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 |
$ {\frac {1}{2}} $ |
1 |
0 |
1 |
0 |
$ {\frac {1}{2}} $ |
0
|
| 1 |
1 |
0 |
$ {\frac {1}{2}} $ |
0 |
$ -{\frac {1}{2}} $ |
0 |
$ -{\frac {1}{2}} $ |
0 |
$ -{\frac {1}{4}} $
|
| Superposition: |
1 |
$ -{\frac {1}{2}} $ |
1 |
$ -{\frac {1}{2}} $ |
$ {\frac {1}{2}} $ |
$ -{\frac {1}{4}} $
|
| $ {\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) $
|
See also
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Time-domain operations
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