Four parameters govern wave propagation in general elastic transversely isotropic media with
a vertical symmetry axis Aki and Richards (1980).
In a recent paper Alkhalifah (1997c), I have showed that only three are needed to totally
characterize *P*-wave propagation in acoustic VTI media. Although an acoustic VTI medium does
not physically exist, it provides, like its isotropic counterpart, exceptionally accurate
kinematic representation of wave propagation in elastic media.
It also accurately characterizes the geometrical amplitude behavior of the wave propagation.
The only drawback of the acoustic assumption is its inaccurate
description of transmission and reflection behavior of waves in true elastic conditions.

The three parameters needed to characterize
the acoustic model in VTI media are the vertical
*P*-wave velocity, *v*_{v}, the normal moveout (NMO) velocity for a horizontal reflector
Thomsen (1986), *v*,
and an anisotropy parameter denoted by . This anisotropy parameter is related to the
horizontal velocity, *v*_{h}, Alkhalifah and Tsvankin (1995) as follows:

(1) |

Conveniently, in a recent paper Alkhalifah (1997d), I derived a wave
equation based on the acoustic assumption for VTI media. Taking *P* as the wavefield,
this equation in two-dimensional (2-D) media is

with

where *f*(*x*,*z*,*t*) is the force function, and *x* and *z* are variables of the 2-D Cartesian
coordinate system.

Equation (2) is fourth order in time and as such, has four solutions. Two of them are the
conventional *P*-wave solutions for incoming and outgoing waves. The other two solutions
represent artifacts of the new formulation and thus are unwanted.
They are also conjugate solutions
that tend to grow exponentially for negative , and behave
like propagating waves for positive .Luckily, these waves, or artifacts, do not travel in isotropic media.
Therefore, they can be eliminated by placing the
receivers in an isotropic layer. For marine-type surveys, the receivers are commonly
placed in the water layer which is conveniently isotropic Alkhalifah (1997d). This is also the case
for the Marmousi model, which
includes a thin shallow water layer.

10/9/1997