Page 247 - Acquisition and Processing of Marine Seismic Data

238 4. FUNDAMENTALS OF DATA PROCESSING

The digitized signal is multiplied by δ(t nΔt), spectrum line up at n/Δt intervals. Because
and multiplication in the time domain corre- the widening in the frequency domain causes
sponds to their amplitude spectra being con- narrowing in the time domain, Δf becomes smal-
volved in the frequency domain. The amplitude ler as Δt gets larger. Therefore, high- and low-
spectrum of a comb function is also a comb func- frequency components of the periodic spectrum
tion sampled at Δf intervals (Fig. 4.25). When the superimpose (Fig. 4.25). Consequently, if we take
observed signal f(t) is digitized, this results in its 1
amplitude spectrum F(ω) being convolved with a Δt > (4.38)
2f N
comb function. As a consequence, F(ω)becomes
periodic along the frequency axis with a period then aliasing occurs in the digitized time series.
of Δf, and periodic recurrences of the discrete As a general rule, if the time sampling interval

FIG. 4.25 Schematic illustration of sampling and aliasing in time (left) and frequency (right) domains. (A) Analog function,
(B) comb function, (C) dense sampling, (D) sampling at Nyquist frequency (f N ), and (E) sparse sampling, which results in
frequency aliasing (red areas). * denotes convolution, Δt and Δf are time and frequency sampling intervals, respectively.