NOAA / NESDIS / National Geophysical Data Center / WDC for SEG, Boulder, Colorado USA

Example 5.1

Use the program s_eigs to calculate the eigenvalues and eigenvectors of the data in ex5.1.

Solution

Type the following:

% s_eigs < ex5.1 > ex5.1a

and the computer responds by printing the eigenvalues and directions of the associated eigenvectors for each record into file ex5.1a. All eigenvectors are mapped to lower hemisphere.

Example 5.2

It is often useful to go ``backwards'' from the eigenparameters to the matrix elements. Use eigs_s to convert the output from Example 5.1 (ex5.1a) back to matrix elements.

Solution

Type:

% eigs_s< ex5.1a> ex5.2

to get back the matrix elements.

Notice how these are slightly different from the original data (ex5.1), because of round-off error; hence the transformation between matrix elements and eigenvalues is rather unstable and should be done as little as possible.

Example 5.3

Anisotropy data come from specimens with an arbitrary orientation. Use s_geo to rotate sets of six matrix elements referenced to an orientation arrow (along X₁) with given azimuth and plunge into geographic coordinates. Use the data in ex5.3a.

Solution

Type:

% s_geo < ex5.3a

The computer responds with ex5.1 from the first example. Now rotate the data in ex5.1 using a strike of 204 and a dip of 25. First attach the desired strike and dip to the data by;

% awk '{print $1,$2,$3,$4,$5,$6,204,25}' ex5.1 > ex5.3b

Then type

% s_tilt < ex5.3b > ex5.3c

to get the data ex5.3c.

Example 5.4

Use k15_s to calculate the best-fit tensor elements and residual error for data in ex5.4. These are: the sample name, azimuth and plunge, and strike and dip, followed by the fifteen measurements made using the scheme outlined in Figure 5.2 of Paleomagnetic Principles and Practice. Calculate the s data in geographic and tilt adjusted coordinates.

Solution

To calculate the matrix elements in specimen coordinates, type

% k15_s < ex5.4

and the computer responds with the data in ex5.4a.

To do just the geographic rotation, type:

% k15_s -g < ex5.4

The computer should respond with the data as in ex5.4b, but with sigma from the previous example at the end of each record.

Finally, to do the geographic and tectonic rotations, type:

% k15_s -t < ex5.4

to get the data in ex5.4c.

Example 5.5

Use k15_hext to calculate statistics for the data in ex5.4 using the linear propogation assumptions. Calculate the following: bulk susceptibility, F, F12, F23, E12, E13, E23 and the assorted directions.

Repeat the same calculations for geographic and tilt adjusted coordinates.

Solution

For specimen coordinates, type

% k15_hext < ex5.4

for: tr245f bulk susceptibility = 998.733
F = 418.84 F12 = 338.36 F23= 194.46
0.33521 256.1 45.9 1.8 165.0 1.0 4.2 74.1 44.1
0.33351 74.1 44.1 3.2 165.0 1.0 4.2 256.1 45.9
0.33127 165.0 1.0 1.8 256.1 45.9 3.2 74.1 44.1

tr245g bulk susceptibility = 1071.67
F = 148.81 F12 = 2.71 F23 = 251.69
0.33604 314.0 32.6 3.4 51.9 12.0 3.7 159.4 54.8
0.33218 159.4 54.8 3.7 314.0 32.6 32.1 51.9 12.0
0.33178 51.9 12.0 3.4 314.0 32.6 32.1 159.4 54.8

tr245h bulk susceptibility = 1225.67
F = 202.61 F12 = 120.34 F23= 133.64
0.33625 72.8 2.5 2.6 342.8 0.8 5.1 234.7 87.4
0.33328 234.7 87.4 5.1 72.8 2.5 5.4 342.8 0.8
0.33047 342.8 0.8 2.6 72.8 2.5 5.4 234.7 87.4
.
.
.

