Structure Lab Book Part 3

Revised November 6, 2008

Plotting and Measuring Using Stereonets

Introduction

Stereonets are graphical calculators that became popular among geologists and mining engineers in the era before the digital computer. They permit the solution of some rather cumbersome problems of 3-D geometry with little effort and no computation. In addition to their ease and utility, they provide a familiar and convenient way to plot and represent the orientation of planar or linear structural fabric.

Stereonets are used for plotting the orientation of lines, vectors and planes. All geometric objects plotted on a stereonet are constrained to pass through the center of an imaginary plotting sphere, so all lines, vectors and planes share at least one point in common.

Lines and vectors appear as points on a stereogram.

Planes trace great-circle arcs on a stereogram, or are represented by the orientation of their vector normals.

Two general types of stereographic projection are heavily used in geology.



The (x, y) coordinates that correspond to a line plotted on a Schmidt net are given by the following:

One important point to remember is that when you need to measure an angle using a stereonet, you should do so by counting the number of degrees along a great-circle. Great-circles function as protractors. So we measure the azimuth/trend of a line using the great circle known as the "primitive circle." We measure the dip angle using a vertical plane; that is, using a straight-line great-circle that passes through the center of the stereonet. And we measure the angle between any two lines (two points on the stereogram) by finding the great-circle arc that contains the two lines (two points...) and counting degrees of arc along that great circle.

Some References

Allmendinger, Richard: Stereonet program for Macintosh and Windows microcomputers, available for download at http://www.geo.cornell.edu/geology/faculty/RWA/programs.html

Leyshon, P.R., and Lisle, R.J., 1996, Stereographic projection techniques in structural geology: Oxford, Butterworth-Heinemann, 104 p., ISBN 0-7506-2450-7.

Phillips, F.C., 1954, The use of stereographic projection in structural geology: London, Edward Arnold.

Pollard, D.D., and Fletcher, R.C., 2005, Fundamentals of structural geology: Cambridge, Cambridge University Press, 500 p., (p. 56-67), ISBN 0-521-83927-0.

Ragan, D.M., 1985, Structural geology, an introduction to geometrical techniques [3rd edition]: New York, John Wiley & Sons, 393 p., ISBN 0-471-08043-8.

Rowland, S.M., Duebendorfer, E. M., and Schiefelbein, I.M., 2007, Structural analysis and synthesis, a laboratory course in structural geology [3rd edition]: Oxford, Blackwell Publishing, 301 p., ISBN 1-4051-1652-8.

van der Pluijm, B.A., and Marshak, S., 2004, Earth structure, an introduction to structural geology and tectonics: New York, W.W. Norton & Co., 656 p., (), ISBN 0-393-92467-X.

Solving Problems

  1. How do I plot a line on a stereonet?

    Approach to solution:

    1. Prepare your tracing paper by putting a dot atop the center of the plotting grid and by placing a tick mark atop the "north pole" of the plotting grid. Label the tick mark with an N, and circle the center dot so that it will be more visible.
    2. Find the trend of the line along the primitive circle. Make a tick mark at the trend azimuth.
    3. Rotate the tracing paper until the trend tick mark is on one of the vertical planes (either the east-west plane or the north-south plane on the plotting grid).
    4. Count down from the primitive circle along the great circle (because great circles function as protractors) by a number of degrees equal to the plunge. Place a small dot to represent the intersection of the line and the lower hemisphere of the plotting sphere.
    5. Optional: label the point with an identifying number or letter.
    6. Rotate the tracing paper back so that the north mark on the plot is positioned over the north pole of the stereonet plotting grid.


    Practice problems

    Plot the following lines, listed as plunge, trend
    22, 1355, 27835, 19586, 138
    70, 1645, 6042 S66W16 N34W
    57 S16E12 N28E80 S14W56 N72E
  2. How do I plot the trace of a plane and the pole to that plane?

