# Time-domain operations

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

Consider a reflectivity sequence represented by the time series (1, 0, 12). Also consider an impulsive source that causes an explosion at t = 0 with an amplitude of 1. The response of the reflectivity sequence to an impulse is called the impulse response. This physical process can be described as in Table 1-1.

 Time of Onset Reflectivity Sequence Source Response 0 1 0 12 1 0 1 0 12 0
 Time of Onset Reflectivity Sequence Source Response 1 1 0 12 0 - 12 0 - 12 0 - 14

One unit time later, suppose that the impulsive source generates an implosion with an amplitude of - 12. This response is described in Table 1-2.

Note that the response in each case is the reflectivity sequence scaled by the impulse strength and delayed by the impulse onset. Since a general source function is considered to be a sequence of explosive and implosive impulses, the individual impulse responses are added to obtain the combined response. This process is called linear superposition and is described in Table 1-3.

 Time of Onset Reflectivity Sequence Source Response 0 1 0 ${\displaystyle {\frac {1}{2}}}$ 1 0 1 0 ${\displaystyle {\frac {1}{2}}}$ 0 1 1 0 ${\displaystyle {\frac {1}{2}}}$ 0 ${\displaystyle -{\frac {1}{2}}}$ 0 ${\displaystyle -{\frac {1}{2}}}$ 0 ${\displaystyle -{\frac {1}{4}}}$ Superposition: 1 ${\displaystyle -{\frac {1}{2}}}$ 1 ${\displaystyle -{\frac {1}{2}}}$ ${\displaystyle {\frac {1}{2}}}$ ${\displaystyle -{\frac {1}{4}}}$ ${\displaystyle {\text{Expressed}}\ {\text{differently}}:\ \left(1,\ 0,\ {\frac {1}{2}}\right)*\left(1,\ -{\frac {1}{2}}\right)=\left(1,\ -{\frac {1}{2}},\ {\frac {1}{2}},\ -{\frac {1}{4}}\right)}$