Time-domain operations

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Seismic Data Analysis
Series Investigations in Geophysics
Author Öz Yilmaz
DOI http://dx.doi.org/10.1190/1.9781560801580
ISBN ISBN 978-1-56080-094-1
Store SEG Online Store

Consider a reflectivity sequence represented by the time series (1, 0, 1/2). Also consider an impulsive source that causes an explosion at t = 0 with an amplitude of 1. The response of the reflectivity sequence to an impulse is called the impulse response. This physical process can be described as in Table 1-1.

Figure 1.1-15  Starting with the zero-phase wavelet (a), its shape is changed by applying constant phase shifts. A 90-degree phase shift converts the zero-phase wavelet to an antisymmetric wavelet (b), while a 180-degree phase shift reverses its polarity (c). A 270-degree phase shift reverses the polarity, while making the wavelet antisymmetric (d). Finally, a 360-degree phase shift does not modify the wavelet (e).
Table 1-1. Response of the reflectivity sequence (1, 0, 1/2) to a zero-delay explosive impulse (1, 0).
Time of Onset Reflectivity Sequence Source Response
0 1 0 1/2 1 0 1 0 1/2 0
Figure 1.1-16  A portion of a seismic section with different degrees of constant phase rotations.
Table 1-2. Response of the reflectivity sequence (1, 0, 1/2) to a unit-delay implosive impulse (0, -  1/2).
Time of Onset Reflectivity Sequence Source Response
1 1 0 1/2 0 -  1/2 0 -  1/2 0 -  1/4

One unit time later, suppose that the impulsive source generates an implosion with an amplitude of -  1/2. This response is described in Table 1-2.

Figure 1.1-17  A linear (as in Figure 1.1-12) combined with a constant phase shift (as in Figure 1.1-14) results in a time-shifted antisymmetric wavelet. The wavelet is represented by the trace on the right (denoted by an asterisk).

Note that the response in each case is the reflectivity sequence scaled by the impulse strength and delayed by the impulse onset. Since a general source function is considered to be a sequence of explosive and implosive impulses, the individual impulse responses are added to obtain the combined response. This process is called linear superposition and is described in Table 1-3.

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Time-domain operations
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