Reflection tomography in $\left(\tau,\xi\right)$

 One of the potential application of the focusing eikonal is reflection tomography. Because both the velocity function and the reflectors are more ``stationary'' in the $\left(\tau,\xi\right)$ domain than in the $\left(z,x\right)$,we speculate that reflection tomography performed in $\left(\tau,\xi\right)$ is more stable than reflection tomography performed in $\left(z,x\right)$.

To perform reflection tomography, in addition to ray tracing, we need to compute the gradient of traveltimes with respect to the velocity function and to handle correctly the reflections at the boundaries. Appendix A shows the relationships between the ray parameters of the incident and reflected $\tau$-rays at a planar interface. In this section we derive the traveltime gradients for $\tau$-rays. The derivation is straightforward and is based on Fermat principle applied to the $\tau$-rays.

The transformation of variables defined in equations (2) and (3) implies the following relationships between the differential quantities $\left(dz,dx\right)$ and $\left(d\tau,d\xi\right)$.

$$\begin{eqnarray} d z & = & \frac{V_m}{2} d \tau - \frac{V_m\sigma_f}{2} d\xi \\ d x & = & d \xi.\end{eqnarray}$$

Applying this transformations to the expression of the time increment along a z-ray, leads to the equivalent expression for the the time increment along a $\tau$-ray,

 

$$dt = \sqrt{\frac{dz^2}{V_m^2} + \frac{dx^2}{V_f^2}} = \sqrt{ \left( \frac{d\tau - \sigma_fd\xi}{2} \right)^2 + S_f^2 d\xi^2, }$$ (15)

where S_f is the focusing slowness. The first derivative of the time increment dt with respect to the focusing slowness is given by

   $$\begin{eqnarray} \frac{d\left(dt\right)}{dS_f} & = & \frac{\widetilde{S_f}d\xi^2... ... d\xi } { 4 \widetilde{dt}, } \frac{d \widetilde{\sigma_f}}{d S_f}\end{eqnarray}$$

where the tildes on the variables indicate that they are evaluated along the raypath.

Applying Fermat principle, the first order perturbations in the traveltimes $\Delta t$caused by perturbations in slowness $\Delta S_f$ are given by the following integral evaluated along the unperturbed raypath $\tau$-ray,

$$\Delta t= \int_{\tau-{\rm ray}} \frac{d\left(dt\right)}{dS_f} \Delta S_f \;\; dl,$$ (18)

where dl is the path-length increment. Notice that the $\tau$-ray is not stationary in the $\left(z,x\right)$ domain, but that the term in equation (17) that includes $\widetilde{\sigma_f}$ takes into account the perturbation of the raypath in the $\left(z,x\right)$ domain.