Introduction

A fold is a distortion of a volume of rock material that manifests itself as a bend or nest of bends in linear or planar elements (Hansen, 1971, p. 8). Many folds involve elements which were originally planar. Sedimentary bedding is the common example, and this is an important case because the geometry of the fold then represents an important indicator of the nature of the deformation. In particular, its features can be correlated with certain aspects of rotation and stretch. However, folds may also develop from originally curved elements, and the problem of relating the features to the deformation is much more severe.

Folding occurs when pre-existing elements are transformed into new curviplanar or curvilinear configurations, whatever their original state. Thus folding is just an inhomogeneous deformation which acts on a body of material containing linear or planar elements (see Fig. 11.1). It is worth noting, however, that a deformation which produces a fold in one situation may not in another. Planar or linear elements may be entirely absent from the rock mass, and therefore there is nothing to mark a fold form. It is also possible that initially curved elements might become planar or linear, or that the elements may be so oriented as to remain planar or linear (Ramsay, 1967, p. 473).

In the following sections, a number of relatively simple geometrical properties of folded surfaces are explored.

Introduction

The emphasis in this book is the geometrical description and analysis of geological structures, especially by graphical means. The basis of much of this is descriptive geometry: the art of accurately drawing three-dimensional objects and of graphically solving associated space problems. It is based on the idea of depicting such objects by means of projections. Everyday examples of projections are shadows and photographs. Both are the result of projecting various parts of an object to a plane by rays of light. These rays are projectors which connect points on the object with the corresponding points on the image plane.

By projecting an object to an image plane, a view of that object is obtained. Alternatively, a view may be thought of as an actual picture of the object obtained along a line of sight perpendicular to the corresponding image plane.

Orthographic projection

The simplest type of projection, and the one used most in engineering as well as for many purposes in geology, is the orthographic projection. Orthographic means “drawn at right angles” and refers to parallel projectors that are perpendicular to an image plane (Fig. A.1a). The most important property of this projection is that the images of objects appear in their true shape.

Definitions

Thickness: the perpendicular distance between the parallel planes bounding a tabular body, as displayed on any section perpendicular to these planes; also called the true or stratigraphic thickness (Fig. 2.1).

Apparent thickness: the distance between the bounding planes measured in some other direction, for example, the perpendicular distance between the traces of the bounding planes on an oblique section, or in some other specified direction, as in a drill hole. It is always greater than true thickness.

Outcrop width: the strike-normal distance between the traces of the parallel bounding planes measured at the earth's surface. It may be measured horizontally or on an incline.

Depth: the vertical distance from a specified level (commonly the earth's surface) downward to a point, line or plane.

Thickness determination

Although geologists may determine the thickness of any stratiform body of rock, most often the concern is with the thickness of layers of sedimentary rocks. In this context “measuring a section” generally refers to a lithologic description of the rock strata as well as a determination of their thicknesses (Kottlowski, 1965; Compton, 1985). Here, the concern is with thickness alone. The thickness of a layer may be determined in a number of ways. In special circumstances it may be possible to measure it directly, otherwise it must be determined from indirect measurements.

Introduction

Deforming a circle into an ellipse clearly demonstrates that the orientations and lengths of lines and the angles between pairs of lines generally change. With suitable material the stretch and angle of shear associated with a line may be determined from measurements made on deformed objects whose original shape or size are known. It may then be possible to determine something of the shape, size and orientation of the strain ellipse. For example, measurement of the deformed shape of an originally spherical oolite yields the shape of the ellipse and its orientation directly. This chapter deals with some additional techniques for extracting two-dimensional strain information from deformed rocks. Many more examples, including some excellent photographs, can be found in the book by Ramsay and Huber (1983). Lisle (1994) gives a good review of more recent developments.

Before describing the full analytical method it will be useful to show that in some situations the shape and orientation of the strain ellipse can be obtained simply and directly using purely graphical means.

Deformed grains

The center points of individual grains in a section through a rock form a grid. In terms of the center-to-center distances the possible geometrical patterns have two end-members. If the distribution is random, the minimum distance between centers is zero and such pattern exhibits clustering. If all the grains are perfectly uniform circles and are closely packed then all distances between centers will be equal in the undeformed state (Fig. 12.1a), and thus are radii of a circle.

The first steps in the study of geological structures are largely geometrical. This is true in the historical development of our knowledge of such structures, in the initial stages of any field investigation, and in the education of a structural geologist. This concern for geometry includes the methods of describing and illustrating the form and orientation of geological structures, and the solution of various dimensional aspects of these structures.

This book attempts to fill a need for an introduction to the geometrical techniques used in structural geology. I have sought an approach which is basic yet modern. The topics covered include well-established techniques, newer approaches which hold promise and an introduction to certain fundamental mechanical concepts and methods. Students who go no further in structural geology should have a working knowledge of the basic geometrical techniques and at least some appreciation of where the field is headed. Those who do go on, either in advanced courses or on their own should have the necessary foundation.

