On anelliptic approximations for qP velocities in TI and orthorhombic media

Examples

To evaluate the accuracy of the proposed approximations, we produce relative error plots and tables, using several sets of normalized stiffness tensor measurements summarized in Table 7, which can be converted to any parameterization scheme (Table 3). The error plots in Figures 4-10 are generated using the standard model (Schoenberg and Helbig, 1997) and are presented as both 3D surfaces and sterographic projections with $\theta$ (or $\Theta$ ) changing radially and $\phi$ (or $\Phi$ ) changing azimuthally. The standard model assumes a shale background with a set of parallel vertical cracks; therefore, we use the following relationship between anelliptic parameters in shales: $q_1 = 0.83734 q_3 +0.1581$ to reduce the number of parameters in the vertical [$n_1$ ,$n_3$ ] and [$n_2$ ,$n_3$ ] planes. The anisotropy in the horizontal plane [$n_1$ ,$n_2$ ], on the other hand, corresponds to a different cause, which in this case is assumed to be vertical fractures. Because we do not know a proper relationship between anelliptic parameters for such feature, we resort to the previously used assumption of $q_{13} = q_{23}$ . Tables 8 and 9 show RMS relative error results of our approximations in comparison with results from some of the previously suggested approximations , which are computed based on

$$= \sqrt{\sum_{\psi_1=0}^{90} \sum_{\psi_2=0}^{90} (v_{exact}(\psi_1,\psi_2)-v_{approx}(\psi_1,\psi_2))^2}~,$$ (61)

where $\psi_1$ and $\psi_2$ denote the zenith and azimuthal phase or group angles as appropriate. The best-performing approximation is denoted in red and bold. In all examples, the proposed approximations appear to be significantly more accurate than the other known approximations.

Sample	$c_{11}$	$c_{22}$	$c_{33}$	$c_{44}$	$c_{55}$	$c_{66}$	$c_{12}$	$c_{23}$	$c_{13}$
1. Standard model	9	9.84	5.938	2	1.6	2.182	3.6	2.4	2.25
2. Tsvankin 1	11.7	13.5	9	1.728	1.44	2.246	8.824	5.981	5.159
3. Tsvankin 2	17.1	13.5	9	1.728	1.44	2.246	9.772	4.580	7.745
4. Alkhalifah 1	1.452	2.016	1	0.25	0.25	0.25	1.089	0.695	0.599
5. Alkhalifah 2	1.452	2.016	1	0.49	0.36	0.49	0.608	0.206	0.378

Table 7. Normalized stiffness tensor coefficients (in $km^2$ /$s^2$ ) from different orthorhombic samples: 1 is from Schoenberg and Helbig (1997), 2 and 3 are from Tsvankin (1997), and 4 and 5 are from Alkhalifah (2003).

Sample	Tsvankin (1997)	Alkhalifah (2003)	Proposed
1	0.5787	0.1742	0.1029
2	0.5918	0.0645	0.0275
3	0 .7104	0.0952	0.0637
4	0.8960	0.1382	0.0293
5	1.0736	0.3274	0.2084

Table 8. RMS relative error (%) from 0 - $90\,^{\circ}$ (both $\theta$ and $\phi$ ) of orthorhombic phase-velocity approximations by Tsvankin (1997), Alkhalifah (2003), and of proposed six-parameter approximation. Bold red highlight indicates the best-performing approximation. In all cases, the proposed approximation appears to be the most accurate.

Sample	Xu-Vasconcelos	Proposed
1	0.8985	0.1446
2	0.6066	0.1354
3	0.7966	0.0311
4	0.4907	0.0387
5	0.4588	0.1729

Table 9. RMS relative error (%) from 0 - $90\,^{\circ}$ (both $\Theta$ and $\Phi$ ) of orthorhombic group-velocity approximations by Xu et al. (2005) and Vasconcelos and Tsvankin (2006), and of proposed six-parameter approximation. Bold red highlight indicates the best-performing approximation. In all cases, the proposed approximation appears to be more accurate.

phaseweak90leglow,phaseweakleglow
Figure 4. Relative error of phase-velocity approximation by Tsvankin (1997). a) from azimuth 0 to $90\,^{\circ}$ . b) from azimuth 0 to $360\,^{\circ}$ .

phaseacoustic90leglow,phaseacousticleglow
Figure 5. Relative error of phase-velocity approximation by Alkhalifah (2003). a) from azimuth 0 to $90\,^{\circ}$ . b) from azimuth 0 to $360\,^{\circ}$ .

phasemshappq90leglownew,phasemshappqleglownew
Figure 6. Relative error of proposed six-parameter phase-velocity approximation. a) from azimuth 0 to $90\,^{\circ}$ . b) from azimuth 0 to $360\,^{\circ}$ .

phasemshtrueq90leglow,phasemshtrueqleglow
Figure 7. Relative error of proposed nine-parameter phase-velocity approximation. a) from azimuth 0 to $90\,^{\circ}$ . b) from azimuth 0 to $360\,^{\circ}$ .

groupxu90leglow,groupxuleglow
Figure 8. Relative error of group-velocity approximation by Xu et al. (2005) and Vasconcelos and Tsvankin (2006). a) from azimuth 0 to $90\,^{\circ}$ . b) from azimuth 0 to $360\,^{\circ}$ .

groupmshappq90leglownew,groupmshappqleglownew
Figure 9. Relative error of the proposed six-parameter group-velocity approximation. a) from azimuth 0 to $90\,^{\circ}$ . b) from azimuth 0 to $360\,^{\circ}$ .

groupmshtrueq90leglow,groupmshtrueqleglow
Figure 10. Relative error of the proposed nine-parameter group-velocity approximation. a) from azimuth 0 to $90\,^{\circ}$ . b) from azimuth 0 to $360\,^{\circ}$ .

On anelliptic approximations for qP velocities in TI and orthorhombic media