DISPERSION RELATIONS FOR SEISMIC WAVES

The equations of motion and their solutions for seismic waves in anisotropic media are well known, and have been derived in many places including Berryman (1979) and Thomsen (1986). The dispersion relations for phase velocities are

$$\begin{eqnarray} \rho\omega_{\pm}^2 = {{1}\over{2}} \left\{(a+l)k_1^2 + (c+l)k_3... ...\sqrt{[(a-l)k_1^2 - (c-l)k_3^2]^2 + 4(f+l)^2k_1^2k_3^2}\right\}, \end{eqnarray}$$ (29)

for quasi-compressional (+) waves and quasi-SV (-) waves (i.e., vertically polarized quasi-shear waves, by which we mean the plane normal to the cross-product of the polarization vector and the propagation vector is vertical) and

$$\begin{eqnarray} \rho \omega_s^2 = mk_1^2 + lk_3^2, \end{eqnarray}$$ (30)

for horizontally polarized shear waves. In these equations, $\rho$ is the overall density (including fluids when present), $\omega$ is the angular frequency, k₁ and k₃ are horizontal and vertical wavenumbers (respectively), and the phase velocities are determined simply by $V = \omega/k$ with $k = \sqrt{k_1^2 + k_3^2}$.Elastically, the SH wave depends only on the two parameters l and m, which are not dependent in any way on layer Lamé parameter $\lambda$ and, therefore, will play no role in the poroelastic analysis. The densities of any fluids present affect all three wave speeds equally, and cannot therefore contribute to shear wave bi-refringence by itself. Thus, we can safely ignore SH except when we want to check for shear wave splitting -- in which case the SH results will be most useful as a baseline for such comparisons.

The dispersion relations for quasi-P- and quasi-SV-waves can be rewritten in a number of instructive ways. One of these that we will choose for reasons that will become apparent shortly is

$$\begin{eqnarray} \rho\omega_{\pm}^2 = {{1}\over{2}} \left[(a+l)k_1^2 + (c+l)k_3^... ...2+ck_3^2)lk^2 + \{(a-l)(c-l)-(f+l)^2\}k_1^2k_3^2]}\right].\qquad \end{eqnarray}$$ (31)

Written this way, it is obvious that the following two relations hold:

$$\begin{eqnarray} \rho\omega_{+}^2 + \rho\omega_{-}^2 = (a+l)k_1^2 + (c+l)k_3^2, \end{eqnarray}$$ (32)

and

$$\begin{eqnarray} \rho\omega_{+}^2\cdot\rho\omega_{-}^2 = (ak_1^2+ck_3^2)lk^2 + [(a-l)(c-l)-(f+l)^2]k_1^2k_3^2, \end{eqnarray}$$ (33)

either of which could have been obtained directly from (29) without the intermediate step of (31).

We are motivated to write the equations in this way in order to try to avoid evaluating the square root in (29) directly. Rather, we would like to arrive at a natural approximation that is quite accurate, but does not involve the square root operation. The desire to do this is not new (Thomsen, 1986), but our goal is different since we must necessarily treat strong anisotropy in this paper. From a general understanding of the problem, it is clear that a reasonable way of making use of (32) is to make the identifications

$$\begin{eqnarray} \rho\omega_{+}^2 \equiv ak_1^2 + ck_3^2 - \Delta, \end{eqnarray}$$ (34)

and

$$\begin{eqnarray} \rho\omega_{-}^2 \equiv lk^2 + \Delta, \end{eqnarray}$$ (35)

with $\Delta$ still to be determined. Then, substituting these expressions into (33), we find that

$$\begin{eqnarray} (ak_1^2 + ck_3^2 - lk^2 - \Delta)\Delta = [(a-l)(c-l)-(f+l)^2]k_1^2k_3^2 \end{eqnarray}$$ (36)

Solving (36) for $\Delta$ would just give the original results back again. So the point of (36) is not to solve it exactly, but rather to use it as the basis of an approximation scheme. If $\Delta$ is small, then we can presumably neglect it inside the parenthesis on the left hand side of (36) -- or we could just keep a small number of terms in an expansion.

The leading term, and the only one we will consider here (but see the Appendix for further discussion), is

$$\begin{eqnarray} \Delta = {{[(a-l)(c-l)-(f+l)^2]k_1^2k_3^2}\over {(a-l)k_1^2 + (... ...simeq {{[(a-l)(c-l)-(f+l)^2]}\over {(a-l)/k_3^2 + (c-l)/k_1^2}}. \end{eqnarray}$$ (37)

The numerator of this expression is known to be a positive quantity for layers of isotropic materials (Postma, 1955; Berryman, 1979). Furthermore, it can be rewritten (without approximation) in terms of Thomsen's parameters as

$$\begin{eqnarray}[(a-l)(c-l)-(f+l)^2] = 2c(c-l)(\epsilon-\delta). \end{eqnarray}$$ (38)

Using the first of the identities noted earlier in (5), we can also rewrite the first elasticity factor in the denominator as $a-l = (c-l)[1+2c\epsilon/(c-l)]$. Combining these results in the limit of $k_1^2 \to 0$ (for relatively small horizontal offset), we find that

$$\begin{eqnarray} \rho\omega_{+}^2 \simeq ck^2 + 2c\delta k_1^2, \end{eqnarray}$$ (39)

and

$$\begin{eqnarray} \rho\omega_{-}^2 \simeq lk^2 + 2c(\epsilon-\delta)k_1^2, \end{eqnarray}$$ (40)

with $\Delta \simeq 2c(\epsilon-\delta)k_1^2$ for very small angles from the vertical. These two equations may be recognized simply as small angle approximations to the weak-anisotropy equations of Thomsen (1986). However, the main thrust of this paper (as we will soon see) requires strong anisotropy and therefore also requires improved approximations, which can be obtained to any desired order with only a little more effort by using (36) instead of the first approximation derived here in (37). Note that Eqs. (39) and (40) were derived without assumptions about the smallness of $\epsilon$ or $\delta$.

Although the approximations being discussed in this section are of some practical interest in their own right, their elaboration at this point would lead us away from the main theme of the paper. So, to avoid further digression here from the issue of fluid effects on shear modulus, we collect our remaining results concerning these dispersion relation approximations in the Appendix.