Convolution

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

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 c_k is given by m+n−1, where m and n are the lengths of the operand array a_i and the operator array b_j, 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		${\frac {1}{2}}$
			${\frac {1}{2}}$	0	1	$-{\frac {1}{4}}$

See also

• Dictionary:convolution theorem
• Analog versus digital signal
• Frequency aliasing
• Phase considerations
• Time-domain operations
• Crosscorrelation and autocorrelation
• Vibroseis correlation
• Frequency filtering
• Practical aspects of frequency filtering
• Bandwidth and vertical resolution
• Time-variant filtering

External links

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