CONTINUOUS REPRESENTATION OF A VELOCITY FIELD

The representation of a velocity field is crucial to the accuracy and efficiency of traveltime and amplitude calculations. Because the traveltime and amplitude calculations are more accurate in a continuous velocity model than in a discontinuous velocity model, I map a given velocity field onto a grid formed by regular triangular cells. Within each triangular cell, velocity varies linearly, as follows:

 

v(x,z) = v_o+v_x(x-x_o)+v_z(z-z_o), (1)

where two components of the velocity gradient are determined by  

$$\left\{ \begin{array} {lll} v_x = \displaystyle{(z_b-z_o)(v_... ...\over (z_b-z_o)(x_a-x_o)-(x_b-x_o)(z_a-z_o)},\end{array}\right.$$ (2)

v_o, v_a and v_b are the velocities at the three corners, (x_o,z_o), (x_a,z_a) and (x_b,z_b), of the triangular cell, respectively. Figure  shows examples of a triangular grid and a triangular cell. With this representation, velocity is continuous everywhere, including the boundaries of adjacent triangular cells. Velocity contrasts at the interfaces of the original velocity model are approximated by triangular cells of large velocity gradients.

The reason for using linearly varying velocities within triangular cells is that seismic waves propagate along circular ray paths in a linearly varying velocity medium. Therefore, local ray-tracing in these triangular cells can be done analytically. An alternative is to use linearly varying sloth (slowness squared) within each triangular cell. The ray paths in such a medium are hyperbolas.

In a manner similar to the finite difference method developed by Vidale (1988), two schemes need to be built for the local ray-tracing method. In the following sections, I first develop the scheme used to propagate a local wavefront through a triangular cell. I then describe the scheme that addresses the order in which the local computations proceed.

 

vptrigrid
Figure 1 Examples of a triangular grid and a triangular cell. The velocity field is specified on the grid points indicated by little circles. Velocities at the three corners of a cell determines the constant velocity gradient within the cell.