Using the method of characteristics, we can derive a system of ordinary differential equations that define the ray trajectories. To do so, we need to transform equation (9) to the following form:
where s is a running parameter along the rays, related to the traveltime t as follows:
Using equation (9), we obtain
and and , and the same holds for and .
To trace rays, we must first identify the initial values x0, , px0, and . The variables x0 and describe the source position, and px0 and are extracted from the initial angle of propagation. Note that, from equation (9),
because =0 at the source position (z=0).
The raytracing system of equations (18-21) describes the ray-theoretical aspect of wave propagation in the -domain, and can be used as an alternative to the eikonal equation. Numerical solutions of the raytracing equations, as opposed to the eikonal equation, provide multi-arrival traveltimes and amplitudes. In the numerical examples, we use raytracing to highlight some of the features of the -domain coordinate system.