitude depends on the square ( P/G = PG) because folds with large growth rates tend to develop ‘‘chevron type’’ fold shapes. root of the strain. Note that this dependence is similar to the relations between the amplitude and the strain (or the axial load) that is known in the engineering show a ‘‘chevron type’’ fold evolution and the arc literature as post-buckling (e.g., Bazant and Cedolin, length method is accurate because for folding of 1991), where axially compressed, elastic beams are layers with large growth rates, the limbs are rigid investigated. and exclusively rotate around the fold hinge (Fig. 5; After substitution of Eqs. (27) and (6) (for n = 1), for a geometrical description see Price and Cosgrove, the finite amplitude solution for strain (Eq. (25)) 1990). depends on two geometric ratios: the amplitude to The new finite amplitude solution derived by wavelength ratio A/l and the initial wavelength to Schmalholz and Podladchikov (2000) is given by: thickness ratio ldd/H0. Therefore, the observable ratios A/l and l/H are chosen as coordinates for the a= 2 a 1= 2 a construction of a strain contour map. Observed wave- L l ð þ Þ A l ð þ Þ e ¼ ln arc 0 þln 0 ; (25) length to thickness ratios are usually < 10 (e.g., n L A arc0 l 0 l Johnson and Fletcher, 1994; Sherwin and Chapple, 1968). Consequently, H/l is employed to represent where Larc0 is the initial arc length of the fold, for small wavelength to thickness ratios on linear coor- which a first order Taylor expansion of Eq. (25) about dinate axes (Fig. 6). The finite amplitude approxima- a 1 gives: tion (Eq. (28)) is valid for small H/l values (which correspond to large ldd/H0 values and large growth Larc l0 en ¼ ln ð26Þ rates). Therefore, the strain increase with A/l can be Larc0 l estimated assuming H/l 0 with Eq. (28). However, this approximation (Eq. (28)) must be corrected for Assuming that l0/Larc0 1, and employing Eq. increasing H/l (or decreasing ldd/H0). Importantly, (23) to relate natural and engineering strains, the strain results of the finite amplitude solution (Eq. (25)) show 204 S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213

Fig. 6. The strain contour map. The measured ratios A/l and H/l of any observed fold shape allow strain and competence contrast estimation using the strain contour map. Lines marked with numbers in percent are contour lines of the corresponding strain in percent.

that all strain contours exhibit a more or less constant, C1 is applied for strain contours above the folding line negative slope of A/l with increasing H/l (Fig. 6). for a viscosity contrast of 50. This folding line was Therefore, A/l in Eq. (28) is replaced by A/l À C1H/ approximated by least squares as: l, where C1 is the average slope of a strain contour. A H The finite amplitude solution is only accurate for À0:22 þ 2:43 ð29Þ viscosity contrasts >50 due to usage of thin-plate l l assumptions (Schmalholz and Podladchikov, 2000). This is confirmed by numerical results of viscous The regression equation permits calculation of the single-layer folding with small viscosity contrasts. value of H/l (termed H50), at which the break in slope These numerical results indicate a change in strain of a strain contour occurs for a given A/l. The strain contour slope, if H/l values are larger than these H/l contour map (Fig. 6) is then constructed by solving values, which belong to the folding line for a viscosity Eq. (28) for strain and correcting A/l by the slopes C1 contrast of around 50 (Fig. 6). A folding line is the and C2. This yields: characteristic line containing all points that are p2Z2 e ¼ defined at different strains through the measured A/l 1 þ Z2ð3 þ p2Þ and H/l values for a fixed viscosity contrast (Fig. 6). The initial point of such a line corresponds to the with initial dominant wavelength and the nucleation amplitude. Numerical results show that the second slope has A=l þ C H=l; if H=l= H50 all strain contours. The folding line for a viscosity contrast of 50 defines the boundary, at which the strain contour slope changes from C1 to C2. The slope H50 ¼ðA=l þ 0:22Þ=2:43; C1 ¼ 0:8; C2 ¼ 0:4 S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213 205