    Approach to solution:

    1. Prepare your tracing paper by putting a dot atop the center of the plotting grid and by placing a tick mark atop the "north pole" of the plotting grid. Label the tick mark with an N, and circle the center dot so that it will be more visible.
    2. Find the trend of the strike along the primitive circle. Make a tick mark at the strike azimuth.
    3. Visualize how the plane will be plotted on the stereonet. You may wish to place some sort of help-mark in the quadrant of the plot that is in the direction of dip.
    4. Rotate the tracing paper until the strike tick mark is positioned on the north or south "pole" of the plotting grid. The help mark that you may have placed on the plot to indicate the dip direction will now be located on either the left or right side of the plotting grid.
    5. Count down from the primitive circle along the east-west vertical great circle (because great circles function as protractors) by a number of degrees equal to the dip angle δ. Trace the great circle that is δ degrees below the primitive circle and that runs from the north to the south poles of the plotting grid.
    6. Find the pole to the plane either (1) by counting 90° from the trace of the plane, using the east-west vertical plane as a protractor, or (2) by counting the dip angle δ from the center of the stereonet grid to find the point that is 90° from the trace of the plane.
    7. Rotate the tracing paper back so that the north mark on the plot is positioned over the north pole of the stereonet plotting grid.


    Practice problems

    Plot the trace of the following planes (expressed as strike and dip) on a stereonet. Also plot to pole to each plane.
    14 35NW138 67NE278 35SW220 85SE
    N70W 14NE
  3. How do I find the orientation of the plane that is defined by two lines?

    Approach to solution:

    1. Prepare your tracing paper by putting a dot atop the center of the plotting grid and by placing a tick mark atop the "north pole" of the plotting grid. Label the tick mark with an N, and circle the center dot so that it will be more visible.
    2. Plot the orientation of each of the two lines.
    3. Rotate the tracing paper until both points are on the same great circle.
    4. Trace the great circle that contains the two points. The points where the great circle intersect the primitive circle correspond to the strike of the plane.
    5. Counting down from the primitive circle, determine the magnitude of the dip angle.
    6. Rotate the tracing paper back so that the north mark on the plot is positioned over the north pole of the stereonet plotting grid.
    7. Determine the strike azimuth by counting along the primitive circle to the "north pole" of the plotting grid.
    8. Determine the dip direction by inspection.


    Practice problems

    Given the following two lines (plunge, trend), find the orientation of the plane that contains the two lines.
    30 4040 115
  4. How do I find the angle between two lines?

    Approach to solution:

    1. Prepare your tracing paper by putting a dot atop the center of the plotting grid and by placing a tick mark atop the "north pole" of the plotting grid. Label the tick mark with an N, and circle the center dot so that it will be more visible.
    2. Plot the orientation of each of the two lines.
    3. Rotate the tracing paper until both points are on the same great circle.
    4. Using that great circle as a protractor, count the number of degrees between the two points. This is the angle between the two lines.


    Practice problem

    Given the following two lines (plunge, trend), find the orientation of the plane that contains the two lines.
    30 4040 115
  5. How do I find the dihedral angle between two planes?

    Approach to solution:

    1. Prepare your tracing paper by putting a dot atop the center of the plotting grid and by placing a tick mark atop the "north pole" of the plotting grid. Label the tick mark with an N, and circle the center dot so that it will be more visible.
    2. Plot the pole to each of the two planes.
    3. Find the angle between the two poles.
      1. Rotate the tracing paper until both points are on the same great circle.
      2. Using that great circle as a protractor, count the number of degrees between the two points. This is the angle between the two planes.


    Practice problem

    Given the following two planes, find the dihedral angle between the two planes.
    N14W 32NEN56E 67NW
  6. How do I determine the orientation of the line of intersection between two planes?

    Approach to solution:

    1. Prepare your tracing paper by putting a dot atop the center of the plotting grid and by placing a tick mark atop the "north pole" of the plotting grid. Label the tick mark with an N, and circle the center dot so that it will be more visible.
    2. Plot the trace of each of the two planes. The plot of the line of intersection is the point where the two great circles cross each other.
    3. Rotate the tracing paper until the intersection point lies along one of the vertical great circles of the plotting grid: the east-west line or the north-south line.
    4. Count down from the primitive circle along the vertical great circle (because great circles function as protractors). This is the plunge angle.
    5. Place a tick mark where the vertical great circle (on which the intersection point is still positioned) meets the primitive circle.
    6. Rotate the tracing paper back so that the north mark on the plot is positioned over the north pole of the stereonet plotting grid.
    7. Determine the trend azimuth by counting along the primitive circle from the "north pole" of the plotting grid to the trend tick mark. Remember that the trend is the azimuth of a vertical plane in the direction that the plunging line is pointed downward.


    Practice problem

    Given the following two planes, determine the orientation of the line of intersection between the two planes.
    274 32NE36 67NW
  7. How do I determine the apparent dip angle in a given direction if I know the strike and dip of the plane?

