Introducing Eikonal equation and future work

In two dimensions, considering one-way time, the wave equation is:  

$$\frac{\partial^2 p}{\partial x^2} + \frac{\partial^2 p}{\partial z^2} = \frac{1}{V^2}\frac{\partial^2 p}{\partial t^2}$$ (3)

where p(x,z,t) is the pressure wave field and V(x,z) is the velocity of the wave propagation in the medium.

In 1991, Vijay Khare from Exxon proposed to use the following formula as the wave equation corresponding to the ray-theoretic interpretation of all-angle time migration he wants to introduce Khare (1991):  

$$\frac{\partial^2 p'}{\partial x'^2} + \frac{1}{\tilde{V^2}... ...2}(x',\zeta)} \frac{\partial^2 p'}{\partial t' \partial \zeta}$$ (4)

where $\zeta(x,z)$ is the time-like vertical coordinate.

As we can express the familiar Eikonal equation for the travel time $\tau(x,z)$ in high frequency domain, V. Khare proposes from equation (4) the following paraxial approximation of Eikonal equation for time migration:  

$$\left( \frac{\partial \tau'}{\partial x'} \right)^2 + \frac... ...}{\tilde{V^2}(x',\zeta)} \frac{\partial \tau'}{\partial \zeta}$$ (5)

The first step of this project is to retrieve and to understand how V. Khare obtains the equations mentioned in his paper and particularly the equation (5) from the wave equation.

Then I will try to apply to the 3-D case all the transformations described in the 2-D case, that is to say, to express the paraxial approximation of Eikonal equation for 3-D time migration and to find the inverse process, the 3-D time de-migration.

The next step will be to use a finite difference method with a synthetic model in order to have a rough but clearest idea of the possibilities I can take out of this equation. At last, it could be interesting to contemplate testing other methods for this process and/or to use real data.