Stereographic Analysis of Structural Geology

Stereographic projection
Stereographic projection comes in very handy for representing points lying on a three-dimensional surface in two dimensions for improved analysis. Stereographic presentations remove one dimension from consideration so that lines or points alone may stand for planes and points for lines. One important limitation of stereographic projections is that they consider only angular relationships between lines and planes; they do not stand for the position or size of the feature.

Stereographic projection A reference sphere in which its equatorial plane is horizontal, and its orientation is fixed relative to north. Planes and lines with specific plunge and trend are positioned in an imaginary sense so that the axis of the feature passes through the center of the reference sphere (Figure 1). The intersection of the feature with the lower half of the reference sphere defines a unique line on the surface of the reference hemisphere. For a plane this intersection of the reference sphere is a circular arc called a great circle, for a line it is just a single point. A stereographic projection of a plane or line is created by rotating the intersection with the reference sphere down to a horizontal surface at the base of the sphere (Figure 2). The rotated lines and points form unique locations on the stereonet that define the dip (plunge) and dip direction (trend) of the feature. In slope stability analysis using stereonets, planes are used to represent both discontinuities and slope faces.

Figure 1: Stereographic representation of plane and line on lower hemisphere of reference sphere: (a) plane projected as great circle; (b) isometric view of line (plunge and trend)

Figure 2: Equal area projections of plane and line: (a) plane projected as great circle and corresponding pole; (b) line projected as pole.

An alternative means of representing the orientation of a plane is the pole to the plane (Figure 2(a)). The pole is the point on the surface of the reference sphere that is pierced by a radial line in a direction normal to the plane. One of the values of the pole projection lies in the fact that a single point may represent the complete orientation of a plane. As explained in section Pole plots and contour plots, the use of poles enables a large number of planes to be analysed with much less work than using great circles.

It can be seen from Figures 1 and 2 that planes and lines with shallow dips have great circles and points which plot near the circumference of the stereonet, while those with steep dips plot near the center. This will be an aid to interpreting the information shown on stereonets. In contrast, the pole of a shallow dipping plane plots close to the center of the circle and the pole of a steep plane plots close to the perimeter. 

                                                 Figure 3: Polar and equatorial projections of a sphere

The two types of stereographic projections used in structural geology are the polar and equatorial projections as shown in Figure 3. The polar net can obviously be used only to plot poles while the equatorial net may be used for plotting both planes and poles as described later. For the equatorial projection, the most common type of stereonet projection is the equal area, or Lambert, Schmidt net. On this net, any area on the surface of the reference sphere projects as an equal area on the stereonet. This property of the net is exploited in the contouring of pole plots to find concentrations of poles that represent preferred orientations, or sets of discontinuities. The other type of equatorial projection is the equal angle or Wulff net; both the Wulff and Lambert nets can be used to examine angular relationships but only the Lambert net can be used to develop contours of pole concentrations.

For hand plotting of structural data the usual procedure is to place tracing paper on the nets and then draw poles and planes on the tracing paper. Since great circles have to be plotted with the tracing paper rotated on the net, as described below, a thumbtack is placed at the center point so that the curves can be plotted without distortion. Further details of stereographic projections are described by Phillips, 1971 who discusses the theoretical background to this technique, and Leyshon and Lisle, 1996 who demonstrated applications of this technique to geological mapping. Goodman and Shi, 1985 demonstrate stereographic techniques for identifying wedges of rock that can slide from the face, or are "removable"; this technique is termed key block theory.

Pole plots and contour plots
As may be seen from Fig. 2 the pole to a plane enables a point to represent the orientation of the plane. Pole plots, in which each plane is represented by a single point, are thus the most convenient means of examining the orientation of a large number of discontinuities. The plot thus immediately suggests a visual image of concentrations of poles that represent the orientations of sets of discontinuities, and the analysis is helped by using different symbols for different types of discontinuities. The plotting of poles by hand on a polar net is illustrated in Figure 4.

On the net, the dip direction scale, 0–360◦, has the zero mark at the bottom of the vertical axis and the 180◦ mark at the top. This is a convenience for plotting such that poles can be plotted directly without the need for rotating the tracing paper; it can be shown that poles plotted on both the polar and equatorial nets are in identical positions. Pole plots are commonly produced by stereographic computer programs, an example of which is shown in Figure 5.

Figure 4: Plotting poles on a polar net. Plot pole of plane oriented at 50/130—locate dip direction of 130◦ clockwise around the circumference of a circle starting at the lower end of the vertical axis. At 130◦ radial line, count 50◦ out from the center of the net, and plot a point at the intersection between 130◦ radial line and 50◦ circle.

