The VTI acoustic wave equation

I have derived a simple equation that relates the vertical slowness, p_z, to the horizontal one, p_r, in VTI media, based on setting the vertical shear wave velocity to zero 1997. In such media, the slowness surface in the horizontal plane is circular (isotropic), and therefore, p_r can be replaced by $\sqrt{p_x^2+p_y^2}$, where the slowness vector, p, has components in the Cartesian coordinates given by p_x, p_y, and p_z. As a result, the migration dispersion relation in 3-D media is  

$$p_z^2 = \frac{v^2}{v_v^2} \left(\frac{1}{v^2}-\frac{p_x^2+p_y^2}{1-2 v^2 \eta (p_x^2+p_y^2)} \right).$$ (3)

Using ${\bf k}=\omega {\bf p}$, where k is the wavenumber vector with components in the Cartesian coordinates (k_x, k_y, k_z), and $\omega$ is the angular frequency, equation (3) becomes  

$$k_z^2 = \frac{v^2}{v_v^2} \left(\frac{\omega^2}{v^2}- \frac{... ...ga^2 (k_x^2+k_y^2)}{\omega^2-2 v^2 \eta (k_x^2+k_y^2)} \right).$$ (4)

Multiplying both sides of equation (4) with the wavefield in the Fourier domain, $F(k_x,k_y,k_z,\omega)$,as well as using inverse Fourier transform on k_z ($k_z \rightarrow -i\frac{d}{dz}$) yields  

$$\frac{d^2 F(k_x,k_y,z,\omega)}{d z^2} = -\frac{v^2}{v_v^2} \... ...\omega^2-2 v^2 \eta (k_x^2+k_y^2)} \right) F(k_x,k_y,z,\omega).$$ (5)

Likewise, using inverse Fourier transform on k_x and k_y ($k_x \rightarrow -i\frac{\partial}{\partial x}$, $k_y \rightarrow -i\frac{\partial}{\partial y}$), we obtain

   $$\begin{eqnarray} v_v^2 \omega^2 \frac{\partial^2 F}{\partial z^2} +2 \eta v^2 v... ...artial y^2} \right) + \nonumber \\ \omega^4 F(x,y,z,\omega) = 0.\end{eqnarray}$$

Finally, applying inverse Fourier transform on $\omega$ ($\omega \rightarrow i\frac{\partial}{\partial t}$), the acoustic wave equation for VTI media is given by

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

This equation is a fourth-order partial differential equation in t. Setting $\eta=0$ yields the acoustic equation for elliptically anisotropic media with vertical symmetry axis

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

Substituting $P=\frac{\partial^2 F}{\partial t^2}$ gives the familiar second-order wave equation for elliptically anisotropic media with vertical symmetry axis,

   $$\begin{eqnarray} \frac{\partial^2 P}{\partial t^2} = v^2 \left(\frac{\partial^2 ... ...P}{\partial y^2} \right) +v_v^2 \frac{\partial^2 P}{\partial z^2}.\end{eqnarray}$$

For isotropic media, v_v=v and equation (9) reduces to the familiar acoustic wave equation for isotropic media,

   $$\begin{eqnarray} \frac{\partial^2 P}{\partial t^2} = v^2 \left(\frac{\partial^2 ... ...al^2 P}{\partial y^2} + \frac{\partial^2 P}{\partial z^2} \right).\end{eqnarray}$$

Rewriting equation (7) in terms of P(x,y,z,t) instead of F(x,y,z,t), yields

   $$\begin{eqnarray} \frac{\partial^2 P}{\partial t^2} = (1+2 \eta) v^2 \left(\frac{... ...tial z^2}+ \frac{\partial^4 F}{\partial y^2 \partial z^2} \right),\end{eqnarray}$$

where

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

In the numerical implementation, for convenience, I rely on the equation (11).

For comparison, the 2-D elastic wave equation, which is best described in VTI media using the density-normalized elastic coefficients, A_ijkl($=C_{ijkl}/\rho$), is given by Aki and Richards (1980)

$$\frac{\partial^2 u_x}{\partial t^2} = A_{1111} \frac{\partia... ...x \partial z} + A_{1313} \frac{\partial^2 u_x}{\partial z^2},$$

and

$$\frac{\partial^2 u_z}{\partial t^2} = A_{3333} \frac{\partia... ...x \partial z} + A_{1313} \frac{\partial^2 u_z}{\partial x^2},$$

where u_x and u_z are the components of the wavefield vector, u, in two dimensions. Solving the elastic wave equation in heterogeneous media requires applying finite-difference computation to two equations (three equations in 3-D media) corresponding to the components of the wavefield. Calculating the wavefield for each component is almost as computationally involved as calculating the acoustic wavefield. This method also incurs the additional expense of input, output, and storage of the wavefield and the corresponding medium parameters in the elastic medium case.

In addition, the solution of the elastic wave equation contains both P- and S-waves, whereas the acoustic equation yields only P-waves. The presence of S-waves in the solution of elastic wave equation makes that equation less desirable when used for modeling P-wave propagation in zero-offset conditions, such as when the exploding reflector assumption is used.