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Stereographic projection – the basics

Plotting the orientation of a plane on a stereonet; finding strike and dip from two apparent dips

This post is part of the How To… series

My first exposure to stereographic projection methods of structural analysis was one of my more baffling moments of undergraduate geological learning. Looking around the class I could see I wasn’t the only one. Our tutor, a master at pointing out the obvious, suggested we try thinking in three dimensions. In retrospect it was probably some of the best advice I ever received. After all, geology is nothing if not an exercise in visualizing the 3- and 4-dimensional world around us.

Visualize a sphere; cut the top half off – we don’t need it. If you look directly down the vertical axis of the hemisphere it will appear to be two-dimensional. When you do this, you are basically projecting the 3-D hemisphere onto a 2-D circle. Once this projection has become embedded in your mind’s eye, everything else will fall into place.

The brief animation below shows a hemisphere rotated from an initial oblique view to the point where you are looking directly down the axis (or radius). At this point the hemisphere is projected as a circle that can then be overlaid by a stereonet grid like the example shown above. Each point and curve on the grid represents a point and curve on the hemisphere.

Stereographic projection is all about representing planes (e.g. bedding, foliation, faults, crystal faces) and lines (e.g. dip and plunge directions, fold axes, lineations) onto the 2-D circle. In geology, we overlay the 2-D projection with a grid of meridians, or great circles (analogous to longitudes), and parallels or small circles (analogous to latitudes). Thus, all compass points are represented. The net-like grid is called a Wulff Net, or stereonet. It is an equal angle grid where meridians and parallels intersect at right angles.

Note the two extreme projection conditions: the circumference is actually the great circle of a horizontal plane; the opposite is a straight line passing through the centre that is the projection of a vertically dipping plane. Thus, shallowing dips tend toward the circumference.

The following animation shows how the great circle changes as the plane rotates from vertical to horizontal dip (in increments of 15°). The great circle corresponding to the dip is in red. The line drawn from the two points on the circumference through the centre is the strike.

In practice we use a transparent overlay pinned to the centre of the stereonet; we rotate the overlay according to the orientation of lines and planes.

Plotting a dipping plane

Follow the next exercise with the short animation. It was made from still images: use the pause and play buttons as you work through the exercise.

Imagine a plane striking 060^o (N60E), dipping 60^o SE; the plane passes through the centre of the sphere. In our vertical 2-D view the intersection of the plane with the hemisphere appears as a curved line, or great circle. Rotate the overlay counter-clockwise 60^o; mark this point on the circumference. The great circle representing the angle of dip is found by counting along a line at right angles to the strike, i.e. along the E-W axis, counting 60^o from the horizontal (the circumference).

Note that in the 2-D view, the great circle meets the circumference at 060^o and 240^o that are two points on a horizontal line across the plane; the great circle is therefore the strike of the plane.

If a plane projects onto the stereonet as a great circle, then a line or lineation on the plane will plot as a point on the same great circle. To illustrate, we will find the true dip and strike of a plane for which we have measured two apparent dips: (1) 30^o at 315^o (N45W) and (2) 32^o at 014^o (N14E). On a transparent overlay, mark the positions of the N-S and E-W axes, and the stereonet centre. For (1), rotate the overlay 45^o clockwise, mark the position at the N pole, then count 30^o from N along the N-S axis. This point is the projection of apparent dip (1). Likewise, repeat for apparent dip (2). Rotate the overlay until the two apparent dip projections lie on the same great circle – read the bearing from N and the true dip along the E-W axis.

Follow this exercise with the short animation. It was made from still images: use the pause and play buttons as you work through the exercise.

Stereographic projection is a powerful method, not just to solve relatively simple (but important) problems of dip and strike, but as an analytical tool for more complex structural geology. There are several good software programs and Apps to automate projections for large data sets. But before you dive into these digital tools, try the simple overlay first – there’s nothing like a hands-on approach to help galvanise an understanding of basic methodology.

