Splitting diffraction and lens terms

The customary numerical solution to the x-domain forms of the equations in Tables .3 and .4 is arrived at by splitting. That is, you march forward a small $\Delta z$-step alternately with the two extrapolators

$$\begin{eqnarray} { \partial Q \over \partial z } \ \ \ &=&\ \ \ \ \rm{lens\ term... ...rtial Q \over \partial z } \ \ \ &=&\ \ \ \ \rm{diffraction\ term}\end{eqnarray}$$ (18) (19)

Formal justification of the splitting process is found in chapter . The first equation, called the lens equation, is solved analytically:  

$$Q ( z_2 ) \eq Q ( z_1 ) \ \exp \, \left\{ \ i \omega \ \int_... ...r v(x,z)} \ -\ {1 \over \bar v ( z ) } \ \right) dz \ \right\}$$ (20)

Migrations that include the lens equation are called depth migrations. The term is often omitted, giving a time migration.

Observe that the diffraction parts of Tables .3 and .4 are the same. Let us use them and equation (19) to define a table of diffraction equations. Substitute $\partial / \partial x$ for i k_x and clear $\partial / \partial x$ from the denominators to get Table .5.

 

$$5^\circ {\strut\partial Q\over \partial z} \eq$$

$$15^\circ {\strut\partial Q\over \partial z} \eq \displaystyle {v(x, z)\over - 2i\omega} {\strut\partial^2 Q\over\partial x^2}$$

$$45^\circ \left[ 1 - \left( \displaystyle {\strut v(x, z)\over - 2i\omega}\right) ^2 \d... ...\displaystyle {v(x, z)\over - 2i\omega} {\strut\partial^2 Q\over\partial x^2}$$