Geoscience Reference
In-Depth Information
A popular method used to compute power spectra in earth sciences is the
method introduced by Blackman and Tukey (1958). h e Blackman-Tukey
method uses the complex Fourier transform X xx ( f ) of the autocorrelation
sequence corr xx ( k ),
where M is the maximum lag and f s the sampling frequency. h e Blackman-
Tukey auto-spectrum is the absolute value of the Fourier transform of the
autocorrelation function. In some i elds, the power spectral density is used
as an alternative way of describing the auto-spectrum. h e Blackman-Tukey
power spectral density PSD is estimated by
where X* xx ( f ) is the conjugate complex of the Fourier transform of the
autocorrelation function X xx ( f ) and f s is the sampling frequency. h e actual
computation of the power spectrum can only be performed at a i nite
number of dif erent frequencies by employing a Fast Fourier Transformation
(FFT). h e FFT is a method of computing a discrete Fourier transform with
reduced execution time. Most FFT algorithms divide the transform into
two portions of size N /2 at each step of the transformation. h e transform
is therefore limited to blocks with dimensions equal to a power of two. In
practice, the spectrum is computed by using a number of frequencies that is
close to the number of data points in the original signal x ( t ).
h e discrete Fourier transform is an approximation of the continuous
Fourier transform. h e continuous Fourier transform assumes an ini nite
signal but discrete real data are limited at both ends, i.e., the signal amplitude
is zero beyond either end of the time series. In the time domain, a i nite signal
corresponds to an ini nite signal multiplied by a rectangular window that has
a value of one within the limits of the signal and a value of zero elsewhere. In
the frequency domain, the multiplication of the time series by this window
is equivalent to a convolution of the power spectrum of the signal with the
spectrum of the rectangular window (see Section 6.4 for a dei nition of
convolution). h e spectrum of the window, however, is a sin( x )/ x function,
which has a main lobe and numerous side lobes on either side of the main
peak, and hence all maxima in a power spectrum leak , i.e., they lose power
on either side of the peaks (Fig. 5.4).
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