Geology Reference
In-Depth Information
0.050
0.055
2500
Original length
Lengthx10
2000
1500
1000
500
0.045
0.05
0.055
0.06
Frequency (1/n)
Figure 4.13
Benefit of zero padding is demonstrated for the test time series in Figure 4.10a. The blue curve is the
Dirichlet periodogram of the time series, which is N = 2048 in length (same as in Figure 4.10b (inset) and
Figure 4.12a), with an FFT mesh defined by Δf = 1/2048, i.e., no frequency bin (multiple of Δf ) coincides with
either of the two signal frequencies (0.050 and 0.055, see vertical dashed red lines). The green curve is the Dirichlet
periodogram of the same test time series, padded with 0s to 10 times its original length (with zeros appended to the
end of the time series prior to Fourier transformation). Now the FFT mesh is defined by Δf = 1/20480, with
frequency bins very close to the true signal frequencies. The measured power is now equal between the two
frequencies, reflecting their equal amplitude contributions in the time series. The Dirichlet “sidelobes” of this
interpolated spectrum reveal the original frequency resolution of the time series.
Importantly, the (prewhitened) time series must be tapered prior to zero
padding, so as not to institute a new source of spectral leakage from sudden
transition to zeros at the end of the time series. In the literature, spectra reported
with high-frequency “multilobed” features, such as those in Figure 4.13, or
embellishing the tops of smoothed spectral peaks may have been zero padded,
and should not be interpreted or used to evaluate the outcome of hypothesis
tests (Section 4.3.6).
4.3.5.5 Indirect Spectral Estimators and the BT Correlogram
The autocorrelation function ρ(n) of a time series equals the squared
modulus of the Fourier transform of the time series (the Wiener-Khinchin
theorem; Blackman and Tukey (1958)). This relationship leads to an
alternative estimate of the power spectrum:
2
M
−
∑
ρ
1
BT
−
i f
2
π
S f)
=
()()
nDne
D
nM
=−
(
−
1
)
where M ≤ N equals the number of autocorrelation “lag” coefficients used in
the estimates and sets the smoothing factor of the estimates. This is the BT
correlogram, also known as the “lag window spectral estimator” or “indirect
