Statistical classification of macroscopic folds as cylindrical, circular conical, or elliptical conical

A folded surface can be represented by the orientation of normals to the surface measured at several locations. When plotted on the unit sphere, the pattern of normals determines the type of fold. Poles from a cylindrical fold give a great circle on the unit sphere, whereas poles of a circular conical fold give a small circle, and poles from an elliptical conical fold give the projection of an ellipse onto the surface of the sphere. Several statistical tests that appear in the literature for classifying folds are discussed and compared. All but one of the tests use quantities obtained from an iterative least-squares procedure that fits the appropriate curve on the sphere. The classification procedure is illustrated with folds from the Canadian Rocky Mountains and uses for examples a cylindrical fold and a circular conical fold from the Smoky River coal field near Grande Cache, Alberta, and an elliptical conical fold near Jasper, Alberta. This methodology has resulted in new coal reserves in the Grande Cache area.This paper was presented at Emerging Concepts, MGUS-87 Conference, Redwood City, California, 13–15 April 1987.     

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.     

A mathematical model for orientation data from macroscopic cylindrical folds

An asymptotic form of Bingham's distribution on the sphere is applied to orientation data from cylindrical folds. Data from cylindrical folds typically form two clusters, one cluster for each fold limb. A bimodal distribution is obtained by fitting a unimodal distribution to each cluster. One parameter of the distribution gives the fold axis, another parameter is directly related to the curvature of the fold limb. Certain tests of hypotheses based on this distribution are the same as tests based on the Dimroth—Watson (symmetric girdle)distribution. One such is the test of whether two folds have the same fold axis.     

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。     

Conical folds and apparent rotations in paleomagnetism (a case study in the Southern Pyrenees)

The so-called apparent rotation was defined as the angular deviation between a local paleomagnetic direction (after the standard bedding correction) and their corresponding paleomagnetic reference [J. Geophys. Res. 85 (1980) 3659]. In this paper, we make a theoretical exploration on this concept and we conclude that (depending on the number, sequence, orientation and magnitude of the deformation axes that have affected to the rock volume) the apparent rotation may be the addition of a vertical-axis rotation plus a spurious rotation. The later is an error whose origin is the inappropriate application of the bedding correction during the restoration (which does not fit the reverse sequence of deformations). Then, Apparent rot. (s.l.) (δ)=Spurious rot. (θ)+Vertical-axis rot. (β).Conical folds are complex geometries that cannot be restored by using the bedding correction. However, appearance of apparent and spurious rotations has not been studied even though the presence of this kind of folds is very common in fold and thrust belts. In this paper, we show a way to restore these structures and its associated paleomagnetic data by means of forward modelling on a stereographic projection. The modelling has to be based on a good characterization of the geometry (fold axis orientation) and understanding of the kinematics of the fold. General modelling has also allowed us to predict the apparent rotation in conical synclines. Its magnitude depends on the semiapical angle and on the degree of development of the fold; the sense of the rotation (clockwise or counter-clockwise) will depend on the sense of rotation the fold axis.The western External Sierras provide an excellent case study of apparent rotations due to the presence of a conical fold in the footwall (Ebro foreland basin) of the South Pyrenean sole thrust. In addition, a vertical-axis clockwise rotation up to 47° (32° in average) has been detected in the hagingwall. An apparent rotation up to 28° (20° in average) is observed in the footwall of the structure when a simple bedding correction is used. This deviation does not fit with the expected Ebro basin direction (reference) and is caused by the effect of the Riglos conical syncline, developed by the flexure of the foot wall ramp of the South Pyrenean sole thrust. The forward modelling carried out considering the geometry and kinematics (non significant rotations in the autochthonous foot wall) of this structure predicts very well the paleomagnetic observations in the field (geographic coordinates) with angular departures of only 5° (in average). The only application of the bedding correction would introduce errors (spurious rotations up to 21°, 12 in average) related to the conical geometry that would not allow the differentiation of these distinct structural units.     

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.     

