A mathematical model for orientation data from macroscopic elliptical conical folds

An iterative least-squares technique to fit circular and elliptical conical surfaces to orientation data from folds is presented. A statistical model is used which assumes that each data point is an observation from a Fisher distribution. The mean of this distribution is assumed to lie on the curve to be fitted. Estimates of variances and covariances for the fitting parameters are calculated, and confidence intervals for the cone axis and half apical angle are estimated from variances and covariances. A normal test with null hypothesis that the cone angle is 90° determines if a conical model fits the data better than a cylindrical model. AnF test is used to determine whether an elliptical cone is a better model than a circular cone. In this fashion, macroscopic folds are classified into cylindrical, circular conical, or elliptical conical folds. Examples of these three types of fold are given. The Wynd Syncline near Jasper, Alberta is the first natural elliptical conical fold described as such.     

构造置换及其控矿规律——以吉林板石沟铁矿为例

 强烈的塑性变形使华北地台东北部太古宙鞍山群中的吉林板石沟铁矿发生强烈的构造置换;造成原始仅有二三层的铁矿褶皱重复,在X(包络线)、Y(枢纽线)方向均被拉断,形成透镜状的复式褶皱勾状体。现有的19个矿组均为这种复式褶皱的转折端,并多呈"Z"型不对称形式。根据以上控矿规律本文提出两个找矿方向,一是包络线方向,另一是枢纽线方向,对1、3矿组具体地做了勘探设计。目前本文的勘探设计已得到勘探验证,新增铁矿储量数千万吨。     

The structural evolution of sheath folds: A case study from Cap de Creus

It has long been recognised that within zones of intense non-coaxial deformation, fold hinges may rotate progressively towards the transport direction ultimately resulting in highly curvilinear sheath folds. However, there is a surprising lack of detailed and systematic field analysis of such “evolving” sheath folds. This case study therefore focuses on the sequential development of cm-scale curvilinear folds in the greenschist-facies El Llimac shear zone, Cap de Creus, Spain. This simple shear-dominated dextral shear zone displays superb three dimensional exposures of sheath folds defined by mylonitic quartz bands within phyllonite. Increasing amounts of fold hinge curvature (δ) are marked by hinge segments rotating into sub-parallelism with the mineral lineation (Lm), whilst the acute angle between the axial-planar hinge girdle and foliation (ω) also displays a sequential reduction. Although Lm bisects the noses of sheath folds, it is also clearly folded and wrapped-around the sheath hinges. Lm typically preserves a larger angle (θ) with the fold hinge on the lower limb (L) compared to the upper (U) limb (θL > θU), suggesting that Lm failed to achieve a steady orientation on the lower limb. Adjacent sheath fold hinges forming fold pairs may display the same sense of hinge arcing to define synthetic curvature, or alternatively opposing directions of antithetic curvature. Such patterns reflect original buckle fold geometries coupled with the direction of shearing. The ratio of long/short fold limbs decreases with increasing hinge curvilinearity, indicating sheath folds developed via stretching of the short limb, rather than migrating or rolling hinge models. This study unequivocally demonstrates that both hinges of fold pairs become curvilinear with sheaths closing in the transport direction recording greater hinge-line curvilinearity compared to adjacent return hinges. This may provide a useful guide to bulk shear sense.     

A mathematical model for orientation data from macroscopic conical folds

Statistical techniques are developed to classify folds into one of three classes: cylindrical, conical, or neither. A translated version of Bingham's distribution on the sphere is applied to orientation data fron conical folds. Iterative least-squares techniques are used to determine the best-fitting small circle (or cone), and confidence intervals for the cone axis and half apical angle are developed. Examples of a cylindrical and a conical fold are given. Another fold is neither cylindrical nor conical and is classified as pseudoconical. Relationships between the Bingham and Fisher distributions are presented.     