For geographic coordinates, type

%k15_hext -g < ex5.4

and you will get:

tr245f bulk susceptibility = 998.733
F = 420.98 F12 = 338.40 F23 = 194.44
0.33521 20.7 38.3 1.8 238.6 45.0 4.2 127.3 20.0
0.33351 127.3 20.0 3.2 238.6 45.0 4.2 20.7 38.3
0.33127 238.6 45.0 1.8 20.7 38.3 3.2 127.3 20.0

tr245g bulk susceptibility = 1071.67
F = 148.82 F12 = 2.71 F23 = 251.69
0.33604 13.1 15.5 3.4 282.4 2.5 3.7 183.3 74.3
0.33218 183.3 74.3 3.7 13.1 15.5 32.1 282.4 2.5
0.33178 282.4 2.5 3.4 13.1 15.5 32.1 183.3 74.3

tr245h bulk susceptibility = 1225.67
F = 203.10 F12 = 120.35 F23= 133.64
0.33625 16.7 6.0 2.6 283.6 26.8 5.1 118.3 62.5
0.33328 118.3 62.5 5.1 16.7 6.0 5.4 283.6 26.8
0.33047 283.6 26.8 2.6 16.7 6.0 5.4 118.3 62.5
.
.
.

For tilt adjusted coordinates, type

k15_hext -t < ex5.4

and you will get:

tr245f bulk susceptibility = 998.733
F= 419.92 F12 = 338.40 F23 = 194.44
0.33521 2.8 32.8 1.8 252.7 28.1 4.2 131.5 44.1
0.33351 131.5 44.1 3.2 252.7 28.1 4.2 2.8 32.8
0.33127 252.7 28.1 1.8 2.8 32.8 3.2 131.5 44.1

tr245g bulk susceptibility = 1071.67
F = 149.35 F12 = 2.71 F23 = 251.69
0.33604 7.6 9.5 3.4 101.5 21.9 3.7 255.7 65.9
0.33218 255.7 65.9 3.7 7.6 9.5 32.1 101.5 21.9
0.33178 101.5 21.9 3.4 7.6 9.5 32.1 255.7 65.9

tr245h bulk susceptibility = 1225.67
F = 202.85 F12 = 120.34 F23= 133.64
0.33625 14.8 2.4 2.6 284.7 2.1 5.1 153.0 86.8
0.33328 153.0 86.8 5.1 14.8 2.4 5.4 284.7 2.1
0.33047 284.7 2.1 2.6 14.8 2.4 5.4 153.0 86.8

.
.
.

Example 5.6

Use k15_hext to calculate the statistics of the whole file ex5.4. Repeat the excercise for geographic and tilt adjusted coordinates.

Solution

Type

% k15_hext -a < ex5.4

The [-a] switch tells the program to average over the whole file and the response is:

F = 2.55 F12 = 2.16 F23 = 1.12

0.33471 265.8 17.6 19.6 356.5 2.1 40.4 93.0 72.2

0.33349 93.0 72.2 31.5 356.5 2.1 40.4 265.8 17.6

0.33180 356.5 2.1 19.6 265.8 17.6 31.5 93.0 72.2

For geographic coordinates, type

k15_hext -ag < ex5.4

and get:

% k15_hext -ag < ex5.4

F = 5.77 F12 = 3.66 F23= 3.55

0.33505 5.3 14.7 13.3 268.8 23.6 25.5 124.5 61.7

0.33334 124.5 61.7 25.1 268.8 23.6 25.5 5.3 14.7

0.33161 268.8 23.6 13.3 5.3 14.7 25.1 124.5 61.7

For tilt adjustment, type

% k15_hext -at < ex5.4

and get:

F = 6.08 F12 = 3.86 F23= 3.74

0.33505 1.1 5.7 13.0 271.0 0.7 24.9 173.8 84.3

0.33334 173.8 84.3 24.6 271.0 0.7 24.9 1.1 5.7

0.33161 271.0 0.7 13.0 1.1 5.7 24.6 173.8 84.3

Example 5.7

Use s_hext to calculate statistics from the averaged matrix elements obtained from k15_s and s_geo in ex5.4.

Solution

The outcome of Example 5.4 was the input for Example 5.1, so

% s_hext < ex5.1

to get:

F = 5.77 F12 = 3.66 F23 = 3.55

N = 8 sigma = 6.41813E-04

0.33505 5.3 14.7 13.3 268.8 23.6 25.5 124.5 61.7

0.33334 124.5 61.7 25.1 268.8 23.6 25.5 5.3 14.7

0.33161 268.8 23.6 13.3 5.3 14.7 25.1 124.5 61

which you will notice is identical to the outcome of Example 5.6. Also, note that while barely anisotropic (F > 3.48), these data fail the discrimination tests F12 and F13, suggesting that in fact tau₁,tau₂, and tau₃ cannot be discriminated.

Example 5.8

Calculate confidence ellipses of Jelinek [1978] for ex5.1 using s_jel78 for the data in ex5.1.

