Geology Reference
In-Depth Information
that F-tests should be interpreted in conjunction with power spectrum struc-
ture (as was done here) and provides additional notes on expected outcomes
of F-testing using cyclostratigraphy as an example.
4.3.7
Time-Frequency Analysis
Natural systems often “drift,” and their frequencies and magnitudes can
change, slowly (e.g., groundwater flow), or suddenly (e.g., seismic events), or
may have quasi-periodic behavior (e.g., Milankovitch-forced climatic
change). Analytical methods that track time-frequency changes are needed
to assess such processes. Here, two representative methods are illustrated:
evolutionary spectrograms and complex signal analysis.
4.3.7.1 The Evolutionary Spectrogram
The evolutionary spectrogram can take on numerous forms and is the
simple application of a spectral estimator with a moving (or “running”)
time window through power with respect to frequency on the x-axis and
time on the y-axis. 3D displays can also be effective (not shown here).
The spectrogram shown in Figure 4.24 displays a running “unsmoothed
periodogram” (Section 4.3.5) of the La2004 astronomical model of the
Earth's orbital eccentricity compared with the two astronomically tuned
versions of the Arguis ARM series (Figure 4.22; see also Chapter 5).
These spectrograms highlight the intricate quasi-periodicity of the
orbital eccentricity parameter, in which the ~100 kyr spectral terms
experience an interval of low power relative to the 405 kyr term in the
interval centered on 37.5 Ma. Experimentation with window length is an
important aspect of the spectrogram; if it is too long, high frequencies
will be smoothed out, and if it is too short, low frequencies will not be
measured adequately.
4.3.7.2 Complex Signal Analysis
Complex (or quadrature) signal analysis is a classical technique used to
estimate instantaneous amplitude, phase, and frequency attributes of a
real signal as a function of time (Taner et al. 1979). The complex repre-
sentation of a real signal g(t) is given as G(t) = g(t) + ig * (t), with i
2
= -1
and “*” denoting complex conjugate, where g * (t) is obtained by Hilbert
transformation of g(t).
The technique is demonstrated in Figure 4.25, in which a 1.47 kyr cycle
(Dansgaard-Oeschger scale) experiences a 40 kyr frequency modulation
and a 20 kyr amplitude modulation over a 100 kyr interval. The complex
signal analysis shows that the amplitude modulation imposes “singularities”
in f(t) whenever the amplitude passes through zero, also producing discon-
tinuities in the instantaneous phase. Such “singularities” can be picked up by
depositional hiatuses or other perturbations not related to amplitude forcing
and can also occur as the result of band-pass filtering.
