3D generalized nonhyperboloidal moveout approximation

Zero-offset ray

The Taylor expansion of equation 1 around the zero offset

$$t^2(x,y) \approx t^2_0 + W(x,y) + \frac{A(x,y)}{2 t_0^2} + ...$$ (14)

allows for a direct evaluation of nine coefficients: $t_0$ , $W_i$ , and $A_i$ by matching equation 14 with the expansion of the exact traveltime in vector offset.

Sample	$c_{11}$	$c_{22}$	$c_{33}$	$c_{44}$	$c_{55}$	$c_{66}$	$c_{12}$	$c_{23}$	$c_{13}$
HTI	5.06	7.086	7.086	2	2.25	2.25	1.033	3.086	1.033
Layer 1	9	9.84	5.938	2	1.6	2.182	3.6	2.4	2.25
Layer 2	11.7	13.5	9	1.728	1.44	2.246	8.824	5.981	5.159
Layer 3	12.6	13.94	8.9125	2.5	2	2.182	2.7	3.425	3.15

Table 1. Normalized stiffness tensor coefficients (in $km^2$ /$s^2$ ) from different anisotropic samples: HTI is from Al-Dajani and Tsvankin (1998), layer 1 is from Schoenberg and Helbig (1997), layer 2 is from Tsvankin (1997), and layer 3 is a modifed sample based on the same fracture model as layer 1.

3D generalized nonhyperboloidal moveout approximation