ical simulations for viscosity contrasts of 10, 25 and 50. Numerically calculated folding lines for viscosity contrasts of 10, 25, 50 and 250 are plotted within the strain contour map (Fig. 6). These folding lines permit estimation of the viscosity contrast for viscous folding or, for unknown rheology, the ‘‘effective’’ viscosity contrast. In Fig. 7, the real strain is plotted versus the strain estimated using the strain contour map for several numerical simulations. The accuracy of the strain predictions justifies the usage of the constant slopes C1 and C2, and the usage of the approximated folding line for a viscosity contrast of 50 (Eq. (29)) as the boundary between the two domains of different strain Fig. 7. Verification of the strain estimation method by numerical contour slopes. simulations. Real strains from viscous folds with different viscosity contrast (vc) are plotted versus strains estimated by the strain contour map (Fig. 6). Estimated strains are close to real strains which justifies usage of only two constant slopes to approximate the 5. Verification of the strain contour map by change of A/l with varying H/l along a fixed strain contour (see numerical and analogue experiments Fig. 6). To verify the strain estimation method, two numerical simulations were performed with our spectral/

The slope C1 of the strain contours above the finite-difference code for folding of layers with initial folding line for a viscosity contrast of 50 is approxi- random perturbations (Schmalholz et al., 2001). The mated by least square fit using the finite amplitude initial amplitude to thickness ratio was 0.02 and the solution. Alternatively, C1 can be derived analytically amplitudes exhibited an uncorrelated (‘‘white noise’’) by the derivative of A/l with respect to H/l by distribution. One layer was pure viscous with a applying the rules of implicit differentiation to the viscosity contrast of 100 and the other layer was finite amplitude solution presented in Eq. (25). The viscoelastic (linear viscoelastic Maxwell material; slope C2 of the strain contours below this folding line e.g., Turcotte and Schubert, 1982; Findley et al., is approximated by least square fit using three numer- 1989; Shames and Cozzarelli, 1997) with a viscosity

Fig. 8. Numerical simulations of fold train evolution for viscous and viscoelastic rheologies. The initial random perturbations corresponded to an uncorrelated (‘‘white noise’’) amplitude distribution. 206 S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213

