Generalized nonhyperbolic moveout approximation |
For a numerical test, we create a one-dimensional velocity model by extracting a depth column out of the anisotropic Marmousi model, created by Alkhalifah (1997a) and shown in Figure 4. We evaluate exact reflection traveltimes by ray tracing (Figure 5). Next, we compare the exact time for different reflection rays with values predicted by different traveltime approximations. As shown in Figure 6, only the proposed generalized approximation is able to predict the true traveltime accurately over the full range of offsets. To define approximation parameters, we used equations 19-21 and 22-23 and the ray with the largest offset as the reference ray.
vz,v,eta
Figure 4. Anisotropic Marmousi model. (a) Vertical velocity. (b) NMO velocity. (c) parameter. |
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vz1
Figure 5. One-dimensional model extracted from the left column of the anisotropic Marmousi model and corresponding reflection rays. |
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hynmo,shnmo,atnmo,fsnmo
Figure 6. Exact moveout from ray tracing in the one-dimensional anisotropic Marmousi model (dots) and different approximations (solid lines). (a) Hyperbolic approximation, (b) Shifted hyperbola approximation, (c) Alkhalifah-Tsvankin approximation, (d) Generalized nonhyperbolic approximation. |
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Generalized nonhyperbolic moveout approximation |