__Stereonet analysis part 2__

Contouring and interpreting orientation data.

In trying to document the character of structural
elements it is not uncommon to have hundreds of orientation readings.
Trying to interpret such data plotted as raw orientations (scatter
plots) on a stereonet, without further data treatment is difficult
and susceptible to bias. In order to guide interpretation it is
standard treatment to contour the orientation data. The resulting
contours show **data density patterns**. Perhaps think of it as a crude cluster
analysis. Minor sub-populations and subtle patterns hidden by
the 'noise' will often be 'brought out' or highlighted by such
data treatment. On the other hand biased sampling may produce
clusters or patterns that do not reflect reality, especially if
your sample size is small (less than 30 readings). Properly done,
contoured stereonets are extremely useful and can tell you such
things as: joint set orientation, cylindrical or non cylindrical
fold form, interlimb angle, fold curvature, conjugate fault geometry,
rotation axes for faults, degree of structural obliquity, deformational
vorticity, and more.

Contouring data is basically a way of mapping
the density distribution of data. It greatly aids in the visual
interpretation of data clouds. Two major 'cloud' patterns often
found are: a) a **bullseye distribution**, and b) a girdle distribution.

A bullseye pattern can be thought of as even dispersion around some average orientation; i.e. as a population or set. A dike swarm, joint set or a simple fault population can often exhibit such a pattern. The amount of spread is also a function of how well behaved the populations is (the natural variation, or dispersion) and the amount of measurement error. Some fractures are much more planar, others anastomose. Often the distribution of scatter around the mean approaches a normal distribution, and standard deviations are typically in the range of 5-15 degrees. In addition, the noise, the other portion of the data is neglected, and an interesting question that can be asked is what percentage of your total data set does not belong to the defined set.

*As an example, the above is a series of stereonet contour plots of poles to fractures in a variety of different rock types from a USGS study (http://pubs.usgs.gov/of/2002/ofr02-013/html/bedrock.html). The details can be found by following the link. Since many of the fractures are subvertical they are also show a form of circular histogram (rose diagram) for the fracture strike azimuths. A key and important finding here is that different fracture sets are vound in different rocks. Another component of such analysis is to identify domains that you develop plots for. In this case the domains are determined by rock type.*

A **great-circle girdle distribution pattern **is often due to some
differential tectonic or strain rotation of a planar element,
such as occurs with folding. Remember that if it is a cylindrical fold the girdle defined by contours of poles to the folded surface should follow a great circle whose pole is itself the fold axis. Sometimes small-circle girdle distributions can
also be found, indicating conical fold forms. Additional information about the fold form exists in how and where the data is concentrated along the great circle best fit. If measurements of the strikes and dips are representative of the fold surface as a whole then distinct clusters likely represent fold limbs (longer portions that should be representatively sampled more). If limbs are well defined then an interlimb angle can be estimated. If the fold style is very angular, then the fold limbs should be well defined. Otherwise for rounded forms there should be more a distribution along the girdle. If the one cluster from the one limb is better developed than another, this may reflect the asymmetry of the fold. It could also reflect that more measurements were taked from one fold than the other, and so one must be cautious in making these types of interpretations.

*Contoured stereonet plot of poles
to bedding from deformed Jurassic shales, Revneset, Spitsbergen.
the different colors represent contoured data density with a 3%
interval. The strongest concentration is greater than 15% of the
data in a 1% area. Note the great circle girdle pattern evident
here. The great circle shown is a computer best-fit solution generated by the program (Allmendinger, http://www.geo.cornell.edu/geology/faculty/RWA/programs.html). **One could speculate that two limbs are evident in the two bullseye, and that the one limb is better represented and thus the folds may be asymmetric. Both limbs would be shallowly dipping (remember that the poles are at 90 degrees and so the poles plunge steeply, and here the NE dipping limb readings are more prevalent.
*

*This is another plot of bedding from Spitsbergen, from Triassic strata involved in a thin-skinned fold-thrust belt. The black lines are the great circle plots of the bedding. The black dots are poles to bedding. The blue lines are contour lines in increments of 3% per 1 percent area. The red box and great circle are the programs statistical best-fit pole girdle and the pole to that girdle, which is the best estimate of the fold axis. The purple line and rectangle is a hand-picked estimate of the equivalent girdle and fold axis. These are relatively upright and tight folds, with subhorizontal axes.*

Part two of the stereonet analysis lab is to
take the data provided below and ** a) create scatter plots,
b) create contour plots, c) interpret your plots as fully as possible**.
The assignment is purposefully open ended. See what you can get
out of the data and don't hesitate to come see me for feedback.
The data was acquired by students during a field trip in 2000,
and are from the west end of the Arbuckle mountains near Turner
Falls and along thruway 35. You should be familiar with the basic
geology a bit from your air photo interpretation exercise. There
is a major Pennsylvanian angular unconformity, and you have readings
from above and below this unconformity. You also have the strikes
and dips of fault planes in the area.

1) First plot all the data on a **equal-area
stereonet.** Plot the data as poles to create a scatter plot.
Make sure you have the right stereonet. Equal angle stereonets
have distorted areas and so a symmetrical, bullseye data population
will appear distorted. If you are dealing with planar data, plot
it as poles (since you can not contour great circles).

