On anelliptic approximations for qP velocities in TI and orthorhombic media

Choice of anisotropic parameters

Previous researchers have shown that in TI and orthorhombic media, only three and six combinations of stiffness coefficients respectively are sufficient to describe qP-wave propagation. The most widely used parameters are Thomsen's parameters (Thomsen, 1986) in TI media and their extension to orthorhombic media (Tsvankin, 1997). Another notable parameterization scheme involves the $\eta$ parameter (Alkhalifah and Tsvankin, 1995) and is commonly used when the so-called acoustic approximation is assumed (Alkhalifah, 1998,2000b,2003,2000a). In this study, following the work of Muir and Dellinger (1985) and Dellinger et al. (1993), we adopt the set of anisotropic parameters, referred to as Muir-Dellinger parameters, which represent different combinations of elastic moduli.

In our notation, the Muir-Dellinger parameters include $w_1$ , $w_3$ , $q_1$ , and $q_3$ , where $w_1$ denotes the horizontal velocity squared and $w_3$ denotes the vertical velocity squared, and $q_1$ and $q_3$ are anelliptic parameters derived from fitting the phase velocity curvatures along the horizontal axis and vertical axis. In terms of density-normalized stiffness tensor coefficients in Voigt notation, $w_1=c_{11}$ , $w_3=c_{33}$ , and

$$q_1 = \frac{c_{55}(c_{11}-c_{55})+(c_{55}+c_{13})^2}{c_{33}(c_{11}-c_{55})}~,$$ (1)

$$q_3 = \frac{c_{55}(c_{33}-c_{55})+(c_{55}+c_{13})^2}{c_{11}(c_{33}-c_{55})}~.$$ (2)

Equations 1 and 2 were derived previously by Muir and Dellinger (1985). Thomsen's parameters and the parameter $\eta$ used by Alkhalifah and Tsvankin (1995) are related to $q_1$ and $q_3$ by the conversion shown in Table 1. Table 2 shows a comparison between parameterization schemes for anisotropic parameters in TI media. Uncertainty analysis for different choices of anisotropic parameters in TI media is discussed in Appendix A.

Anelliptic	Thomsen
$q_1 = 1+\frac{2(R-1)(\delta-\epsilon)}{R-(1+2\epsilon)}$	$\delta = \frac{(q_3-q_1)+R(q_1q_3-2q_3+1)}{2(q_1-q_3 +R (q_3-1))}$
$q_3 = \frac{1+2\delta}{1+2\epsilon} = \frac{1}{1+2\eta}$	$\epsilon = \frac{(q_1-q_3)(R-1)}{2(q_1-q_3 +R (q_3-1))}$

Table 1. Conversion table between anelliptic parameters, Thomsen's parameters (Thomsen, 1986) and the time-processing parameter $\eta$ (Alkhalifah and Tsvankin, 1995). $R=V^2_{S0}/V^2_{P0}$ denotes the ratio between vertical velocity squared of qP-wave and qS-wave.

Schemes	Parameters	Elliptical Ani.	Acoustic Approx.
Thomsen (1986)	$V_{P0}$ , $V_{S0}$ , $\epsilon$ , and $\delta$	$\epsilon = \delta$	$V_{S0}=0$
Alkhalifah (1998)	$V_{P0}$ , $V_{S0}$ , $V_{nmo}$ and $\eta$	$\eta=0$	$V_{S0}=0$
Muir and Dellinger (1985),
Fomel (2004), and proposed

Table 2. Comparison of four-parameter parameterization schemes for qP-wave anisotropic parameters.

To study correlations between anisotropic parameters, we compiled laboratory measurements of stiffness tensor elements or other equivalent measurements of anisotropy (Jones and Wang, 1981; Thomsen, 1986; Vernik and Liu, 1997; Wang, 2002; Schoenberg and Helbig, 1997). Computing $q_1$ and $q_3$ according to equations 1 and 2, we discover an empirical relationship between anelliptic parameters shown in Figure 1. The results suggest that $q_1$ and $q_3$ exhibit a linear relationship ( $q_1 = a q_3 + b$ ) that appears to depend solely on lithology, regardless of the geographical location of the samples. We can also observe that the proportionality constant $a$ is close to 1 for nearly isotropic rocks (carbonates and sandstones) but deviates from 1 for highly anisotropic rocks (shales). We assume that each relationship between $q_1$ and $q_3$ given in Figure 1 is valid for that type of media and therefore, the proportionality constant $a$ and the intercept $b$ are not new independent parameters.

