Notation for VTI media

We begin by introducing 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 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)

Certain interpretations are allowed for these parameters when they are small enough. 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 $(c/l)(\epsilon - \delta)$ 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, and the interpretations mentioned are valid for weakly anisotropic media such that all these parameters are relatively small (< 1). However, the definitions are also useful outside the range of these constraints, and we will use the same definitions (and also continue to call them the ``Thomsen parameters'') even when the smallness condition is violated; there is no fundamental problem doing this as long as it is recognized that the interpretations already mentioned in this paragraph are not necessarily valid any more when the parameters are large. This generalization of the Thomsen parameters will however require us to be careful in our subsequent usage of the parameters, as they cannot always be assumed to be small here as is usual in other treatments. Unless explicitly stated otherwise, the parameters $\epsilon$, $\gamma$, and $\delta$ are not small quantities in this paper.

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)

where smallness of $\delta$ was in fact assumed in the third expression. In TI media, c and l are directly related to 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.