Page 246 - Acquisition and Processing of Marine Seismic Data

4.10 SAMPLING THEORY 237

FIG. 4.23 Sinusoid with a 100-Hz frequency sampled at (A) 2 ms, and (B) 8 ms sampling rate (left) and their corresponding
amplitude spectra (right). Δt is the sampling interval and f N represents the Nyquist frequency.

FIG. 4.24 Digitization of a time series at regular Δt time intervals equals the multiplication of the time signal with a comb
function sampled at Δt.

Δt ¼ 8ms (f N ¼ 62.5 Hz) sampling rate, fre- onto the lower frequency part of the amplitude
quency aliasing occurs and the amplitude of the spectrum. Aliasing occurs because: when digi-
input signal appears at 25-Hz frequency tized at Δt intervals, the analog f(t) signal is
(Fig. 4.23B). multiplied by a unit-amplitude comb function δ-
Disruption of the spectrum because of the (t nΔt) in the time domain (Fig. 4.24). This
sparse sampling of a time signal is termed alias- multiplication produces a discrete time series
ing. If the time sampling interval Δt is not spec- f r , which consists of a series of amplitude values,
ified sufficiently small, the digitized signal mathematically expressed as
cannot represent the original analog signal accu-
ð
rately and high-frequency components are f r ¼ fnΔtð Þ ¼ ftðÞ δ t nΔtÞ
∞
missed. These missing components, however, X
ð
ð
¼ f n nΔtÞ δ t nΔtÞ (4.37)
do not simply disappear, but they are aliased n¼ ∞