Difference between revisions of "Time-domain operations"

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Consider a reflectivity sequence represented by the time series (1, 0, {{sfrac|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.
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Consider a reflectivity sequence represented by the time series (1, 0, {{sfrac|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 of the velocity-stack operator|''impulse response'']]. This physical process can be described as in Table 1-1.
 
 
[[file:ch01_fig1-15.png|thumb|{{figure number|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).]]
 
  
 
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|0 ||1 ||0 ||{{sfrac|1|2}}||1 ||0 ||1 ||0 ||{{sfrac|1|2}}||0
 
|0 ||1 ||0 ||{{sfrac|1|2}}||1 ||0 ||1 ||0 ||{{sfrac|1|2}}||0
 
|}
 
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[[file:ch01_fig1-16.png|thumb|{{figure number|1.1-16}} A portion of a seismic section with different degrees of constant phase rotations.]]
 
  
 
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One unit time later, suppose that the impulsive source generates an implosion with an amplitude of {{sfrac|- |1|2}}. This response is described in Table 1-2.
 
One unit time later, suppose that the impulsive source generates an implosion with an amplitude of {{sfrac|- |1|2}}. This response is described in Table 1-2.
  
[[file:ch01_fig1-17.png|thumb|{{figure number|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).]]
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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.
  
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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{|class="wikitable" style="text-align:center; width:300px; height:200px; "border="1"
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|+ '''Table 1-3.''' Linear superposition of the two responses described in Tables 1-1 and 1-2.
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|-
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|Time of Onset ||colspan="3"|Reflectivity Sequence ||colspan="2"|Source ||colspan="4"|Response
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|-
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|0 ||1 ||0 ||<math>\frac{1}{2}</math> ||1 ||0 ||1 ||0 ||<math>\frac{1}{2}</math> ||0
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|-
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|1 ||1 ||0 ||<math>\frac{1}{2}</math> ||0 ||<math>-\frac{1}{2}</math> ||0 ||<math>-\frac{1}{2}</math> ||0 ||<math>-\frac{1}{4}</math>
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|-
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|colspan="4"|Superposition: ||1 ||<math>-\frac{1}{2}</math> ||1 ||<math>-\frac{1}{2}</math> ||<math>\frac{1}{2}</math> ||<math>-\frac{1}{4}</math>
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|-
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|colspan="10"|<math>\text{Expressed}\ \text{differently}:\ \left( 1,\ 0,\ \frac{1}{2} \right)*\left( 1,\ -\frac{1}{2} \right)=\left( 1,\ -\frac{1}{2},\ \frac{1}{2},\ -\frac{1}{4} \right)</math>
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|}
  
 
==See also==
 
==See also==

Latest revision as of 13:49, 17 September 2014

Seismic Data Analysis
Seismic-data-analysis.jpg
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.

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
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.

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.

Table 1-3. Linear superposition of the two responses described in Tables 1-1 and 1-2.
Time of Onset Reflectivity Sequence Source Response
0 1 0 1 0 1 0 0
1 1 0 0 0 0
Superposition: 1 1

See also

External links

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Time-domain operations
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