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Next: USING 3-D EIKONAL EQUATION Up: Berlioux: 3-D Eikonal equation Previous: INTRODUCTION

INTRODUCTION OF EIKONAL EQUATION FOR TIME-MIGRATION

In two dimensions, considering one-way time, the wave equation is  
 \begin{displaymath}
\frac{\partial^2 p}{\partial x^2} \: + \:
 \frac{\partial^2 ...
 ...\;
 \frac{1}{V^2 \, (x,z)} \, \frac{\partial^2 p}{\partial t^2}\end{displaymath} (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:  
 \begin{displaymath}
p \, (x,z,t) \; = \; \hat{p} \, (x,z) \:
 e^{\: i \, \omega \, (t \, - \, \tau \, (x,z) \, )}\end{displaymath} (2)
expressing the pressure wave field in term of amplitude $\hat{p}(x,z)$ and phase $\hbox{{<tex2html_image_mark\gt ... , 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:  
 \begin{displaymath}
\left( \frac{\partial \tau}{\partial x} \right)^2 \: + \:
 \...
 ...ial \tau}{\partial z} \right)^2 \; = \;
 \frac{1}{V^2 \, (x,z)}\end{displaymath} (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):  
 \begin{displaymath}
\frac{\partial^2 p'}{\partial x'^2} \: + \:
 \frac{1}{\tilde...
 ...zeta)} \, 
 \frac{\partial^2 p'}{\partial t' \, \partial \zeta}\end{displaymath} (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{displaymath}
\begin{array}
{lcl}
 x' & = & x \\  t' & = & t \: + \: \zeta\end{array}\end{displaymath} (5)
with the time-like vertical coordinate $\zeta$ Lowenthal et al. (1985),  
 \begin{displaymath}
\zeta \; = \; \int_{0}^{z} \: \frac{dz}{V \, (x,z)}\end{displaymath} (6)

If we make the following approximation of the wave field in the new coordinate system, equivalent to the equation (2):  
 \begin{displaymath}
p' \, (x',\zeta,t') \; = \; \hat{p'} \, (x',\zeta) \:
 e^{\: i \, \omega \, (t' \, - \, \tau' \, (x',\zeta) \, )}\end{displaymath} (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:  
 \begin{displaymath}
\left( \frac{\partial \tau'}{\partial x'} \right)^2 \: + \:
...
 ...e{V}^2 \, (x',\zeta)} \,
 \frac{\partial \tau'}{\partial \zeta}\end{displaymath} (8)

Using equations (5) and (6), we obtain the 2-D Eikonal equation for time migration in the physical coordinate system:  
 \begin{displaymath}
\left( \frac{\partial \tau}{\partial x} \: + \:
 {\cal A} \,...
 ...ial \tau}{\partial z} \right)^2 \; = \;
 \frac{1}{V^2 \, (x,z)}\end{displaymath} (9)
with  
 \begin{displaymath}
{\cal A} \, (x,z) \; = \; V \, (x,z) \: 
 \int_{0}^{z} \: \f...
 ...z \; = \;
 - \, V \, (x,z) \, \frac{\partial \zeta}{\partial x}\end{displaymath} (10)


previous up next print clean
Next: USING 3-D EIKONAL EQUATION Up: Berlioux: 3-D Eikonal equation Previous: INTRODUCTION
Stanford Exploration Project
11/17/1997