On anelliptic approximations for qP velocities in TI and orthorhombic media

Moveout approximation

To convert the proposed group-velocity approximation (equation 47) to the corresponding moveout approximation, we apply again the general expression given in equation 31. Adopting the same notation rules, the moveout approximation takes the form:

$$t^2 = H_{ortho}(x,y)(1-\hat{S}) + \hat{S}\sqrt{F}~,$$ (54)

$$F = H^2_{ortho}(x,y)+ \frac{2}{\hat{S}}\bigg(\frac{(\hat{Q}_1-1)t^2_0... ...2}}+ \frac{(\hat{Q}_3-1)x^2y^2}{(Q_{31}V^2_{nmo_1})(Q_{32}V^2_{nmo_2})}\bigg)~,$$

where

$$\hat{S} = \frac{\frac{\hat{S}_1}{Q_{32}V^2_{nmo_2}}x^2 + \frac{\h... ...}{\frac{1}{Q_{32}V^2_{nmo_2}}x^2 + \frac{1}{Q_{31}V^2_{nmo_1}}y^2 + t^2_0}~,\\$$ (55)

$$\hat{Q}_1 =\frac{\frac{Q_{21}}{Q_{31}V^2_{nmo_1}}y^2 + Q_{31}t^2_... ...{nmo_1}}y^2}{\frac{1}{Q_{32}V^2_{nmo_2}}x^2 + \frac{1}{Q_{31}V^2_{nmo_1}}y^2}~,$$ (56)

$$\hat{S}_1 = \frac{\frac{S_{13}}{Q_{31}V^2_{nmo_1}}y^2 + S_{12}t^2... ...{nmo_1}}y^2}{\frac{1}{Q_{32}V^2_{nmo_2}}x^2 + \frac{1}{Q_{31}V^2_{nmo_1}}y^2}~,$$ (57)

$x$ denotes the offset in $N_1$ direction, $y$ denotes the offset in $N_2$ direction, $V_{nmo_2} = \sqrt{1/W_1Q_{32}}$ denotes the NMO-velocity in $N_1$ direction, $V_{nmo_1} = \sqrt{1/W_2Q_{31}}$ denotes the NMO-velocity in $N_2$ direction, and $H_{ortho}(x,y)$ denotes the hyperboloidal part of reflection traveltime squared given below,

$$H_{ortho}(x,y) = t^2_0 + \frac{x^2}{Q_{32}V^2_{nmo_2}} + \frac{y^2}{Q_{31}V^2_{nmo_1}}~.$$ (58)

We apply the same strategy to reduce the number of parameters with an approximation on $Q_{ij}$ for $S_{ij}$ as in equation 35.

For small offset, the Taylor expansion of equation 54 is

$$t^2(x) \approx t^2_0 +\frac{x^2}{V^2_{nmo_2}} + \frac{y^2}{V^2_{nmo_1}} -$$ (59)

$$\frac{1-2S_{32}(Q_{12}+1)+Q_{32}(4S_{32}+Q_{32}-2)}{2S_{32}Q_{32}^2t_0^2V^4_{nmo_2}} x^4 -$$

$$\frac{1-2S_{31}(Q_{21}+1)+Q_{31}(4S_{31}+Q_{31}-2)}{2S_{31}Q_{31}^2t_0^2V^4_{nmo_1}} y^4 +$$

$$\frac{S_{31}(Q_{32}-1)^2-S_{32}(Q_{32}-1)(Q_{32}+2Q_{31}-3)-2S^2_... ..._{31}-Q_{13}-1)}{2S^2_{32}Q_{31}Q_{32}t_0^2V^2_{nmo_1}V^2_{nmo_2}} x^2y^2 + ...$$

The asymptote of this expression for unbounded offsets $x$ and $y$ is given by

$$\frac{1}{Q_{32} V^2_{nmo_2}} = \frac{1}{w_1}~~~ ~~~\frac{1}{Q_{31} V^2_{nmo_1}} = \frac{1}{w_2}~,$$ (60)

which denote the horizontal velocities squared along $N_1$ and $N_2$ directions respectively.

On anelliptic approximations for qP velocities in TI and orthorhombic media