The acoustic wave equation for orthorhombic media

Using the dispersion relation of equation 17, we can derive an acoustic wave equation for orthorhombic media as shown in Appendix A. The resultant wave equation is sixth-order in time given by,

$$\begin{eqnarray} \frac{\partial^6 F}{\partial t^6}= V_1^2 \frac{\partial^6 F}{\p... ...ight) \frac{\partial^6 F}{\partial x^2 \partial y^2 \partial z^2}.\end{eqnarray}$$ (18)

This equation is two time-derivative orders higher than its VTI equivalent and 4 orders higher than the conventional isotropic acoustic wave equation. Setting $\eta_1=\eta_2$, v₁=v₂, and $\gamma=1$, the conditions necessary for the medium to be VTI, equation (18) reduces to

$$\begin{eqnarray} \frac{\partial^2}{\partial t^2} \left(\frac{\partial^4 F}{\part... ...l^4 F}{\partial y^2 \partial z^2} \right) \right) \nonumber \\ =0.\end{eqnarray}$$ (19)

Substituting $M=\frac{\partial^2 F}{\partial t^2}$ gives us the acoustic wave equation for VTI media derived by Alkhalifah (1997a),

$$\begin{eqnarray} \frac{\partial^4 M}{\partial t^4} - (1+2 \eta) v^2 \left(\frac{... ...tial z^2}+ \frac{\partial^4 M}{\partial y^2 \partial z^2} \right).\end{eqnarray}$$ (20)

Thus, as expected, equation (18) reduces to the exact VTI form when VTI model conditions are used, and subsequently it will reduce to the exact isotropic form (the conventional second-order acoustic wave equation) when isotropic model parameters are used (i.e., $\delta=0$, $\eta_1=\eta_2=0$ and v₁=v₂=v_v).

Equation (18) can be solved numerically using finite difference methods. However, such solutions require complicated numerical evaluations based on sixth-order approximation of derivative in time. Substituting $M=\frac{\partial^2 F}{\partial t^2}$ into equation (18) yields,

$$\begin{eqnarray} \frac{\partial^4 M}{\partial t^4}= V_1^2 \frac{\partial^4 M}{\p... ...ight) \frac{\partial^6 F}{\partial x^2 \partial y^2 \partial z^2},\end{eqnarray}$$ (21)

which is a fourth order equation in derivates of t. In addition, substituting $P=\frac{\partial^2 M}{\partial t^2}$ into equation (21) yields

$$\begin{eqnarray} \frac{\partial^2 P}{\partial t^2}= V_1^2 \frac{\partial^2 P}{\p... ...ight) \frac{\partial^6 F}{\partial x^2 \partial y^2 \partial z^2},\end{eqnarray}$$ (22)

which is now second order in derivates of t. Equation (22) also clearly displays the various levels of parameter influence on the wave equation. For example, if $\eta_1=\eta_2$ and v₁=v₂ the last term in equation (22) drops.