圆锥状褶皱的几何分析

长期以来,地质学家习惯于把不同类型和不同成因的褶皱的基本形态当作圆柱状几何性状处理,建立了褶曲构造的基本几何要素,并且以此作为划分构造均匀区段的几何准则。从最近十余年来关于区域变质造山带的大量观测研究证明,还有另一种褶皱,即圆锥状褶皱,广泛发育于褶皱复杂的变质岩系之中,在不同尺度上显示其控制作用和对区域构造格架的影响。     

Layer shortening and fold-shape development in the buckling of single layers

The progressive development of folds by buckling in single isolated viscous layers compressed parallel to the layering and embedded in a less viscous host is examined in several ways; by use of experiments, an analogue model to simulate simultaneous buckling and flattening and by an application of finite-element analysis.The appearance of folds with a characteristic wavelength in an initially flat layer occurs in the experiments for viscosity ratios (μ_layer/μ_host = μ₁/μ₂) of between 11 and 100; progressive fold development after the initial folds have appeared is similar in the experiments and in the finite-element models. Except for the finite-element model for μ₁/μ₂ = 1,000 layer-parallel shortening occurs in the early stages of folding and a stage is reached where little further changes in arc length occur. The amount of layer-parallel shortening increases with decreasing viscosity contrast, and becomes relatively unimportant after the folds have attained limb dips of about 15°–25°.Thickness variations with dip are only significant here for the finite-element model with μ₁/μ₂ = 10, and in experiments for μ₁/μ₂ = 5 where the layer is initially in the form of a moderate-amplitude sine wave. The variations range from a parallel to a near-similar fold geometry, and in general depend on the viscosity contrast, the degree of shortening and the initial wavelength/thickness ratio. They are very similar to the variations predicted by the analogue model of combined buckling and flattening. The difference between the thickness/dip variations in a fold produced by buckling at low viscosity contrast and one produced by flattening a parallel fold is marked at high limb dips and very slight at low limb dips.Many natural folds in isolated rock layers or veins show thickness/dip relationships expected for a flattened parallel fold, and some show relationships expected for buckling at low viscosity contrasts. Studies of the wavelength/thickness ratios in natural folds have suggested that competence contrast is often low. Many folds in isolated rock layers or veins whose geometry may vary between parallel and almost similar, and may be indistinguishable from those of flattened parallel folds, have probably developed by a process of buckling at low viscosity contrasts.     

Mega arrowhead interference patterns in the central part of the Yanshan Orogenic Belt,North China

The Chengde-Pingquan region is located in the central part of the Yanshan Orogenic Belt (YOB). At Daheishan and Pingquan in the central YOB, thrusts and folds of variable trends are displayed in 2 km-scale fold interference patterns. Detailed field mapping was conducted to decipher the geometry of these two superimposed structures. Map-view geometry and stereonet plots for outcrop-scale folds indicate that the superimposed structures form arrowhead interference pattern where NW-SE-trending F₁ folds are refolded by later ENE-WSW F₂ folding. After remove the effects of later faulting, restored map-views of the superimposed structures show that when the F₁ folds have inclined axial surfaces but with no an overturned limb, an arrowhead interference pattern (here called modified type-2 pattern) can form. Our field data and reinterpretation of the findings of previous studies suggest that five major shortening phases have occurred in the Chengde-Pingquan region. The first two phases, which formed the superimposed folds, occurred earlier than the Late Triassic (D1) and during the Late Triassic to Early Jurassic (D2). These two phases were followed by three deformation phases that are mainly characterized by thrusting and strike-slip faulting, which strongly modified the large-scale fold interference patterns.     

An energy calculation concerning the roundness of folds

The calculation estimates the energy consumed in a representative fold in an extended layered sequence of two alternating lithologies, during buckling through finite amplitudes (limb inclinations up to 70°). By estimating separately the energy quantities comsumed in different parts of a fold, it is concluded:

1. (1) that angular folds (chevrons, kinks etc.) do not require brittleness or any localized material abnormality for their development; they have the most economic profile as regards energy-consumption even in strictly Newtonian materials, particularly at high limb inclinations;

2. (2) where a fold profile approximates a circular-arc hinge section and a straight limb, the ratio (length of curved part/total length of profile) is a useful parameter, that can be linked in a systematic way to the mechanical properties of the assemblage at time of folding;

3. (3) measurements so far made in outcrop are compatible with the calculations and with published opinions about possible mechanical properties of rocks.

The discussion presented is merely exploratory; further study of the ratio mentioned is likely to throw light on the folding process.     