2) Second, construct a counting circle. The counting circle has a radius one tenth that of the stereonet you are using. As a result it includes within it 1% of the entire stereonet's area. Note that you need two counting circles whose centers are separated by a line equal in length to the diameter of the stereonet you are using. This is so that when you are counting on the stereonet's margin and part of the counting circle is outside of the stereonet you can count the 'missing' portion which actually lies diametrically opposed on the other side of the stereonet.

3) Overlay a counting grid on the stereonet with the plotted data. The grid spacing is not necessarily fixed, but must be < the radius of the counting circle. Try to think of why this is so.

4) Position the counting circle with its center
at a grid intersection. Count all the data points within the counting
circle and put that number at that intersection. Repeat for every
intersection until the all the grid intersections within the stereonet
have been 'counted'. Remember to count the data points on the
other side when part of your counting circle falls outside of
the stereonet (i.e. use both counting circles). You are basically
asking over and over the question - " **how many data points
fall within this 1% area of the stereonet projection, hence this
can be considered a data density map**." Since you are
mapping data density you are free to place the counting circle
center anywhere if you want to get a better control on the density
distribution.

5) Transform the count numbers into percentages (count # / total number of data points * 100).

6) Contour these percentage areas following the normal rules of contouring a continuum (the continuum representing the presumed infinite population of possible structural element orientations which has been sampled, and represents, a continuous, smooth surface). You will need to decide on a contour interval. Look at your percentage values and their pattern and decide which interval will serve the best.

7) The fun part - interpret the data. Pick out different populations or girdles. You can also compare your hand plot versus a computer plot (after calculating poles chose the contour option.

Kamb plots are where the departure from a random distribution of points is mapped. As the number of data points goes up the Kamb and standard 1% data density plots look more and more alike. I encourage you to use such plots, especially if n<100 or so. They can greatly aid in preventing overinterpretation of the data (something humans are demonstrably prone to). Their disadvantage is simply that this is not the conventional approach, and you may have to explain their use to some people.

Hints on using Allmendinger's software (available on computers in Cart Lab):

- Make sure you set your data format first
(
**Data -> Set Data Format**) before you enter data. - Save your entered data file so you can replot it if necessary.
- In order to plot poles to planes, you need
to have the program calculate them first (
**Operations -> Poles**). - To contour data use the standard 1% option
(
**Plot -> 1% Area Contour**), but you can also do the Kamb plot for comparisons sakes. - You can ask the computer for a statistically
computed great circle girdle estimate or the average orientation
for a simple bullseye population (
**Plot -> Cylindrical Best Fit**or**Plot Mean Vector**). - You can also visually pick your own girdle
by using
**Plot -> Pick Great Circle**. - You can't print plots directly from the program. You need to save the plot (it will be in a vector based Mac format known as pict), and then open it in Adobe Photoshop or Illustrator, or some other program, and plot it from there. At this stage you can also label and alter the image.

