Notation for VTI media

We begin by recalling some notation needed in the remainder of the paper. For transversely isotropic media with vertical symmetry axis, the relationship between components of stress $\sigma_{kl}$and strain $e_{ij} = {\textstyle {1\over2}}(u_{i,j} + u_{j,i}) = {\textstyle {1\over2}}\le... ...\partial u_i}\over{\partial x_j}} + {{\partial u_j}\over{\partial x_i}}\right)$ (where u_j is the jth component of the displacement vector) is given by

$$\begin{eqnarray} \left( \begin{array} {c}\sigma_{11} \ \sigma_{22} \ \sigma_... ... e_{33} \ e_{23} \ e_{31} \ e_{12} \ \end{array}\right), \end{eqnarray}$$ (1)

where a = b + 2m (e.g., Musgrave, 1970; Auld, 1973), with i,j,k,l obviously each ranging from 1 to 3 in Cartesian coordinates. The matrix describes isotropic media in the special case when $a=c=\lambda+ 2\mu$, $b=f=\lambda$, and $l=m=\mu$.

The Thomsen (1986) parameters $\epsilon$, $\delta$, and $\gamma$ are related to these stiffnesses by

$$\begin{eqnarray} \epsilon &\equiv& {{a-c}\over{2c}}, \end{eqnarray}$$ (2)

$$\begin{eqnarray} \delta &\equiv& {{(f+l)^2-(c-l)^2}\over{2c(c-l)}}, \end{eqnarray}$$ (3)

$$\begin{eqnarray} \gamma &\equiv& {{m-l}\over{2l}}. \end{eqnarray}$$ (4)

For P-wave propagation in the earth near the vertical, the important anisotropy parameter is $\delta$. For SV-wave propagation near the vertical, the combination $(\epsilon - \delta)$ plays essentially the same role as $\delta$ does for P-waves. For SH-waves, the pertinent anisotropy parameter is $\gamma$. All three of the Thomsen parameters vanish for an isotropic medium.

It is also useful to note for later reference that

$$\begin{eqnarray} a = c(1+2\epsilon), \qquad m = l(1+2\gamma), \qquad\hbox{and}\qquad f \simeq c(1+\delta) - 2l. \end{eqnarray}$$ (5)

In TI media, c and l are the velocities normal to the layering. Then, $\epsilon$, $\gamma$, and $\delta$ measure the deviations from these normal velocities at other angles. We present the relevant details of the phase velocity analysis later in the paper.