compare well to the real strains determined from the analogue experiment (Fig. 10). A case illustrated by Hudleston (1973) (his Fig. 3B), consists of two folds within a layer (viscosity contrast = 24) that was shortened by a bulk deformation of 6 (around 59%). In this case, the estimated strains are about 10% less than the real strains, a deviation that is attributed to the unknown initial geometry and the small viscosity contrast of the experimental configuration. However, this discrepancy is small in comparison to the error from the arc length method that yields a strain of about 30%. In Lan and Hudleston (1995b), their Fig. 1 was considered, which shows two ductile folds produced by finite element simulations. The two ductile folds exhibited a power-law exponent n =1 and Fig. 9. Verification of the strain estimation method by numerical n = 10, and were shortened 40%. Lan and Hudleston simulations (Fig. 8). Each fold limb within the fold trains was used an initial amplitude to thickness ratio of 0.1 (A0/ treated as a fold limb of a fold that is symmetric to the fold axial H0 = 0.1) for a viscosity contrast of 100. This initial plane. The measured ratios A/l and H/l were used to estimate strains using the strain contour map. These estimates are compared amplitude is larger than the corresponding nucleation with real strains of each fold limb, and good coincidence is amplitude (AN). However, a comparison between real obtained. strain and strain estimated by our method is possible, if the real strain is corrected. The strain that would contrast of 2500 and R = 2 (Fig. 8). The method is applied to every individual fold limb, which is treated as half of a fold that is symmetric with respect to the fold axial plane. For the purpose of strain estimation, the wavelength is measured as two times the horizontal distance between two neighboring fold hinges. This distance does not correspond to a wavelength in the strict mathematical sense (cf., Fletcher and Sherwin, 1978). However, the good correlation (Fig. 9) between the real strains versus strains estimated by Eq. (30) shows that (i) the use of the horizontal distance between neighboring hinges as wavelength is acceptable and (ii) the strain estimation method is applicable to folds with natural, asymmetric shape. Furthermore, the strain estimation method was applied to results of analogue and numerical experiments performed by other authors. Perturbation C in Fig. 10. Verification of the strain estimation method by analogue Fig. 9 of Abbassi and Mancktelow (1992) was con- (Abbassi and Mancktelow, 1992; Hudleston, 1973) and numerical sidered. The real strain of the fold, which developed in (Lan and Hudleston, 1995a,b; Mancktelow, 1999) experiments of the middle part of the shortened layer, was determined other authors. The solid line represents a perfect fit. Estimated using the experimentally deformed grid. The layer at strains correspond well with real strains except for the analogue 2.1% shortening was considered as initial geometry. experiments of Hudleston (1973). The reason is the low viscosity contrast of 24 and the unknown initial geometry. However, the arc The shapes of the middle fold at 8.6%, 12.9% and length method estimates for this case a strain of around 30%, which 22.5% bulk strain were used to measure the ratios A/l is considerably more inaccurate than our strain estimation method. and H/l. The estimated strains using our method See the text for a more detailed description of the comparison. S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213 207 have been accommodated during folding from AN up The application of the newly developed strain to A0/H0 = 0.1 is calculated using Eq. (21). AN is estimation method to analogue and numerically pro- determined by Eqs. (4) and (9). The calculations yield duced fold shapes demonstrates that the method gives that a strain of around 6% for n = 1 and around 4% for reliable strain estimates and can be used to estimate n = 10 would have been accommodated between AN strains from natural fold shapes. and A0/H0 = 0.1. The viscous fold (n = 1) accommodated around 46% strain and the ductile fold (n = 10) around 44% strain, if AN is considered as initial 6. Strain partitioning in fold trains amplitude. The estimated strains are close to the real strains (Fig. 10). Finally, the strain estimation method In the following the term strain partitioning refers was applied to finite element simulations of single- to the difference in strain that is accommodated by the layer folding performed by Mancktelow (1999). The whole fold train (bulk strain) and the individual fold deformed layer in his Fig. 12b at a logarithmic strain limbs within the fold train (individual strain). To of À 1.112 (around 67% strain) was considered. For investigate strain partitioning we employ the numer- each individual fold, the ratios A/l and H/l were ical simulations of fold trains presented in Fig. 8. The measured, and the estimated strains are again plotted bulk strain of the total layer is plotted versus individ- versus the real strain of 67%. ual strains that are accommodated by individual folds

Fig. 11. Strain partitioning in numerically simulated viscous and viscoelastic fold trains. The individual strains of individual folds within the fold train are plotted versus the layer bulk strain. If all individual strains would lie on the dotted line, no strain partitioning is present because all individual folds accommodate the same strain than the fold train. The viscous fold train shows slightly increasing strain partitioning, whereas the viscoelastic fold train shows a strong increase in strain partitioning already at the beginning of shortening. 208 S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213

Fig. 12. (A) Individual strains of folds within the numerically simulated viscous and viscoelastic fold trains (see Fig. 8) at a bulk strain of 13% are plotted versus the wavelength to thickness ratios (l/H) of the corresponding folds. The viscous fold train shows little strain partitioning and little variation in the l/H ratios. The viscoelastic fold train shows strong strain partitioning where larger l/H ratios correspond to smaller individual strains. (B) The same individual strains are plotted versus the restored initial wavelength to thickness ratios (l0/H0). The l/H ratios were restored using Eq. (33).

