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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 timeaxis
stretching or compression.
For example, replacing Z by Z^{2} 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:
2c 

2cSubstitutions for ZTransform
Variable Z 

2c[all preserve C(Z)=A(Z)B(Z)] 

2c 



Exponential growth 


() 




Time expansion () 





(Inverse) Constant Q dissipation 



EXERCISES:

Use a table of integrals to show that a seismic source
with spectrum implies a divergence
correction .

Assuming that t^{2} is a suitable divergence correction
for field profiles,
what divergence correction should be applied to CDP stacks?

How is the t^{2} correction altered for water of travel
time depth t_{0}?
Assume the Q of water is infinite.

Consider a source spectrum .How is the t^{2} correction altered?

The spectrum in Figure 3 shows high frequencies smoother
than low frequencies.
Explain.

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