Ray tracing in $\left(\tau,\xi\right)$

The solutions to the focusing eikonal can be computed using current methods for solving the standard eikonal, either directly by modern eikonal solvers Fomel (1997); Sethian and Popovici (1997), or by ray tracing. We chose a ray tracing solution, because for reflection tomography is handier to have rays than traveltime maps.

To derive the ray-tracing system for the focusing eikonal we begin by writing its associated Hamiltonian as a function of the ray parameters $p_\tau$ and $p_\xi$,

$$H\left(\tau,\xi,p_{\tau},p_{\xi}\right)= \frac{1}{2} \left\{... ...^2+ V_f^2 \left[ p_{\xi}+ \sigma_f p_{\tau} \right]^2 \right\}.$$ (10)

The associated ray-tracing equation are:

   $$\begin{eqnarray} \frac{d \xi}{d t} =& \;\;\frac{\partial H}{\partial p_\xi} =& ... ... p_{\tau} \right) \frac{\partial \sigma_f}{\partial \tau} \right].\end{eqnarray}$$

Rays can be traced in $\left(\tau,\xi\right)$ by solving the ray-tracing equations in (11) by a standard ODE solver. The appropriate initial conditions for the ray parameters $p_\tau$ and $p_\xi$when the source is at $\left(\tau_0,\xi_0\right)$and the take-off angle is $\theta_\tau$ are:

   $$\begin{eqnarray} p_{\tau_0} &=& \frac { \cos \theta_\tau } {2} \nonumber \\ p_{\... ..._0\right)} - \sigma_f\left(\tau_0,\xi_0\right) p_{\tau_0} \right].\end{eqnarray}$$

To test the accuracy of our derivations we numerically solved the ray tracing equations (11) for a heterogeneous velocity function, and compared the results with a ray-tracing solution of the standard eikonal equation. As expected, $\tau$-rays map exactly into z-rays, for all velocity fields. Figure  and Figure  show an example of the ray field when the velocity function is a Gaussian-shaped negative velocity anomaly superimposed onto a constant velocity background. Notice that the focusing eikonal handles correctly the caustic and wavefront triplication below the anomaly. Figure  shows the effects of neglecting the differential mapping factor $\sigma_f$.It shows the $\tau$-rays computed setting $\sigma_f$to zero, and remapped into $\left(z,x\right)$.The wavefronts are distorted compared to the true wavefronts shown in Figure 

 

Raytau-z-an
Figure 1 Ray field in $\left(z,x\right)$ domain with a negative velocity anomaly in constant background.

 

Raytau-an
Figure 2 Ray field in $\left(\tau,\xi\right)$ domain with a negative velocity anomaly in constant background. This ray field maps exactly into the one shown in Figure .

 

Raytau-ell-z-an
Figure 3 Ray field in $\left(z,x\right)$ domain computed assuming the differential mapping factor $\sigma$ equal to zero. This ray field is different than the one shown i n Figure .