    Approach to solution:

    1. Prepare your tracing paper by putting a dot atop the center of the plotting grid and by placing a tick mark atop the "north pole" of the plotting grid. Label the tick mark with an N, and circle the center dot so that it will be more visible.
    2. Plot the trace of the given plane.
    3. For each of the apparent dip determinations, place a tick mark along the primitive circle at the point corresponding to the azimuth of the vertical plane.
    4. Rotate the tracing paper until the tick mark is positioned by one of the vertical planes on the plotting grid (east-west or north-south).
    5. Count down from the primitive circle along the vertical great circle (because great circles function as protractors) to find the magnitude of the apparent dip.


    Practice problem

    Given a plane oriented 34 57SE, find the apparent dip angle in vertical planes trending 140°, 85° and 200°.
  8. Problem: Economically valuable hydrothermal mineralization is discovered at the surface, where a fault truncates a limestone bed. A mining company stakes a claim to this lead, intending to explore the bed-fault intersection zone at depth. The planar fault zone is oriented 37 83NW, and the limestone bed is oriented 347 47NE.

    In what orientation should the mine adit be dug so that it follows the intersection of the fault and the limestone bed?

    Approach to solution:

    1. Prepare your tracing paper by putting a dot atop the center of the plotting grid and by placing a tick mark atop the "north pole" of the plotting grid. Label the tick mark with an N, and circle the center dot so that it will be more visible.
    2. Plot the great-circle trace of the fault zone. Do the same for the limestone bed. The plot of the line of intersection is the point where the two great circles cross each other.
    3. Rotate the tracing paper until the intersection point lies along one of the vertical great circles of the plotting grid: the east-west line or the north-south line.
    4. Count down from the primitive circle along the vertical great circle (because great circles function as protractors). This is the plunge angle.
    5. Place a tick mark where the vertical great circle (on which the intersection point is still positioned) meets the primitive circle.
    6. Rotate the tracing paper back so that the north mark on the plot is positioned over the north pole of the stereonet plotting grid.
    7. Determine the trend azimuth by counting along the primitive circle from the "north pole" of the plotting grid to the trend tick mark. Remember that the trend is the azimuth of a vertical plane in the direction that the plunging line is pointed downward.
  9. Problem: A planar bed is oriented 52 45NW. A vertical roadcut trending 278° will be made through that bed. What will be the apparent dip of that bed in the plane of the roadcut?

    Approach to solution:

    1. Prepare your tracing paper by putting a dot atop the center of the plotting grid and by placing a tick mark atop the "north pole" of the plotting grid. Label the tick mark with an N, and circle the center dot so that it will be more visible.
    2. Plot the trace of the given planar bed.
    3. Place a tick mark along the primitive circle at the point corresponding to the azimuth of the vertical roadcut: 278°.
    4. Rotate the tracing paper until the tick mark is positioned by one of the vertical planes on the plotting grid (east-west or north-south).
    5. Count down from the primitive circle along the vertical great circle (because great circles function as protractors) to find the magnitude of the apparent dip.
  10. Problem: A planar coal seam is oriented N68E 34SE. The mining company wants to run an adit (tunnel) down the seam as steeply as possible. What should the plunge and trend of that adit be?

    Approach to solution:

    1. Prepare your tracing paper by putting a dot atop the center of the plotting grid and by placing a tick mark atop the "north pole" of the plotting grid. Label the tick mark with an N, and circle the center dot so that it will be more visible.
    2. Plot the great-circle trace of the plane on the stereonet.
    3. Along the primitive circle, count 90° from strike in the direction of dip. The direction of dip is 90° from the strike, and is the direction of greatest inclination along the inclined plane.
    4. You can prove that the direction with the greatest inclination along the inclined plane is 90° from the strike. Rotate the tracing paper, and watch the east-west great circle on the plotting grid as the great-circle trace of the inclined plane moves over it. When the strike is aligned with the east-west great circle, the angular difference between the primitive circle (horizontal) and the great-circle trace of the inclined plane is 0°. The angular difference (the apparent dip) increases as the tracing paper is rotated, until it reaches a maximum when the strike is aligned with the north-south axis of the plotting grid.
  11. Problem: The mining company wants to establish a couple of drainage adits along a planar coal seam that is oriented N68E 34SE. The adits must slope 5° to ensure proper drainage. In what two directions should the adits be dug?

