Geology Reference
In-Depth Information
with local maxima at the signal frequencies, but never reaching the
maximum of 14 dof. SSA-MTM Toolkit (Ghil et al. 2002) applies adaptive
weights to the power spectrum, but does not estimate υ(f ) for hypothesis
testing, instead using only maximum dof (e.g., the horizontal dashed line at
14 in Figure 4.17c).
Analyseries
(Paillard et al. 1996) does not calculate
adaptive weights and reports unweighted MTM power spectra, known as
“high-resolution” spectra (Thomson 1982).
4.3.6
Hypothesis Testing and Noise Modeling
The usual question asked of cyclostratigraphy is whether or not a signal with
Milankovitch cycles is present. The power spectrum is consulted to assess
narrow frequency bands of elevated power with nonrandom variance at the
Milankovitch frequencies (“signal”). In natural data, uncorrelated, random
variance (“noise”) is also present and must be distinguished from the signal.
Noise tends to occur at all frequencies, also known as the “continuum,” and
has a power level that is generally lower than that of the signal. The simulated
time series in Figure 4.17 are typical of signal + noise combinations found in
nature; in fact, in these two series, the noise level is quite high and challeng-
ing to distinguish from the embedded signal, as demonstrated below.
The signal versus noise problem in spectral analysis is approached through
modeling and hypothesis testing. Signal is modeled as a process that is char-
acterized by a frequency, or set of frequencies, of variation with persistent
magnitude and constant phase. Noise is characterized by variation with con-
tributions at all resolvable frequencies, and serves as the “null model” in the
following hypothesis testing procedure:
1
The “null hypothesis” H
0
is that the data results from a random pro-
cess, here represented by the noise model (see the following sections
for commonly used models).
2
The “alternative hypothesis” H
A
is that the data represents a
combination of a nonrandom process and random chance.
3
A “test statistic” is used to assess the validity of H
0
. In this case, the
test statistic is the power spectrum.
4
The probability P is computed for the test statistic to evaluate whether
it is at least as significant as the case for a true H
0
. Smaller P is stronger
evidence against H
0
.
5
Compare P to a significance value α, e.g., 0.05; if P ≤ α, H
0
is ruled out
and H
A
is accepted.
4.3.6.1 Spectral Noise: The Null Model
Two statistical models of noise are in wide use today: autoregressive
(Markovian) noise and power law (1/f
α
) noise. These models recognize that
noise spectra can have different frequency distributions depending on the
process under consideration and its measurement. White noise has equal
