The amplitude transport equation

To obtain the transport equation for this new $(x-\tau)$-domain coordinate system, we use a ray-theoretical model of the image,

$$F(x,\tau,t)= A(x,\tau) f[t-\tilde{t}(x,\tau)],$$

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, A(x,y,z), of wave propagation. Substituting the trial solution into the partial differential equation (23) and considering only the terms with the highest asymptotic order (those containing the fourth-order derivative of F) yields the eikonal equation (9). The next asymptotic order (third-order in derivatives of F) gives us a linear partial differential equation of the amplitude transport, as follows:

$$\begin{eqnarray} 2\,{v^2}\,{A_x}\,\left( {{{\tilde{t}}}_x} + \sigma \,{{{\tilde{... ...}}}_{\tau }}}^2} \right) {{{\tilde{t}}}_{x x}} \right)=0.\,\,\,\,\end{eqnarray}$$ (37)

Setting $\eta=0$, yields the corresponding transport equation for elliptically anisotropic media,

$$\begin{eqnarray} 2\,{v^2}\,{A_x}\,\left( {{{\tilde{t}}}_x} + \sigma \,{{{\tilde{... ...}\,{{\sigma }^2} \right) \,{{\tilde{t}}}_{\tau \tau } \right)=0.\end{eqnarray}$$ (38)

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., $\frac{\partial^2 t}{\partial x \partial \tau}$), which result from the cross-dependent nature of the new coordinate system. Some of these terms are caused by anisotropy Alkhalifah (1997b). When $\sigma=0$ in equation (38) all such cross-derivate terms drop out.