Eccentricity modulation of precession

Eq. 2.5 shows that the sum of two cosines can be re-written as an amplitude modulated signal; in this case giving

˜

One further modification is now necessary to relate the amplitude modulation of ˜µ(t) to the frequency modulation,w⁰. The amplitude modulation term, cos(^2πt₂₀₀), becomes negative whereas the instantaneous amplitude is typically defined as a positive quan-tity. The absolute value of the amplitude modulation term can be written,

AM =

where the relationship cos(f)² = cos(2f) + 1 was used. Thus, both the amplitude modulation, AM, and frequency modulation, w⁰, terms are positive and periodic at 100KY. Figure 2-4 shows the excellent correspondence when both these terms are plotted against one another.

2.2.3 Eccentricity modulation of precession

For the simple periodic modulation of a cosine by another cosine it was shown that narrow-band-pass filtering can generate an amplitude modulation with a period sim-ilar with the original frequency modulation. Now we turn to the more complicated precession signal. It was shown earlier that both the amplitude and precession of the climatic precession parameter are modulated by the eccentricity variability. In this

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Figure 2-4: Top leftthe frequency modulated signalµ(t) given by Eq. 2.9. Note there is no amplitude modulation. Top right shows the periodogram of µ(t) with power concentrated at the carrier frequency of 1/23KY and side-bands at 1/23±k/100KY, k ={0,1,2...}. Middle leftapplying a narrow-band-pass filter toµ(t) gives ˜µ(t) with amplitude modulation. Middle rightshows the periodogram of ˜µ(t). The pass-band filter cut-off frequencies at 1/23KY and 1/16KY are indicated by the vertical dashed lines. A small amount of white noise was added for plotting purposes. Bottom shows that the instantaneous amplitude of ˜µ(t) (black) is strongly correlated with the instantaneous frequency ofµ(t) (red).

section it is shown that, in direct analogy with the previous simple example, eccentric-ity amplitude modulation can be generated from precessional frequency modulation alone.

Figure 2-5 shows the frequency modulated precession signal sin$; note there is no eccentricity amplitude modulation. The periodogram of sin$contains concentrations of energy at 1/23 and 1/19KY as well as concentrations of energy at a variety of

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Figure 2-5: Similar to Figure 2-4 but now for the precession signal. Top left the precession signal sin$ without eccentricity amplitude modulation. Top right The periodogram of sin$ showing excesses of energy near 1/23 and 1/19KY but also significant energy in numerous side-bands. Upper left is sin$ after narrow-band-pass filtering, and upper right is the associated periodogram. The filtering cut-off frequency are indicated by the vertical dashed lines at 1/25 and 1/18KY. Lower left shows the strong similarity between the instantaneous amplitude of the filtered precession signal (black) and the eccentricity (red), each plotted on their own scales.

Bottom is again the instantaneous amplitude (black) but also the instantaneous frequency (red) of the filtered precession signal. The horizontal dashed lines indicate the cut-off frequencies used for filtering the precession signal. Note that when the instantaneous frequency strays outside the cut-off frequencies, the amplitude of the filtered precession signal tends to be small.

bands. For the signal µ(t), the distribution of side-band energy was particularly simple [see Figure 2-4]; by contrast, the distribution of precession side-band energy is complicated by the more abrupt and episodic changes in the frequency of precession.

Nonetheless, Figure 2-5 shows that after narrow-band-pass filtering sin$ between cut-off frequencies of 1/25 and 1/18KY, an eccentricity-like amplitude modulation appears.

The instantaneous amplitude of the filtered sin$ signal closely corresponds with the eccentricity variability with a cross-correlation of 0.87. A qualitative explanation for the close resemblance between the filtered signal’s amplitude and the eccentricity is that during times of low eccentricity, precession tends to have significant deviations in instantaneous frequency [see Eq. 2.4]. These deviations in frequency manifest as the side-band energy in the periodogram of sin($). The removal of this side-band energy by narrow-band-pass filtering also removes the energy associated with the anomalously high or low frequencies in sin($), tending to give a reduced amplitude during times of low eccentricity. Thus the amplitude of the filtered precession vari-ability corresponds in phase and magnitude with the eccentricity induced frequency modulations.

The further question arises of whether an eccentricity amplitude modulation can be built into a signal, absent any true precession energy. In Appendix C of Chap-ter 3, it is demonstrated that orbitally-tuning a noisy signal to precession and then narrow-band-pass filtering over the precession band does, in fact, generate a pre-cession period signal with eccentricity-like amplitude modulation, consistent with the results ofNeeman[1993]. Thus, contrary to assertions made elsewhere, it is concluded that the appearance of eccentricity-like amplitude modulation in orbitally-tuned and pass-band-filtered paleoclimate records does not provide evidence for orbital control.

In summary, it was shown that the frequency of the precession parameter under-goes large deviations when the eccentricity is small. It was also shown that narrow-band-pass filtering a frequency modulated signal can generate an amplitude mod-ulation similar in period to the original frequency modmod-ulation. Because it is well established that orbital-tuning can build frequency modulation into a signal [e.g.

Shackleton et al, 1995], it is expected and shown that eccentricity amplitude modula-tion will appear in records which are tuned to precession and then narrow-band-pass filtered. Thus, the presence of eccentricity amplitude modulation in records tuned to precession does not provide evidence for orbital climate control.

2.3 Rectification and precession signals in the