    Approach to solution:

    1. Prepare your tracing paper by putting a dot atop the center of the plotting grid and by placing a tick mark atop the "north pole" of the plotting grid. Label the tick mark with an N, and circle the center dot so that it will be more visible.
    2. Plot the great-circle trace of the inclined coal seam. Be particularly careful to be accurate in your plotting near the strike directions (i.e., near the north and south "poles" of the plotting grid).
    3. Find one of the points on the plotting grid that corresponds to a line plunging 5°. There are four such points on the plotting grid, for lines trending north, south, east or west. We'll call this the control point.
    4. Rotate the tracing paper until the great-circle trace of the inclined plane lies on top of the control point. Mark this point on the tracing paper. Continue rotating the tracing paper to find the second place along the great-circle trace of the inclined plane where the trace lies on top of the control point.
    5. Find the trends of the two lines (plotted points) you have identified on the tracing paper. Their plunge is 5°.
  12. Problem: On a joint oriented 316 76NE, a lineation pitches 14° from the SW. What is the plunge and trend of that lineation?

    Approach to solution:

    1. Prepare your tracing paper by putting a dot atop the center of the plotting grid and by placing a tick mark atop the "north pole" of the plotting grid. Label the tick mark with an N, and circle the center dot so that it will be more visible.
    2. Plot the great-circle trace of the plane on the tracing paper.
    3. Label the northwest and southeast ends of the strike line.
    4. With the tracing paper still aligned so that the strike of the plane lies along the north-south axis of the plotting grid, count the pitch angle (14°) along the great-circle trace of the inclined plane from the direction indicated (SW). Mark the point corresponding to the lineation.
    5. Determine the plunge and trend of the point that you just plotted on the tracing paper.
  13. Problem: On a joint oriented 93 72NE, a bedding-plane lineation pitches 42° from the NW. On an adjacent joint oriented 283 68SW, a bedding-plane lineation pitches 38° from the SE. What is the strike and dip of the bedding plane?

    Approach to solution:

    1. Prepare your tracing paper by putting a dot atop the center of the plotting grid and by placing a tick mark atop the "north pole" of the plotting grid. Label the tick mark with an N, and circle the center dot so that it will be more visible.
    2. Plot the two lineations on the stereonet.
    3. Find the plane (great circle) that contains the two lineations.
  14. Problem: Precambrian current ripples indicate water flow in a direction that has a pitch of 50° from the NW in the plane 283 84NE. Overlying an unconformity are Paleozoic beds oriented 42 60SE. Overlying another unconformity are Mezozoic beds oriented 278 28 NE. Sequentially untilt all of the beds to find the direction the water flowed locally in the Precambrian.

    Approach to solution:

    1. Prepare your tracing paper by putting a dot atop the center of the plotting grid and by placing a tick mark atop the "north pole" of the plotting grid. Label the tick mark with an N, and circle the center dot so that it will be more visible.
    2. Plot the great-circle trace of the Precambrian bed, as well as the pole to that bed.
    3. With the strike of the Precambrian bed aligned with the "north pole" of the plotting grid, measure 50° from the primitive circle at the northwest end of the strike line along the great-circle trace of the bed (because great circles function as protractors). Plot the current direction vector as a point along the great-circle trace of the bed.
    4. Plot the poles to the Paleozoic and Mesozoic beds in the usual way.
    5. We will now begin to untilt the beds in sequence, starting with the youngest bed and ending with the oldest bed.
      1. Rotate the tracing paper until the pole to the Mesozoic bed lies on the east-west great circle (i.e., the equator of the plotting grid).
      2. Measure the angle θmz between the Mesozoic pole and vertical, using the east-west great circle as a protractor. Now, move the current-direction vector and the poles to all the beds around the north-south pole of the plotting grid, in the same direction by the same number of degrees (θmz). In effect, you will be rotating the Mesozoic bed around its strike line until it is horizontal, and rotating all of the lower beds around the same axis by the same angle. Note that the poles that do not lie on the east-west great circle follow small-circle paths during the rotation.
      3. Now that the Mesozoic beds are back to horizontal, we can ignore them during future rotations. We have rotated all of the older beds to their orientation at the time the first of the Mesozic sediments were accumulating atop the erosional unconformity.
    6. Follow the procedures in step 4, first for the Paleozoic beds and then for the Precambrian beds.
    7. After the Precambrian beds have been rotated back to horizontal, the current-direction vector will lie along the primitive circle. You can then read the azimuth in which the Precambrian current flowed directly, along the primitive circle.
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