into the larger time interval Tnew. But the number of periods
N may be calculated by dividing the original time interval
T0 by the original period estimate for wave 2, which we here
call P0. Thus, if we know the original period P0 for wave 2,
we can estimate the new period Pnew:
Eq. ( 5) may also be used to estimate the new periods for
the other two waves comprising the composite signal.
Here we have made an assumption that is generally not
strictly correct, but which is ‘good enough’ for our purposes.
We assume that the frequency of a dominant spectral peak
corresponds exactly to the frequency of one of the individual
waves comprising the resultant signal. Because the signal
is composed of a finite number of waves, this is not really
correct—an estimated peak frequency often falls ‘between’
two of the discrete frequencies in the power spectrum.
Nevertheless, for a power spectrum with a reasonably large
number of discrete frequencies within a finite frequency
band, we expect a particular peak frequency to be quite
close to one of those discrete frequencies. This means that
we can also use Eq. ( 5) to estimate the new periods (after
stretching) of the spectral peaks, provided that we know the
original periods for those peaks. In the following discussion,
we treat the Blackman–Tukey method of obtaining those
original period estimates as a ‘black box’ and accept as a
given that the original period estimates are accurate.
Confirming the results
My original SIMPLEX values of P0 (obtained using my
reconstructed Pacemaker data and the Blackman–Tukey
method) are shown in tables 2 and 3. For instance, the
middle section of table 2 lists my original periods for the
three dominant RC11-120 δ18O spectral peaks. The period
of the smallest spectral peak was 23. 8 ka. Eq. ( 5) implies
that the estimated value for the new (stretched) period is
Pnew = (308.75 ka ÷ 273.00 ka) × 23. 8 ka = 26. 9 ka. This
compares favourably with the new period of 27. 1 ka I
obtained using the Blackman–Tukey method. The agreement
between these estimated periods and those obtained by the
B–T method is generally poorer for the longer (~ 100 ka)
periods (i.e. the estimated uncertainty in the new period
estimate is larger for larger periods). The reason for this is
given in the online appendix,
43 which provides a means of
estimating this uncertainty.
Unlike the SIMPLEX timescales, the ELBOW timescale
for the PATCH Composite ‘core’ did not have a perfectly
constant slope versus depth; the radiocarbon age of 9. 4 ka
(table 2 in the Pacemaker paper) remained the same both
before and after stretching of the timescale, and a third
anchor point (at 8. 25 cm) was used in the E49-18 section
of the PATCH core. Hence, this shortcut method was not
strictly valid, and I did not calculate error estimates for
this particular case. Nevertheless, there was still generally
good agreement between periods calculated using this ‘easy’
method and the B–T method (table 4).
These comparisons were obtained using my estimates
for the periods of the spectral peaks, calculated using
reconstructed data. Although these results are generally
in good agreement with the original published Pacemaker
results, there are some discrepancies, likely due to subtle
errors in the values of the reconstructed data. However,
one can also use this method and the original published
Pacemaker results to estimate the periods that the Pacemaker
authors would have themselves obtained had they used
the currently accepted age of 780 ka for the B–M reversal
boundary in their calculations.
Since the earth’s inferred orbital cycles are quasi-periodic,
the frequencies expected from Milankovitch theory will not
be exactly the same before and after the stretching of the
timescales for the cores. However, one typically expects
periods of lengths 100, 41, and 19–23 ka to result from
such orbital calculations. The new results are generally
in poor agreement with Milankovitch expectations. This
is especially true for the new E49-18 and PATCH results
(tables 3 and 4).
Figure 10. A stretching of the timescale over which a waveform has been
sampled will also stretch the component waves comprising the signal.
This fact enables one to quickly estimate the new periods of the prominent
spectral peaks, provided that the original periods of those spectral peaks
are already known.