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 -rays at a planar interface. In this section we derive the traveltime gradients for -rays. The derivation is straightforward and is based on Fermat principle applied to the -rays.
The transformation of variables defined in equations (2) and (3) implies the following relationships between the differential quantities and .
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 -ray,
where the tildes on the variables indicate that they are evaluated along the raypath.
Applying Fermat principle, the first order perturbations in the traveltimes caused by perturbations in slowness are given by the following integral evaluated along the unperturbed raypath -ray,