Depth-variable velocity

Identification of i k_z with $\partial / \partial z$ converts the dispersion relations of Table

.2 into the differential equations of Table

.3.

**Table 3:** Extrapolation equations when velocity depends only on depth.

$5^\circ$	$\displaystyle {\strut \partial P\over \partial z} \eq i\left( \displaystyle {\strut\omega\over v} \right) P$


$15^\circ$	$\displaystyle {\strut \partial P\over \partial z} \eq i \left( \displaystyle {\omega\over v} - {\strut v k_x^2\over 2\omega} \right) P$


$45^\circ$	$\displaystyle {\strut\partial P\over \partial z} \eq i \left( \displaystyle {... ...\displaystyle {2\,{\omega\over v} - {\strut v k_x^2\over 2\omega}}} \right) P$

The differential equations in Table 4.3 were based on a dispersion relation that in turn was based on an assumption of constant velocity. So you might not expect that the equations have substantial validity or even great utility when the velocity is depth-variable, v = v(z). The actual limitations are better characterized by their inability, by themselves, to describe reflection.

Migration methods based on equation (9) or on Table

.3 are called phase-shift methods.