Stopping phase-shift migration wraparound (S. Levin)

Figure 4 shows a family of hyperbolas.

 

hyptrunc Figure 4 Hyperbolas truncated at a particular time.

Notice that these hyperbolas do not extend to infinite time but they truncate at a cut-off time t_c. A Fourier method will be described to create such time-truncated data. The method leads to a phase-shift migration program without wraparound artifacts.

When the fast Fourier transformation algorithm first came into use people noticed that it could be used for filtering. Transient filtering could be done exactly in the periodic Fourier domain if signals and filters were surrounded by enough zero padding. The same concept applies with migration. If field data and migration hyperbolas are surrounded by enough zeroes in the time- and space-domain then migration can be done in the Fourier domain with no wraparound. The trick is to see how the truncated hyperbolas in Figure 4 can be constructed in the Fourier domain.

To have truncations at time t_c, special point sources must be used. The deeper the source, the narrower must be its angular aperture. Take a hyperbola with first arrival at time t₀ to be truncated at some time t_c. The propagation angle $\theta$ of energy at the cutoff is given by $\cos \,\theta = t_0 / t_c$.So exploding reflectors have their k_x-spectrum truncated at $\sin \theta = v k_x \omega$.A 90$^\circ$ aperture implies echoes with an infinite time delay. Here is a sketch of the program.

# Modeling with time truncation at tc 
Model$(k_x,z)\ =\$FT[model(x,z)] 
For all $\omega$ and all kx 
		 $U(\omega,k_x)\ =\ 0$.     
For $z\ =\ z_{\rm max},\ z_{\rm max} -\triangle z,\$ ..., 0 {
		 For all $\omega$ {
		 		 For all $\mid k_x\mid < \mid \omega\mid /v$ {
		 		 		 if ($z\ < v\ t_c$) {
		 		 		 		 sine$\ =\ \sqrt{1 - z^2 /v^2 t_c^2}$ 
		 		 		 		 if( $\mid v\ k_x\mid\ < \ \mid\omega\mid\$sine  )
		 		 		 		 		 aperture$\ =\ 1$.		 		 		 		 else
		 		 		 		 		 aperture$\ =\ 0$.		 		 		 		 }
		 		 		 else
		 		 		 		 aperture$\ =\ 0$.		 		 		$U(\omega,k_x)=U(\omega,k_x)e^{-i\Delta z\omega\sqrt{v^{-2}-k_x^2/\omega^2}}+\$aperture$\times$Model(kx,z) 
		 		 		 }
		 		 }
		 }  
$u(t, x)\ =\$FT2D$[U(\omega, k_x)]$

The above modeling program may be converted to a migration program by running the depth z loop down instead of up and by multiplying the downward continued data by the aperture function. The modifications to the program not only improve the quality of the migration, but the calculation is faster.