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Dip filtering

Dip filtering is conveniently incorporated into the wave extrapolation equations. Instead of initializing the Muir expansion with $i k_z = -i \omega r_0$ we use $i k_z =\epsilon-i \omega r_0$.(Recall from chapter [*] that r0 is the cosine of an exactly fitting angle). For the 15$^\circ$ equation we have  
 \begin{displaymath}
i\,k_z^{ (15) }\,v \eq -\,i\, \omega \ +\
{v^2 \, k_x^2 
\over \epsilon\ -\ i\, \omega \, ( r_0 \ +\ 1 ) }\end{displaymath} (6)
For the 45$^\circ$ equation we have  
 \begin{displaymath}
i\,k_z^{ (45) }\,v \eq -\,i\, \omega \ +\
{v^2 \,k_x^2 \over...
 ...2 \, k_x^2 
\over \epsilon\ -\ i\, \omega \, ( r_0 \ +\ 1 ) }
}\end{displaymath} (7)

Figures 5 and 6 show hyperbolas of diffraction for the 15$^\circ$ and 45$^\circ$ equations with and without the dip filtering parameter $\epsilon$.

 
hyp15
hyp15
Figure 5
Diffraction hyperbolas of the 15$^\circ$ equation without dip filtering (left), and with dip filtering (right).


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hyp45
hyp45
Figure 6
Diffraction hyperbolas of the 45$^\circ$ equation without dip filtering (left), and with dip filtering (right).


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previous up next print clean
Next: Gain control does dip Up: COSMETIC ASPECT OF WAVE Previous: Accentuating faults
Stanford Exploration Project
10/31/1997