INTRODUCTION OF EIKONAL EQUATION FOR TIME-MIGRATION

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

$$\frac{\partial^2 p}{\partial x^2} \: + \: \frac{\partial^2 ... ...\; \frac{1}{V^2 \, (x,z)} \, \frac{\partial^2 p}{\partial t^2}$$ (1)

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

We can make the following approximation of the wave field:  

$$p \, (x,z,t) \; = \; \hat{p} \, (x,z) \: e^{\: i \, \omega \, (t \, - \, \tau \, (x,z) \, )}$$ (2)

expressing the pressure wave field in term of amplitude $\hat{p}(x,z)$ and phase , where $\tau$ is the traveltime along the ray paths. Then replacing (2) in equation (1) and making the high frequency approximation, we obtain the 2-D Eikonal equation:  

$$\left( \frac{\partial \tau}{\partial x} \right)^2 \: + \: \... ...ial \tau}{\partial z} \right)^2 \; = \; \frac{1}{V^2 \, (x,z)}$$ (3)

Among other uses, this equation is useful to compute traveltimes of rays in a given earth model.

V.J. Khare introduced the following equation for the ray-theoretic interpretation of all-angle time migration Khare (1991):  

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

where $p' \, (x',\zeta,t')$ and $\tilde{V} \, (x',\zeta)$ are respectively the pressure wave field and the wave propagation velocity in the time-retarded coordinate system Claerbout (1976),  

$$\begin{array} {lcl} x' & = & x \\ t' & = & t \: + \: \zeta\end{array}$$ (5)

with the time-like vertical coordinate $\zeta$ Lowenthal et al. (1985),  

$$\zeta \; = \; \int_{0}^{z} \: \frac{dz}{V \, (x,z)}$$ (6)

If we make the following approximation of the wave field in the new coordinate system, equivalent to the equation (2):  

$$p' \, (x',\zeta,t') \; = \; \hat{p'} \, (x',\zeta) \: e^{\: i \, \omega \, (t' \, - \, \tau' \, (x',\zeta) \, )}$$ (7)

we can express the 2-D Eikonal equation for time migration giving the traveltime $\tau'$ in the frequency domain in the time-retarded coordinate system:  

$$\left( \frac{\partial \tau'}{\partial x'} \right)^2 \: + \: ... ...e{V}^2 \, (x',\zeta)} \, \frac{\partial \tau'}{\partial \zeta}$$ (8)

Using equations (5) and (6), we obtain the 2-D Eikonal equation for time migration in the physical coordinate system:  

$$\left( \frac{\partial \tau}{\partial x} \: + \: {\cal A} \,... ...ial \tau}{\partial z} \right)^2 \; = \; \frac{1}{V^2 \, (x,z)}$$ (9)

with  

$${\cal A} \, (x,z) \; = \; V \, (x,z) \: \int_{0}^{z} \: \f... ...z \; = \; - \, V \, (x,z) \, \frac{\partial \zeta}{\partial x}$$ (10)