In Table 1-3, the asterisk denotes convolution. The response of the reflectivity sequence (1, 0, 1/2) to the source wavelet (1, - 1/2) was obtained by convolving the two series. This is done computationally as shown in Table 1-4. A fixed array is set up from the reflectivity sequence. The source wavelet is reversed (folded) and moved (lagged) one sample at a time. At each lag, the elements that align are multiplied and the resulting products are summed.
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 |
1/2 |
1 |
0 |
1 |
0 |
1/2 |
0
|
| 1 |
1 |
0 |
1/2 |
0 |
- 1/2 |
0 |
- 1/2 |
0 |
- 1/4
|
| Superposition: |
1 |
- 1/2 |
1 |
- 1/2 |
1/2 |
- 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) $
|
Table 1-4. Convolution of the source wavelet (1, $ -{\frac {1}{2}} $) with the reflectivity sequence (1, 0, $ {\frac {1}{2}} $).
| Reflectivity Sequence |
Output Response
|
|
1 |
0 |
$ {\frac {1}{2}} $ |
|
|
| $ -{\frac {1}{2}} $ |
1 |
|
|
|
1
|
|
$ -{\frac {1}{2}} $ |
1 |
|
|
$ -{\frac {1}{2}} $
|
|
|
$ -{\frac {1}{2}} $ |
1 |
|
$ {\frac {1}{2}} $
|
|
|
|
$ -{\frac {1}{2}} $ |
1 |
$ -{\frac {1}{4}} $
|
The mechanics of convolution are described in Table 1-5. The number of elements of output array ck is given by m+n−1, where m and n are the lengths of the operand array ai and the operator array bj, respectively.
When the roles of the arrays in Table 1-4 are interchanged, the output array in Table 1-6 results. Note that the output response is identical to that in Table 1-4. Hence, convolution is commutative — it does not matter which array is fixed and which is moved, the output is the same.
Table 1-5. Mechanics of the convolutional process.
| Fixed Array:
|
| a0, a1, a2, a3, a4, a5, a6, a7
|
| Moving Array:
|
| b0, b1, b2
|
| Given two arrays, ai and bj:
|
| Step 1 : Reverse moving array bj.
|
| Step 2 : Multiply in the vertical direction.
|
| Step 3 : Add the products and write as output ck.
|
| Step 4 : Shift array bj one sample to the right and repeat Steps 2 and 3.
|
| Convolution Table:
|
|
|
a0 |
a1 |
a2 |
a3 |
a4 |
a5 |
a6 |
a7 |
|
|
Output
|
| b2 |
b1 |
b0 |
|
|
|
|
|
|
|
|
|
c0
|
|
b2 |
b1 |
b0 |
|
|
|
|
|
|
|
|
c1
|
|
|
b2 |
b1 |
b0 |
|
|
|
|
|
|
|
c2
|
|
|
|
b2 |
b1 |
b0 |
|
|
|
|
|
|
c3
|
|
|
|
|
b2 |
b1 |
b0 |
|
|
|
|
|
c4
|
|
|
|
|
|
b2 |
b1 |
b0 |
|
|
|
|
c5
|
|
|
|
|
|
|
b2 |
b1 |
b0 |
|
|
|
c6
|
|
|
|
|
|
|
|
b2 |
b1 |
b0 |
|
|
c7
|
|
|
|
|
|
|
|
|
b2 |
b1 |
b0 |
|
c8
|
|
|
|
|
|
|
|
|
|
b2 |
b1 |
b0 |
c9
|
| where
|
| $ {c_{k}}=\sum \limits _{j=0}^{n}{{a_{k-j}}\;{b_{j}},}\quad k=0,\;1,\;2,\;\cdots \;,m\;+\;n\;-\;1. $
|
Table 1-6. Convolution of the reflectivity sequence (1, 0, $ {\frac {1}{2}} $) with the source wavelet (1, $ -{\frac {1}{2}} $).
| Source Wavelet |
Output Response
|
|
|
1 |
$ -{\frac {1}{2}} $ |
|
|
|
| $ {\frac {1}{2}} $ |
0 |
1 |
|
|
|
1
|
|
$ {\frac {1}{2}} $ |
0 |
1 |
|
|
$ -{\frac {1}{2}} $
|
|
|
$ {\frac {1}{2}} $ |
0 |
1 |
|
$ {\frac {1}{2}} $
|
|
|
|
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \frac{1}{2}
|
0 |
1 |
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): -\frac{1}{4}
|
See also
External links
find literature about
Convolution
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