Geology Reference
In-Depth Information
50
0
-50
-100
0
0.025
0.05
Frequency (1/n)
0.075
0.1
Figure 4.16
2π MTM estimated spectrum of the test time series of Figure 4.10, using
spectrum.mtm.m
, with the lower
and upper 95% CL shown for 6 dof (see also Figure 4.15 for display with linear y-scale). While there are four tapers that
qualify for use (and potentially 2 × 4 = 8 dof ), MATLAB drops the last taper owing to its relatively low eigenvalue
(0.7219). These results may be compared with the 95% CL of single-tapered estimates displayed in Figure 4.12.
units as the single-taper estimates in Figure 4.12. This example demonstrates
the strength of the multitaper estimator: the adjustable bandwidth W solves
many resolution problems while maintaining the high statistical stability (mul-
tiple dof ). Thomson (1982) also describes a “harmonic line” F-test that exam-
ines the spectrum at the resolution of Δf for the presence of significant sinusoids
(see Section 4.3.6.2).
4.3.5.7 Adaptive Weighting
In natural data spectra, power is usually unevenly distributed with frequency.
Frequencies with high power generally (but not always) host more information
than bands with low power. Moreover, high-order eigentapers tend to con-
tribute more bias that preferentially affects the low power spectral estimates.
To counter these problems, a “data adaptive weighting” strategy was devel-
oped to down-weight low power spectral estimates based on the data spec-
trum and bias from the
k
th eigentaper (Section V in Thomson (1982)). The
resultant adaptive weights, d
k
(f ), are unique to the input data time series and
eigentapers being used in the spectral estimation. The weights are applied to
the eigenspectra prior to the final averaging. The d
k
(f ) also provide an
empirical estimate of the effective dof as a function of frequency, assigning
fewer dof to spectral regions with lower power:
K
−
∑
1
2
υ(f)
=
2
d (f
k
)
k
=
0
This contrasts with mathematical statistics that provide a single estimate
of dof to be applied across the entire spectrum, for single tapered estimates
such as those in Figure 4.12, as well as for unweighted multitapered spectra
such as in Figure 4.16. The effect on adaptive weighting and υ(f ) from dif-
ferent distributions of power is illustrated in Figure 4.17. In these examples,
signal is deeply embedded in noise. The noiseless signal has υ(f ) with the
maximum number of 14 dof (i.e., 7 eigenspectra contribute 2 dof each) in
the bands of width W surrounding the two signal frequencies; elsewhere
υ(f ) is at a minimum of 2. The power spectra of the two noisy time series just
resolve the two signal frequencies; υ(f ) tracks the estimated power level,