Parasitic folds with wrong vergence: How pre-existing geometrical asymmetries can be inherited during multilayer buckle folding

Parasitic folds are typical structures in geological multilayer folds; they are characterized by a small wavelength and are situated within folds with larger wavelength. Parasitic folds exhibit a characteristic asymmetry (or vergence) reflecting their structural relationship to the larger-scale fold. Here we investigate if a pre-existing geometrical asymmetry (e.g., from sedimentary structures or folds from a previous tectonic event) can be inherited during buckle folding to form parasitic folds with wrong vergence. We conduct 2D finite-element simulations of multilayer folding using Newtonian materials. The applied model setup comprises a thin layer exhibiting the pre-existing geometrical asymmetry sandwiched between two thicker layers, all intercalated with a lower-viscosity matrix and subjected to layer-parallel shortening. When the two outer thick layers buckle and amplify, two processes work against the asymmetry: layer-perpendicular flattening between the two thick layers and the rotational component of flexural flow folding. Both processes promote de-amplification and unfolding of the pre-existing asymmetry. We discuss how the efficiency of de-amplification is controlled by the larger-scale fold amplification and conclude that pre-existing asymmetries that are open and/or exhibit low amplitude are prone to de-amplification and may disappear during buckling of the multilayer system. Large-amplitude and/or tight to isoclinal folds may be inherited and develop type 3 fold interference patterns.     

雅砻江锦屏水电站地区褶皱分期及其对岩石地层的影响研究

根据褶皱构造的分类、分期原则,首次将雅砻江锦屏水电站地区褶皱划分为3期:第1期为印支造山期主褶皱;第2期为印支期后层内流变褶皱,第3期为燕山期—喜山期叠加褶皱。这3期褶皱在类型、规模和特征上各具特色,晚期褶皱对早期褶皱还存在叠加关系,从而对本区构造格局的形成以及地层岩石的展布、形态和厚度变化等方面起着直接的控制作用。印支期主褶皱在宏观上控制了坝区地层岩石的总体分布,第二期和第三期褶皱造成坝区岩层厚度、形态、岩性组合的明显变化。     

朝鲜平南盆地叠加褶皱构造特征

对于朝鲜平南盆地沉积盖层内发育的倒转褶皱,过去一般认为是直立褶皱的次级从属褶皱。通过对平南盆地内倒转褶皱发育区详细的地质调查,结合煤炭开发过程中获得的地质及钻探资料,提出平南盆地内叠加褶皱的主要识别标志为：地质平面图上呈现不同类型的两组褶皱脊线的交叉;倒转褶皱的轴面被直立褶皱改造弯曲;直立褶皱的两翼发育的倒转褶皱表现为两组牵引褶皱。查明了平南盆地存在3个阶段的褶皱构造：第一阶段为东西向的倒转褶皱,形成于印支期;第二阶段褶皱为东西向的直立褶皱,形成于早燕山期;第三阶段褶皱为北北东向的直立褶皱,形成于晚燕山期。     

新型锥形PDC齿犁削破岩理论研究

目前关于锥形PDC齿的研究主要集中于数值模拟、现场及室内试验,而关于其切削力学特性的理论分析未见相关报道。通过数值模拟阐述了锥形PDC齿拉伸剪切破岩机理,以及切削载荷分布特性;根据能量平衡原理,推导了锥形PDC齿切削载荷理论公式。结果表明,相比于常规PDC齿,锥形PDC齿破岩过程更加稳定,其切削载荷受岩石性质、齿的形状参数、切削深度以及切削角度的影响,且切削载荷随着锥顶半径、切削角度以及吃入岩石深度的增加而增加。研究成果可为锥形PDC齿及钻头的设计提供理论支撑。      

The glaciotectonic deformation of Quaternary sediments by fault-propagation folding

Fault-propagation folding is an important yet seldom recognised structural style within sediments affected by glacier-induced deformation. Fault-propagation folds develop in the hanging wall of low angle thrust faults and compensate part of the slip along the fault. Field examples are recognised across northern Europe, in glaciotectonic complexes of north Germany, Wales and the Isle of Man. The recognition of the fault-propagation fold mechanism in glaciotectonic deformation is important because resultant structures are related to exactly the same phase of deformation (i.e. the same phase of ice advance), and thus play a critical role in analyses of the temporal and spatial evolution of glacier-induced deformation. Some field examples show monoclinal geometries that are in good agreement with predictions of trishear kinematic theory. The trishear approach is appropriate to model these structures because the structures analysed in the field and simulated below show characteristics that are compatible with fault-propagation folds that were produced by trishear kinematics. The curved forelimb and the monocline geometry of the fault-propagation folds fit to the trishear model. The occurrence of footwall synclines is also in good agreement with trishear kinematics. These synclines show the typical thickening of the strata in the hinge. With respect to the modelling output, most important factors for the structural evolution of the fault-propagation folds is the ramp angle of the thrust, the position of the tip line and the propagation-to-slip ratio along the fault. This fits to observations made by previous studies at large scale fault-propagation folds in fold-and-thrust belts.     

