# Convolution

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, 12) to the source wavelet (1, - 12) 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.

Time of Onset | Reflectivity Sequence | Source | Response | ||||||

0 | 1 | 0 | 12 | 1 | 0 | 1 | 0 | 12 | 0 |

1 | 1 | 0 | 12 | 0 | - 12 | 0 | - 12 | 0 | - 14 |

Superposition: | 1 | - 12 | 1 | - 12 | 12 | - 14 | |||

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 *c _{k}* is given by

*m+n*−1, where

*m*and

*n*are the lengths of the operand array

*a*and the operator array

_{i}*b*, respectively.

_{j}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*.

Fixed Array: | ||||||||||||

a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}
| ||||||||||||

Moving Array: | ||||||||||||

b_{0}, b_{1}, b_{2}
| ||||||||||||

Given two arrays, a and _{i}b:
_{j} | ||||||||||||

Step 1 : Reverse moving array b.
_{j} | ||||||||||||

Step 2 : Multiply in the vertical direction. | ||||||||||||

Step 3 : Add the products and write as output c.
_{k} | ||||||||||||

Step 4 : Shift array b one sample to the right and repeat Steps 2 and 3.
_{j} | ||||||||||||

Convolution Table: | ||||||||||||

a_{0} |
a_{1} |
a_{2} |
a_{3} |
a_{4} |
a_{5} |
a_{6} |
a_{7} |
Output | ||||

b_{2} |
b_{1} |
b_{0} |
c_{0}
| |||||||||

b_{2} |
b_{1} |
b_{0} |
c_{1}
| |||||||||

b_{2} |
b_{1} |
b_{0} |
c_{2}
| |||||||||

b_{2} |
b_{1} |
b_{0} |
c_{3}
| |||||||||

b_{2} |
b_{1} |
b_{0} |
c_{4}
| |||||||||

b_{2} |
b_{1} |
b_{0} |
c_{5}
| |||||||||

b_{2} |
b_{1} |
b_{0} |
c_{6}
| |||||||||

b_{2} |
b_{1} |
b_{0} |
c_{7}
| |||||||||

b_{2} |
b_{1} |
b_{0} |
c_{8}
| |||||||||

b_{2} |
b_{1} |
b_{0} |
c_{9}
| |||||||||

where | ||||||||||||

Source Wavelet | Output Response | |||||

1 | ||||||

0 | 1 | 1 | ||||

0 | 1 | |||||

0 | 1 | |||||

0 | 1 |

## 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