3D generalized nonhyperboloidal moveout approximation

Layered orthorhombic model from SEAM Phase II unconventional model

For a complex numerical test, we create a one-dimensional layered orthorhombic model (Figure 6) by extracting a depth column out of the SEAM Phase II unconventional model (Oristaglio, 2015). We assume no azimuthal rotation in the sublayers. Therefore, this model represents an example of complex layered orthorhombic model with aligned symmetry planes. The reflection traveltimes and offsets can be computed from ray tracing and are shown in Figure 7. Note that since this is a layered orthorhombic medium with aligned symmetry planes, it is sufficient to show the results only in one quadrant defined by the two slowness components: $p_x$ and $p_y$ as they remain symmetric in all other possible quadrants. Following the similar process as in previous examples, Figure 8 shows a performance comparison of various moveout approximations. The reference rays for the proposed generalized moveout approximation (equation 1) are shot at $P_{x1} = 0.124$ and $P_{y1}=0.0$ along $x$ -axis, at $P_{x2}=0.0$ and $P_{y2}=0.133$ along $y$ -axis, at $P_{x3}=0.084$ and $P_{y3}=0.0983$ along $x=y$ , and at $P_{x4}=0.085$ and $P_{y4}=-0.0996$ along $x=-y$ . The proposed generalized approximation shows again the highest accuracy with the maximum traveltime error of 6.66 $ms$ and RMS error of 0.083 $ms$ . The maximum traveltime error and RMS error for other approximations are 309.75 $ms$ and 18.86 $ms$ for the NMO ellipse, 66.33 $ms$ and 2.975 $ms$ for the approximation by Al-Dajani et al. (1998), and 110.87 $ms$ and 3.829 $ms$ for the approximation by Xu et al. (2005).

vpseam2cut
Figure 6. Vertical P-wave velocity in $km/s$ of the SEAM Phase II unconventional model.

rayx,t,r
Figure 7. One-dimensional model construct from the first column of SEAM Phase II unconventional model and a) example reflection rays at constant $p_x = 0.0908$ and varying $p_y$ from 0 to 0.09425. The exact traveltime and the magnitude of offset $r=\sqrt {x^2+y^2}$ are shown in b) and c) for increasing values of slowness components $p_x$ and $p_y$ .

nmoerr,alerr,xuerr,gmaerr
Figure 8. Error plots in the one-dimensional layered orthorhombic model from of SEAM Phase II unconventional model 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 a) is plotted with 120 $ms$ clipping, whereas, b), c), and d) are plotted under the same color scale with 25 $ms$ clipping. The scale bars show the total range of errors for different approximations. The proposed approximation achieves the highest accuracy with the maximum traveltime error of 6.66 $ms$ and RMS error of 0.083 $ms$ .

3D generalized nonhyperboloidal moveout approximation