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  

$$i\,k_z^{ (15) }\,v \eq -\,i\, \omega \ +\ {v^2 \, k_x^2 \over \epsilon\ -\ i\, \omega \, ( r_0 \ +\ 1 ) }$$ (6)

For the 45$^\circ$ equation we have  

$$i\,k_z^{ (45) }\,v \eq -\,i\, \omega \ +\ {v^2 \,k_x^2 \over... ...2 \, k_x^2 \over \epsilon\ -\ i\, \omega \, ( r_0 \ +\ 1 ) } }$$ (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
Figure 5 Diffraction hyperbolas of the 15$^\circ$ equation without dip filtering (left), and with dip filtering (right).

 

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