On anelliptic approximations for qP velocities in TI and orthorhombic media |
Parameters | ||||
87.11 | 0.467 | 105.39 | 25.19 | |
0.005 | 0.649 | 0.20 | ||
181.74 | 22.54 | |||
13.16 |
Parameters | ||||
0.576 | 0.143 | 0.651 | 0.599 | |
0.538 | 0.257 | 1.061 | ||
1.325 | 1.279 | |||
4.124 |
Table 10 shows a significantly larger correlation between the change in phase velocity with in comparison with that of , which agrees with the general assumption of the independency of in qP velocities approximations. Likewise, the effect from has a higher correlation with the change of phase velocity than because the exact qP phase-velocity formula (equation 9) can be expressed in terms of Thomsen parameters with corresponding to the lower order of than . Moreover, and also have high correlation with , which is apparent from their definitions.
Table 11 shows relatively similar correlations from and to the change in exact phase velocity suggesting a more symmetric contribution from both parameters. The dimensionless anelliptic parameters and exhibit a strong correlation, which is consistent with the relationships shown in Figure 1.
By ignoring the effect of in the case of Thomsen parameters or using the relationship between and (Figure 1) to reduce the number of parameters to three, we can transform the matrix from to ( ). Note that the matrix for Thomsen parameters is similar to Table 10 with the omittance of the row and column associated with . Table 12 shows the three-parameter matrix for anelliptic parameters with similar behavior of relatively equal correlations from and as before.
Parameters | ||||
0.578 | 0.144 | 1.166 | ||
0.534 | 1.286 | |||
7.411 |
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 and plot its contour at a given amount of change in the exact phase velocity expression,
(66) |
thomsenmatrix,zonematrix
Figure 14. Ellipsoids obtained from the quadratic form of in the case of a) Thomsen parameters b) anelliptic parameters. |
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On anelliptic approximations for qP velocities in TI and orthorhombic media |