Page 267 - Acquisition and Processing of Marine Seismic Data

258 5. PREPROCESSING

FIG. 5.16 (A) A box-car shaped amplitude spectrum in the frequency domain, and (B) its time domain counterpart, the sinc
function.

FIG. 5.17 (A) A truncated sinc function (upper panel) and its real amplitude spectrum (lower panel). (B) If both flanks of the
pass-band region are inclined (indicated by S 1 and S 2 ), and (C) the edges indicated by I, II, III, and IV are smoothed, a narrower
filter operator can be obtained in the time domain, and the Gibbs effects are eliminated in the frequency domain.

car using a finite number of coefficients in the makes the operator much narrower in the time
time domain (Fig. 5.17A). These oscillations domain with fewer nonzero filter coefficients.
are normally undesired, since some of the ampli- These inclined parts in the pass-band region
tudes are weakened while some others are are known as the transition band. In practice,
strengthened during the filtering. Furthermore, the slope at the higher frequency side of the trap-
these oscillations also result in passing of ampli- ezoid is suggested to be larger than that at the
tudes in the reject band. In order to avoid the lower frequency side. In addition, since the Fou-
Gibbs effect in the frequency domain, instead rier transform is valid only for continuous func-
of defining a rectangular pass-band region, both tions, a smoothing is applied to the edge points I,
flanks of the box-car are inclined (Fig. 5.17B) so II, III, and IV in Fig. 5.17C in order to avoid sharp
that the pass-band becomes a trapezoid, which edges in the spectrum.