There are two ways to derive the eikonal equation from the above formulas. One way is to substitute , , and into the dispersion relation [equation (3)] directly. A second method is based on using a ray-theoretical model of the image,

as a trial solution to equation (7). This approach yields the Eikonal equation as well as the transport equation that describes amplitude behavior,
Again setting and *v*_{v}=*v* gives the isotropic eikonal equation

(17) |

The next asymptotic order (third-order in derivatives of *F*) gives us a
linear partial differential equation
of the amplitude transport, as follows:

The various derivatives of *t* are computed, as in the case
of an isotropic medium, from the solution of the
eikonal equation.
Despite the apparent complexity of this transport equation, it is linear,
and it is also of the first order in derivates of
*A*. Setting and *v*_{v}=*v* in equation (18)
gives us the transport equation for isotropic
media Babich and Buldyrev (1989),

(19) |

Another method for calculating traveltimes is ray tracing. Using the method of characteristics, acoustic ray tracing equations for VTI media are derived and shown in Appendix A. Ray tracing is an additional efficient method for calculating traveltime and amplitudes based on the high-frequency approximation. Its main advantage over numerically solving the eikonal equation, is the ability to compute multiarrival traveltimes and amplitudes.

10/9/1997