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Singular value decomposition

The singular value decomposition (SVD), or equivalently the eigenvalue decomposition in the case of a real symmetric matrix, for (6) is relatively easy to perform. We can immediately write down four eigenvectors:
{c} 0 0 0 1 0 0 \\ end{array}\right),\qqua...
{c} 1 -1 0 0 0 0 \\ end{array}\right),
 \end{eqnarray} (22)
and their corresponding eigenvalues, respectively 2l, 2l, 2m, and a-b = 2m. All four correspond to shear modes of the system. The two remaining eigenvectors must be orthogonal to all four of these and therefore both must have the general form
1, & 1, & \Omega, & 0, & 0, & 0 \\ end{array}\right)^T
 \end{eqnarray} (23)
with corresponding eigenvalue
\omega = a + b + f \Omega.
 \end{eqnarray} (24)
The remaining condition that determines both $\Omega$ and $\omega$is
\omega\Omega = 2f + c \Omega,
 \end{eqnarray} (25)
which, after substitution for $\omega$, yields a quadratic equation having the solutions
\Omega_{\pm} = {{1}\over{2}}\left(\left[{{c - a - b}\over{f}}\right]
\pm\sqrt{8 + \left[{{c - a - b}\over{f}}\right]^2}\right).
 \end{eqnarray} (26)
Then (26) and (28) imply that
\omega_+\omega_- = (a + b)c - 2f^2
 \end{eqnarray} (27)
\omega_+ + \omega_- = a + b + c = 2(a-m) + c,
 \end{eqnarray} (28)
which are two identities that will be used repeatedly later.

The ranges of values for $\Omega_\pm$ are $0 \le \Omega_+ \le \infty$ and (since $\Omega_- = -2/\Omega_+$) $-\infty \le \Omega_- \le 0$.The interpretation of the solutions $\Omega_\pm$ is simple for the isotropic limit where $\Omega_+ = 1$ and $\Omega_- = -2$, corresponding respectively to pure compression and pure shear modes. For all other cases, these two modes have mixed character, indicating that pure compression cannot be excited in the system, and must always be coupled to shear. Some types of pure shear modes can still be excited even in the nonisotropic cases, because the other four eigenvectors in (24) are not affected by this coupling, and they are all pure shear modes. Pure shear (in strain) and pure compressional (in stress) modes are obtained as linear combinations of these two mixed modes according to
\alpha \left(
{c} 1 1 \Omega_+ 0 0 0 \\ end{array...
{c} 1 1 -2 0 0 0 \\ end{array}\right),
 \end{eqnarray} (29)
with $\alpha = -2(\Omega_+-1)/[\Omega_+(\Omega_+ + 2)]$for pure shear, and
{c} 1 1 \Omega_+ 0 0 0 \\ end{array}\right...
{c} 1 1 1 0 0 0 \\ end{array}\right),
 \end{eqnarray} (30)
and with $\beta = \Omega_+(\Omega_+ - 1)/(\Omega_+ + 2)$for pure compression.

To understand the behavior of $\Omega_+$ in terms of the layer property fluctuations, it is first helpful to note that the pertinent functional $f(x) = {{1}\over{2}}\left[-x + \sqrt{8 + x^2}\right]$ is easily shown to be a monotonic function of its argument x. So it is sufficient to study the behavior of the argument x = (a+b-c)/f.

Exact results in terms of layer elasticity parameters

Combining results from Eqs.(11)-(14), we find after some work on rearranging the terms that
{{a+b-c}\over{f}} = \left<{{\lambda}\over{\lambda + 2\mu}}\righ...
 ... - \left<{{\mu}\over{\lambda + 2\mu}}\right\gt^2\right\}\right],
where $\Delta\mu \equiv \mu - \left<\mu\right\gt$ is the deviation of the shear modulus from the layer-averaged shear modulus m. Note that the term in curly brackets in (33) is in Cauchy-Schwartz form (i.e., $\left<\alpha^2\right\gt\left<\beta^2\right\gt
- \left<\alpha\beta\right\gt^2 \ge 0$) and therefore is always non-negative. This term is also effectively quadratic in the deviations of $\mu$from its layer average, and thus is of higher order than the term explicitly involving $\Delta\mu$. This fact, together with the fact that $\left<\Delta\mu/\mu\right\gt = 1 - \left<\mu\right\gt\left<1/\mu\right\gt
\le 0$, suggests that the dominant corrections to unity (since the leading term is exactly unity) for this expression will be positive if $\lambda$ and $\mu$ are positively correlated throughout all the layers, but the correction could be negative in cases where there is a strong negative correlation between $\lambda$ and $\mu$. If the fluctuations in shear modulus are very large throughout the layered medium, then the quadratic terms can dominate, in which case the result could be less than unity. Numerical examples developed by applying a code of V. Grechka [used previously in a similar context by Berryman et al. (1999)] confirm (and, in fact, motivated) these analytical results.

Our main conclusion is that the shear modulus fluctuations giving rise to the anisotropy due to layering are (as expected) the main source of deviations of (33) from unity. But, there are some other more subtle effects present having to do with the interplay between $\lambda$, $\mu$ correlations as well as the strength of the $\mu$fluctuations that ultimately determine the magnitude of the deviations of (33) from unity.

A fifth effective shear modulus?

From what has gone before, we know that there are four eigenvalues of the system that are easily identified with effective shear moduli (two are l and two are m). The bulk modulus K of the simple system is well-defined, and the bounds on Keff for polycrystals are quite simple to apply and interpret. But we are still missing an important element of the overall picture of this system, and that is how the remaining degree of freedom is to be interpreted. It seems clear that it should be imterpreted as an effective (quasi-)shear mode, since we have already accounted for the bulk mode. It is also clear that analysis of this remaining degree of freedom is not so easy because it is never an eigenfunction of the elasticity/poroelasticity tensor except in the cases that are trivial and therefore of interest to us here only as the isotropic baseline for comparisons.

Although it seems problematic to define a new shear modulus arbitrarily, we will now proceed to enumerate a number of possibilities and then, by a process of elimination, arrive at what appears to be a useful result.

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Stanford Exploration Project