Thomsen's $\epsilon$

An important anisotropy parameter for quasi-SV-waves is Thomsen's parameter $\epsilon$, defined in equation (2). Formula (18) for a may be rewritten as

$$\begin{eqnarray} a = \left<{{(\lambda+2\mu)^2 - \lambda^2}\over{\lambda+2\mu}}\right\gt + c\left<{{\lambda}\over{\lambda+2\mu}}\right\gt^2, \end{eqnarray}$$ (20)

which can be rearranged into the convenient and illuminating form

$$\begin{eqnarray} a = \left<\lambda+2\mu\right\gt - c\left[\left<{{\lambda^2}\ove... ...ight\gt -\left<{{\lambda}\over{\lambda+2\mu}}\right\gt^2\right]. \end{eqnarray}$$ (21)

This formula is very instructive because the term in square brackets is in Cauchy-Schwartz form [$\left<\alpha^2\right\gt\left<\beta^2\right\gt \ge \left<\alpha\beta\right\gt^2$], so this factor is nonnegative. Furthermore, the magnitude of this term depends mainly on the fluctuations in the $\lambda$ Lamé parameter, largely independent of $\mu$, since $\mu$ appears only in the weighting factor $1/(\lambda+2\mu)$.Clearly, if $\lambda= constant$, then this bracketed factor vanishes identically, regardless of the behavior of $\mu$. Large fluctuations in $\lambda$ will tend to make this term large. If in addition we consider Thomsen's parameter $\epsilon$ written in a similar fashion as

$$\begin{eqnarray} 2\epsilon = \left[\left<\lambda+2\mu\right\gt\left<{{1}\over{\l... ...ight\gt -\left<{{\lambda}\over{\lambda+2\mu}}\right\gt^2\right], \end{eqnarray}$$ (22)

we find that the term enclosed in the first bracket on the right hand side is again in Cauchy-Schwartz form showing that it always makes a positive contribution unless $\lambda+ 2\mu= constant$, in which case it vanishes. Similarly, the term enclosed in the second set of brackets is always non-negative, but the minus preceding the second bracket causes this contribution to make a negative contribution to $2\epsilon$unless $\lambda= constant$, in which case it vanishes. So, the sign of $\epsilon$ is indeterminate. The Thomsen parameter $\epsilon$ may have either a positive or a negative sign for a TI medium composed of arbitrary thin isotropic layers.

Helbig and Schoenberg (1987) discuss an interesting case where large fluctuations in $\mu$ combined with large fluctuations in $\lambda$,including $\lambda< 0$ for one component, lead to wavefronts with anomalous polarizations in layered TI media. Schoenberg (1994) also discusses several shale examples having large fluctuations in both $\lambda$ and $\mu$.

Fluctuations of $\lambda$ in the earth have important implications for oil and gas exploration. As we recalled in our earlier discussion, Gassmann's well-known results (Gassmann, 1951) show that, when isotropic porous elastic media are saturated with any fluid, the fluid has no mechanical effect on the shear modulus $\mu$, but -- when these results apply -- it can have a significant effect on the bulk modulus $K = \lambda+ {2\over3}\mu$, and therefore on $\lambda$. Thus, observed variations in layer $\mu$ should have no direct information about fluid content, while observed variations in layer $\lambda$, especially if they are large variations, may contain important clues about variations in fluid content. So the observed structure of $\epsilon$ in (22) strongly suggests that small positive and all negative values of $\epsilon$ may be important indicators of significant fluctuations in fluid content.

From (21), we can infer in general that

$$\begin{eqnarray} a \le \left<\lambda+2\mu\right\gt, \end{eqnarray}$$ (23)

so we have an upper bound on $\epsilon$ in finely layered media stating that

$$\begin{eqnarray} 2\epsilon \le \left(\left<\lambda+2\mu\right\gt\left<{{1}\over... ...rho v_p^{2}\right\gt\left<\rho^{-1}v_p^{-2}\right\gt - 1\right), \end{eqnarray}$$ (24)

where $\rho$ is the density.