Fig. 4.18. Proportions of interval lengths (probability densities) of gamma distributions plotted against the interval length (scaled to the mean length). Plots are for the Poisson distribution (k = 1) and for k = 2, 3, and 4. Note the relatively large number of short intervals for the Poisson distribution. After Merrill etal. (1996).

immediately apparent that a gamma process with k > 1 has far fewer very short intervals than a Poisson process.

Obtaining a reliable GPTS from the marine magnetic anomaly record is not a trivial matter (see chapter 5). The major problems relate to accurate dating of the individual events and reliable recognition of the shorter intervals. A small error in the dating of a reversal leads to small consequences in the analysis of the reversal sequence. However, if a short polarity interval is missed then this is a much more significant error. As shown by the examples given in Fig. 4.19, when a short interval is missed it is combined with the preceding and succeeding intervals of the opposite polarity. This means that the short interval is missed from its own polarity sequence and incorrectly produces a long interval of the opposite polarity, which is then the sum of at least three intervals.

McFadden (1984a) has shown that if n observations from a gamma process with index k have been joined together then the resulting interval appears to have come from a gamma process within index nk. Naturally this is the case for a Poisson process (k= 1) as well. Therefore, if a short interval is missed from a Poisson process then the resulting long interval of the opposite polarity looks like an observation from a gamma process with k = 3. This has two immediate

Fig. 4.18. Proportions of interval lengths (probability densities) of gamma distributions plotted against the interval length (scaled to the mean length). Plots are for the Poisson distribution (k = 1) and for k = 2, 3, and 4. Note the relatively large number of short intervals for the Poisson distribution. After Merrill etal. (1996).

1 2 Interval length/Mean length

1 2 Interval length/Mean length

Fig. 4.19. Reduction in the number of polarity intervals caused by preferential filtering of very short intervals, (a) Three intervals reduced to one, (b) four intervals reduced to two, and (c) five intervals reduced to one. From Merrill and McElhinny (1983).

Fig. 4.19. Reduction in the number of polarity intervals caused by preferential filtering of very short intervals, (a) Three intervals reduced to one, (b) four intervals reduced to two, and (c) five intervals reduced to one. From Merrill and McElhinny (1983).

consequences since it is likely that some short intervals have been missed. First, it is convenient to analyze the reversal process as a gamma process because this automatically accounts for any filtering in the observed sequence. Second, the parameter k can be a fairly sensitive indicator of polarity intervals that have been missed (McFadden and Merrill, 1984). This means that estimates of the parameter k can vary by a large amount over small intervals of time without necessarily implying that the process itself has changed much.

McFadden and Merrill (1984) concluded that k is about 1.25 for the period from about 80 Ma to the present. This suggested that several short intervals in the actual reversal process had not been identified in the GPTS. However, subsequent polarity time scales did not change this situation, and so McFadden and Merrill (1993) discussed the alternative interpretation that the reversal process itself is actually gamma. They showed that this would imply the existence of an interval of about 50 kyr immediately following a reversal during which the probability for another reversal would be depressed. The observed sequence is similar (statistically) to that which would be expected from a Poisson process that had been filtered to join into surrounding intervals any interval less than 30 kyr in duration. This is consistent with the conclusion of Parker (1997) that the maximum resolution of the GPTS from marine magnetic anomalies is about 36 kyr. Both interpretations remain viable, with the hope being that new techniques with higher resolution (see §5.1.3) will resolve the matter.

If the rate at which reversals occurs is variable, then the process is referred to as nonstationary. This is of central geophysical interest because the existence of nonstationarity implies a change in the properties of the origin of the process. The variation in the reversal rate, using the reversal chronology of Cande and Kent (1995) for the interval 0-118 Ma and that of Kent and Gradstein (1986) for the interval 118-160 Ma, is shown in Fig. 4.20. The use of sliding windows is necessary because of the variation in reversal rate (nonstationarity), which precludes considering all of the intervals of the past 160 Myr as a random sample from a single distribution. Consequently Fig. 4.20 has been constructed using a sliding window containing 50 polarity intervals (using constant length windows is inappropriate, see below). Because the Cretaceous Superchron is not part of the surrounding reversal process (§4.5.4), the sliding window does not run across the superchron. The characteristic time of these changes is the same as that associated with mantle processes, leading to suggestions that nonstationarities in the reversal record are associated with changes in the core-mantle boundary conditions (e.g., Jones, 1977; McFadden and Merrill, 1984, 1993, 1995, 1997; Courtillot and Besse, 1987). Suggestions that spatial variations in the core-mantle boundary conditions affect reversals are now supported by some dynamo theory (Glatzmaier et al., 1999).

Gallet and Hulot (1997) proposed an alternative nonstationarity for the reversal rate. They suggest a model in which the rate is essentially constant from

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