Interpretation of a set of faults across the hinge and a limb of a large-scale flexure in the Mount Isa district, Queensland, Australia, in terms of fractures related to the folding process

A revised interpretation of a number of faults across the hinge and western limb of a large-scale anticlinal flexure in the Mount Isa district has been made in terms of the faults following earlier-formed be joints. Such joints often develop in weakly or moderately folded competent sediments, as a result of either tensile stresses that were active at a late stage during folding or the influence of residual stresses generated during tectonic uplift. The joints are oriented such that on a stereographic projection their poles plot parallel to the a axis of a fabric cross and at 90° to the fold axis (b). bc joints are thus approximately normal to bedding and contain the fold axis, and hence they fan around the axial plane of the fold containing them. Across the hinge and western limb of a steeply N-plunging large-scale F₂ flexure in the Mount Isa district, a number of faults at high angles to bedding fan about the axial plane. Making use of the fold geometry and local bedding orientation it is possible to predict the orientation of ideal bc fractures at locations within the fold. These predictions fit well with the observed fault pattern. The movement on the faults, although apparently complex, appears consistent with continued shortening perpendicular to an axial-plane cleavage during the D₂ deformation or as part of a later D₂ deformation.     

An analysis of noncylindrical and incongruous fold pattern from the eo-cambrian rocks of söröy, northern Norway: II. The significance of synfold stretching lineation in the evolution of noncylindrical folds

Many folds of both deformation phases on Söröy exhibit a prominent lineation brought out by elongate micas or spindleform-quartz grains. This lineation displays a variety of patterns, depending on ils orientation relative to the fold axis. This dependence confirms the primary nature of the axial noncy lindrism.The lineation is a product of layer-parallel extension and represents the long axis of the sectional strain ellipse (X¹), for the particular attitude of the layering. The formation of the lineation is confined to those layer orientations which cut the extension field of the strain ellipsoid. If the noncylindrical fold axes curve from normal into parallelism with the lineation, the patterns of lineation orientation developed will reveal the attitude of the major extension axis (X) and indicate the nature of the strain.The lineation developed in the earlier stages of fold growth and became a passive marker in later fold modification.The regional pattern of the lineation is similar to that for minor fold axes, and together they are held to identify the XY-plane of the bulk finite strain-field.     

Structures in the banded iron-formation of the southeastern Bababudan Hills, Karnataka, India

The banded iron-formation in the southeastern Bababudan Hills display a macroscopic synformal bend gently plunging towards WNW. The bedding planes in smaller individual sectors show a cylindrical or conical pattern of folding. The dominant set of minor folds has WNW-ESE trending axial planes and the axes plunge towards WNW at gentle to moderate angles, though there is considerable variation in orientation of both axes and axial planes. A later set of sporadically observed folds has N-S trending axial planes. The macroscopic synformal bend within the study area forms the southeastern corner of a horseshoe shaped regional synformal fold closure which encompasses the entire Bababudan range. The minor folds are buckle folds modified to a varying extent by flattening. In some examples the quartzose layers appear to be more competent than the ferruginous layers; in others the reverse is true. The folds are frequently noncylindrical and the axes show curvature with branching and en echelon patterns. Such patterns are interpreted to be the result of complex linking of progressively growing folds whose initiation is controlled by the presence of original perturbations in the layers. Domes and basins have at places developed as a result of shortening along two perpendicular directions in a constrictional type of strain. Development of folds at different stages of progressive deformation has given rise to nonparallelism of fold axes and axial planes. The axes and axial planes of smaller folds developed on the limbs of a larger fold are often oriented oblique to those of the latter. Progressive deformation has caused rotation and bending of axial planes of earlier formed folds by those developed at later stages of the same deformational episode. Coaxial recumbent to nearly reclined fold locally encountered on the N-S limb of the macroscopic fold may belong to an earlier episode of deformation or to the early stage of the main deformation episode. The E-W to ESE-WNW strike of axial plane of the regional fold system in the Bababudan belt contrasts with the N-S to NNW-SSE strike of axial planes of the main fold system in the Chitradurga and other schist belts of Karnataka.     

Early formed regional antiforms and synforms that fold younger matrix schistosities: their effect on sites of mineral growth

In the metamorphic cores of many orogenic belts, large macroscopic folds in compositional layering also appear to fold one or more pervasive matrix foliations. The latter geometry suggests the folds formed relatively late in the tectonic history, after foliation development. However, microstructural analysis of four examples of such folds suggests this is not the case. The folds formed relatively early in the orogenic history and are the end product of multiple, near orthogonal, overprinting bulk shortening events. Once large macroscopic folds initiate, they may tighten further during successive periods of sub-parallel shortening, folding or reactivation of foliations that develop during intervening periods of near orthogonal shortening. Reactivation of the compositional layering defining the fold limbs causes foliation to be rotated into parallelism with the limbs.Multiple periods of porphyroblast growth accompanied the multiple phases of deformation that postdated the initial development of these folds. Some of these phases of deformation were attended by the development of large numbers of same asymmetry spiral-shaped inclusion trails in porphyroblasts on one limb of the fold and not the other, or larger numbers of opposite asymmetry spirals on the other limb, or similar numbers of the same asymmetry spirals on both limbs. Significantly, the largest disparity in numbers from limb to limb occurred for the first of these cases. For all four regional folds examined, the structural relationships that accompanied these large disparities were identical. In each case the shear sense operating on steeply dipping foliations was opposite to that required to originally develop the fold. Reactivation of the folded compositional layering was not possible for this shear sense. This favoured the development of sites of approximately coaxial shortening early during the deformation history, enhancing microfracture and promoting the growth of porphyroblasts on this limb in comparision to the other. These distributions of inclusion trail geometries from limb to limb cannot be explained by porphyroblast rotation, or folding of pre-existing rotated porphyroblasts within a shear zone, but can be explained by development of the inclusion trails synchronous with successive sub-vertical and sub-horizontal foliations.     

Side-stepping of axial surface traces in superposed folding

During the refolding of an early non-isoclinal fold (say,F ₁) we may find an offset or side-stepping of the axial surfaces of the later folds (say,F ₂). The offsets can be seen in both type 2 and type 3 interference patterns. An analysis of the shear fold model shows that there is a maximum limit for the magnitude of side-stepping. The side-stepping is larger for larger interlimb angles ofF ₁. It decreases with progressive tightening ofF ₂. By recognizing such side-stepping we can predict on which side the F₁ hinge should lie even if the hinge is unexposed or lies outside the domain of observation. The general rule for the sense of side-stepping is the same for shear folds, flexural slip folds and buckling folds. However, the side-stepping in buckling folds should be used with caution, sinceF ₂ folds on buckled single-layers may show an offset whose sense is opposite to that predicted by the general rule.     

Noncylindrical flexural slip folds in nature and experiment

Some naturally formed folds in North Cornwall, England, show the following geometrical features:

1. (a) each fold is noncylindrical;

2. (b) the profile shape varies along the hinge-line (chevron-shaped at culmination, rounded at terminations);

3. (c) hinge-lines and axial surfaces of some folds curve strongly in certain restricted areas. Micro-structures indicate that the folds formed by geometric bending and flexural slip.

The progressive development of folds like these has been simulated by subjecting multilayered plasticine models to layer-parallel compression in coaxial stages. This technique allows folds to be observed and measured after each deformation stage. The folds initiate at irregularities in the layering or stress-field. As each fold amplifies, it acquires a characteristic three-dimensional shape: generally the profile is chevron-shaped at the point of maximum amplitude and is rounded at the terminations. Growth of a fold also involves a lengthening of the hinge-line by propagation of the terminations (e.g. into areas of previously unfolded layering). Individual folds also tend to trigger the development of new folds at either side, eventually forming a fold complex with regular wavelength. Propagating fold complexes may interfere by processes of linking and blocking that are not strictly the same as interference effects in other wavelike processes. The interference of fold complexes is described with reference to layer surfaces and crosssections. Causes of noncylindricity are examined. Geometrical aspects of naturally formed folds are analyzed in the light of the experimental findings.     

