# Construct a Fold Cross-Section Using the Arc (Busk) Method

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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## How the Arc (Busk) Method Works

This method approximates folds as a series of circular arcs. This method was published by H.G. Busk in 1929, so it is sometimes called the Busk Method.

 1. Given the two dips shown, how do we approximate the fold as circular arcs? We cannot assume the measurements are on the same bed - they almost certainly are not. 2. The problem is to find concentric circles tangent to the two dip measurements. 3. Radii of circles are always perpendicular to the tangent where the radius hits the circle. 4. Therefore, we construct perpendiculars to each dip, and the intersection of the two perpendiculars is the center of the desired arcs.

We find the centers of curvature between adjacent dip measurements and construct the arcs for each. The arcs are bounded by the perpendiculars for each pair of dip measurements.

## A Common Problem

 1. It's quite common in this method for perpendiculars to dip measurements to intersect far off the diagram. 2. Locate the bisector of the angle between the perpendiculars. One way is to construct lines parallel to each side the same distance in, so that the lines intersect. Then bisect that angle. 3. From each dip datum, draw a line perpendicular to the bisector and extend it to the opposite side. This locates the other end of the arc. 4. Construct the dips on opposite sides of the sector. Dips along any one side of the sector are all equal and parallel. 5. If you extend the dips in to the bisector, the arcs must lie within the yellow triangles. 6. Sketch the arcs. They will approximately bisect the center line of each triangle. A good visual approximation is sufficient. 7. The completed arcs. 8. Data from adjacent sectors can be carried across (tick marks in red) by measuring distances relative to the already plotted arcs .

The earth will not fall out of orbit if the arcs in a sector like that shown above are approximate. What matters most is the end points of the arc, because they determine relative stratigraphic position from one side to the other. Within the sector, between the dip datum points, there is no data, and the arc is only an approximation to the true (and unknown) exact shape of the fold.

If the dips are exactly equal, then the perpendiculars will be parallel, and the center will be at infinity. No problem - the "arcs" become straight lines.

## Example

 In the example at left, dip data are shown. We want to construct a cross-section that satisfies the data. The stratigraphic units are colored here but will not be colored for most of the remaining diagrams. It is often better not to consider stratigraphy until after the cross-section is drawn. We first find the center for concentric circles tangent to dips 1 and 2.  All the circles tangent to dip 1 have their centers on a line perpendicular to dip 1. All the circles tangent to dip 2 have their centers on a line perpendicular to dip 2 Therefore, the intersection C12 is the center of concentric circles tangent both to dip 1 and to dip 2. Using C12 as a center, draw arcs tangent to dips 1 and 2 as shown. Draw the arcs only between the two perpendiculars. Now locate center C23, the center of concentric circles tangent to dips 2 and 3. You already have a line perpendicular to dip 2, so you only need to draw a line perpendicular to dip 3. Note that, as often happens, the center is off the diagram. Draw the arcs tangent to dips 2 and 3. Again, draw them only between the two perpendiculars. We can now exchange information with sector 1-2. Extend the arc from dip 1 into sector 2-3, using C23 as a center (lower red arc). Extend the arc from dip 3 into sector 1-2, using C12 as a center (upper red arc). In general, as we complete the cross-section, we will extend data from one sector to the next like this. We can now construct center C34 by drawing a perpendicular to dip 4. We already have the perpendicular to dip 3. We extend the arcs from sector 2-3 into sector 3-4 as shown. Note that the arc that starts at dip 2 passes very close to dip 4. We don't need an arc through every dip, even though we may use that dip to construct a center of curvature. So we won't bother drawing an arc for dip 4. Construct center C45 by drawing a perpendicular to dip 5. We already have the perpendicular to dip 4.  Extend the arcs from sector 3-4 into sector 4-5 as shown. Note that the arc that starts at dip 1 passes very close to dip 5. Again, we need not bother drawing an arc for dip 4.

Note that center C45 lies very near dip measurement 4. This is purely coincidental and has no significance.

Sector 4-5 presents a problem. The arc from dip 2 passes just about through C45, and the arc from dip 3 passes on the opposite side of C45 than does the arc from dip 1. When concentric folds have tight curvature, something has to give in the middle. If an arc passes on the wrong side of the center of curvature, do not draw it.

 Construct center C56 by drawing a perpendicular to dip 6. We already have the perpendicular to dip 5. Note that the intersection is now beneath the surface. This is no problem. It means the fold is now concave downward (an anticline) Construct the arc tangent to dip 6 as shown. Since this point is stratigraphically lower than all the other datum points, we continue the arc back through all the other sectors as well (shown in red). Construct arcs to connect with all the previously-constructed arcs as shown in red.

Sectors 6-7, 7-8, 9-9 and 9-10 are handled the same way, so the remaining illustrations simply show the results for each sector.

 Sector 6-7 completed Sector 7-8 completed. Sector 8-9 completed. Since point 9 falls between two already drawn arcs, there is no real need to construct another arc for it, at least for now. Note that centers C67, C78 and C89 are all close together. This simply means the fold has fairly uniform curvature over that interval. Sector 9-10 completed. Since point 10 falls very close to an already drawn arc, there is no real need to construct another arc for it.

## Tying the Diagram to Reality

It is virtually certain when you draw a cross section using strictly geometric methods that the contacts will not match exactly with their predicted positions. There are many reasons why not:

• The units will not be uniform in thickness
• There are small construction errors
• Dips are not uniform from place to place
• Dip measurements have small errors
• Folds do not have ideal geometrical shapes.
 Here we have indicated the stratigraphy. It is virtually certain  when you draw a cross section using strictly geometric methods that the contacts will not match exactly with their predicted positions. What we need to do now is redraw the folds so the cross-section matches both the dips and the stratigraphy. Here all the construction has been removed and the arcs are subdued.  Most of the time you can modify the fold shapes by hand to match the stratigraphy without too much trouble. Modified contacts are in black.

Do not get distracted by your dip symbols or stratigraphic colors. The only requirement is that the stratigraphy and dips match on the surface. Be prepared to modify the colors and depart from the dips below the surface if it's called for. Compare the two diagrams above to see that this was actually done.