In consideration of a linear relationship ( $q_1 = a q_3 + b$ ) above, the ratio of veritical qP- and qSV-wave velocites squared $R=V^2_{S0}/V^2_{P0}$ can be expressed as,

$$R = \frac{(1+2\epsilon)[(1+2\delta)(a-1) + b(1+2\epsilon)]}{a(1+2\delta)+(1+2\epsilon)(b-1-2(\delta-\epsilon))}~.$$ (3)

Therefore, if $a=1$ and $b=0$ , $R=0$ . This result agrees with that of Fomel (2004), who previously showed that if $w_1 \ne w_3$ , $q_1 \ne 1$ , and $q_3 \ne 1$ , setting $q_1=q_3$ is equivalent to the assumption used in the acoustic approximation (Alkhalifah, 1998). In other words,

$$\lim_{R \to 0} q_1 = q_3~.$$ (4)

A similar condition is discussed by Fowler (2003).

qshalelegnew,qsandlegnew,qcarbonateleg
Figure 1. Relationship between $q_1$ and $q_3$ in different lithology. Data are obtained from various publications on laboratory measurements. Dashed line indicates graph of $q_1=q_3$ in each case.

Because the in-plane qP-wave propagation in orthorhombic media behaves identically to the case of TI media (Tsvankin, 2012), we can extend the Muir-Dellinger parameters from 2D to 3D appropriately for studies of orthorhombic media. The full set of parameters in 3D includes $w_1$ , $w_2$ , $w_3$ , $q_{21}$ , $q_{31}$ , $q_{12}$ , $q_{32}$ , $q_{13}$ , and $q_{23}$ , where $w_i$ denotes the velocity squared in the $n_i$ direction and $q_{ij}$ denotes the anelliptic parameters derived from fitting the phase velocity curvatures along the $n_i$ axis in the symmetry plane defined by the $n_j$ axis (Figure 3b). For example, in plane [$n_1$ ,$n_3$ ], we consider $q_{12}$ and $q_{32}$ because we can find $q$ either by fitting along $n_1$ or $n_3$ axis with $n_2$ as the axis defining the symmetry plane. The expressions for the anelliptic parameters are as follows:

$$q_{21} = \frac{c_{44}(c_{22}-c_{44})+(c_{44}+c_{23})^2}{c_{33}(c_{22}-c_{44})}~,$$ (5)

$$q_{31} = \frac{c_{44}(c_{33}-c_{44})+(c_{44}+c_{23})^2}{c_{22}(c_{33}-c_{44})}~,$$ (6)

$$q_{13} = \frac{c_{66}(c_{11}-c_{66})+(c_{66}+c_{12})^2}{c_{22}(c_{11}-c_{66})}~,$$ (7)

$$q_{23} = \frac{c_{66}(c_{22}-c_{66})+(c_{66}+c_{12})^2}{c_{11}(c_{22}-c_{66})}~.$$ (8)

Expressions for $q_{12}$ and $q_{32}$ are equivalent to expressions for $q_1$ and $q_3$ in equations 1 and 2. For subscript convention on Muir-Dellinger parameters and associating expressions in 3D, we reserve $i$ , $j$ , and $k$ specifically to indicate the fitting location in each symmetry plane. The summation convention for repeating indices is not assumed. Alternatively, we adopt the notation that, for each combination of $i$ , $j$ , and $k$ , the first digit indicates the index of the fitting axis and the second digit indicates the index of the axis defining the symmetry plane. Therefore, in our notation, $i$ , $j$ , and $k$ are integers between 1 and 3 and in each expression, they must be different from one another. This set of $q_{ij}$ parameters may also be considered as elements of a 3$\times$ 3 matrix with the diagonal elements of this matrix absent. Table 3 shows a comparison between parameterization schemes for anisotropic parameters in orthorhombic media.

Schemes	Parameters	Acoustic Approx.
	$V_{P0}$ , $V_{S0}$ , $\delta^{(3)}$ ,
			$\epsilon^{(1)}$ , $\delta^{(1)}$ , $\gamma^{(1)}$ ,
			$\epsilon^{(2)}$ , $\delta^{(2)}$ , $\gamma^{(2)}$
	$V_v$ , $V_{S1}$ , $V_{S2}$ ,	$V_{S1}=0$
		$V_{S3}$ , $V_{nmo_1}$ , $V_{nmo_2}$ ,	$V_{S2}=0$
		$\eta_1$ , $\eta_2$ , $\delta$	$V_{S3}=0$
	$w_1$ , $w_2$ , $w_3$ ,	$q_{21} = q_{31}$ ,
		$q_{12}$ , $q_{21}$ , $q_{13}$ ,	$q_{12} = q_{32}$
		$q_{31}$ , $q_{23}$ , $q_{32}$	$q_{13} = q_{23}$

Table 3. Comparison of parameterization schemes for qP-wave anisotropic parameters and the assumption for acoustic approximation in orthorhombic media.

On the basis of the Muir-Dellinger parameters in TI and orthorhombic media introduced in this section, we propose novel phase- and group-velocity approximations for qP waves in the subsequent sections. We first suggest a symmetric extension of the velocity approximations in TI media by Fomel (2004) and also extend the expressions to the orthorhombic case. We then utilize the relationships from Figure 1 to reduce the number of independent parameters in the proposed approximations to three and six in TI and orthorhombic media respectively so that they require the similar number of parameters as the other previously suggested approximations.

On anelliptic approximations for qP velocities in TI and orthorhombic media