# Analysis of Reversal Sequences Probability Distributions

Cox (1968) recognized the apparent randomness of interval lengths (§4.2.1) and developed a probabilistic model in which he assumed that polarity changes occur as a result of interaction between random processes and a steady oscillation in the intensity of the axial dipole field. However, the observed distribution of field intensities is inconsistent with that predicted by Cox's model (Kono, 1972; McFadden and McElhinny, 1982). Also, Laj et al. (1979) noted that this model suffered from mechanical problems arising from the different time constants of the random fluctuations of the nondipole field and the steady oscillations of the dipole field. Nevertheless, Cox (1968) ushered in the era of statistical characterization of the reversal process and the use of such characterization to attempt to understand processes occurring in the deep Earth.

The time taken for a polarity transition is, in geological terms, short (§4.4.4), and is short relative to the typical length of a polarity interval. Thus within the GPTS it is a reasonable first order approximation to think of a reversal as a rapid, and indeed almost instantaneous, event. Together with the apparent randomness, this leads naturally to consideration of the reversal process as a general renewal process. This would require that the lengths of individual polarity intervals be independent. Such independence was claimed early on (Cox, 1968, 1969; Nagata, 1969) but was subsequently challenged by Naidu (1974). The later work of Phillips et al. (1975) and Phillips and Cox (1976) shows that there is no significant correlation between the lengths of polarity intervals. Thus it is a reasonable first-order approximation to regard geomagnetic reversals as a random process with independent intervals. This is quite different from some other systems, such as for the Sun in which reversals occur about every 11 years.

Recognizing that a reversal is a rare event, Cox (1969) showed that his model led to a Poisson process and that the relevant distribution of interval lengths was therefore the exponential distribution. For a Poisson process the probability density p(x) of interval lengths .r is given by p(x)dx = ¡UT^dx , (4.5.1)

where X is the rate of the process and |i = MX is the mean interval length. Despite a reasonable fit to the observations, there were fewer very short intervals than required by the Poisson process. Naidu (1971) showed that a gamma distribution provided a good fit to the then observed intervals of the Cenozoic time scale and Phillips (1977) confirmed this in an extensive study of geomagnetic reversal sequences. For a gamma process the probability density p(x) of interval lengths x is given by p(x)cLc = — (kA)k x(k-%~kAj< dx rW (4.5.2)

where the mean interval length is now given by (i = k/X = 1/A.

From (4.5.1) and (4.5.2) it is apparent that the gamma process leads to a family of distributions depending on the value of k, and that the Poisson process is simply the special case of a gamma process with k = 1. Appropriately scaled probability densities for this family of distributions are shown in Fig. 4.18. It is