EFFICIENT COMPUTATIONS OF IMAGE RAYS

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:

 

$$\left({\partial\tau_0 \over \partial x}\right)^2+ \left({\partial\tau_0 \over \partial z}\right)^2 = {1 \over v^2(x,z)},$$ (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:

 

$$\left({\partial\tau_0 \over \partial x}\right) \left({\parti... ...}\right) \left({\partial\lambda_0 \over \partial z}\right) = 0,$$ (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:

$$\begin{array} {l} {\tt \ for \ each} \ x_j \\ \ \left[ \... ...) \end{array} \right. \\ \end{array} \right. \\ \end{array}$$

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

 

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

and

 

$$\left({\partial\tau \over \partial x}\right) \left({\partial... ...rtial z}\right) \left({\partial p \over \partial z}\right) = 0,$$ (15)

are solved with the point-source initial conditions. The implementation of image-ray corrections for prestack imaging is similar to poststack imaging.