Local ray-tracing

The local ray-tracing method propagates a wavefront locally through triangular cells of linearly varying velocities by following local rays. Because the whole process of wavefront propagation involves tracing many local rays, it is necessary to have an efficient ray-tracing algorithm. Fortunately, the analytical solutions to the ray equations for a linearly varying velocity field are well known, and the analytical solutions of R and J for such a velocity field are also known. Here I merely quote the results. The details of the derivations can be found in Telford et. al. (1976) and Cervený (1981abc).

Suppose we know the attributes of a wavefront, ${\bf \eta}_0=\{\tau_0,(u_0,w_0),\gamma_0,R_0,J_0\}$, at an initial point (x₀,z₀). By tracing a ray, we can find the attributes of the wavefront, ${\bf \eta}=\{\tau,(u,w),\gamma,R,J\}$, at an arbitrary point (x,z) on the ray. In a linearly varying velocity field, the two components of the velocity gradient are constant, (v_x,v_z). It is known that in such a velocity field the ray follows a circular trajectory. The traveltime is determined by integrating the inverse of velocity along the ray, which yields

 

$$\tau = \tau_0+{1 \over g}\log\left( {\sin i(1+\cos i_0) \over \sin i_0(1+\cos i)}\right),$$ (4)

where $g=\sqrt{v^2_x+v^2_z}$ is the magnitude of the velocity gradient, and i₀ and i are the ray angles shown in Figure . It can be shown that

$$\left\{ \begin{array} {lll} \cos i_0 & = & \displaystyle{v_0... ...& \displaystyle{v_0 \over g}(v_zw_0 +v_xu_0),\end{array}\right.$$ (5)

where v₀ is the velocity at the initial point. The ray angle, i, at point (x,z) is related to the arc-length of the ray path as follows:

$$i = i_0+(s-s_0){g \cos i_0 \over v_0}.$$ (6)

where s-s₀ is the arc length of the ray between two points. The two components of the traveltime gradient at the point (x,z) are determined by

 

$$\left\{ \begin{array} {lll} u & = & \displaystyle{1 \over vg... ...isplaystyle{1 \over vg}(v_z\sin i +v_x\cos i)\end{array}\right.$$ (7)

where v is the velocity at point (x,z). Because the take-off angle of the ray does not change along the ray,

 

$$\gamma = \gamma_0.$$ (8)

Cervený (1981a) shows that in a velocity field of zero second order spatial derivatives, the curvature radius of a wavefront at a point on a ray is linearly proportional to the integration of the velocity field along the ray. Because a linearly varying velocity field has zero second-order spatial derivatives, the integration of the velocity along the circular ray path gives

 

$$R = {v_0 \over v}\left[R_0+{v_0 \over g\cos i_0}(1-{\cos i \over \cos i_0}). \right]$$ (9)

Similarly, we can show that  

$$J = J_0{vR \over v_0R_0}.$$ (10)

where R is defined in equation (9).

 

vpraytrc Figure 2 Ray-tracing in a linearly varying velocity field. The dash arrow shows the direction of the velocity gradient. The solid arrows show the directions of traveltime gradients that are tangential to the ray. Ray angles are measured between the direction of the velocity gradient and the directions of the ray.

When a ray reaches the boundary of two adjacent triangular cells, the phase matching method can be used to trace the ray across the boundary (Cervený, 1981c). Traveltime $\tau$ and take-off angle $\gamma$ are always continuous across the boundary. Because of the continuous velocity representation, there is no discontinuity of velocity across the boundary. Therefore traveltime gradient (u,w) and geometrical spreading factor J are also continuous at the boundary. However, curvature radius R is generally not continuous at the boundary because of the change of the velocity gradient across the boundary. Cervený (1981c) derives a general formula that relates the curvature radii of wavefronts on the two sides of the boundary. This general formula can be greatly simplified by considering that the velocity field is continuous at the boundary and that the boundary has a zero curvature. The result is

$$R^+ = {R^- \over 1+vIR^-}$$ (11)

where R⁺ and R^- are the curvature radii before and after crossing the boundary, respectively, I is a factor defined as follows:

As shown in Figure , $\theta$ is the incident angle of the ray to the normal of the boundary, $\phi$ is the angle between the orientation of the boundary and the z-axis, and (v^-_x,v^-_z) and (v⁺_x,v⁺_z) are the velocity gradients of two adjacent cells, respectively.

 

vpbound Figure 3 Ray-tracing across a boundary of two triangular cells. Because the velocity is continuous at the boundary, the angle of incidence is equal to the angle of refraction.

Finite difference methods propagate local wavefronts in cells of constant velocity. Because wavefronts propagate along straight rays in a constant velocity field, local ray-tracing is trivial. The local-plane-wave assumption implies that $R \rightarrow \infty$. All these simplifications make finite difference methods more efficient than the local ray-tracing method. Of course, they also reduce the accuracy of the computations.