Amplitude correction in variable velocity

The results presented in the preceding subsection are strictly valid in constant velocity. When velocity varies, the propagation operator ($\bold L$) is not only a constant magnitude phase-shift multiplication, but also includes an amplitude term that changes with depth and horizontal location. Therefore, the integrals in Equation (5) cannot be interpreted as Fourier transforms anymore. Clayton and Stolt (1981) show how this issue can be theoretically side-stepped by datuming the data just above each reflector, and by approximating the velocity as constant in the imaging interval. In practice, this issue is taken care of by evaluating the Jacobian and the reflection operator G, using the local velocity at the imaging location.

When the continuation equation includes the amplitude term ($\bold \AA$), Equation (7) becomes  

$$\bold d= \bold L\bold \AA\bold i= \bold L\bold \AA\bold G\bold r.$$ (9)

Since conventional migration is the adjoint of modeling, Equation (8) becomes  

$$\widehat{\bold i}= \bold \AA^{*} \bold L^{*} \bold d= \bol... ...\bold \AA\bold i= \bold \AA^{*} \bold W\bold \AA\bold G\bold r,$$ (10)

In layered media, the frequency-wavenumber amplitude term of the continuation operator can be computed using the WKBJ approximation as a diagonal real operator Clayton and Stolt (1981):  

$$\AA=\sqrt{ \frac{{k_{\rm zs}}\left (z \right )}{{k_{\rm zs}}... ..._{\rm zr}}\left (z \right )}{{k_{\rm zr}}\left (z=0 \right )}},$$ (11)

where ${k_{\rm zs}}$ and ${k_{\rm zr}}$ are the depth wavenumbers associated with source and receiver.