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The mapping functions for the time-to-depth conversion are previously computed by tracing image rays. The method gives the mapping functions along the image rays. Mapping time-migrated image along the image rays subsequently gives the depth-migrated image that has to be interpolated onto a regular grid. For complicated velocity models, image ray may cross each other or not penetrate shadow zones; the interpolation is thus cumbersome and computationally expensive. In this section, I describe an efficient algorithm for computing the mapping functions of the image-ray corrections on a regular grid,

To compute the functions $\tau_0(x,z)$ and $\lambda_0(x,z)$ in equation (1), we consider two partial differential equations. Because all image rays shoot vertically when leaving the surface, they are associated with a vertically incident plane wavefront. The value of function $\tau_0(x,z)$ is the traveltime for the wavefront to propagate from the surface to a subsurface point (x,z), which satisfies the eikonal equation:

\left({\partial\tau_0 \over \partial x}\right)^2+
\left({\partial\tau_0 \over \partial z}\right)^2 =
{1 \over v^2(x,z)},\end{displaymath} (12)
with the initial condition $\tau_0(x,0)=0$.The value of function $\lambda_0(x,z)$ is the initial surface position of the image ray that eventually reaches a subsurface point (x,z). This function is thus constant along each image ray. Using the orthogonal relation between rays wavefronts in a isotropic medium, we can show that this function satisfies the partial differential equation as follow:

\left({\partial\tau_0 \over \partial x}\right)
\left({\partial\lambda_0 \over \partial z}\right) = 0,\end{displaymath} (13)
with the initial condition $\lambda_0(x,0) = x$.

Two partial differential equations (12) and (13) can be solved simultaneously by using an up-wind finite difference method (Zhang, 1991), and the results are samples of the two mapping functions on a regular grid. Thus, the image-ray corrections can be implemented with an algorithm summerized, as follow:

 {\tt \ for \ each} \ x_j \\  \ \left[ 
 \end{array} \right. \\  \end{array} 
 \right. \\ \end{array}\end{displaymath}

The functions $\tau(x,z;x_s)$ and p(x,z;xs) in equations (6) and (11) can be computed in a similar way. Two partial differential equations

\left({\partial\tau \over \partial x}\right)^2+
\left({\partial\tau \over \partial z}\right)^2 =
{1 \over v^2(x,z)},\end{displaymath} (14)

\left({\partial\tau \over \partial x}\right)
 ...rtial z}\right)
\left({\partial p \over \partial z}\right) = 0,\end{displaymath} (15)
are solved with the point-source initial conditions. The implementation of image-ray corrections for prestack imaging is similar to poststack imaging.

previous up next print clean
Next: CONCLUSIONS Up: Zhang: Time-to-depth conversion Previous: Constant offset imaging
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