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An important anisotropy parameter for qP-waves is Thomsen's parameter $\epsilon$, defined by

= a-c2c.   Formula (avea) for a may be rewritten as

a = <(+2)^2 - ^2+2> + <+2>^2 <1+2>^-1,   which can be rearranged into the convenient and illuminating form

a = <+2> - [<^2+2> <1+2> -<+2>^2] <1+2>^-1.   This formula is very instructive because the term in square brackets is again in Cauchy-Schwartz form, so this factor is nonnegative. Furthermore, the magnitude of this term depends principally on the fluctuations in the $\lambda$ Lamé constant, largely independent of $\mu$. Clearly, if $\lambda= constant$,then this factor vanishes identically, regardless of the behavior of $\mu$. Large fluctuations in $\lambda$ will tend to make this term large. If in addition we now consider the combination

2= [<+2><1+2> - 1] -[<^2+2><1+2> -<+2>^2],   we find that the first bracket on the right hand side is again in Cauchy-Schwartz form, which makes it always positive unless $\lambda+ 2\mu= constant$, in which case it vanishes. Similarly, the second term always makes a negative contribution unless $\lambda= constant$, in which case it vanishes.

The properties of the isotropic layers making up the anisotropic rocks depend on mineralogy, porosity, and fluid content. We can gain some insight into how the relative sizes of $\lambda$ and $\mu$ are related to mineralogy by considering the measured values of these stiffnesses for quartz and calcite. Simmons and Brace (1965) report measured values of about 28 to 31 GPa for the shear modulus of fused quartz, and measured values of about .027 to .032 GPa-1 for the compressibility, corresponding to values of about 13 to 16 GPa for $\lambda$. (These measurements were made for confining pressures of 0.1 MPa to 10 GPa.) Wilkens et al. (1984) gave ultrasonic velocity values measured in single crystals of quartz and calcite that yield values of about 44 GPa for the shear modulus of quartz, 8.4 GPa for $\lambda$ for quartz, 31 GPa for the shear modulus of calcite, and 55 GPa for $\lambda$ for calcite. (These measurements were made without accounting for the anisotropy in quartz and calcite crystals. See Aleksandrov and Ryzhova (1961) for measured values of all the elastic stiffnesses, for single calcite crystals and quartz crystals.) Both the static and the dynamic measurements show that $\mu$ values in quartz and calcite are similar, whereas the $\lambda$ values differ considerably for the two minerals, with $\lambda$ being larger than $\mu$ for calcite and smaller than $\mu$ for quartz. We also know that vp/vs ratios in sandstones are close to 1.5, and vp/vs ratios are about 1.8 to 2.0 for sedimentary rocks containing calcite, dolomite, or feldspar (e.g., Wilkens et al., 1984; Castagna et al., 1985). If $\lambda$ is approximately equal to $\mu$, the vp/vs ratio is 1.7. Rocks having $\lambda$ larger than $\mu$ have vp/vs ratios larger than 1.7, and rocks having $\lambda$ smaller than $\mu$ have vp/vs ratios smaller than 1.7.

Although the stiffnesses of clay minerals can be found in the literature (e.g., Katahara, 1996), it is more useful to consider instead the velocities of rocks containing clays (e.g., Tosaya and Nur, 1982; Han et al., 1986) because the clays themselves typically contain micropores and the location of the clays in the microstructure (e.g., lining pores, or between grains) is more important than mineralogy, for elastic properties (Caruso et al., 1985; Wilkens et al., 1986). Porosity does not directly affect $\lambda$ and $\mu$ values of the grains, but since it affects the velocities, it affects the vp/vs ratio. Clay content and the presence of pores both tend to increase vp/vs ratios (Castagna et al., 1985). If randomly-oriented flat cracks are present, they will also tend to increase the vp/vs ratio, since vs is more sensitive than vp to flat cracks (Wilkens et al., 1984).

Fluctuations in $\lambda$ in the earth have important implications for oil and gas exploration. 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 can have a significant effect on the bulk modulus $K = \lambda+ {2\over3}\mu$, and therefore on $\lambda$. Thus, observed variations in $\mu$ have no direct information about fluid content, while observed variations in $\lambda$, especially if they are large variations, may contain imporant clues about variations in fluid content. So the observed structure of $\epsilon$ in (Pwavep) strongly suggests that small positive and all negative values of $\epsilon$ may be important indicators of significant fluctuations in fluid content.

If the finely layered medium is composed of only two distinct types of isotropic elastic materials and they appear in the layering sequence with equal spatial frequency, then we find that

2= (_2-_1)(_2-_1) + (_2-_1) (_1+2_1)(_2+2_2).   This result agrees with Postma (1955) except for an obvious typographical error in the denominator of his published formula. This formula shows clearly that if $\mu_1=\mu_2$ then Thomsen parameter $\epsilon$ is identically equal to zero as expected. Also, if $\mu_1 \ne \mu_2$ but $\lambda_1 = \lambda_2$, then (Postma) implies $\epsilon \ge 0$, as we inferred from (Pwavep).

Now, we can use this formula to deduce the smallest possible value of the right hand side of (Postma). The shear moduli must not be equal, so without loss of generality we suppose that $\mu_2 \gt \mu_1$.Then, the numerator is seen to become negative by taking $\lambda_2$towards negative values and $\lambda_1 \to + \infty$. The smallest value $\lambda_2$ can take is determined by the bulk modulus bound $\lambda_2 + {2\over3}\mu_2 \ge 0$. So we may set $\lambda_2 = - {2\over3}\mu_2$in both the numerator and denominator. This choice also makes the factor $\lambda_2 + 2\mu_2 = {4\over3}\mu_2$ as small as possible in the denominator, thus helping to magnify the effect of the negative numerator as much as possible. The result so far is that

2= 34(_2-_1_2) (-_1+_2/3-_1_1+2_1).   The parameter $\lambda_1$ may vary from $-{2\over3}\mu_1$ to plus infinity. At $\lambda_1 = -{2\over3}\mu_1$, the second expression in parentheses is positive; but, this expression is also a monotonically decreasing function of $\lambda_1$ and approaches -1 as $\lambda_1 \to + \infty$.Thus, the smallest value of Thomsen's parameter $\epsilon$ is given by

= - 38_2-_1_2 -38.   This result differs by a factor of 2 from the corresponding result of Postma (1955), which was obtained improperly by allowing three of the four elastic constants to vanish and also using a physically motivated but unnecessary restriction that both $\lambda_1$ and $\lambda_2$ must be nonnegative. If we had used the nonnegativity constraint on the $\lambda$'s, the present result would have changed to

= 1 - _2-_12_2 -14,   which is the same inequality as that found by Postma, but his equality differed from that in (restricted) and was in fact improperly obtained.

As a final point about the formula (arearranged), we note that it implies in general that

a <+2>,   so we have an upper bound on $\epsilon$ in finely layered media stating that

2 (<+2><1+2>-1) = (<v_p^2><v_p^-2> - 1),   where the final form is true for layered media having constant density.

Before concluding this section, we want to note one further identity for $\epsilon$.The general formula can be rearranged to give

= 2[<(+)+2> <1+2> -<++2> <+2>].   This formula is not in Cauchy-Schwartz form, but is nevertheless probably the simplest form of the result for this anisotropy parameter. In particular, it is easy to see from this form that if either $\mu= constant$ or $\lambda+ \mu= constant$, then the right hand side vanishes identically. The first result is well-known and the second has been known since Postma's (1955) work to be true for two-constituent periodic layered media [also see (Postma)]. The present result generalizes Postma's observation in this case.

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