The theory

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 $\eta$. This anisotropy parameter is related to the horizontal velocity, v_h, Alkhalifah and Tsvankin (1995) as follows:  

$$\eta = \frac{1}{2} \left( \frac{v_h^2}{v^2} -1\right).$$ (1)

If $\eta$=0 the medium is elliptically anisotropic, and if we set v_v=v, the medium eventually become isotropic. Typical $\eta$ values in the subsurface range from 0 to 0.3 Alkhalifah (1997a).

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

   $$\begin{eqnarray} \frac{\partial^2 P}{\partial t^2} = (1+2 \eta) v^2 \frac{\parti... ... \eta v^2 v_v^2 \frac{\partial^4 F}{\partial x^2 \partial z^2} +f,\end{eqnarray}$$

with

   $$\begin{eqnarray} P=\frac{\partial^2 F}{\partial t^2},\end{eqnarray}$$

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 $\eta$, and behave like propagating waves for positive $\eta$.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.