3D generalized nonhyperboloidal moveout approximation

Appendix A: Alternative derivation of the moveout approximation in equation 12

On the basis of perturbation theory for a general horizontal homogeneous weakly anisotropic media, we consider the quartic coefficients in equations 1 and 2 $A_i$ to be (Farra et al., 2016; Grechka and Pech, 2006)

$$A_1 = -\frac{4\eta_2}{V^4_{\mbox{ref}}}$$ (22)

$$A_2 = \frac{8(\chi_{13}-\chi_{11})}{V^4_{\mbox{ref}}}$$

$$A_3 = -\frac{4(\eta_1+\eta_2-\eta_3)}{V^4_{\mbox{ref}}}$$

$$A_4 = \frac{8(\chi_{13}-\chi_{12})}{V^4_{\mbox{ref}}}$$

$$A_5 = -\frac{4\eta_1}{V^4_{\mbox{ref}}}~,$$

where

$$\chi_{11} = \frac{c_{16}}{V_{\mbox{ref}}} \qquad, \quad\chi_{12} ... ..._{\mbox{ref}}} \qquad, \quad\chi_{13} = \frac{c_{36}+2c_{45}}{V_{\mbox{ref}}}~,$$ (23)

and $V_{\mbox{ref}}$ denotes the P-wave velocity in the chosen isotropic background. These expressions are appropriate for horizontal homogeneous weakly anisotropic media of any symmetry. A more general form form dipping layer is also discussed by Grechka and Pech (2006). In the specific case of orthorhombic media, the general quartic coefficients (equation A-1) can be simplified to

$$A_r(\alpha) = -\frac{4\eta(\alpha)}{V^4_{\mbox{ref}}}~,$$ (24)

where $\eta(\alpha)$ is given in equation 13. Therefore, the resulting moveout approximation takes the form of

$$t^2(r,\alpha) \approx t^2_0 + W_r(\alpha)r^2-\frac{2\eta(\alpha)}{t^2_0 V^4_{\mbox{ref}}}r^4~.$$ (25)

Subsequently, by setting

$$V^2_{\mbox{ref}} = V^2_{nmo} (\alpha) = (W_r(\alpha))^{-1}~,$$ (26)

we obtain the moveout approximation of the form proposed by Xu et al. (2005) and Vasconcelos and Tsvankin (2006):

$$t^2(r,\alpha) \approx t^2_0 + W_r(\alpha)r^2-\frac{2\eta(\alpha)}{t^2_0 V^4_{nmo}(\alpha)}r^4~,$$ (27)

without the long-offset normalization, which corresponds to the choice of $A_r(\alpha) = -4 \eta(\alpha)/V^4_{nmo}(\alpha)$ , $B_i=0$ , and $C_i=0$ from equation 2. The additional long-offset normalization factor can be included based on the same scheme as in equation 10 with

$$A^*_r(\alpha) = \frac{1+2\eta(\alpha)}{t_0^2V^2_{nmo}(\alpha)}~.$$ (28)

As a result, we obtain the same expression of the moveout approximation in equation 12 by Xu et al. (2005) in the main text.

3D generalized nonhyperboloidal moveout approximation