Hale's constant-offset dip moveout

Hale [1983] found a Fourier representation of dip moveout. Refer to the defining equations in Table 1.

 

$$t \rightarrow t_n t\ =\ \sqrt{t_n^2 + 4h^2 v^{-2}}$$ NMO

$$t \rightarrow t_0 t\ =\ \sqrt{t_0^2 + 4h^2 v^{-2}\cos^2\alpha}$$ Levin's NMO

$$t_n \rightarrow t_0 t_n\ =\ \sqrt{t_0^2 - 4h^2 v^{-2}\sin^2\alpha}$$ DMO

To use the dip-dependent equations in Table 7.1 it is necessary to know the earth dip $\alpha$.The dip can be measured from a zero-offset section. On the zero-offset section in Fourier space, the sine of the dip is $v k_y / 2 \omega$.To stress that this measurement applies only on the zero -offset section, we shall always write $\omega_0$. 

$$\sin \, \alpha \eq { v \, k_y \over 2 \, \omega_0 }$$ (9)

In the absence of dip, NMO should convert any trace into a replica of the zero-offset trace. Likewise, in the presence of dip, the combination of NMO and DMO should convert any constant-offset section to a zero-offset section. Pseudo-zero-offset sections manufactured in this way from constant-offset sections will be denoted by p₀ (t₀ , h, y). First take the midpoint coordinate y over to its Fourier dual k_y. Then take the Fourier transform over time letting $\omega_0$ be Fourier dual to t₀.  

$$P_0 ( \omega_0 , h , k_y ) \eq \int \ dt_0 \ e^{ i \omega_0 t_0 } \ P_0 ( t_0 , h , k_y )$$ (10)

Change the variable of integration from t₀ to t_n.  

$$P_0 ( \omega_0 , h , k_y ) \eq \int \ d t_n \ {\ d t_0 \ove... ...\ e^{ i \omega_0 t_0 ( t_n ) } \ P_0 ( t_0 ( t_n ) , h , k_y )$$ (11)

Express the integrand in terms of NMOed data P_n. This is done by means of $P_n ( t_n , h , k_y )\ =$P₀ ( t₀ ( t_n ) , h , k_y ).  

$$P_0 ( \omega_0 , h , k_y ) \eq \int \ d t_n \ {\ d t_0 \ove... ...t_n } \ \ e^{ i \omega_0 t_0 ( t_n ) } \ P_n ( t_n , h , k_y )$$ (12)

As with Stolt migration, the Jacobian of the transformation, dt₀ / d t_n scales things but doesn't do time shifts. The DMO is really done by the exponential term.

Omitting the Jacobian (which does little), the over-all process may be envisioned with the program outline:

P(ky) =FT[P(y)] 
Pn(tn) = NMO[P(t)] 
for all ky  {   # three nested loops,              interchangeable   
for all h  {   # three nested loops,              interchangeable   
for all $\omega_0$  {   # three nested              loops, interchangeable
		 sum = 0
		 for all tn  {
		 		 sum = sum + $\exp \left( i\omega_0 \ 
\sqrt{ t_n^2 + {h^2 k_y^2\over \omega_0^2} } \ 
 \right) P_n(t_n, h, k_y)$ 
		 		 }
		 $P_0(\omega_0, h, k_y)$ = sum
		 } } }  
p0(t0, h, y) =FT2D$[P_0(\omega_0, h, k_y)]$

Notice that the exponential in the inner loop in the program does not depend on velocity. The velocity in the DMO equation in Table 7.1 disappears on substitution of $\sin\,\alpha$ from equation (9). So dip moveout does not depend on velocity. An example of processing with dip moveout is in Figure 16.

 

dmoproc
Figure 16 Processing with dip moveout. (Hale)

The procedure outlined above requires NMO before DMO. To reverse the order would be an approximation. This is unfortunate because we would prefer to do the costly, velocity-independent DMO step once, before the iterative, velocity-estimating NMO step.