Geology Reference
In-Depth Information
(Appendix). The dof of the Bartlett or Hann-tapered periodogram are less
straightforward owing to the loss of information from the tapering in the
time domain, but gain in information from a wider central lobe (and lower
sidelobes) in the frequency domain. Numerous topics and papers report the
“effective” dof 's of smoothed periodograms for a variety of data windows
(tapers), including the three discussed here (e.g., Table 6.6 in Jenkins and
Watts (1968), Table 6.2 in Priestley (1981), Table 269 in Percival and Walden
(1993)). For our three data windows, the effective dof per Fourier coefficient
are Dirichlet (n = 1), Bartlett (n = 3), and Hann (n = 8/3 = 2.667).
Thus, for example, the Dirichlet-windowed (unsmoothed) periodogram,
with n = 2, S
D
(f ) has a 95% chance of falling within interval defined by the
so-called lower and upper 95% confidence limits (CL): [n·S
D
(f )/upper CL,
n·S
D
(f )/lower CL] = [0.002·S
D
(f ), 9.5238·S
D
(f )]; the Bartlett-windowed peri-
odogram, with n = 2(3) = 6, gives a 95% CL interval of [6·S
D
(f )/14.4, 6·S
D
(f )/
1.2373] = [0.41667·S
D
(f ), 4.8493·S
D
(f )].
These low dof do not translate into reasonable constraints for spectrum
estimates. However, averaging over adjacent frequencies (also known as “fre-
quency merging”) increases the dof and decreases the CL interval. At the same
time, averaging reduces frequency resolution. Averaging can be carried out in
the frequency domain, but it is usually done in the time domain by dividing
the data time series into segments, either end-to-end and independent
(“Bartlett's method,” using the Dirichlet data window; Bartlett (1948, 1950)) or
overlapping (“Welch-Overlapped-Segment-Averaging or WOSA method”;
Welch (1967); now adapted for other data windows, for example, Schulz and
Stategger (1997)), and subsequently averaging the segment periodograms in
the frequency domain.
In the WOSA method, the data time series of length N is divided into n
50
seg-
ments each of length N
seg
that overlap each other by 50%, i.e., n
50
= 2N/N
seg
− 1.
The effective dof of WOSA estimators depend further on the data window that
is applied: n
eff
= n
50
/(1 + 2c
50
2
- 2c
50
2
/n
50
), where c
50
is the 50% overlap correlation
(e.g., 50% for the Dirichlet window, 25% for the Bartlett window, and 16.7% for
the Hann window, see Table 1, rightmost column, of Harris (1978)). WOSA
applied to the test time series (Figure 4.10) for different segment lengths shows
improved leakage control and CL, although much of the improvement occurs
with application of the Hann window, with diminishing returns for increased
averaging, which broadens the spectral bandwidth, seriously encroaching the
definition of the two frequencies (Figure 4.12).
4.3.5.4 Zero Padding
A common practice in spectrum estimation is zero padding. Many practi-
tioners are unaware that it is being applied to their data, when, for example,
the FFT used in the spectral estimation requires that the data time series have
a length that is a power of 2, for example, in the SSA-MTM Toolkit (Ghil et al.
2002), and the default settings of MATLAB's FFT. The other motivation for
zero padding is to interpolate the FFT mesh of spectrum estimates, as illus-
trated in Figure 4.13. Zero padding can also be used to estimate uncertainties
on spectral peak frequency identification (e.g., Abe & Smith 2004).