The first few chapters apply the methods of orthographic projection to the solution of simple structural problems. An introduction or review of these methods is given in Appendix A. Application to geological and topographical maps are included and extensive use is made of Mackin's powerful method of visualization – the down-structure view of geological maps.

The method of stereographic projection and the stereonet, together with the methods of plotting and solving angular problems are introduced fairly early.

Map symbols

Just as structural planes intersect the earth's surface to give characteristic outcrop patterns, equally distinctive though more complex patterns result from the intersection of folds with the topographic surface. In this chapter we treat some important aspects of the patterns made by folds which are essentially cylindrical.

If exposures are good, it is generally possible to locate and measure the attitudes of both the hinge and hinge surface of a fold in the field. On the other hand, if these features can not be directly observed, then the attitude information and the location of the hinge surface must be found using indirect methods.

In any case, the fold is then identified on the map with a line marking the trace of the hinge surface at the earth's surface, together with symbols indicating the attitude of the hinge and hinge surface and the direction of closure (Fig. 17.1).

Outcrop patterns

Horizontal folds have the simplest type of outcrop pattern. On a horizontal exposure plane, the map pattern of such a fold is essentially the sum of the patterns of the inclined limbs, that is, a series of parallel outcrop bands (Fig. 17.2a). If the relative age or correlation of the units, or the attitude at several points is not known, it may not be possible to interpret the pattern as being part of a fold at all.

Introduction

In some folds the thicknesses of layers remain essentially uniform with the result that they display constant orthogonal thickness. Such folds are parallel. They commonly have gentle to close shapes and are typically developed in well-bedded sedimentary rocks. There are two end-member shapes (see Fig. 14.4):

1. Rounded forms have smoothly curved limbs and broad hinge zones.

2. Angular forms have straight limbs and narrow hinge zones.

We seek ways of reconstructing both types of such folds in profile from field data. Here we consider only the case of horizontal folds, that is, folds whose profile planes are vertical. The methods will not work for plunging folds and the additional steps required to construct their profiles are treated in Chapter 17.

Rounded folds

First, we treat the case of smoothly rounded folds, sometimes referred to as concentric folds.

The requirement of strict constant orthogonal thickness within a folded layer or packet of layers severely limits the states of strain which can exist in parallel folded layers. There are two mechanisms involved: shear and extension.

Some insight into possible states can be obtained by considering simple models. Once again we resort to a deck of thin cards. To produce a fold hold the deck with both hands and press inward on the ends (it helps to give the center of the deck a slight upward nudge with the thumbs).

Introduction

Especially in the early stages of an investigation of the geology of an area, much attention is paid to determining and recording the location and orientation of various structural elements. Planes are the most common of these. They are also a useful starting point in the introduction to the geometrical methods of structural geology.

Definitions

Plane: a flat surface; it has the property that a line joining any two points lies wholly on its surface. Two intersecting lines define a plane.

Attitude: the general term for the orientation of a plane or line in space, usually related to geographical coordinates and the horizontal (see Fig. 1.1). Both trend and inclination are components of attitude.

Trend: the direction of a horizontal line specified by its bearing or azimuth.

Bearing: the horizontal angle measured east or west from true north or south.

Azimuth: the horizontal angle measured clockwise from true north.

Strike: the trend of a horizontal line on an inclined plane. It is marked by the line of intersection with a horizontal plane.

Structural bearing: the horizontal angle measured from the strike direction to the line of interest.

Inclination: the vertical angle, usually measured downward, from the horizontal to a sloping plane or line.

True dip: the inclination of the steepest line on a plane; it is measured perpendicular to the strike direction.

Apparent dip: the inclination of an oblique line on a plane; it is always less than true dip.

Introduction

As we have seen in Chapter 11, the study of deformation is concerned solely with a comparison of a body of rock in its initial and final configurations: the translation compares the initial and final places, the rotation compares the initial and final orientations, and the stretch compares the initial and final shapes and sizes (see Fig. 11.2). No consideration is given to intermediate configurations or to a particular sequence of configurations (Mase, 1970, p. 77).

However, the motion or flow by which a particular deformed state is attained is also of considerable interest if we are to understand the processes involved in the formation of geological structures. Kinematics is the branch of mechanics concerned with the motion of bodies without regard to any associated forces.

In this chapter we first treat the basic elements of a kinematic analysis by describing the measured velocity field in a tectonically active area and the information that can be derived from it. Second, we consider an approach to the more difficult problem of understanding the flow responsible for old structures. Third, by considering the progressive geometrical evolution of structures we can gain some insight into the geometrical nature of geological flow patterns. Finally, after treating some important theoretical matters, we use these results to consider briefly an alternative approach to estimating the time rates of deformation.

Introduction

Vectors play a prominent role in many geometrical and physical applications. We have seen several simple examples in Chapters 1 and 3 where the attitudes of planes and lines were represented and manipulated using two-dimensional vectors. We now extend the treatment to three dimensions, and to several additional applications.

As we have seen, the stereonet is a useful way of displaying and manipulating structural lines and planes easily and directly in a three-dimensional setting. For the same reasons, we can use the stereonet to introduce an analytical approach involving vectors which is a powerful method for solving these same types of problems (see also Sprenke, 1992).