An alternative interpretation for the formation of doubly plunging folds in sandstone terrains

Folds are marvellous features of mountain terrains, but despite extensive research, many fundamental problems have still not been solved. In terrains of sandstone, fold hinges are rarely straight lines but curved, forming a pattern characterized by doubly plunging, elliptical dome‐and‐basin structures. Such structures are an obvious manifestation of coeval or successive shortening deformation in two orthogonal principal directions in the horizontal plane. Based on an anatomic investigation of the fold pattern of sandstone beds at the rocky beaches of Saint‐Jean‐Port‐Joli (Quebec, Canada), we propose that the doubly plunging folds may result from a transition from a plane deformation to a constrictional deformation due to auxetic effects of quartz‐rich rocks. The sandstone beds possessed potentially such negative values of Poisson's ratio that, when placed under compression in one direction, they become contracted in the transverse direction, producing a series of doubly plunging folds. Further work is needed to approve or disapprove the interpretation.     

The geometrical classification of noncylindrical folds

The majority of naturally occurring folds are noncylindrical if definitions are strictly applied. A new classification of noncylindrical folds using a triangular plot and based on measurements of interlimb angle and hinge angle is proposed. The end-members of the triangular plot are planes, cylindrical isoclines, and isoclinal domes. An infinite range of cylindrical and noncylindrical plane fold shapes may be represented. Noncylindrical nonplane folds may be represented on the plot using proportional circles to signify the degree of non-planarity. The triangular diagram is used to classify large-scale folds from north Norway and their origin is discussed.     

褶皱构造的曲率分析及其裂缝估算--以江汉盆地王场褶皱为例

曲面曲率的几何意义可以反映曲面一点邻近区域的具体形态和变形特征,因而可以用来研究褶皱层面的变形特征、构造样式及其可展性.褶皱面裂缝发育主要为一组张裂系,张裂隙走向与最大主曲率的方向垂直.褶皱构造的主曲率在一定程度上可以反映与褶皱有关的张裂隙的发育情况.对于圆柱状褶皱可以估算与褶皱有关的裂缝孔隙度和渗透率.以王场褶皱构造面为例,通过计算二次趋势面的特征值和特征向量得到褶皱构造面主曲率的大小和方向,确定该褶皱为圆柱状褶皱,并对与之相关的裂缝孔隙度和渗透率进行了估算.     

Regional flow perturbation folding within an exhumation channel: A case study from the Cycladic Blueschists

Kilometre-scale cylindrical folds and associated parasitic folds that trend at small angles to the transport lineation are analysed along a 100-km-long transport-normal segment of the Cycladic Blueschists in an attempt to reconstruct the 3D structural architecture within an exhumation channel. Reversals in the polarity of both fold vergence and the hinge/lineation obliquity occur in a flow-normal direction, defining transport-parallel culmination and depression surfaces that root downwards onto an underlying detachment. Fold patterns generated around culmination and depression surfaces support models of flow-perturbation folding where folds initiate at small angles or sub-parallel to transport in response to wrench-dominated differential shearing. Successive culmination and depression surfaces are separated from one another by along strike distances of ∼20 km, although atypical fold geometries developed in the flanks of major culmination and depressions follow their own patterns, revealing that smaller perturbations occur within the larger scheme. Major culminations are interpreted to reflect regions of surging flow marked by increased velocity during exhumation, whilst the opposite is true for depressions. This behaviour implies that on a regional scale, differential shear varies laterally in an irregular-sinusoidal manner defining areas of relative high and relative low displacement within the exhumation channel.     

