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 , 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

(3) |

(4) |

(5) |

Finally, applying inverse Fourier transform on (), the acoustic wave equation for VTI media is given by

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

Substituting gives the familiar second-order wave equation for elliptically anisotropic media with vertical symmetry axis,

For isotropic media, *v*_{v}=*v* and equation (9) reduces to the familiar acoustic wave
equation for isotropic media,

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

where

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}(),
is given by Aki and Richards (1980)

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.

10/9/1997