Because the orientations of planes are defined by their poles, we can represent all structural elements by lines. There are two types of such lines.

1. Axes have orientation but no sense. Lineations in metamorphic rocks, lines of intersection and poles of fracture planes are examples.

2. Vectors have both orientation and sense. Examples include some linear sedimentary structures and paleomagnetic directions.

Some structural lines may be treated in either way. For purely geometrical purposes the pole of sedimentary bedding is commonly treated as an axis, but for other purposes the pole in the direction of younging has sense and therefore is a true vector.

Introduction

In a number of geological situations structural lines and planes have been rotated from some initial orientation. One of our tasks is to describe such rotations and this can be done with the aid of the stereonet. Every rigid rotation can be defined by an angle and sense of rotation about a specified axis.

The most general case involves rotation about an inclined axis, but we start with the simpler cases of rotations about vertical and horizontal axes. We do this because it is a good way to introduce the techniques of rotations and because a sequence of such rotations is equivalent to a rotation about a single inclined axis. In all cases, the sense of rotation is described as clockwise or anticlockwise when looking along the rotation axis, whether horizontal, inclined or vertical.

Basic techniques

As an aid to visualization consider a turntable (Fig. 6.1). As the base rotates about its axis R through some angle ω the locus of an oblique line L through its center O is a right-circular cone of rotation Angle ϕ between R and L is the semi-vertex angle of this cone. The intersection of this cone with the sphere will, in general, be a small circle, one in the lower and one in the upper hemisphere. There are, however, two limiting cases: if ϕ = 0 the surface degenerates to a line and if ϕ = 90° it becomes a plane.

Exposures on horizontal surfaces

In Chapter 2 the simplest examples of the determination of thickness assumed that the earth's surface was a horizontal, geometrically perfect plane. The intersection of inclined layers with this surface results in an outcrop pattern. Represented in map view this pattern is a simple geological map.

In this case the width of the outcrop bands depends on two factors: the actual thickness of the layers and the angle of dip of each layer. The separate effect of each of these factors is shown in Fig. 4.1. In essence, these same relationships also apply to less-than-perfect real horizontal topographic surfaces.

In the special case of a vertical layer, the outcrop width in map view is equal to the thickness of the layer. This unique relationship results from the fact that the map shows such a layer in edge view, that is, a line of sight in viewing the map coincides with a line which is parallel to the vertical layer. In estimating thickness of tabular objects, one instinctively seeks just such a view.

In the more general case of an inclined layer, one line of sight is always identifiable on a geological map; it is in the dip direction. For inclined layers an auxiliary view perpendicular to this line could easily be constructed that would show the layers in edge view and therefore in true thickness (Fig. 4.2a).

Definitions

Fault: a surface along which appreciable displacement has taken place; this surface may be planar or curviplanar.

Fault zone: a zone containing a number of parallel or anastomosing faults.

Echelon faults: relatively short, parallel faults of a zone which display an overlapping or staggered pattern.

Shear zone: a zone across which two blocks have been displaced in fault-like manner, but without development of visible fractures.

Footwall: the surface bounding the body of rock immediately below a non-vertical fault. The body of rock itself is called the footwall block.

Hangingwall: the surface bounding the body of rock immediately above a non-vertical fault. The body of rock itself is called the hangingwall block.

Cut-off line: the trace of a displaced plane on the fault surface; these lines occur in pairs, one on the footwall and one on the hangingwall.

Slip: the relative displacement of formerly adjacent points (Fig. 8.1a); also called the net slip. It is represented by the relative slip vector, usually of the hangingwall block relative to the footwall block. Dip slip and strike slip are components. This total displacement is the result of the accumulation of a number of small slip events, which are not necessarily parallel to the slip vector.

Separation: the distance between two displaced planes. It may be measured on the fault perpendicular to the cut-off lines, but more commonly in the strike or dip directions (Fig. 8.1b).

Introduction

Processes acting within the earth at various times and places cause bodies of rocks to be displaced from the sites of their origin. After such displacements the bodies have different locations and orientations, and also commonly different shapes and sizes. Such bodies are said to be deformed. Although it is ultimately necessary to treat these changes in a full three-dimensional setting, many situations can be approached from a consideration of just two, and this also serves as a useful way to introduce the subject.

Even in two dimensions, the complete geometrical description of the deformed state for a relatively simple structure may be quite involved. Imagine a rectangular block composed of sedimentary strata (Fig. 11.1a). As the result of a general deformation, original lines become curvilinear, original planes curviplanar and original parallel lines and planes are no longer parallel (Fig. 11.1b). Such a deformation is inhomogeneous.

The description of a deformation consists of comparing the initial and final configurations of the body with emphasis on the changes which have occurred. To facilitate this comparison we refer the body in its initial place, orientation, shape and size to material coordinates (x, y) and its final place, orientation, shape and size to spatial coordinates (x′, y′). Every material point or particle P(x, y) in the body is displaced to spatial point P′ (x′, y′). Each point P′ is related to its particle P by a displacement vectoru.