Discussion

A natural question is why H^θ₀ (that glacial cycles are independent of obliquity) is rejected, while H^p₀ and H^e₀ are not rejected. The first and simplest reason is that the obliquity nonlinear coherence with the terminations (0.70) is higher than that of precession (0.43) or eccentricity (0.66). The second reason is that the obliquity critical value (0.60) is lower than that of eccentricity (0.84). To address why there is a difference in the critical values, one must consider the glacial timescale built into the random walk glacial model.

Figure 4-6 shows a histogram of the duration between threshold crossings derived from a long run of the stochastic ice-volume model. The mean time between consec-utive terminations is 100KY with an approximately normal distribution and±20KY standard deviation, whose spread agrees with other estimates of the deviation in glacial cycle length [Raymo, 1997]. Note if the bias term, b= 1, were not included in Eq 4.3 the distribution would more resemble a Poisson distribution. The magnitude of this standard deviation suggests that the relative phasing between terminations and obliquity will be nearly uniform, while the relative phasing between terminations and eccentricity will be more structured. In general, the more structured the phase distribution, the higher the expected nonlinear coherence. Thus, the requirements for establishing a significant phase coupling between eccentricity and the terminations (i.e. the critical value) is higher because eccentricity has a timescale similar to the terminations.

The above discussion may seem to turn the argument on its head: typically, a similarity between timescales is cited as evidence for a relationship between two phenomena. But consider a counter example in which two unrelated signals are both periodic at 100KY. Then, the relative phasing at the first event must also be the relative phasing at all subsequent events. In this case, even though the signals

are unrelated, there is only one degree of freedom in the system and a nonlinear coherence of one is assured. Oppositely, when two signals have differing periods and are unrelated (barring the case of one signal being a harmonic of the other) one expects the phase to be more uniformly distributed and the nonlinear coherence to be lower. The average duration between eccentricity maxima is similar to average duration between terminations resulting in fewer DOF and a higher critical value relative the obliquity test.

There are also some qualitative observations which argue against a coupling be-tween eccentricity and the glacial cycles. First, the most significant band of variability in eccentricity is near 1/400KY, but a concentration of 1/400KY variability is absent from Pleistocene climate variability [e.g. Imbrie et al., 1993]. Second, terminations lead eccentricity maxima by an average of 20KY, or 68^◦. This indicates that termi-nations would have to be triggered by moderate values of eccentricity. It would seem more physical for glacial termination to be triggered during maximum rather than intermediate values of the eccentricity. That the lead is equivalent to a full preces-sion cycle is also important, as this argues against glacial pacing by the eccentricity amplitude modulation of the climatic precession. Third, there is a trend whereby ec-centricity maxima lag terminations by smaller values as time progresses. The change in lag averages 6KY per 100KY (see Figure 4-3), and it is difficult to conceive of physical mechanisms which would drift in this way. Finally, the eccentricity varia-tions only cause weak changes in insolation forcing. Taken together with the inability to reject H^e₀, these observations make eccentricity appear an unlikely candidate for pacing the ice-ages.

A second question is whether excluding either of the termination 3 events would change the obliquity test results? Figure 4-2 and Table 4.1 shows that termination 3a contributes to obliquity’s high nonlinear coherence; if it is excluded, the nonlinear coherence decreases from 0.70 to 0.66. Decreasing the number of terminations from seven to eight also increases the critical value from 0.60 to 0.64. Thus the nonlinear coherence remains greater than the critical values so that the null hypothesis would still be rejected and the results unchanged.

In Section 4.3 the difficulty of testing the significance of the Titius-Bode “Law”

was discussed. In particular, it appeared that test results were sensitive to the null-hypothesis used in testing the Titius-Bode “Law”. This raises the final question considered here; how robust are the obliquity test results to modifications of the

0 50 100 150 200 250 0

500 1000 1500 2000 2500

termination intervals (KY)

occurences

Figure 4-6: A histogram of the time between consecutive terminations derived from the random walk ice-volume model (Eq. 4.4). The selected parameterizations are a noise amplitude a=2, a drift b=1, and an ice-volume threshold ξ=100. The mean interval between terminations is 100KY with a ±20KY standard deviation. The distribution is close to Gaussian but with a slightly elongated tail towards longer intervals.

null-hypothesis? To answer this question, it is useful to investigate whether some other plausible formulation of H^θ₀ would increase the critical value above 0.70. One approach (H^θ₀⁰) is to phase randomize EOF1, identify terminations in the phase ran-domized record, and compute the newly realized nonlinear coherence. A more so-phisticated approach, which accounts for the non-Gaussian distribution of EOF1, is to use the phase shuffling algorithm of Schreiber and Schmitz [2000]. Experiments were also performed in which the non-Gaussian distribution of the rates of change of EOF1 was preserved. In all cases, the PDF resulting from these phase randomized approaches was qualitatively very similar to the PDF associated with the random walk accumulation mode (H^θ₀) and the obliquity null-hypothesis was invariably re-jected. Assuming a uniform phase distribution between obliquity and terminations also leads to rejection of the null-hypothesis.

Another approach to exploring whether the obliquity null-hypothesis can be re-jected is to build other simple models, akin to the random walk ice-volume model presented earlier. A simple model was formulated (H^θ₀⁰⁰) which requires terminations to follow a Poisson distribution. Pursuing the idea that termination identification

times could be biased by the existence of a linear response to orbital forcing, obliquity variability was superimposed on the results of the Poisson model prior to identifying termination initiation times. It is found that H^θ₀⁰⁰ can be rejected at the 5% signifi-cance level even when the obliquity variability is made to account for an unreasonably large 40% of the total record variance. On the basis of three separate formulations, it is concluded that the null-hypothesis of no coupling between terminations and the obliquity variability can be safely rejected. It should, however, be noted that this result does not constitute proof of the obliquity pacing theory. For proof, one wants an ironclad physical mechanism, a much stronger statistical test, or preferably both.