RANGE OF THE ANISOTROPY PARAMETERS

From (SHdispersion), I know that the SH-wave in finely layered VTI media has an elliptical surface for velocity squared. Furthermore,

m/l = <><1> 1   follows from (avel), (avem), and (CSineqform). So the horizontal shear wave velocity for SH-waves is always greater than or equal to the vertical velocity. I define the ratio m/l to be the SH-wave anisotropy parameter, and have the simple universal result that this parameter is always greater than or equal to unity.

The qP-wave does not always have an elliptical dispersion relation, but it is nevertheless always true that if k₃ = 0 then $\rho\omega_+^2 = ak_1^2$ and if k₁ = 0 then $\rho\omega_+^2 = ck_3^2$.Thus, I may reasonably define the P-wave anisotropy parameter to be a/c and seek to determine what the range of this parameter might be. 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 I now consider the combination

ac - 1 = [<+2><1+2> - 1] -[<^2+2><1+2> -<+2>^2],   I find that the first bracket on the right hand side is again in Cauchy-Schwartz form showing that it always makes a positive contribution 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.

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 I find that

ac - 1 = (_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 show clearly that if $\mu_1=\mu_2$ then the P-wave anisotropy parameter is identically equal to unity as expected. Also, if $\mu_1 \ne \mu_2$ but $\lambda_1 = \lambda_2$, then (Postma) implies $a/c \ge 1$, as I inferred from (Pwavep).

Now, I 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 I 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

ac - 1 = 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 the P-wave anisotropy is given by

ac = 1 - 34_2-_1_2 14.   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 I had used the nonnegativity constraint on the $\lambda$'s, the present result would have changed to

ac = 1 - _2-_12_2 12,   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), I note that it implies in general that

a <+2>,   so I have a general upper bound on the P-wave anisotropy parameter stating that

ac <+><1+2>.  

Before concluding this section, I want to note one further identity for the P-wave anisotropy parameter. The general formula can be rearranged to give

ac - 1 = 4[<(+)+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.