In this paper, we develop an accurate and robust 3-D dynamic ray tracing
system. Our implementation is independent of the model gridding scheme and
is easy to be incorporated into 3-D velocity model representation methods.
We use a triangle smoothing filter to smooth the model to make the model
more ``ray valid'' and the numerical solution more stable.
The Fortran90 version Runge-Kutta solver *rksuite90* has the ability
to adapt its step-length to fit the requirement of the local gradient of
the slowness.
The results in this paper verifies that our program is correct and also
demonstrates our method's capability of handling complex model.

We have not calculated the phase-shift along the central ray. It is
straight-forward after we get **Q**, **P**, and **M**, as shown in
equation (15). Theoretically, dynamic ray tracing theory can
be further categorized into *paraxial ray tracing* and *Gaussian
beam approximation*. Gaussian beam approximation can deal with complex
structures more accurately, but it is also more time-consuming. In the past,
researchers always considered these two approximations separately. We will
investigate the possibilities of combining them together to achieve an
accurate result with reasonable costs.

Our final aim for this project is to generate a regularly gridded 3-D Green's function for complex structures, which is essential to 3-D imaging and velocity analysis. The result from dynamic ray tracing is irregularly distributed on the model. So we need some interpolation schemes to bridge the gap between spatial regularly-sampled Green's function and spatial irregularly-sampled ray tracing result. Our idea is to first find the nearest-neighborhood ray tracing results for each grid point. One possible candidate for the searching scheme is K-D tree, which is an extension to binary tree data structure. Then we design a linear operator to link the regular grids and the ray tracing results together. This is actually a large-scale inverse problem. We can choose different iterative solvers to interpolate the kinematic and dynamic information on the grid from the information on the raypaths.

11/11/1997