Some other useful posts in this series:

Measuring dip and strike

Solving the three-point problem

Stereographic projection – unfolding folds

Stereographic projection – poles to planes

The Rule of Vs in geological mapping

Folded rock; some terminology

Here are some good website links:

Visible Geology

Rick Allmendinger’s Stereo 10 – there’s also a mobile App version

Innstereo (open source)

Links to several programs in The Structural Geology Page

Solving the three-point problem

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A graphical method for solving the three-point problem

This post is part of the How To… series

Determining the orientation of a plane given the points for three intersecting wells.

Mapping is the essence of geology. Geology maps provide the wherewithal to decipher the time and space organization of Earth’s solid and fluid spheres. We map the outer veneer by directly observing rocks and fluids, ‘walking out’ rock units, measuring, sampling and imaging as we go. More recent tools include all manner of remotely sensed data and satellite imagery (seismic, Lidar, radar, Landsat). We apply the same tools to map other planetary surfaces, although the walking-out is done by remotely controlled rovers.

Subsurface mapping is the essence of all explorations for aquifers, hydrocarbons, minerals, geothermal energy and geotechnical constructions. Subsurface mapping provides us with a deeper (sic) understanding of how the Earth works. Subsurface mapping is entirely dependent on remote sensing (e.g. seismic, gravity, radar) and borehole probing.

The orientation of geological planes in the subsurface is no less important than in surface mapping, but the database is commonly one-dimensional (e.g. borehole depths and lithologies). For example, a zone of mineralization at depth lies beneath an unconformity; knowing the orientation of the unconformity plane will give us more confidence predicting the trend of mineralization (assuming the unconformity is reasonably flat). To solve the problem, we need depths from three borehole intersections with the plane. The solution is commonly referred to as the 3-point problem. It is based on an understanding of dip and strike.

Geometric calculation of dip and strike in a 3-point problem

A graphical solution is shown in the animation. You need paper, a ruler and a protractor. This method requires the horizontal and vertical (depth or elevation) scales to be the same (no vertical exaggeration). Normally the construction would be done on the plane itself (i.e. 2-dimensional) – here the 3-dimensional view has been added to help you visualize the problem.

The animation was made from still images: use the pause and play buttons as you work through the exercise.

Some other useful posts in this series:

Measuring dip and strike

Stereographic projection – the basics

Stereographic projection – unfolding folds

The Rule of Vs in geological mapping

Folded rock; some terminology

Measuring dip and strike

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How to measure dip and strike

This post is part of the How To… series

If we had to designate one set of measurements that is fundamental to all geology, it would have to be Dip and Strike. These simple measures define uniquely the orientation (compass bearings and angles) of a planar surface – any plane: bedding, faults, axial planes, mineralized veins, dykes and sills. Armed with dips and strikes, a geologist can project planes and the rocks they encompass across valleys, through mountains and deep beneath the Earth’s surface. They are fundamental to deciphering Earth structures.

Strike: The compass bearing of an imagined horizontal line across a plane. If the plane is flat there is an infinite number of strike lines, all having the same dip (zero) but different bearings. If the plane is curved (e.g. a plunging fold) the bearing may change systematically over the fold.

Azimuth, or compass bearing is recorded as either (for example) 035^o or N35E, or its counterpart 215^o and S35W.

Dip: Dip is the angle of inclination measured from a horizontal line at right angles to strike. The angle is measured by placing a compass on the line of dip and rotating the inclinometer to the point where a spirit level indicates horizontal. The direction of dip need not be measured (it can be calculated directly from the strike bearing), but an approximate direction should always be recorded to avoid ambiguity, as in 48^oNW.

The inclination measured at right angles to strike is the true dip. Inclinations measured at other angles on the plane will always be less than true dip – these are called apparent dips.

The animation below was made from still images: use the pause and play buttons as you work through the exercise.

Dip and strike indicate the orientation of a plane at a specific location. Dips and strikes of folded strata will tend to show systematic changes at different locations. In the example below the fold axis is horizontal and axial plane vertical. Strikes at any point on the fold limbs will all have the same azimuth, but dips will change progressively from one limb to the other. Dips and strikes recorded on geological maps can be used to reconstruct the 2- and 3-dimensional structure of deformed strata.