(Fig. 11). Each fold limb is considered to be the limb Substituting H0 = l0/(l0/H0) into Eq. (31) and of a fold that is symmetric with respect to the fold rearranging yields: axial plane. Fig. 11 shows that the strain partitioning in the viscoelastic layer is much stronger than in the l0 l expð2enÞ pure viscous layer. ¼ : ð33Þ H H L =l Individual strains at a layer bulk strain of 13% 0 arc are plotted versus the corresponding wavelength to The ratio of L /l can be expressed through the thickness ratio (l/H) of individual folds (Fig. 12A). In arc ratio of A/l (see Eq. (27)) and the natural strain can be the viscous case, all folds show more or less the same replaced by the engineering strain using Eq. (23). strain and similar l/H ratios. In contrast, individual Therefore, l /H can be calculated through l/H, A/l folds in the viscoelastic layer accommodate very 0 0 and the strain. In Fig. 11B, the average restored values different amounts of strain and l/H ratios vary of l /H for individual folds in the viscous layer is strongly. The initial l/H ratio (l /H ) can be restored 0 0 0 0 14, and comparable to the theoretical dominant if it is assumed that the area of the fold does not wavelength (Eq. (4)) of 16 for a viscosity contrast change during folding and that the initial arc length is of 100. approximately equal to the initial wavelength. The The strain estimation method is used to investigate initial fold area L H is then equal to the current fold arc0 strain partitioning within natural fold trains from area L H and the equation for the conservation of arc Ramsay and Huber (1987) (Fig. 13A) and Weiss area (or mass) is: (1972) (Fig. 13B). The measured data, the estimated strains and the restored initial wavelength to thickness l0H0 ¼ LarcH: ð31Þ ratios are presented in Tables 2 and 3. The fold train of Under pure shear shortening, the initial wavelength Fig. 13A (Table 2) has approximately constant strain is related to the current wavelength through the of 70% for all individual folds, whereas in the fold equation: train of Fig. 13B (Table 3) strains vary from 35% to 62%. In Fig. 14A, the estimated strains of individual folds within three different fold trains are plotted l0 ¼ lexpðenÞ: ð32Þ versus the restored ratio of l0/H0. The three fold S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213 209 trains are the train shown in Fig. 13A, the train other fold trains are plotted versus the restored ratio of produced by the numerical simulation of Mancktelow l0/H0. The fold trains are the train shown in Fig. 13B (1999) (the same fold train used for the verification of and the train with viscoelastic rheology produced by our method) and the train generated by our numerical our numerical simulation presented in Fig. 11. The simulation for viscous rheology (Fig. 11). All three strain partitioning within these fold trains exhibits the fold trains show little strain partitioning. In Fig. 14B, same characteristic relation to the restored ratio of l0/ the estimated strains of individual folds within two H0, namely that increasing values of l0/H0 correspond to decreasing strains.

7. Discussion and conclusion

It this paper, a new method is proposed for the strain and competence contrast estimates as a function of two easily measurable geometrical parameters: the thickness to wavelength (H/l) ratio and the amplitude to wavelength (A/l) ratio (Fig. 6, Eq. (30)). Analytical solutions for folding of ductile and viscoelastic layers are combined using linear viscoelastic folding theory (Schmalholz and Podladchikov, 1999, 2001) to enable the applicability of our method to natural folds without knowing the deformation mechanism. Our method reliably estimates strain from observed fold shapes and is a considerable improvement to the arc length method. The major shortcoming of the method is that the nucleation amplitude AN must be defined as initial amplitude corresponding to zero strain. If the natural (or real) initial amplitude is greater or less than AN, the method over- or underestimates, respectively, the strain. For purpose of comparison, the value of AN for a viscosity contrast of 50 provides a reasonable reference value to define small or large initial amplitude. AN for a viscosity contrast of 50 is 0.03. For viscosity contrasts < 50 thin-plate approximations are inaccurate, because layer thickening influences the deformation of the layer. In general, smaller AN leads to larger growth rates (cf. Eq. (8)). In the initial folding stages large growth rates cause exponential amplitude growth within small amounts of strain. Therefore, if a natural layer is characterized by initial amplitudes >AN, the error in the strain estimate is, in general, small. Also, further improvements of the precision of the strain estimates can easily be made Fig. 13. Natural examples of fold trains. (A) Folded layer (thin using Eq. (22) to correct for the discrepancy in strain section, picture from Ramsay and Huber, 1987) where five fold limbs were used to estimate strain (see Table 1). (B) Folded layer between the real initial amplitude and the nucleation (picture from Weiss, 1972) where six fold limbs were used to amplitude, if the former can be constrained independ- estimate strain (see Table 2). ently. 210 S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213

Table 2 Table 3

Measured data, estimated strains and restored l0/H0 ratios from the Measured data, estimated strains and restored l0/H0 ratios from the fold train in Fig. 13A fold train in Fig. 13B

