Page 229 - Acquisition and Processing of Marine Seismic Data

220 4. FUNDAMENTALS OF DATA PROCESSING

TABLE 4.3 Mathematical Properties of Fourier Because the real seismic traces are composed
Transform, Where * Denotes Convolution, a and b are of amplitudes at all frequency components,
Scalars, n is an Integer, and Y*(ω) is Complex Conjugate amplitude spectra of actual seismic traces are
of Y(ω) continuous and contain a combination of several
successive amplitudes which exist at every fre-
Property Mathematical Expression
quency values.
Linearity af(t)+ bg(t) $ aF(ω)+ bG(ω)
Fig. 4.7 graphically shows a Fourier analysis
1
ω
Scaling fatðÞ $ F
of an input time signal. The signal, indicated
a jj a
1 f
$ FaωÞ by (*), is expanded into Fourier series to obtain
t
ð
a jj a
several sinusoids with different phase and
Time shift f(t a) $ e iωa F(ω)
amplitudes. The analysis in Fig. 4.7 is obtained
Frequency shift e ita f(t) $ F(ω + a) by plotting these sinusoids side-by-side on a
horizontal axis of their corresponding frequen-
Symmetry f(ω) $ F(t)
cies in increasing order. Hence, the horizontal
Time reversal f( t) $ F( ω)
axis of the plot is frequency of the sinusoids.
n
Differentiation (nth df tðÞ n The signal marked by (*) can be reconstructed
dt n $ iωðÞ F ωðÞ
order) n d F ωðÞ if all these sinusoids are simply summed up. It
n
ð itÞ ftðÞ $
dω n
can be observed that amplitudes of the sinusoids
Integration R ∞ F ωðÞ
ftðÞdt $ iω with approximately 40, 80 and 150 Hz frequency
∞
R ω (marked with A, B, and C) are relatively higher
ftðÞ F ωðÞdω
$
it than the others.
ω
Fig. 4.8 shows the same information in spec-
Convolution f(t) g(t) $ F(ω)*G(ω) trum display form. Considering the analysis in
f(t)*g(t) $ F(ω) G(ω)
Fig. 4.7, the amplitude spectrum of (*) signal
Crosscorrelation R xy (τ) $ X(ω) Y*(ω) can be obtained by plotting the amplitudes of
Autocorrelation R xx (τ) $ X(ω) each individual sinusoids with respect to their
2
X*(ω) ¼jX(ω)j ¼ R xx (ω) corresponding frequencies (Fig. 4.8B), whereas
the phase spectrum of the (*) signal is achieved
by plotting the phase angles (Fig. 4.8C) as a func-
tion of frequency of the sinusoids. Every single
samples exactly equals to an order of the closest point on the amplitude spectrum corresponds
n
2 . This process is termed zero padding. to the amplitude of that component shown in
If we compute Fourier transform of a mono- Fig. 4.7, and each point on the phase spectrum
chromatic sinusoid, we get only one amplitude indicates the phase of the component at that fre-
value in the amplitude spectrum appeared quency. Amplitude values in the amplitude
exactly at the frequency value of the input spectrum in Fig. 4.8B are also higher at the spe-
mono-frequency sinusoid. Fig. 4.6A and B show cific frequency values of approximately 40,
amplitude spectra of two sinusoids with 40 and 80 and 150 Hz (around A, B, and C).
100 Hz frequencies, respectively. Their ampli- As a specific case, if the peak amplitudes of all
tude spectra consist of only two amplitude sinusoids constituting the time signal are
values at 40 and 100 Hz, respectively. Similarly, aligned to time zero (t ¼ 0), the signal has a zero
the amplitude spectrum of a time signal phase spectrum, and is termed a zero phase sig-
obtained by the superposition of two sinusoids nal. In Fourier analysis of a zero phase signal,
with 40 and 80 Hz frequency consists of two suc- amplitudes of the harmonics are constructively
cessive amplitudes at 40 and 80 Hz (Fig. 4.6C). summed up and a maximal peak amplitude