Geology Reference
In-Depth Information
Components of a Fourier pair are interchangeable,
such that, if
G
(
f
) is the Fourier transform of
g
(
t
), then
g
(
t
) is the Fourier transform of
G
(
f
). Figure 2.8 illus-
trates Fourier pairs for various waveforms of geophysical
significance. All the examples illustrated have
zero phase
spectra
; that is, the individual sine wave components of
the waveforms are in phase at zero time. In this case
f
(
f
)
= 0 for all values of
f
. Figure 2.8(a) shows a spike func-
tion (also known as a
Dirac function
), which is the shortest
possible transient waveform. Fourier transformation
shows that the spike function has a continuous frequency
spectrum of constant amplitude from zero to infinity;
thus, a spike function contains all frequencies from zero
to infinity at equal amplitude.The 'DC bias'waveform of
Fig. 2.8(b) has, as would be expected, a line spectrum
comprising a single component at zero frequency. Note
that Fig. 2.8(a) and (b) demonstrate the principle of
interchangeability of Fourier pairs stated above (equa-
tion (2.4)). Figures 2.8(c) and (d) illustrate transient
waveforms approximating the shape of seismic pulses,
together with their amplitude spectra. Both have a band-
limited amplitude spectrum, the spectrum of narrower
bandwidth being associated with the longer transient
waveform. In general, the shorter a time pulse the wider
is its frequency bandwidth and in the limiting case a spike
pulse has an infinite bandwidth.
Waveforms with zero phase spectra such as those illus-
trated in Fig. 2.8 are symmetrical about the time axis
and, for any given amplitude spectrum, produce the
maximum peak amplitude in the resultant waveform. If
phase varies linearly with frequency, the waveform re-
mains unchanged in shape but is displaced in time; if the
phase variation with frequency is non-linear the shape of
the waveform is altered. A particularly important case in
seismic data processing is the phase spectrum associated
with
minimum delay
in which there is a maximum con-
centration of energy at the front end of the waveform.
Analysis of seismic pulses sometimes assumes that they
exhibit minimum delay (see Chapter 4).
Fourier transformation of digitized waveforms is
readily programmed for computers, using a '
fast Fourier
transform
' (FFT) algorithm as in the Cooley-Tukey
method (Brigham 1974). FFT subroutines can thus be
routinely built into data processing programs in order to
carry out spectral analysis of geophysical waveforms.
Fourier transformation is supplied as a function to
standard spreadsheets such as Microsoft Excel. Fourier
transformation can be extended into two dimensions
(Rayner 1971), and can thus be applied to areal distribu-
tions of data such as gravity and magnetic contour maps.
Frequency
Frequency
Fig. 2.7
Digital representation of the continuous amplitude and
phase spectra associated with a transient waveform.
thin frequency slices, with each slice having a frequency
equal to the mean frequency of the slice and an ampli-
tude and phase proportional to the area of the slice of the
appropriate spectrum (Fig. 2.7). This digital expression
of a continuous spectrum in terms of a finite number of
discrete frequency components provides an approximate
representation in the frequency domain of a transient
waveform in the time domain. Increasing the sampling
frequency in the time domain not only improves the
time-domain representation of the waveform, but also
increases the number of frequency slices in the frequen-
cy domain and improves the accuracy of the approxima-
tion here too.
Fourier transformation
may be used to convert a time
function
g
(
t
) into its equivalent amplitude and phase
spectra
A
(
f
) and
f
(
f
), or into a complex function of
frequency
G
(
f
) known as the
frequency spectrum
, where
Gf
)
=
Af
e
i
f
(
f
)
(
(
)
(2.3)
The time- and frequency-domain representations of a
waveform,
g
(
t
) and
G
(
f
), are known as a
Fourier pair
,
represented by the notation
gt
()
ยด
G f
()
(2.4)