Limb A/l H/le[%] l0/H0 Limb A/l H/le[%] l0/H0 AB 0.70 0.36 71 11 AB 0.22 0.07 38 25 BC 0.92 0.26 72 15 BC 0.29 0.12 50 20 CD 0.69 0.44 72 9 CD 0.19 0.09 35 21 DE 0.61 0.43 70 10 DE 0.34 0.33 62 11 EF 0.56 0.51 70 8 EF 0.31 0.23 58 14 FG 0.25 0.30 55 11

Measured A/l and H/l ratios that lie below the layer exhibiting large power-law exponents (Fig. 10). folding line for a viscosity contrast of 50 indicate It appears that the observable differences in fold shape small competence contrast (Fig. 6) and that the natural for viscous and strongly power-law rheologies do not initial amplitudes may have been smaller than AN. reflect significant differences in strain. This behaviour Therefore, the strain estimate is a minimum value, and suggests that A/l and H/l are in fact the appropriate the real strain may be considerably larger than esti- coordinates for strain contour maps. mated. However, for folding with small competence Recently performed numerical simulations of sin- contrast the arc length method is inaccurate (Fig. 5), gle-layer folding showed, that for both simple and and because our method yields more accurate strain pure shear conditions, the evolution of the ratio A/l estimates than the arc length method the error is versus strain is similar. Despite the strong asymmetry unlikely to be important. AN is only likely to under- of the folds developed under simple shear, the finite estimate natural initial amplitudes for layers with amplitude solution (Schmalholz and Podladchikov, small competence contrast because the corresponding 2000), used for constructing the strain contour map, AN is relatively large. Under this condition, the strain predicted the fold amplification successfully. For the contour map allows estimation of the competence folds developed under simple shear, the amplitude contrast. Also, the strain contour map provides accu- was measured as half the vertical distance between the rate strain estimates for folding of a nonlinear viscous highest and lowest point of the top layer boundary and

Fig. 14. Two characteristic types of strain partitioning. Estimated strains of individual folds within fold trains are plotted versus restored initial wavelength to thickness ratios (l0/H0, see Eq. (33)). (A) The fold trains from Fig. 13A (Ramsay and Huber, 1987), from a numerical simulation of Mancktelow (1999) (see text) and the viscous fold train (Fig. 8) show little strain partitioning and little variation in the restored l0/H0 ratio. (B) The fold trains from Fig. 13B (Weiss, 1972) and the viscoelastic fold train (Fig. 8) show strong strain partitioning and increasing restored l0/ H0 ratios with decreasing individual strain. S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213 211

