On anelliptic approximations for qP velocities in TI and orthorhombic media

Appendix A: Uncertainty Analysis

In this appendix, we study the variation of the phase-velocity expression in TI media with respect to different choices of parameters, particularly Thomsen's parameters and the proposed Muir-Dellinger parameters. To study resolution, we use the general formula,

$$R_{ij} = \int_0^{\pi/2} \frac{\partial V^2}{\partial m_i} \frac{\partial V^2}{\partial m_j} d\theta~,$$ (65)

where $V$ is the exact phase-velocity expression (equation 9), $m_i$ and $m_j$ are two of the four parameters present in the expression, and $\theta$ is the phase angle measured from vertical. The matrix $R_{ij}$ , in both cases, is computed based on the stiffness tensor of Greenhorn shales given in Table 6. The results are shown in Tables 10 and 11. Note that the matrix is symmetric, so the values are shown only on one side of the diagonal.

Parameters	$V_{P0}$	$V_{S0}$	$\epsilon$	$\delta$
$V_{P0}$	87.11	0.467	105.39	25.19
$V_{S0}$		0.005	0.649	0.20
$\epsilon$			181.74	22.54
$\delta$				13.16

Table 10. $R_{ij}$ of the exact phase-velocity expression with the two considered Thomsen parameters are denoted in each row and column.

Parameters	$w_1$	$w_3$	$q_1$	$q_3$
$w_1$	0.576	0.143	0.651	0.599
$w_3$		0.538	0.257	1.061
$q_1$			1.325	1.279
$q_3$				4.124

Table 11. $R_{ij}$ of the exact phase-velocity expression with the two considered anelliptic parameters are denoted in each row and column.

Table 10 shows a significantly larger correlation between the change in phase velocity with $V_{P0}$ in comparison with that of $V_{S0}$ , which agrees with the general assumption of the independency of $V_{S0}$ in qP velocities approximations. Likewise, the effect from $\epsilon$ has a higher correlation with the change of phase velocity than $\delta$ because the exact qP phase-velocity formula (equation 9) can be expressed in terms of Thomsen parameters with $\epsilon$ corresponding to the lower order of $\sin \theta$ than $\delta$ . Moreover, $\epsilon$ and $\delta$ also have high correlation with $V_{P0}$ , which is apparent from their definitions.

Table 11 shows relatively similar correlations from $w_1$ and $w_3$ to the change in exact phase velocity suggesting a more symmetric contribution from both parameters. The dimensionless anelliptic parameters $q_1$ and $q_3$ exhibit a strong correlation, which is consistent with the relationships shown in Figure 1.

By ignoring the effect of $V_{S0}$ in the case of Thomsen parameters or using the relationship between $q_1$ and $q_3$ (Figure 1) to reduce the number of parameters to three, we can transform the matrix $R_{ij}$ from $4\times4$ to $3\times3$ ( $\tilde{R}_{ij}$ ). Note that the matrix for Thomsen parameters is similar to Table 10 with the omittance of the row and column associated with $V_{S0}$ . Table 12 shows the three-parameter matrix for anelliptic parameters with similar behavior of relatively equal correlations from $w_1$ and $w_3$ as before.

Parameters	$w_1$	$w_3$	$q_3$
$w_1$	0.578	0.144	1.166
$w_3$		0.534	1.286
$q_1$			7.411

Table 12. $\tilde{R}_{ij}$ of the exact phase-velocity expression with the two considered anelliptic parameters are denoted in each row and column.

To better visualize the variational effect from the change in the three parameters in both cases, we follow the approach of Osypov et al. (2008), compute the quadratic form of $\tilde{R}_{ij}$ and plot its contour at a given amount of change in the exact phase velocity expression,

$$\Delta V^2= \mathbf{x}^T \tilde{R}_{ij} \mathbf{x}~,$$ (66)

where $\mathbf{x}$ denotes the vector of parameter variations: [ $\Delta V_{P0}$ , $\Delta \epsilon$ , $\Delta \delta$ ]$^T$ or [ $\Delta w_1$ , $\Delta w_3$ , $\Delta q_3$ ]$^T$ and $\tilde{R}_{ij}$ is computed at the known values of the anisotropic parameters of the model (Greenhorn shales). The resultant plots are shown in Figure A-1. For Thomsen's parameters, Figure A-1a shows a strongly oblate ellipsoid with high degree of deviation (stretch) from a sphere for all three parameters. On the contrary, Figure A-1b shows oblate ellipsoid with smaller deviation suggesting that the Muir-Dellinger parameters may represent a more orthogonal parameterization scheme than Thomsen's parameters. This observation is important for the problem of estimating anisotropic parameters, which goes beyond the scope of this paper.

thomsenmatrix,zonematrix
Figure 14. Ellipsoids obtained from the quadratic form of $\tilde{R}_{ij}$ in the case of a) Thomsen parameters b) anelliptic parameters.

On anelliptic approximations for qP velocities in TI and orthorhombic media