Figure 5: Example of pole plot of 421 planes comprising bedding, joints and faults

This is a lower hemisphere, equal angle projection of 421 original poles mapped over an area of about a square kilometer, at a site where the rock is a bedded limestone. The rock includes four discontinuity sets that embrace the bedding, as well as two sets of joints and a number of faults, generally coincident with the bedding. For each of the three types of discontinuity, there is a different symbol on Figure 5. While there is consid-erable scatter in the pole orientations, careful examination of this plot shows some clustering particularly in the southwest quadrant. In order to identify discontinuity sets with considerable scatter on pole plots such as these, it is necessary to prepare contours of the pole density as described in the next section. 

Pole density

All natural discontinuities show some variation in their orientations which mark the pole plots. If the plot includes poles from a number of discontinuity sets, it can be difficult to distinguish between the poles from the different sets, and to find the most likely orientation of each set. Contouring the plot, however, presents a better definition of the most highly concentrated areas of poles. In most cases, the same conventional contouring package in a stereographic projection can be contained within computer programs. Countouring can also be carried out manually by placing a counting net such as the Kalsbeek net that comprises of mutually over- lapping hexagons, each of area 1/100 of the total planer area of the stereonet (Leyshon and Lisle, 1996); a Kalsbeek counting net is presented on Appendix I. Countouring is done by placing the counting net on the pole plot and counting the number of pokes in each square. For instance, if there are eight poles out of 421 poles in a total square then the concentration of this square is 2%. When the percent concentration has been obtained for each square, contours can be drawn.

Figure 6: Contoured plot of data shown in Figure 5, with great circles corresponding to mean orientation of bedding and two orthogonal joint sets, and lines of intersection between planes

Figure 6 demonstrates a contour map that has been drawn by poles plotted in Figure 5. It can be seen in the contour plot that the orientation of the bedding has relatively little scatter. It can be seen that the mean orientation of the bedding has a dip of 74◦ with a dip direction of 050◦. By contrast the joint orientations show more scatter and on the pole plot it is difficult to identify discontinuity sets. However, the two sets of orthogonal joints can be clearly distinguished on the contoured plot. Set A is shallow dipping with a dip of approximately 26◦ while its dip direction is approximately 219◦, and it is directed 180◦relative to the bedding.

Set B dips almost vertically and has an average dip direction of 326◦, at approximately rightangles to Set A. The poles for set B are concentrated in two main concentrations, on opposite sides of the contour plot, because some dip steeply to the north-west and others dip steeply to the south-east.

In Figure 6, the different pole concentrations are shown with varying symbols for each 1% contour interval. The percentage concentration is that number of poles which fall within each 1% area of the surface of the lower hemisphere.

Figure 7: Pole plot of faults selected from data plotted in Figure 5

Still another use of the stereographic projection program in structural data analysis is in the production of plots of selected data from the total data collected. For example, joints with lengths that are only a small fraction of the slope dimensions are likely to have a significant influence on stability. However, faults usually have greater persistence and lower friction angle than joints. Therefore, it would facilitate design to prepare a stereographic plot showing only faults (Figure 7). This plot shows that only 33 discontinuities are faults, and that their orientations are similar to those of the bedding. Selections can also be made, for example, of discontinuities that have a certain type of infilling, or are slickensided, or show evidence of seepage, provided that the mapping identifies this level of detail for each surface.

Assignment of poles into discontinuity sets is typically accomplished by a combination of contouring, visual examination of the stereonet, and knowledge of the geological conditions on the site that will frequently show expected trends in orientation of the sets. It is also possible to identify discontinuity sets by rigorous and less subjective analysis of clusters in orientation data. Mahtab and Yegulalp (1982) presented a technique that can be carried out to identify clusters in random distributions of orientations using the Poisson's distribution. However, this and any similar procedure tends to identify more than about four concentrations. A result should be carefully scrutinised before using it in design.

Great circles

Once the orientation of the discontinuity sets, together with significant single discontinuities such as faults, have been identified on the pole plots, the second step in the analysis is to determine whether these discontinuities form potentially unstable blocks in the slope face. The analysis is completed by plotting great circles for every one of the discontinuity set orientations along with the orientation of the face. In this way the orientation of all the surfaces that have an influence on stability are represented on a single diagram. Figure 2.11 shows the great circles of the three discontinuity sets identified by contouring the pole plot in Figure 2.10. It is usually possible to have no more than about five or six great circles on a plot, because with a greater number it is difficult to tell where all the circles intersect.

