Next: ANISOTROPY DISPERSION IN MIGRATION Up: COSMETIC ASPECT OF WAVE Previous: Exponential scaling

## The substitution operator

The operator has been defined as the substitution .The main property of this operator is that if C=AB, then C=(A)(B). This property would be shared by any algebraic substitution for Z, not just the one for exponential gain. Another simple substitution can be used to achieve time-axis stretching or compression. For example, replacing Z by Z2 stretches the time axis by two. Yet another substitution, which has a deeper meaning than either of the previous two, is the substitution of the constant Q dissipation operator .In summary:

 2|c|Substitutions for Z-Transform Variable Z 2|c| 2|c|[all preserve C(Z)=A(Z)B(Z)] 2|c| Exponential growth () Time expansion () (Inverse) Constant Q dissipation

## EXERCISES:

1. Use a table of integrals to show that a seismic source with spectrum implies a divergence correction .
2. Assuming that t2 is a suitable divergence correction for field profiles, what divergence correction should be applied to CDP stacks?
3. How is the t2 correction altered for water of travel time depth t0? Assume the Q of water is infinite.
4. Consider a source spectrum .How is the t2 correction altered?
5. The spectrum in Figure 3 shows high frequencies smoother than low frequencies. Explain.
6. Propose some criteria that can be used in the selection of the cutoff parameters and for the filter (8).

Next: ANISOTROPY DISPERSION IN MIGRATION Up: COSMETIC ASPECT OF WAVE Previous: Exponential scaling
Stanford Exploration Project
10/31/1997