Page 228 - Acquisition and Processing of Marine Seismic Data

4.5 1D FOURIER TRANSFORM 219

f(t) function is (Fig. 4.5). However, n should be can be reconstructed by inverse Fourier trans-
infinite for an exact approximation. form by
Z ∞
1
iωt
1 F ωðÞe dω (4.12)
4.5 1D FOURIER TRANSFORM ℑ f F ωðÞg ¼ ftðÞ ¼ 2π
∞
Although the periodic functions can be Here, f(t) and F(ω) are termed Fourier transform
approximated by Fourier series expansion, pair and represented by f(t) $ F(ω). Fourier
non-periodic functions cannot be expressed as transform F(ω) is a complex function of fre-
the summation of sine and cosine functions quency and expressed as
using Fourier series. In that case, non-periodic
functions are regarded as periodic that their F ωðÞ ¼ a ωðÞ ib ωðÞ (4.13)
period is infinitely large. When the period is infi- or
nite, then the frequency becomes infinitely
small, and the frequency components of such F ωðÞ ¼ F ωðÞje iϕωðÞ (4.14)
j
functions can be obtained directly by Fourier where a(ω) and b(ω) are real and imaginary com-
transform using Fourier integral instead of Fou- ponents, and jF(ω)j and ϕ(ω) are amplitude and
rier series expansion. phase spectra, respectively, which can be
The spectrum of a time series of a sequence of obtained by
amplitudes as a function of time is obtained by
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
calculating the amplitude and phase spectra. 2 2
j F ωðÞj ¼ a ωðÞ + b ωðÞ
The basis of taking the spectrum of a time func- (4.15)
b ωðÞ
tion is to find out the amplitudes and phase ϕωðÞ ¼ atan
a ωðÞ
values of the sinusoids which constitute the
original function when summed up. The ampli- Fourier transform is the mathematical process
tudes of these sinusoids as a function of their which allows us to analyze frequency contents
corresponding frequency forms the amplitude of the time series represented by a series of
spectrum, while plotting the phase values as a amplitude values as a function of time in the
function of frequency constitutes the phase spec- time domain as like seismic traces. Some of the
trum. In amplitude spectrum, amplitudes of the mathematical properties of Fourier transform
sinusoids are lined up with respect to their cor- are given in Table 4.3.
responding frequency. Square of the amplitude The seismic traces typically consist of billions
spectrum denotes power spectrum which equals of discrete time samples on millions of traces,
the Fourier transform of autocorrelation which make the Fourier transform computation-
function. ally expensive because it takes excessive time for
Fourier transform F(ω) of a non-periodic f(t) huge data volumes such as seismic datasets.
time signal is analytically obtained by Fourier Today, discrete Fourier transform is computed
integral as much faster by using a specific algorithm known
∞ as fast Fourier transform (FFT). FFT algorithm
Z works only for time series whose number of time
f
ℑ ftðÞg ¼ F ωðÞ ¼ ftðÞe iωt dt (4.11) n
samples are on the order of 2 (where n is an
∞ integer). If the input time series do not exactly
p ffiffiffiffiffiffiffi n
where i ¼ 1 and ω ¼ 2πf is angular frequency. contain 2 discrete amplitude samples, then
Fourier transform is reversible, and for a given the data is extended by appending additional
Fourier transform F(ω), original time signal f(t) zeroes at the end of the series so that it has time