the most strain. It is assumed that during compression the dominant wavelength is selected and locked when the amplitude of the layer perturbation is approximately equal to AN. Individual fold amplitudes within a layer need not grow to AN simultaneously. In the pure viscous case, the dominant wavelength exclusively depends on the viscosity contrast. This dependence is unchanged regardless of whether the individual folds grow simultaneously. Therefore, the development of the same dominant wavelength is expected, which explains the observed clustering of dominant wavelengths (Figs. 12 and 14A). However, if the layer has a viscoelastic or power-law rheology, the situation may be different. In the viscoelastic cases, as in the viscous case, an individual fold selects its wavelength Fig. 15. Growth rates for ductile and elastic layers are plotted versus when its fold amplitude is equal to AN,butthe the wavelength to thickness ratios (l0/H0) of the layers. Growth dominant wavelength depends on the layer-parallel rates for ductile layers (Eq. (6)) increase with increasing l0/H0 stress. The amplification of an individual fold may ratios for different power-law exponents (n). Growth rates for elastic reduce the layer-parallel stress within an entire viscoe- layers (Eq. (13))˙ decrease with increasing l0/H0 ratios for different lastic layer by increasing the fold arc length (Schmal- values of P/(m e). l holz and Podladchikov, 2000). A later fold may then evolve under different stress conditions than the initial the wavelength was measured as the horizontal dis- fold. For power-law layers, the dominant wavelength tance between two neighboring, concave upward fold depends on the effective viscosity contrast. However, hinges. These new results indicate that the presented for power-law materials the ‘‘effective’’ viscosity is a strain estimation method can also be successfully function of the stresses within the layer and matrix applied to asymmetric fold shapes that developed (m = m(s); Fletcher, 1974). The stresses within the under simple shear conditions. whole fold train may alter due to asynchronous growth There are two different types of strain partitioning of individual folds. The effective viscosity contrast patterns of individual folds within fold trains (Fig. 14): may therefore change during shortening of power-law (i) little variation in strain and restored l0/H0 ratio, and layers. Consequently, individual folds that have differ- (ii) strong variation in strain, where restored values of ent amplitudes may select wavelengths of different l0/H0 vary inversely with strain. The two patterns are size. observed, respectively, in numerically simulated pure Also, strong wavelength selectivity, predicted by viscous and viscoelastic (R = 2) fold trains. We spec- the linear theories (e.g., Biot, 1961), decreases strongly ulate that strain partitioning within fold trains is related after only a few percent of strain for layers with to rheology. To justify this speculation, ldd/H0 and lde/ relatively large growth rates (i.e., >75) (Schmalholz H0 is replaced by l0/H0 within the ductile growth rate and Podladchikov, 2000). This means that the develop- (Eq. (6)) and the elastic growth rate (Eq. (13)), ment of a single dominant wavelength within layers respectively. These growth rates have different rela- exhibiting large growth rates is improbable. In such tionships to l0/H0 (Fig. 15). In the ductile case (n =1,5 layers wavelengths of different size may develop. and 10), increasing l0/H0 ratios cause increasing These wavelengths can be shorter or longer than the growth˙ rate. In contrast, in the elastic case ( P/ dominant wavelength, and are expected to amplify mle_ = 50, 250 and 500) increasing l0/H0 ratios cause according to the finite amplitude growth rate spectra decreasing growth rate. Therefore, in ductile layers, (Schmalholz and Podladchikov, 2000). Consequently, larger wavelengths grow faster, whereas in elastic we speculate that periodic folds with approximately layers shorter wavelengths grow faster. In both cases, the theoretical dominant wavelength only develop the fastest growing wavelength should accommodate when growth rates are small ( < 75). Folds that grow 212 S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213 rapidly are expected to exhibit heterogeneous wave- Findley, W.N., Lai, J.S., Onaran, K., 1989. Creep and Relaxation of length spectra. Nonlinear Viscoelastic Materials. Dover Publications, New York. Fletcher, R.C., 1974. Wavelength selection in the folding of a single The typical patterns of strain partitioning between layer with power-law rheology. Am. J. Sci. 274 (11), 1029–1043. individual folds in fold trains can thus be used to Fletcher, R.C., Sherwin, J., 1978. Arc lengths of single layer folds: a discriminate deformation mechanisms: (i) folding with discussion of the comparison between theory and observation. small growth rates (relatively constant wavelengths), Am. J. Sci. 278, 1085–1098. (ii) folding with large growth rates due to viscoelastic Hudleston, P.J., 1973. An analysis of ‘‘single-layer’’ folds developed experimentally in viscous media. Tectonophysics 16, rheology (shorter wavelengths accommodate more 189–214. strain), and (iii) folding with large growth rates due Hudleston, P.J., Stephansson, O., 1973. Layer shortening and fold- to power-law rheology (longer wavelengths accom- shape development in the buckling of single layers. Tectono- modate more strain). Therefore, measured strain par- physics 17, 299–321. titioning patterns of natural fold trains and our strain Hunt, G., Mu¨hlhaus, H., Hobbs, B., Ord, A., 1996. Localized folding of viscoelastic layers. Geol. Rundsch. 85, 58–64. estimation method could be used to assess the defor- Johnson, A.M., Fletcher, R.C., 1994. Folding of Viscous Layers. mation mode (ductile or viscoelastic) and the magni- Columbia Univ. Press, New York. tude of the competence contrast during the initiation of Lan, L., Hudleston, P.J., 1995a. The effects of rheology on the strain folding. distribution in single layer buckle folds. J. Struct. Geol. 17 (5), 727–738. Lan, L., Hudleston, P.J., 1995b. A method of estimating the stress Acknowledgements exponent in the flow law for rocks using fold shape. PAGEOPH 145 (3/4), 621–635. Mancktelow, N.S., 1999. 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