Convolution

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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


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
Table 1-4. Convolution of the source wavelet (1, ) with the reflectivity sequence (1, 0, ).
Reflectivity Sequence Output Response
1 0
1 1
1
1
1

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
Table 1-6. Convolution of the reflectivity sequence (1, 0, ) with the source wavelet (1, ).
Source Wavelet Output Response
1
0 1 1
0 1
0 1
0 1

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Convolution
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