Figure 8: Construction of great circles and a pole representing a plane with orientation 50 (dip)/130 (dip direction) on an equal area net: (a) with the tracing paper located over the stereonet by means of the center pin, trace the circumference of the net and mark the north point. Measure off the dip direction of 130◦ clockwise from north and mark this position on the circumference of the net; (b) rotate the net about the center pin until the dip direction mark lies on the W–E axis of the net, that is, the net is rotated through 40◦ counterclockwise. Measure 50◦ from the outer circle of the net and trace the great circle that corresponds to a plane dipping at this angle. The position of the pole, which has a dip of (90–50), is found by measuring 50◦ from the center of the net as shown, or alternatively 40◦ from the outside of the net. The pole lies on the projection of the dip direction line which, at this stage of the construction, is coincident with the W–E axis of the net; (c) the tracing is now rotated back to its original position so that the north mark on the tracing coincides with the north mark of the net. The final appearance of the great circle and the pole representing a plane dipping at 50◦ in a dip direction of 130◦ is as illustrated.

While computer generated great circles are convenient, hand plotting is of value in developing an understanding of stereographic projections. Figure 2.13 illustrates the procedure for drawing great circles on an equal area net. As shown in Figure 2.8, the procedure involves overlaying the stereonet with tracing paper on which the great circles are plotted.

It is the shape of blocks formed by the intersection of discontinuities, and the direction in which they may slide that is usually the primary purpose of plotting great circles of discontinuity sets in a slope. For example, in Figure 2.1 the slope failures only occurred for conditions where single discontinuities, Figure 2.1(a), or pairs of intersecting discontinuities, Figure 2.1(b), dip out of the face. It is, of course, important to identify such potential failures before movement and collapse occurs. This requires an ability to visualize the three-dimensional shape of the wedge from the traces of the discontinuities on the face of the original slope. Stereographic projection is a convenient means of carrying out the required three-dimensional analysis, keeping in mind that this procedure examines only orientation of discontinuities, not their position or dimensions. For instance, if the stereonet shows the occurrence of a possibly unstable block, examination of a geologic map showing the location of the discontinuities would help to determine whether they intersect a slope.

Figure 9: Determination of orientation (plunge and trend) of line intersection between two planes with orientations 50/130 and 30/250: (a) the first of these planes has already been drawn in Figure 8. The great circle defining the second plane is obtained by marking the 250◦ dip direction on the circumference of the net, rotating the tracing until the mark lies on the W–E axis and tracing the great circle corresponding to a dip of 30◦; (b) the tracing is rotated until the intersection of the two great circles lies along the W–E axis of the stereonet, and the plunge of the line of intersection is measured as 20.5◦; (c) the tracing is now rotated until the north mark coincides the north point on the stereonet and the trend of the line of intersection is found to be 200.5◦

Lines of intersection

The intersection of two planes defines a line in space that is characterised by a trend (0–360◦) and plunge (0–90◦). In the stereographic projection, this line of intersection is defined as the point where the two great circles cross (Figure 9). The two planes can form a wedge-shaped block and the trend of the line of intersection defines the direction in which this block may slide. The presence of two intersecting great circles on the stereonet does not, however, automatically imply that a wedge failure will take place. The factors that influence the stability of the wedge include the direction of sliding relative to the slope face, the dip of the planes relative to the friction angle, external forces such as ground water, and whether or not the planes are located so that they actually intersect behind the face. 

Figure 9 The method of measuring the trend and plunge of a line of intersection of two planes on the equal area stereonet, while Figure 10 shows the method for measuring the angle between two planes. These two measurements have the value that the bearing of this line of intersection has a relationship to the direction of sliding, while the angle between planes gives indication for wedging action where two planes intersect. If the angle between the planes is small, a narrow, tight wedge will be formed with a higher factor of safety compared to a wide, open wedge in which the angle between the planes is large. For the data shown on Figures 5 and 6, intersections occur between the bedding and joint set A, and between the bedding and joint set B, and between joint sets A and B. The orientations of the three lines of intersection are shown on Figure 7. Intersection line I3 has a trend of 237◦ and a plunge of 24◦, and joint sets A and B could together form a wedge failure that would slide in the direction of the trend. Intersection line I1 is nearly horizontal, and thus the wedge formed by the bedding with the joint set A is unlikely to slide, while intersection line I2 is near vertical and would form a thin wedge in the face.

Figure 10: Determination of angle between lines with orientations 54/240 and 40/140: (a) the points A and B that define the poles of these two lines are marked on the stereonet as described in Figure 8 for locating the pole; (b) the tracing is rotated until the two poles lie on the same great circle on the stereonet. The angle between the lines is determined by counting the small circle divisions between A and B, along the great circle; this angle is found to be 64◦. The great circle on which A and B lie defines the plane that contains these two lines. The dip and direction of this plane are 60◦ and 200◦ respectively.