Geology Reference
In-Depth Information
4.3.4
Digital Filters
Frequency domain filters have proven to be the most effective in
cyclostratigraphy. The data are Fourier transformed into the frequency
domain, then multiplied by the filter, and inverse-Fourier transformed
back to the time domain. The filters must be continuous and differentiable
to avoid generating artifacts in the data when transforming back to the time
domain. In addition, the filters cannot impose a time delay, i.e., they must
be “zero-phase” filters. Here we discuss the Gaussian (Klapper & Harris
1959) and Taner (Taner 2000) filters that perform extremely well in
cyclostratigraphy.
The frequency and phase responses of these two filters (Figure 4.6) indi-
cate that strict zero phase is maintained through the passband. The Gauss
filter passes power outside of the edges of the cutoff frequencies, whereas the
Taner filter has a steep slope. Application of the two filters shows excellent
recovery of precession index variation from the 4-myr-long insolation series
(Figure 4.7); differences are most noticeable where the amplitude modula-
tions of the precession are low, i.e., when the local frequency of the precession
is furthest from the average precession frequency. Both filters faithfully
reproduce the phasing of the precession index, although it should be noted
that the filtered series are in opposite polarity to the precession as is expected
in summer insolation (see Chapter 5).
4.3.5
Spectral Analysis
The cornerstone of univariate time series analysis is the power spectrum,
which is the distribution of time series variance (power) as a function of
frequency. The power spectrum can be estimated with a variety of non-
parametric (Fourier based) or parametric (model based) techniques; each
“spectrum estimator” has its own statistics, accuracy (bias), and resolution
issues. Spectrum estimators assume stationarity, i.e., the basic statistics of
the time series, for example, the mean and the variance, do not change over
time. Stationarity is generally not valid for natural time series, especially
cyclostratigraphy, which can undergo significant changes in accumulation
rate and proxy behavior within a single section. The usual remedy is to apply
a spectral estimator as an evolutionary or “running” application through the
time series (Section 4.3.7).
Three nonparametric (nonmodel based) spectrum techniques are dis-
cussed: the smoothed periodogram, Blackman-Tukey (BT) correlogram,
and Thomson multitaper estimator. In particular, the multitaper technique—
which includes a harmonic line (amplitude) spectral estimator, an average
power spectral estimator, coherency and cross-phase estimators, and other
higher-order spectral estimators—provides a “full-service” toolbox of flex-
ible, high resolution, statistically robust estimators that outperform the other
nonparametric techniques.
