Fig. 3.3. North pole Stereographic projection of VGPs observed in 508 lava flows that lie between 35° and 40° north or south latitude and have ages <5 Ma. The expected mean VGP is the North Geographic Pole. From McElhinny and McFadden (1997).
where f0 and fe are the observed and expected frequencies within the range. If the observations were drawn from a Fisher distribution then X2 will be chi-square distributed with v degrees of freedom, %\ , where v = /M-l-n (3.2.29)
and Ft is the number of distribution parameters that had to be replaced by their maximum likelihood estimates in order to calculate the observed frequencies.
An example of this test is given in Fig. 3.3 for the observed VGPs from lava flows that lie within the latitude band [35—401° on the Earth's surface and have ages <5 Ma. The details of the test for a Fisher distribution are given in Table 3.1. For both the azimuthal (longitude) and radial (colatitude) variation the 508 observations were divided into 10 cells of equal expectation, so m = 10. Typically it would be necessary to estimate the true mean direction (i.e., two parameters) and the precision k so that II would be 3 for testing the radial (vy) distribution and II would be 2 for testing the azimuthal (<(>) distribution (the precision is not needed to test the azimuthal distribution). In this case the expected mean VGP is known because it is the North Geographic Pole. Hence, only the precision needs to be estimated and so the two values of II are 1 and 0. The calculated (observed) value of X2 in each case is less than the critical value at the 5% level. Therefore, there is no reason to reject the hypothesis that the observations conform with a Fisher distribution.
Graphical methods may also be used to test the goodness-of-fit of the Fisher distribution to a set of data (see e.g., Lewis and Fisher, 1982; Fisher et al., 1987).
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