The Migif–Hafafit gneissic complex of the Egyptian Eastern Desert: fold interference patterns involving multiply deformed sheath folds

The Wadi Hafafit Complex (WHC) is an arcuate belt of orthogneisses, migmatites and other high-grade metamorphic rocks, which marks the boundary between the Central Eastern and the South Eastern Deserts of Egypt. In the WHC, gneissic meta-gabbro outlines macroscopic fold interference patterns characterized by elliptical to irregular culminations cored by gneissic meta-tonalite to meta-trondhjemite. The five main culminations of the WHC have previously been labeled A (most northerly), B, C, D and E (most southerly). A detailed structural investigation of B, C, D and E reveals that these structures are a result of the interference of four macroscopic fold phases, the first three of which may represent a single deformation event. The first folding involved sheath-like fold nappes, which were transported to the N or NW, assisted by translation on gently dipping mylonite zones. The regional gneissosity and mineral extension lineations formed during this folding event. The fold nappes were deformed by mainly open upright small macroscopic and mesocopic folds with approximately NE-trending hinges. As a probable continuation of the latter folding, the sheaths were buckled into large macroscopic folds and monoclines with the same NE-trends. The fourth macroscopic folding resulted from shortening along the NE–SW direction, producing mainly NW–SE-trending upright gently plunging folds. Gravitative uplift is disputed as a component of the deformation history of the WHC. The peculiarities of the fold interference pattern result from the interesting behaviour of sheath folds during their refolding.     

Regional significance of kilometric-scale north-east vergent recumbent folds associated with east to south-east directed shear on the southern border of the Central Iberian Zone (Hornachos-Oliva region,Variscan belt,Iberian Peninsula)

On the southern border of the Central Iberian Zone there are two sectors with different styles of deformation. To the south-west, in the Hornachos sector, large-scale recumbent folds associated with ductile shearing can be seen. This shearing is characterized by a direction of movement parallel to the fold axes and can be correlated for 150 km along strike. The K-values of the strain ellipsoid range from 0.8 to 2.0. Stretching in the X direction, parallel to the recumbent fold axes, is more than 100%. To the north-east, in the Oliva sector, first-phase folds are upright and the strain intensity is lower than in the Hornachos sector. Metamorphic, geometric and kinematic considerations lead us to conclude that the shearing in the Hornachos sector is better explained as conjugate to a main shear zone along which the southern border of the Central Iberian Zone is moved onto the Ossa-Morena Zone. This main thrust is at present obliterated by a left-lateral extensional shear zone that affects a high pressure exotic unit located between the Central Iberian and the Ossa-Morena Zones. This high pressure unit constitutes a suture of the Variscan belt in the Iberian Peninsula.     

Flow pattern within a Permian submarine slump recorded by oblique folds and deformed fossils, Ulladulla, south-eastern Australia

The flow pattern within a slump in Permian marine rocks of the southern Sydney Basin, Australia, is recorded by folds and deformed fossils. Abundant brachiopod and bryzoan fossils in the slumped rocks are relatively undeformed, but fossil crinoid stems have been deformed by relative rotation of individual ossicles. Measurement of the strain indicates that the deformation of the crinoids is consistent with flexural flow folding within the slump. Previous models assume that curved slump fold axes remain parallel to the enveloping bedding surface of a slump sheet. Detailed measurements of the orientation of slump folds in this study found fold axes to be oblique to bedding, which is interpreted as a result of folds plunging downward towards the flanks of the slump or slump lobes. In the present model, fold axes are not generally parallel to the strike of the fold axial surface, and this can explain differences between the orientations of slump fold axes and axial surfaces when these are used as directional indicators of slump movement.     

Structural control on gold mineralization at Walhalla,Victoria

Gold mineralization associated with quartz reefs is related to the structural history of the Early Devonian, Walhalla Group. These reefs are situated in the Walhalla Synclinorium, developed during the Middle to Late Devonian Tabberabberan Orogeny. A pervasive north‐south‐trending axial planar cleavage and two styles of folding were produced during regional east‐west compression. The first are upright, open to close folds with sub‐horizontal fold axes. The second are plunging inclined, close to tight folds with fold axes that plunge steeply to the north and south. An extensional event is associated with the emplacement of the Woods Point Dyke swarm and a set of normal faults that offset all earlier structures. High‐angle reverse faults, which post‐date the folding and the emplacement of the dykes, were utilized as conduits for hydrothermal fluids and preferentially localize mineralization to laminated quartz veins. En echelon vein arrays formed during initial stages of reverse faulting became deformed during prolonged shearing to produce ptygmatic veins. Laminated quartz veins within high‐angle reverse faults contain arsenopyrite and pyrite in vein margins and gold in fractures that cross‐cut continuous quartz crystals. Gold, galena, chalcopyrite and sphalerite may also be deposited adjacent to and within fractured arsenopyrite and pyrite. Late‐stage, cross faults developed in a regime of north‐south compression and post‐date the laminated quartz veins and mineralization.