The X-TAU acoustic wave equation

Following the approach of Alkhalifah (1997b), an acoustic wave equation is simply derived from the eikonal equation using Fourier transformations. The addition of $\sigma$ results in a more intriguing wave equation than the one derived by Alkhalifah. Instead of the symmetric form of the familiar Laplacian in isotropic media, two sources of unsymmetry are introduced into the new wave equation. One is caused by the unbalanced new coordinate system with one axis given in time and the other in position. The second, caused by anisotropy, is similar to that which Alkhalifah described.

Using $k_x=\omega \frac{\partial t}{\partial x}$, and $k_{\tau}=\omega \frac{\partial t}{\partial \tau}$, where k_x is the horizontal component of the wavenumber vector, $k_{\tau}$ is the vertical-time-normalized component of the wavenumber vector, and $\omega$ is the angular frequency, we can transform equation (9) to  

$${v^2}\,\left( 1 + 2\,\eta \right) \,{\left(\frac{k_x}{\omega... ...\omega}+ \frac{k_{\tau}}{\omega} \sigma \right)^2} \right)=1.$$ (22)

Multiplying both sides of equation (22) with the wavefield in the Fourier domain, $F(k_x,k_{\tau},\omega)$,as well as using inverse Fourier transform on $k_{\tau}$, k_x and $\omega$ ($k_{\tau} \rightarrow -i\frac{d}{d\tau}$, $k_x \rightarrow -i\frac{\partial}{\partial x}$, and $\omega \rightarrow i\frac{\partial}{\partial t}$), we obtain the acoustic wave equation in this new vertical-velocity-independent coordinate system,

   $$\begin{eqnarray} \frac{\partial^4 F}{\partial t^4} =-8\,\frac{\partial^4 F}{\par... ...t( 4 + {v^2}\,\left( 1 + 2\,\eta \right) \,{{\sigma }^2} \right).\end{eqnarray}$$

This equation is a fourth-order partial differential equation. Unlike, the acoustic wave equation for VTI media of Alkhalifah (1997b), equation (23) has odd-order derivatives caused by the unsymmetry of the coordinate system. Setting $\sigma=0$ [v(z)=0], we obtain a similar equation, with $\partial z$replaced by $v_v\partial \tau$ as follows:

   $$\begin{eqnarray} \frac{\partial^4 F}{\partial t^4} =-8\,\frac{\partial^4 F}{\par... ...ta \right)+ 4 \frac{\partial^4 F}{\partial t^2 \partial \tau^2}.\end{eqnarray}$$

Setting $\eta=0$ in equation (23) yields the acoustic equation for elliptically anisotropic media:

   $$\begin{eqnarray} \frac{\partial^2}{\partial t^2} \left(\frac{\partial^2 F}{\part... ...artial \tau^2}\,\left( 4 + {v^2}\,{{\sigma }^2} \right) \right)=0.\end{eqnarray}$$

Substituting $P=\frac{\partial^2 F}{\partial t^2}$, we obtain the second-order wave equation for elliptically anisotropic media:

   $$\begin{eqnarray} \frac{\partial^2 P}{\partial t^2} = {v^2}\, \left(\frac{\partia... ...al^2 P}{\partial \tau^2}\,\left( 4 + {v^2}\,{{\sigma }^2} \right).\end{eqnarray}$$

Rewriting equation (23) in terms of P(x,y,z,t) rather than F(x,y,z,t), wherever possible, yields

   $$\begin{eqnarray} \frac{\partial^2 P}{\partial t^2} =-8\,\frac{\partial^4 F}{\par... ... {v^2}\,\left( 1 + 2\,\eta \right) \,{{\sigma }^2} \right),\,\,\,\end{eqnarray}$$

where

$$F(x,y,z,t)= \int_0^t dt' \int_0^{t'} P(x,y,z,\tau) d\tau.$$

Because of its second-order nature in time, equation (27) is simpler to use in a numerical implementation than equation (23). The acoustic wave equation in $(x-\tau)$-domain is clearly independent of the vertical velocity when $\sigma$ is given by equation 12 and $\alpha$ is laterally invariant.