Presentation of the 3-D case

In the 3-D case, the wave equation corresponding to the ray-theoretic interpretation of all-angle time migration is  

$$\frac{\partial^2 p'}{\partial x'^2} \: + \: \frac{\partial^... ...\zeta)} \, \frac{\partial^2 p'}{\partial t' \, \partial \zeta}$$ (11)

in the 3-D time-retarded coordinate system  

$$\begin{array} {lcl} x' & = & x \\ y' & = & y \\ t' & = & t + \zeta\end{array}$$ (12)

with the new time-like vertical coordinate $\zeta$, 

$$\zeta \; = \; \int_{0}^{z} \: \frac{dz}{V \, (x,y,z)}$$ (13)

Following the same scheme as for the 2-D case and using equations (12) and (13), we finally obtain the 3-D Eikonal equation for time migration:  

$$\left( \frac{\partial \tau}{\partial x} \: + \: {\cal A} \,... ...l \tau}{\partial z} \right)^2 \; = \; \frac{1}{V^2 \, (x,y,z)}$$ (14)

with

$${\cal A} \, (x,y,z) \; = \; - \, V \, (x,y,z) \, \frac{\partial \zeta}{\partial x}$$ (15)

and

(16)