The principles for how to make an equal area projection are shown in Figure B.1. The point P corresponds to a D of 40∘ and I of 35∘. D is measured around the perimeter of the equal area net and I is transformed as follows:
where Lo = 1∕.
The principles for plotting directions on an equal area net are shown in Figure B.3. Print out the equal area net provided in Figure B.2. Then poke a thumbtack through the center of the diagram and place a piece of tracing paper over the thumbtack. Mark the top of the stereonet as N and the declination of the direction at Dir in Figure B.3a. Then rotate the mark around the thumbtack such that the declination is at the top of the diagram (Figure B.3b). Count in from the outer ring the number of degrees equal to the inclination - the grid provided is in 2∘ intervals. Mark the direction (star in Figure B.3d). In paleomagnetism, the convention is for solid symbols to represent downward directions and open symbols to be up.
Performing structural corrections can also be done with an equal area net. If samples have been collected from sites where strata have been tilted by tectonic disturbance, a bedding tilt correction is required to determine the NRM direction with respect to paleohorizontal. Structural attitude of beds at the collecting site (strike and dip, or dip angle and direction) must be determined during the course of field work.
The bedding-tilt correction is accomplished by rotating the NRM direction about the local strike axis by the amount of the dip of the beds. Several examples are shown in Figure B.4, and the reader is strongly encouraged to follow through these examples. An intuitive appreciation of these geometrical operations will prove invaluable in understanding many paleomagnetic techniques and applications.
Print out the equal-area grid provided in Figure B.2. Poke a thumb tack through the center and place a piece of tracing paper over it. The graphical procedure for the bedding- tilt correction is as follows:
Ternary diagrams are triangles with the three corners representing a composition (e.g., A,B,C or Fe, FeO, Fe2O3). In Figure B.5a we show only the A component. To get the percentage of this component, we count up from the base of the triangle and find that the star is 60% of the way toward the apex, indicating that the compound is 60% A in composition. The percentage of composition B is shown in Figure B.5b (15%) and similarly C is shown in Figure B.5c (25%).
When does a data set conform to a particular distribution? One way to assess this is through the use of Quantile-quantile, or Q-Q, plots (see Fisher et al., 1987 for a more complete discussion.) In a Q-Q plot, data are graphed against the value expected from a particular distribution. The data ζi are plotted against a value zi that is expected from the distribution; data compatible with the chosen distribution plot along a line. First, we will develop the Q-Q plot for the uniform and exponential functions required for a Fisher distribution. Then we will explain how make a Q-Q plot for a normal distribution.
In order to make Q-Q plot for Fisher distributions, we proceed as follows (Figure B.6):
and where F-1 is the inverse function to F. If the data are uniformly distributed (and constrained to lie between 0 and 1), then both F(x) and F-1(x) = x. For an exponential distribution F(x) = 1 - e-x and F-1(x) = -ln(1 - x).
For a uniform distribution F(x) = x, so Mu is calculated by first calculating DN+ as the maximum of [i∕N -ζi] and DN- as the maximum of [ζi - (i - 1)∕N]. The Kuiper’s statistic V n is DN+ + DN- and Mu is given by:
(see Fisher et al., 1987). A value of Mu > 1.207 can be grounds for rejecting the hypothesis of uniformity at the 95% level of certainty. Similarly, DN+ and DN- can be calculated for the inclination data (using ζi = 90 -Ii) as [i∕N - (1 -e-ζi)] and maximum of [(1 -e-ζi) - (i- 1)∕N] respectively. The Kolmogorov-Smirnov statistic Dn is the largest of the two. The test statistic for exponentially distributed data Me is given by:
Values of Me larger than 1.094 allow rejection of the exponential hypothesis at the 95% level of confidence. If either Mu or Me exceed the critical values, the hypothesis of a Fisher distribution can be rejected.
In order to calculate the appropriate values for zi assuming a normal distribution (see Abramowitz and Stegun, 1970):
The values of zi calculated in this way for a simulated Gaussian distribution are plotted as the “normal quantile” data and will plot along a line if the data are in fact normally distributed. To test this in a more quantitative way, we can calculate DN+ and DN- as follows: