To obtain the transport equation for this new -domain coordinate system, we use a ray-theoretical model of the image,

as a trial solution to the wave equation (23). This procedure yields the eikonal equation as well as the transport equation that describes amplitude behavior,

Setting , yields the corresponding transport equation for elliptically anisotropic media,

Both transport equations include first-and second-order derivatives of time with respect to position,
calculated from the solution of the eikonal equation. Despite the apparent
complexity of the transport equations, they are linear, and contain only first-order derivates of
*A*. As expected, amplitudes depend on second-order derivatives of traveltime, or
wavefront curvature. The dynamic raytracing equations behave similarly.
Also, we can see that equations (37)
and (38) include terms corresponding to cross
derivates of traveltime (i.e., ), which
result from the cross-dependent nature of the new coordinate system. Some of these terms
are caused by anisotropy Alkhalifah (1997b). When in equation (38) all such
cross-derivate terms drop out.

10/9/1997