Geology Reference
In-Depth Information
Blackman-Tukey (B-Tukey), maximum entropy, and MTM, and they all
generate power spectra, but in different ways. We have had the most success
with MTM analysis, which is described theoretically in Chapter 4. MTM
analysis can be conducted by both the Analyseries and SSA-MTM toolkits.
The MATLAB script
ftestmtm.m
available in the Appendix can be used for
harmonic line detection by MTM analysis while the MATLAB script
pmtm.m
can be used to generate an MTM power spectrum. Chapter 4 shows that the
user can select different multitaper taper families by choosing different
averaging bandwidths (i.e., 2π, 3π, 4π, 5π, etc.). In most cases, a 2π band-
width which results in a maximum of three data windows (or a sequence of
three Slepian tapers applied to the data series) should yield good results,
although changing the bandwidth to be larger or smaller will alter the reso-
lution and confidence in the resulting power spectrum. Different band-
widths should be tested to see the effects on the power spectrum. Using
more tapers increases the confidence in the spectral peaks, but causes more
smoothing, hence reduces the resolution of the peaks. Therefore, there is a
trade-off between resolution (fewer tapers) and confidence (more tapers).
The power spectrum should be calculated up to the Nyquist frequency
(1/2*sampling interval).
7.5.3
Significance of the Spectral Peaks
One of the important goals of the time series analysis of a rock magnetic
data series is to obtain an estimate of the statistical significance of the
spectral peaks that emerge in the power spectrum. The question being asked
is whether the spectral peak has emerged, statistically, above the background
noise in the data series. Analyseries generates harmonic F-tests for MTM
spectra that determine the probability that spectral density at any given fre-
quency is associated with true periodic behavior isolated only at that
particular frequency compared with the average power of the surrounding
frequencies (within the averaging bandwidth). Since F-tests are calculated
for all frequencies in the spectrum and are typically plotted on top of the
power spectrum, the resulting combined plot can be confusing. It is better
to just indicate F-tests for spectral peaks with significant amplitudes. As
discussed in Chapter 4, harmonic F-testing can have numerous “false posi-
tives” and careful assessment of the power spectrum and F-testing is needed
(important guidelines are given in Thomson (2009)).
A more general approach for determining spectral peak significance is the
generation of a red noise spectrum for the data series and different confidence
levels above that red noise. The SSA-MTM toolkit will generate a robust red
noise (Mann & Lees 1996) spectrum with 90, 95, and 99% confidence inter-
vals as part of its MTM spectral estimation. Spectral peaks that rise, for in-
stance, above the 95% confidence interval for the red noise are statistically
distinct from the background red noise model with 95% confidence. The red
noise model assumes that the noise in the data has a memory by one time
