The substitution operator

The $\uparrow$ operator has been defined as the substitution $Z \ \to\ Z\, e^{\alpha}$.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 $( -i \omega )^{\gamma}$.In summary:

$$Z\rightarrow Z_e^\alpha$$ Exponential growth

$$i\omega\rightarrow i\omega+\alpha$$

$$\alpha \gt 1 Z\rightarrow Z^\alpha$$

$$-i\omega\rightarrow (-i\omega)^\gamma$$ (Inverse) Constant Q dissipation

EXERCISES:

1. Use a table of integrals to show that a seismic source with spectrum $\vert \omega \vert^{\beta}$ implies a divergence correction $t^{{2+} \beta}$.
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 $e^{ - \beta \vert \omega \vert }$.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 $\alpha$ and $\beta$ for the filter (8).