# Convolution

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 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 ${\displaystyle {\text{Expressed}}\ {\text{differently}}:\ \left(1,\ 0,\ {\frac {1}{2}}\right)*\left(1,\ -{\frac {1}{2}}\right)=\left(1,\ -{\frac {1}{2}},\ {\frac {1}{2}},\ -{\frac {1}{4}}\right)}$
 Reflectivity Sequence Output Response 1 0 ${\displaystyle {\frac {1}{2}}}$ ${\displaystyle -{\frac {1}{2}}}$ 1 1 ${\displaystyle -{\frac {1}{2}}}$ 1 ${\displaystyle -{\frac {1}{2}}}$ ${\displaystyle -{\frac {1}{2}}}$ 1 ${\displaystyle {\frac {1}{2}}}$ ${\displaystyle -{\frac {1}{2}}}$ 1 ${\displaystyle -{\frac {1}{4}}}$

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.

 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 ${\displaystyle {c_{k}}=\sum \limits _{j=0}^{n}{{a_{k-j}}\;{b_{j}},}\quad k=0,\;1,\;2,\;\cdots \;,m\;+\;n\;-\;1.}$
 Source Wavelet Output Response 1 ${\displaystyle -{\frac {1}{2}}}$ ${\displaystyle {\frac {1}{2}}}$ 0 1 1 ${\displaystyle {\frac {1}{2}}}$ 0 1 ${\displaystyle -{\frac {1}{2}}}$ ${\displaystyle {\frac {1}{2}}}$ 0 1 ${\displaystyle {\frac {1}{2}}}$ ${\displaystyle {\frac {1}{2}}}$ 0 1 ${\displaystyle -{\frac {1}{4}}}$