Thomsen's $\epsilon$

An important anisotropy parameter for quasi-SV-waves (which is our main interest in this paper) is Thomsen's parameter $\epsilon$, defined in equation (2). Formula (12) for a may be rewritten as

$$\begin{eqnarray} a = \left<{{(\lambda+2\mu)^2 - \lambda^2}\over{\lambda+2\mu}}\right\gt + c\left<{{\lambda}\over{\lambda+2\mu}}\right\gt^2, \end{eqnarray}$$ (14)

which can be rearranged into the convenient and illuminating form

$$\begin{eqnarray} a = \left<\lambda+2\mu\right\gt - c\left[\left<{{\lambda^2}\ove... ...ight\gt -\left<{{\lambda}\over{\lambda+2\mu}}\right\gt^2\right]. \end{eqnarray}$$ (15)

This formula is very instructive because the term in square brackets is in Cauchy-Schwartz form [$\left<q^2\right\gt\left<Q^2\right\gt \ge \left<qQ\right\gt^2$], so this factor is nonnegative. Furthermore, the magnitude of this term depends mainly on the fluctuations in the $\lambda$ Lamé parameter, and is 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

$$\begin{eqnarray} 2\epsilon = \left[\left<\lambda+2\mu\right\gt\left<{{1}\over{\l... ...ight\gt -\left<{{\lambda}\over{\lambda+2\mu}}\right\gt^2\right], \end{eqnarray}$$ (16)

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, in general 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. Thomsen (2002) states that $\epsilon \gt 0$ if K and $\mu$ are positively correlated. But (16) shows that such correlations only produce $\epsilon \gt 0$ with certainty if they are also supplemented by the stronger condition that $\lambda \simeq const$[in fact, $\lambda \simeq const$ implies that there is a positive correlation between K and $\mu$, but the reverse does not necessarily hold unless we also assume that the fluctuations in $\mu$ are quite small -- an assumption that we do not make here].

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 (high spatial frequency) variations in layer shear modulus $\mu$ should have no direct information about fluid content, while such variations observed in layer Lamé parameter $\lambda$, especially if they are large variations, may contain important clues about variations in fluid content. So the observed structure of $\epsilon$ in (16) strongly suggests that small positive and all negative values of $\epsilon$ may be important indicators of significant fluctuations in fluid content (Berryman et al., 1999).