MODEL PRESENTATION

3-D dynamic ray tracing is a forward problem. It requires that a physical model should be known in advance. Constructing a 3-D slowness model is an open topic. Different model gridding schemes have different characteristics. In order to be more general and portable, our dynamic ray tracing program is designed to be independent of the model gridding scheme. It only requires the local velocity information in the vicinity of the central ray be known. We can choose different types of gridding schemes, such as cube, layer, or tetrahedron, according to the characteristics of the geological structure. We use cubic scheme in our application only because it is easy to implement. we use slowness instead of velocity in this paper.

Since the model has already been gridded, the model construction is transferred into an expression of the slowness within each small cube. We select a trilinear interpolation function.  

$$s(x,y,z)=s_0+s_{\rm x}x+s_{\rm y}y+s_{\rm z}z+s_{\rm xy}xy+s_{\rm xz}xz+s_{\rm yz}yz+s_{\rm xyz}xyz$$ (1)

where x, y, z are Cartesian coordinates. The advantage of trilinear functions is that the slowness is continuous from one cube to another. Therefore the direction of the raypaths are continuous when crossing the cube boundary. The coefficients of (1) can be solely determined from the eight slowness value of the cube. In our application, we first associate the ray point with eight grid points which construct a cube containing the ray point. From the slowness values of eight grid points, we get a trilinear expression for this cube. Then we can estimate the slowness at any point inside the cube.