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Analysis for isotropic layers

The analysis presented in the previous subsection is general for all VTI elastic media. But we can say more by assuming now that the anisotropy arises due to layers of isotropic elastic (or possibly poroelastic) media. Then, using (8)-(12), we have the following relations
f + 2l = c\left<{{\lambda + 2l}\over{\lambda + 2\mu}}\right\gt,
 \end{eqnarray} (51)
c - f - 2l = 2c \left<{{\mu - l}\over{\lambda + 2\mu}}\right\gt,
 \end{eqnarray} (52)
a - f - 2l = 2c \left\{\left<{{2m-\mu -l}\over{\lambda + 2\mu}}...
- \left<{{\mu}\over{\lambda + 2\mu}}\right\gt^2\right]\right\}.
 \end{eqnarray} (53)
Eq. (51) is an easy consequence of the Backus averaging formulas. Then, (52) shows that c differs from f + 2l only by a term that measures the difference in the weighted average of $\mu$ and l. Eq. (53) shows that a differs from f + 2l in a more complicated fashion that depends on the difference in the weighted average of (2m-l) and $\mu$, as well as a term that is higher order in the fluctuations of the layer $\mu$ values. Combining these results, we have
G_{eff} = m - {{4c}\over{3}}\left[\left<{{\mu^2}\over{\lambda +...
- \left<{{\mu}\over{\lambda + 2\mu}}\right\gt^2\right],
 \end{eqnarray} (54)
showing that all the interesting behavior (including strong $\mu$fluctuations in the layers together with $\lambda$ dependence) is collected in Geff. Since the product of (52) and (53) is clearly of higher order in the fluctuations of the layer shear moduli, it is not hard to see that, to leading order when these fluctuation effects are small,
{\cal A} \simeq (c-l)(3G_{eff} + m - 4l).
 \end{eqnarray} (55)
To give a quick estimate, note that if all the layers have the same value of Poisson's ratio, then the ratio $r = \lambda/\mu$ is constant. Then, it is easy to show that Geff = m - 4(m-l)/3(2+r). Since $-2/3 \le r \le \infty$, the effective shear modulus for this class of models lies in the range $l \le G_{eff} \le m$.From this fact, we can conclude that the important coefficient in (40) is given to a good approximation by
2c(\epsilon-\delta) \simeq 3G_{eff} + m - 4l,
 \end{eqnarray} (56)
and ranges from $2l\gamma$ to $8l\gamma$.

To study the fluid effects, the drained Lamé parameter $\lambda$ in each layer should be replaced under undrained conditions by
\lambda^* = K^* - 2\mu/3,
 \end{eqnarray} (57)
where K* was defined by (6). Then, for small fluctuations in $\mu$, Eq. (56) shows that the leading order terms due to these shear modulus variations contributing to $\epsilon - \delta$ actually do not depend on the fluids at all (since m-l does not depend on them). With no fluid in the pores, there is a contribution to the shear wave speed for SV in layered media, just due to the fluctuations in the shear moduli. One part of the contribution is always independent of any fluids that might be present, but the magnitude of this contribution (which is always positive) is small whenever the difference m-l is also small. If m-l is large, then the magnitude of the additional increase due to liquids in the pores can be very substantial as we will see in the following examples. So the effects of liquids on Geff will generally be weak when the fluctuations in $\mu$ are weak, and strong when they are strong.

Furthermore, when the product $\alpha B \ne 0$,we first choose to define
ratio_{\alpha B} = {{m-G_{eff}}\over{m-l}}.
 \end{eqnarray} (58)
so that, for all possible layered models, we have $0 \le ratio_{\alpha B} \le 1$. Then, we consider plotting the quantity $1 - ratio_{\alpha B}/ratio_0$ versus $\gamma$ (which we treat as a simple quantitative measure of the fluctuations in the layer shear moduli). To generate a class of 900 models for each of three choices of $\alpha$ (treated as a single constant for all layers in each individual model) in order to illustrate the behavior of these quantities, I made use of a code of V. Grechka [used previously in a joint publication (Berryman et al., 1999)]. This code chooses layer parameters randomly from within the following (arguable, but generally reasonable) range of values: $1.5 \le V_p \le 5.0$ km/s, $0.1 \le V_s/V_p \le 0.8$, and $1.8 \le \rho \le 2.8$ $\times 10^3$ kg/m3. The results are displayed in Figure 1 for $\alpha = 0.5$, 0.8, and 0.9. We find empirically that (for B = 1) the values never exceed $\alpha$ for any set of choices for the layer model parameters. This apparent fact (as determined by these computer experiments) does not appear to be easy to prove from the general formula. But one simple though nontrivial calculation we can do is based again on an assumption that the bulk moduli in the layers are always proportional to the shear modulus, so $K = s\mu$, for some fixed value of of the proportinality factor s > 0. Then, for a given model, we find that
1 - {{ratio_{\alpha B}}\over{ratio_0}} = 
{{\alpha B}\over{1 + 4(1-\alpha B)/3s}} \le \alpha B,
 \end{eqnarray} (59)
in agreement with the empiricial result from the synthetic data shown in Figure 1.

Figure 1
Scatter plot illustrating how Geff varies over a physically sensible range of layered isotropic media (see text for details) with 2700 distinct models and B = 1 [see Eq. (58) in the text for the definition of $ratio_\alpha$]. Blue dots are for $\alpha = 0.9$, red for $\alpha = 0.8$,and green for $\alpha = 0.5$. Note, that in each case, all the points for a particular choice of $\alpha$ are bounded above precisely by the value of $\alpha$.(A general proof of this empirical observation is currently lacking.)


To check the corresponding result for P-waves, we need to estimate $\delta$.Making use of (50), we have
c\delta = -2c\left<{{\mu - l}\over{\lambda + 2\mu}}\right\gt
 \end{eqnarray} (60)
Working to the same order as we did for the final expression in (56), we can neglect the second term in the square brackets of (60). What remains shows that pore fluids would have an effect on this result. The result is
c^*\delta^* \simeq -2c^*\left<{{\mu - l}\over{\lambda^* +
 \end{eqnarray} (61)
If desired, a similar replacement can also be made for Geff in (44) using the fact that $2(\mu_3^* - l) = c - f - 2l$. Eq. (61) shows that, since c* and $\delta^*$both depend on the $\lambda^*$'s (although in opposite ways, since one increases while the other decreases as $\lambda^*$ increases), the product of these factors will have some dependence on fluids. The degree to which fluctuations in $\lambda^*$ and $\mu$ are correlated, or anticorrelated, as they vary from layer to layer will also affect these results in predictable ways.

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