SUMMARY OF RESULTS ON THOMSEN PARAMETER BEHAVIOR

The results concerning the signs of the Thomsen parameters obtained in the preceding sections are summarized in Table 1.

1.2

TABLE 1.Behavior of anisotropy parameters as the layer material elastic parameters vary. All parameters vanish when $\mu= const$.

$$0.15in] \begin{tabular} {\vert c\vert c\vert c\vert c\vert ... ...space & $\ge 0$\space & $\ge 0$\space \hline\hline\end{tabular}$$

The nonnegativity of $\gamma$ and $\epsilon - \delta$ for layered models is well-known. The fact that $\epsilon$ can be either positive or negative, and the circumstances leading to negative values have been little appreciated before. A quick glance at the Table seems to indicate that $\delta$ is either zero or negative and this is perhaps why there has been so much confusion about the possibility of $\delta \gt 0$ for layered models. We have shown that the correct inference about positive $\delta$ follows rather simply from the result expressed in the last column of the Table. Since $\delta = 0$ when Poisson's ratio is constant or when $\mu= constant$, it is clear that we must have (for example) $\mu_1/\mu_2 \gt 1$ and $\lambda_1/\mu_1 \ne \lambda_2/\mu_2$ if $\delta$ is to be positive. The second expression can be rearranged to $\lambda_1/\lambda_2 \ne \mu_1/\mu_2 \gt 1$. Now we see that there are two possible cases, either $\lambda_1/\lambda_2 \gt \mu_1/\mu_2 \gt 1$ or $\lambda_1/\lambda_2 < \mu_1/\mu_2$. The first case results in positive $\delta$, while the second case does not when $\mu_1/\mu_2 \gt 1$.This also shows that in perfectly random layered media one expects $\delta \gt 0$ to account for 50% of the models. In our simulation we found $\delta \gt 0$ accounted for only about 25% of the models produced, but this apparent discrepancy has been traced to a bias in the particular algorithm we used to generate the models. In any case, the earth does not have to obey perfectly random statistics and there is no reason to suppose that real layered earth will conform to these statistical considerations. Our main point is merely that $\delta \gt 0$is entirely possible and quite understandable for layered earth models, and so it is not at all surprising that $\delta \gt 0$ is often observed in real data examples.