Geology Reference
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the statistical stability (maximize the dof ), minimize bias (due to the prefer-
ential sampling of the center part of the time series), and retain the highest
(narrowest) possible frequency resolution. The quest to satisfy all of these
criteria led to the concept of “multitapers.” Families of functions known as
“discrete prolate spheroidal sequences (DPSS),” each function independent
from the others, and when summed together approximate a Dirichlet data
window (Slepian 1978). DPSS are also known as “Slepian sequences” in
honor of the mathematician David Slepian (1923-2007). If the Slepian
sequences in a family are individually applied as tapers to a time series, and
each Slepian-tapered series is then Fourier transformed, the set of Fourier
transforms may be averaged together to produce a smoothed spectral esti-
mator, with each transform nominally contributing 2 dof (Thomson 1982):
2
1
K
1
N
1
K
K
K
S XX
i
22nf
π
(f)
=
(f)and
()
f
=
x(n)D (n)e
k
x,D
(
K
+
1
)
D
D
k
=
0
n
=
0
where x(n) is the time series and D k (n) are Slepian sequences. The Slepian
sequences maximize variance (power) within a narrow band of frequencies W
spanning several Δf with respect to the total resolvable band of frequencies
in  a time series of length N, i.e., ±f Nyq . The Slepian sequences are solved as
eigenvectors with associated eigenvalues indicating the bias of each eigen-
vector. The eigenvectors with eigenvalues closest to 1 are retained as the
“minimum-bias” set of “eigentapers” to be applied to the time series. The time-
bandwidth product is selected as P = NW, and the maximization problem is
solved (e.g., Park et al. 1987) to yield a set of Pπ prolate eigentapers (usually
shortened to “tapers”), of which the lowest orders 0 through K-1, where K = 2P,
have eigenvalues very close to 1. The Fourier transform of an “eigentapered”
time series is an “eigenspectrum.” Usually the eigentaper of order K - 1 has an
eigenvalue that is significantly lower than 1 and is dropped from the set. For
cyclostratigraphy, P usually ranges from 2 to 6 and can be fractional (e.g., 3.3
and 4.8). The Fourier transforms of the individual tapers from a single family
reveal that as taper order increases, the central lobe of width W is sampled
increasingly toward the outer edge of W (e.g., Figure 2 in Thomson (1982);
Figure 3 in Park et al. (1987)). In the time domain, each taper in a family sam-
ples a different part of the time series (Figure 4.15a-d).
Multitaper spectral estimates of the test time series of length N = 2048 are
compared with the unsmoothed (Dirichlet) periodogram (Figure 4.15e). For
example, “5π,” refers to the averaging bandwidth imposed by the application of
a set of nine 5π Slepian tapers (i.e., orders 0-8), where the value P = 5 has been
selected by the practitioner. The “5” corresponds to an averaging bandwidth
W = 5/2048 = 0.0024414. The bandwidth W also sets the dof listed next to the
displayed spectra on the right, nominally as 2NW (with further discussion in
Section  4.3.5.7). All of the displayed multitaper estimates have high consis-
tency (many dof ), and the 2π estimator in particular shows high definition of
the two signal sinusoids, with 6 dof at only twice the frequency resolution com-
pared to the 2 dof and Δf resolution of the unsmoothed (Dirichlet) periodo-
gram. Figure 4.16 shows the 95% CL of the 2π multitaper spectrum in the same
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