Thomsen's $\delta$

Thomsen's parameter $\delta$ defined by Eq. (3) is pertinent for near vertical quasi-P-waves and can also be rewritten as

$$\begin{eqnarray} \delta = - {{(c+f)(c - f - 2l)}\over{2c(c-l)}}. \end{eqnarray}$$ (17)

This parameter is considerably more difficult to analyze than either $\gamma$ or $\epsilon$ for various reasons, some of which we will enumerate shortly. Thomsen (2002) provides some insight into the behavior of $\delta$ by noting that its sign depends only on the variations of the ratio V_s/V_p. This can be seen to be true from its definition by noting that

$$\begin{eqnarray} c - f - 2l = 2cl\left[\left<{{1}\over{\mu}}\right\gt\left<{{\mu... ...ver{\mu}}\cdot\Delta\left({{V_s^2}\over{V_p^2}}\right)\right\gt, \end{eqnarray}$$ (18)

where

$$\begin{eqnarray} \Delta\left({{V_s^2}\over{V_p^2}}\right) \equiv {{V_s^2}\over{V_p^2}} - \left<{{V_s^2}\over{V_p^2}}\right\gt. \end{eqnarray}$$ (19)

Because of a controversy surrounding the sign of $\delta$ for finely layered media (e.g., Levin, 1988; Thomsen, 1988; Anno, 1997), Berryman et al. (1999) performed a series of Monte Carlo simulations with the purpose of establishing the existence or nonexistence of layered models having positive $\delta$.Those simulation results should be interpreted neither as modeling of natural sedimentation processes nor as an attempt to reconstruct any petrophysical relationships. The main goal was to develop a general picture of the distribution of the sign of $\delta$ using many choices of constituent material properties. This analysis established a similarity in the circumstances between the occurrence of positive $\delta$ and the occurrence of small positive $\epsilon$ (i.e., both occur when Lamé $\lambda$ is fluctuating greatly from layer to layer). The positive values of $\delta$ are in fact most highly correlated with the smaller positive values of $\epsilon$. We should also keep in mind the fact that $\epsilon - \delta \ge 0$ is always true for models with isotropic layers (Postma, 1955; Berryman, 1979) and this fact also plays a role in these comparisons, determining the unoccupied upper left hand corner of a $\delta$ vs. $\epsilon$ plot.