Geology Reference
In-Depth Information
of the presence of lines, at ultra-high-frequency
resolution. The method also provides averaged
spectral coherency and cross-phase estimators
that outperform others, harvesting more informa-
tion (degrees of freedom) from the data, and
with less bias, than other methods (see Hinnov &
Goldhammer (1991) for an example pertaining to
the Latemar cycles).
The multitaper method harmonic analysis of
the Mendola and Latemar sections (Fig. 9a,b)
shows that some of the bundling frequencies are
shared by the two sections, although their
relative amplitudes are quite different. This may
be due to different, very low-frequency tectonic
infl uences on the development of the Mendola
versus Latemar cycles or the presence of differ-
ential cycle thicknesses resulting from normal
sedimentary processes (e.g. lateral sediment ero-
sion and transport) on Mendola tidal fl ats. The
lowest frequency bundling term, Mendola at
1/18.9 and Latemar at 1/27, while measured with
a high
F
-test in both sections, is based on only
two repetitions of the term in the 36-cycle-long
Mendola section and less than two repetitions
for the Latemar section, thus may not represent
a true, sustained bundling at this frequency. The
next to lowest bundling term, Mendola at 1/8.5
and Latemar at 1/9.8, is only 1/8.5-1/9.8 = 1/64
apart, i.e. less than the elementary bandwidth
of the series. The magnitude-squared coherency
spectra measure correlation coeffi cients between
time-series as a function of frequency. To maxi-
mize the degrees of freedom of magnitude-squared
coherency estimates, averaging over two or more
sampled frequencies is necessary. Cross-phase
spectra indicate the phasing of frequency compon-
ents between the two series. For example, when
cross-phase registers 0° at a particular frequency,
the two series are said to be 'in-phase' at that
frequency.
In Fig. 9c, again 2
multitapers have been
applied, which preserves the narrowest possible
averaging bandwidth (
f
), while gathering a
substantial 5-6 degrees of freedom across most of
the frequency range (see degrees of freedom curves,
Fig. 9a,b). To be on the conservative side, the sig-
nifi cance levels for zero coherence for 5 degrees
of freedom. Three bands of non-zero coherence
are present, centred at 1/10, 1/5 and 1/2.4. The
double-peak nature of the magnitude-squared
coherency at 1/10 could be artefacts from exces-
sive weighting imposed by one of the higher order
2
multitapers on the outer edges of the averag-
ing bandwidth
and magnifi ed by the frequency
interpolation (zero-padding). The other striking
magnitude-squared coherency peak is centred at
f
= 1/5, which indicates that 5:1 bundling is pres-
ent and coordinated between the series at a stat-
istically signifi cant level.
The cross-phase spectrum (Fig. 9d) can be inter-
preted at frequencies where magnitude-squared
coherency values indicate signifi cant non-zero
coherence. The cross-phase of 20° at 12:1 bun-
dling may indicate that the two series are phase-
shifted relative to each other at this frequency. In
the 'time' domain this would correspond to a shift
of (20°/360°)
f
= 1/36. Thus, this line may represent the same
process in both sections.
Examination of the Fischer plots (Fig. 8) indic-
ates that one source of this peak in both of the
sections appears to involve a single bundle of
8-9 cycles (cycles 7-15). Surprisingly, the 5:1 bun-
dling frequency has a very low
F
-test (60-75%)
in both sections. In both, the line is slightly
broader than
f
, and in the Latemar data it has
a substantially lower amplitude (Latemar 0.2 m,
Mendola 0.35 m). This suggests that this bundling
component, while quite visible in the Fischer
plots of both series, is somewhat variable about
5:1, causing the
F
-test to fail. Finally, both sec-
tions show a bundling at 1/2.4. Again, these
bundles can be observed in the Fischer plots
(Fig. 8, e.g. Mendola cycles 19-20, or Latemar
cycles 14-15) and some of the power at this fre-
quency could also be a spectral artefact from
asymmetry in the 5:1 bundling.
Magnitude-squared coherency and cross-phase
spectral analysis identify signifi cant frequency
components shared by two time series. For
example, frequencies present in both series that
are consistently coordinated over the duration
12 = 0.67 cycles, i.e. less than a one
cycle lead of the Latemar series over the Mendola
series. Similarly, at 9:1, the Latemar series leads
the Mendola series by around 1 cycle; at 5:1,
Mendola leads the Latemar series by around 0.5
cycles; and at 2.4:1, the two series are statistic-
ally in-phase. Taken together, the Mendola and
Latemar series show close coordination to within
+1 cycle at all signifi cant bundling frequencies,
constituting powerful evidence of a broadband
statistical fi t and phase lock between them.
RESULTS
The following results were derived from the
analysis of the cyclic successions.