Solution

Type

% s_jel78 < ex5.1

to get:

N = 8

0.33505 5.3 14.7 15.1 268.8 23.6 16.8 124.5 61.7

0.33334 124.5 61.7 15.8 268.8 23.6 16.8 5.3 14.7

0.33161 268.8 23.6 15.1 5.3 14.7 15.8 124.5 61.7

Example 5.9

Calculate bootstrap statistics for the data in ex5.4 (transformed into geographic coordinates) using bootams. Repeat using the parametric bootstrap option.

Solution

First the data in ex5.4 must be converted to matrix elements, by the command:

k15_s -g < ex5.4 > ex5.9

as in Example 5.4. Then, type

% bootams < ex5.9

to get:

0.33334 0.00021 124.5 61.7 6.0 225.6 5.9 18.1 318.7 27.5

0.33161 0.00014 268.8 23.6 11.9 2.3 8.1 12.9 110.1 64.9

For a parametric bootstrap, type:

bootams -p < ex5.9

to get:

0.33334 0.00021 124.5 61.7 7.2 224.4 5.3 18.3 317.1 27.7

0.33161 0.00016 268.8 23.6 12.3 10.3 24.6 13.6 140.7 54.7

These measurements have very low signal/noise ratios, hence the outcomes of the simple and parametric bootstraps are similar.

Also note that, according to the standard deviations of the bootstrapped tau values, the three eigenvalues are significantly different, in contrast to the F test results from Examples 5.6 and 5.7

Example 5.10

1) Compare the four methods of calculating confidence ellipses (parametric and non-parametric bootstrap, Hext [1963], and Jelinek [1978]) using plotams on the data from Example 5.9. 2) Plot the distribution of eigenvectors obtained from the site parametric bootstrap in equal area projection.

Solution

1) Type

% plotams -pxj < ex5.9 | plotxy

to get the postscript file mypost. Compare these ellipses with the outcomes of s_hext, s_jel78, bootams, bootams -p in previous examples. The two bootstrap methods are quite similar, the parametric one being slightly fatter in the V₁ ellipse. The Jelinek [1978] ellipses are rounder than the bootstrapped ones. Those of Hext [1963] are much larger than the others.

2) Now type

% plotams -Pv < ex5.9 | plotxy

to get mypost.

Example 5.11

Check if the anisotropy data from two chilled margins of a dike indicate imbrication using plotams and s_hist. The eastern margin data are those in Example 5.4 and the western margin data are in ex5.11a.

Solution

Perform the geographic corrections and specimen averaging using k15_s by

% k15_s -g < ex5.11a > ex5.11b

(The data from Example 5.4 should already be in ex5.9)

To examine these new data, use plotams as before to get mypost. Compare this figure with mypost. Can the eigenvectors be distinguished on the basis of confidence ellipses alone? Another way to consider the problem is to compare histograms of the the mean eigenvectors of interest (in this case the principal one) generated during bootstrapping to see if any humps can be distinguished. s_hist will do this with the appropriate switches.

To see the options available, check the on-line documentation for s_hist, or type:

% s_hist -h

To select a parametric bootstrap, since we have the sigma values already, we choose -p. To compare data from two files, we select the -c option and supply the two file names. To plot the principal eigenvector, we select the -1 option and to plot the 95% confidence bounds for the respective bootstrapped eigenparameters, we select the -b option. Thus, we type:

% s_hist -pcb1 ex5.9 ex5.11b | plotxy

and get this mypost file. The 95% confidence intervals for the x₂ component do not overlap, hence the two data sets are discrete, indicating imbrication and allowing interpretation of flow direction.

Example 5.12

Use s_hist to see what average shape the data in ex5.9 are. Test whether tau₁>tau₂>tau₃.

Solution

We want to plot the eigenvalues generated by the parametric bootstrap and the 95% confidence intervals for each eigenvalue. To do this type:

% s_hist -ptb < ex5.9 | plotxy

and get mypost.

Example 5.13

Use s_flinn to plot Flinn and Ramsey diagrams using a parametric bootstrap of the data in ex5.9. Use s_pt to make a Jelinek diagram of the same data.

Solution

For a Flinn diagram (parametric bootstrap) as shown in mypost, type

% s_flinn -p < ex5.9 | plotxy

For a Ramsey diagram (log Flinn) as shown in mypost, type

% s_flinn -pl < ex5.9 | plotxy

For a Jelinek diagram as shown in mypost, type

% s_pt -p < ex5.9 | plotxy