strike | dip |

270 | 20 |

334 | 15 |

270 | 80 |

280 | 78 |

270 | 15 |

270 | 32 |

270 | 75 |

307 | 46 |

296 | 64 |

315 | 60 |

280 | 72 |

278 | 32 |

275 | 82 |

271 | 81 |

276 | 74 |

300 | 12 |

282 | 14 |

316 | 74 |

326 | 60 |

301 | 4 |

330 | 34 |

315 | 60 |

304 | 70 |

330 | 32 |

278 | 83 |

278 | 78 |

296 | 12 |

316 | 38 |

292 | 42 |

300 | 62 |

320 | 60 |

315 | 32 |

310 | 66 |

286 | 70 |

298 | 64 |

345 | 20 |

275 | 78 |

245 | 10 |

312 | 64 |

276 | 78 |

274 | 74 |

245 | 1 |

285 | 10 |

305 | 68 |

320 | 48 |

312 | 58 |

315 | 65 |

286 | 70 |

300 | 80 |

335 | 32 |

285 | 82 |

280 | 80 |

320 | 40 |

282 | 14 |

306 | 66 |

314 | 88 |

311 | 66 |

300 | 71 |

286 | 70 |

345 | 35 |

314 | 15 |

275 | 82 |

264 | 80 |

266 | 4 |

292 | 80 |

280 | 68 |

245 | 10 |

292 | 42 |

284 | 20 |

300 | 14 |

282 | 14 |

326 | 60 |

330 | 30 |

300 | 72 |

335 | 32 |

278 | 83 |

276 | 78 |

328 | 38 |

292 | 42 |

320 | 60 |

291 | 74 |

276 | 78 |

264 | 82 |

245 | 10 |

330 | 30 |

312 | 64 |

286 | 70 |

330 | 32 |

240 | 18 |

286 | 70 |

330 | 32 |

278 | 32 |

296 | 72 |

252 | 22 |

264 | 70 |

218 | 30 |

216 | 68 |

266 | 40 |

266 | 6 |

298 | 60 |

270 | 19 |

124 | 67 |

207 | 6 |

266 | 6 |

266 | 40 |

220 | 38 |

266 | 40 |

201 | 38 |

252 | 22 |

216 | 32 |

215 | 20 |

234 | 14 |

248 | 68 |

120 | 40 |

90 | 20 |

90 | 28 |

90 | 30 |

80 | 44 |

74 | 40 |

42 | 25 |

28 | 40 |

100 | 72 |

120 | 26 |

92 | 22 |

10 | 45 |

20 | 40 |

30 | 48 |

28 | 40 |

80 | 44 |

100 | 72 |

84 | 70 |

74 | 40 |

112 | 27 |

42 | 72 |

60 | 18 |

106 | 78 |

114 | 46 |

98 | 24 |

141 | 22 |

117 | 46 |

119 | 40 |

174 | 26 |

167 | 28 |

132 | 24 |

122 | 20 |

134 | 68 |

134 | 50 |

129 | 71 |

134 | 63 |

142 | 64 |

114 | 68 |

115 | 80 |

141 | 84 |

121 | 87 |

132 | 16 |

120 | 18 |

137 | 80 |

136 | 28 |

112 | 18 |

132 | 21 |

128 | 20 |

121 | 40 |

130 | 74 |

138 | 68 |

134 | 68 |

129 | 70 |

109 | 68 |

120 | 80 |

145 | 86 |

140 | 40 |

115 | 75 |

115 | 72 |

138 | 52 |

130 | 74 |

105 | 74 |

16 | 18 |

140 | 80 |

120 | 15 |

84 | 20 |

136 | 49 |

125 | 40 |

124 | 50 |

138 | 58 |

90 | 30 |

60 | 38 |

120 | 24 |

120 | 60 |

130 | 74 |

124 | 70 |

120 | 80 |

120 | 75 |

136 | 80 |

120 | 58 |

112 | 68 |

134 | 16 |

104 | 204 |

134 | 30 |

172 | 18 |

120 | 58 |

105 | 65 |

140 | 70 |

135 | 65 |

137 | 17 |

115 | 75 |

170 | 85 |

112 | 32 |

100 | 32 |

120 | 40 |

128 | 45 |

120 | 40 |

134 | 68 |

126 | 78 |

130 | 74 |

138 | 74 |

115 | 75 |

117 | 84 |

140 | 55 |

140 | 83 |

32 | 30 |

112 | 56 |

84 | 20 |

111 | 20 |

132 | 32 |

118 | 22 |

162 | 58 |

137 | 80 |

124 | 16 |

121 | 40 |

121 | 5 |

108 | 68 |

130 | 72 |

129 | 71 |

115 | 75 |

145 | 86 |

142 | 55 |

120 | 15 |

140 | 54 |

110 | 50 |

123 | 46 |

122 | 24 |

126 | 20 |

132 | 68 |

105 | 76 |

119 | 72 |

148 | 85 |

134 | 16 |

136 | 50 |

112 | 10 |

141 | 44 |

128 | 58 |

146 | 68 |

124 | 22 |

120 | 40 |

126 | 45 |

118 | 18 |

118 | 45 |

132 | 68 |

114 | 74 |

118 | 76 |

120 | 80 |

38 | 55 |

40 | 34 |

bedding strike | dip |

288 | 20 |

278 | 13 |

328 | 20 |

219 | 28 |

265 | 14 |

307 | 22 |

316 | 12 |

296 | 20 |

68 | 38 |

282 | 15 |

320 | 18 |

305 | 30 |

290 | 20 |

330 | 48 |

298 | 55 |

280 | 15 |

315 | 21 |

315 | 26 |

307 | 22 |

315 | 60 |

284 | 24 |

316 | 12 |

296 | 20 |

282 | 15 |

318 | 21 |

80 | 20 |

8 | 55 |

113 | 20 |

112 | 18 |

117 | 11 |

124 | 60 |

119 | 36 |

120 | 15 |

60 | 38 |

120 | 50 |

123 | 48 |

125 | 85 |

120 | 22 |

120 | 24 |

118 | 40 |

117 | 11 |

101 | 18 |

120 | 15 |

118 | 40 |

126 | 63 |

60 | 38 |

110 | 22 |

120 | 20 |

fault strike | dip |

0 | 60 |

314 | 88 |

284 | 28 |

357 | 80 |

268 | 80 |

280 | 64 |

54 | 84 |

282 | 14 |

214 | 45 |

214 | 84 |

280 | 64 |

268 | 87 |

82 | 80 |

276 | 74 |

245 | 10 |

244 | 68 |

134 | 44 |

112 | 68 |

248 | 68 |

248 | 68 |

52 | 84 |

114 | 32 |

42 | 72 |

70 | 34 |

42 | 72 |

106 | 70 |

58 | 86 |

52 | 60 |

74 | 85 |

28 | 30 |

65 | 75 |

42 | 72 |

112 | 18 |

136 | 28 |

108 | 38 |

130 | 74 |

100 | 34 |

108 | 40 |

110 | 72 |

100 | 80 |

60 | 38 |

10 | 30 |

84 | 20 |

186 | 90 |