Evolution of thrust system deformation in an intracratonic chain: Study of the Southern-Central Tunisian Atlas

The development of belt structures in intracratonic chains is guided by the convergence system. In the Southern-Central Tunisian Atlas, several parameters control the evolution of thrust folds during different tectonic phases. One of these phases is tectonic inheritance, which leads to the reactivation of pre-existing normal faults during compressive phases. The angle between the direction of these faults and the shortening axis (NNW-SSE) is the most important parameter for interpreting the mode of the evolution of thrust folds. Jebel Elkebar is an example of a structure developed on NW-SE-oriented faults that is perpendicular to the shortening axis. Based on the geometry of its folds, Jebel Elkebar is interpreted as a 'Fault Related Fold'. The E-W-oriented Orbata structure is oblique to the direction of the shortening axis and is interpreted through the model 'Fault Propagation Fold' with 'Breakthrough'. The Gafsa Fault, which is parallel to the shortening axis, is a transpressional fault interpreted through the 'Strain Partitioning' mode, which is associated with the oblique ramp fold. The development of various thrust folds requires the presence of a basal decollement level during the Triassic succession. In the Southern-Central Tunisian Atlas, the deformation is variable (geometry of fold closure) and is correlated with the depth of the decollement level; indeed, the intensity of deformation is proportional to the depth of the decollement level. Consequently, the most important deformation is in the higher successions and is a vertical migration of the decollement level associated with thin-skinned deformation.     

The establishment of recumbent folds in the Lower Palaeozoic near Queanbeyan,New South Wales

Lower Palaeozoic sedimentary and volcanic rocks east of Queanbeyan, N.S.W., have undergone multiple deformation resulting in four systems of folds. The first of these consists of large isoclinal, recumbent folds (F1). The second generation folds (F2) are the most pronounced; they consist of flattened flexural‐slip folds with well developed axial‐plane slaty cleavage. Minor variants of this system are associated with meridionally‐trending faults. Third and fourth generation folds are minor kink systems.

The existence of first generation folds was established on the basis of F2 fold‐facing determinations, and their likely form was deduced from the geometrical variations of F2 folds. It is thought that all fold phases developed during the Late Silurian Bowning Orogeny.     

构造置换及其控矿规律——以吉林板石沟铁矿为例

强烈的塑性变形使华北地台东北部太古宙鞍山群中的吉林板石沟铁矿发生强烈的构造置换;造成原始仅有二三层的铁矿褶皱重复,在X(包络线)、Y(枢纽线)方向均被拉断,形成透镜状的复式褶皱勾状体。现有的19个矿组均为这种复式褶皱的转折端,并多呈"Z"型不对称形式。根据以上控矿规律本文提出两个找矿方向,一是包络线方向,另一是枢纽线方向,对1、3矿组具体地做了勘探设计。目前本文的勘探设计已得到勘探验证,新增铁矿储量数千万吨。     

Fractal Simulation and Classification of Folds

Abstract: The method of fractal simulation and classification of folds is firstly studied here to describe various types of complex fold patterns in quantitative analysis. Based on the characteristics of natural folds with a fractal pattern, the fold patterns are simulated to describe various types of folds quantitatively by means of fractal interpolation. The major factors affecting the fold pattern are elucidated in fractal simulation of folds, i.e. positions of interpolation points (x, y) and the disturbance coefficient d of folds (-1<d<1). The bigger the value d for a fold simulation is, the more complex or disturbed the folds are and the better developed the relative secondary folds are. If d>0, folds are upconvex. IF d<0, they are down-convex. |d|=0, |d|=0.25 and |d|=0.5 represent three conspicuous turning states. If |d|=0, the points will be joined by a straight line. If |d|=0.25, the points will be joined smoothly. If |d|<0.25, there will be complex secondary folds between the points. If |d| >0.5, there will be more complex secondary folds between the points. The complex degrees of the fold pattern, therefore, can be classified by the disturbance coefficient d and by the discongruent degree Δ d. In nature, most folds are self-affine fractal folds.     

Reverse modelling of detachment folds; application to the Pico del Aguila anticline in the South Central Pyrenees (Spain)

A simple method to estimate fold-amplification and thrust-movement rates for detachment folds is documented and illustrated by its application to a symmetrical detachment fold in the Southern Pyrenees, Spain. The technique provides a complete record of the kinematic evolution of detachment folds and is based on the application of equations for detachment folds involving limb rotation. The method uses the stratal pattern of the syntectonic sediments and assumes that these growth strata were deposited horizontally, that the folds involve a homogeneous competent unit detached over a ductile horizon, and that the folds can be represented by chevronkink bands. The procedure is applicable to any detachment fold with associated growth strata that display wedge geometries (‘progressive unconformities’) indicating limb rotation through time. This method can be used for both detachment folds formed with constant limb length or variable limb length, and it can also accommodate undecompacted or decompacted growth strata.