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Splitting again

The customary numerical solution to the x-domain forms of the equations in Tables [*].4 and [*].5 is arrived at by splitting. That is, you march forward a small $\Delta z$-step alternately with the two extrapolators
      \begin{eqnarray}
{ \partial Q \over \partial z } \ \ \ &=&\ \ \ \ \rm{lens\ term...
 ...rtial Q \over \partial z } \ \ \ &=&\ \ \ \ \rm{diffraction\ term}\end{eqnarray} (71)
(72)
The first equation, called the lens equation, is solved analytically:  
 \begin{displaymath}
Q ( z_2 ) \eq Q ( z_1 ) \ \exp \,
\left\{ \ i \omega \ \int_...
 ...r v(x,z)} \ -\ 
{1 \over \bar v ( z ) } \ \right) dz \ \right\}\end{displaymath} (73)

Migration that includes the lens equation is called depth migration. Migration that omits this term is called time migration.

Observe that the diffraction parts of Tables [*].4 and [*].5 are the same. Let us use them and equation ([*]) to define a table of diffraction equations. Substitute $ \partial / \partial x $ for i kx and clear $ \partial / \partial x $ from the denominators to get Table [*].6.

 
Table: Diffraction equations for laterally variable media.
   
$5^\circ$ $\displaystyle {\strut\partial Q\over
 \partial z} \eq $zero
   
   
$15^\circ$ $ \displaystyle {\strut\partial Q\over
 \partial z} \eq 
 \displaystyle {v(x, z)\over - 2i\omega}
 {\strut\partial^2 Q\over\partial x^2}$
   
   
$45^\circ$ $\left[ 1 - \left( \displaystyle 
 {\strut v(x, z)\over - 2i\omega}\right) ^2
\d...
 ...\displaystyle {v(x, z)\over - 2i\omega} 
 {\strut\partial^2 Q\over\partial x^2}$
   

You may wonder where the two velocities v(x,z) and $\bar v(z)$ came from. The first arises in the wave equation, and it must be x-variable if the model is x-variable. The second arises in a mathematical transformation, namely, equation ([*]), so it is purely a matter of definition.
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Next: Time domain Up: END OF CHAPTER FOR Previous: Lateral velocity variation again
Stanford Exploration Project
12/26/2000