3D generalized nonhyperboloidal moveout approximation

Layered orthorhombic model with azimuthal rotation in sublayers

For a more complex model, we introduce azimuthal rotation of 50 $\,^{\circ}$ and 30 $\,^{\circ}$ in the middle layer (Layer 2) and the bottom layer (Layer 3) of the three-layer model from the previous example respectively. The angle measurement is done with respect to the top layer. Similarly to before, the effective coefficients for previously proposed approximations are calculated by the VTI averaging relationship. The reference rays for the proposed approximation (equation 1) are shot at $P_{x1} = 0.254$ and $P_{y1}=0.005$ along $x$ -axis, at $P_{x2} = 0.029$ and $P_{y2}=0.24$ along $y$ -axis, at $P_{x3}=0.18$ and $P_{y3}=0.198$ along $x=y$ , and at $P_{x4}=0.2$ and $P_{y4}=-0.184$ along $x=-y$ . The proposed generalized approximation shows again the highest accuracy (Figure 5).

onelayernmoazi,onelayeralazi,onelayerxuazi,onelayergmaazi
Figure 3. Error plots in the 30 $\,^{\circ}$ rotated homogeneous orthorhombic layer (Layer 1) of a) NMO ellipse b) approximation by Al-Dajani et al. (1998) c) approximation by Xu et al. (2005) , and d) the proposed approximation. Note that b), c), and d) are plotted under the same color scale and $H$ denotes the reflector depth.

layerednmo,layeredal,layeredxu,layeredgma
Figure 4. Error plots in the aligned three-layer orthorhombic model of a) NMO ellipse b) approximation by Al-Dajani et al. (1998) c) approximation by Xu et al. (2005), and d) the proposed approximation. Note that b), c), and d) are plotted under the same color scale and $H$ denotes the reflector depth.

layerednmoazi5030,layeredalazi5030,layeredxuazi5030,layeredgmaazi5030
Figure 5. Error plots in the three-layer orthorhombic model with azimuthal rotation of sublayers (50 $\,^{\circ}$ in the middle layer and 30 $\,^{\circ}$ in the bottom layer) of a) NMO ellipse b) approximation by Al-Dajani et al. (1998) c) approximation by Xu et al. (2005), and d) the proposed approximation. Note that b), c), and d) are plotted under the same color scale and $H$ denotes the reflector depth.

3D generalized nonhyperboloidal moveout approximation