Generalized nonhyperbolic moveout approximation |

Approximation is more of an art than a science. We don't have a justification for suggesting equations 1 or 2 other than pointing out that they reduce to known forms with particular choices of parameters.

The choice of a proper functional form is important for the
approximation accuracy. Suppose that we try to replace the
five-parameter approximation in equation 2 with the
four-parameter equation

A proper selection of the reference ray for equations 22 and 23 is also important for approximation accuracy. If this ray is taken not at the largest possible offset, the accuracy will deteriorate. As an extreme example, suppose that we try to define and by fitting subsequent terms of the Taylor expansion 18 near the zero offset rather than the behavior of the approximation at large offsets. Figure 8 shows the result for the case of a linear sloth model: the approximation is more accurate than alternatives (shown in Figure 2) but not nearly as accurate as the generalized approximation fitted at the critical offset.

Possible extensions of this work may include nonhyperbolic approximations for diffraction traveltimes (for use in prestack time migration) and reflection surfaces (for use in common-reflection-surface methods) as well as approximations for anisotropic phase and group velocities in ray tracing and wave extrapolation.

linsloth2
Relative absolute error of Padé approximation
in equation 37 as a function of velocity contrast and
offset/depth ratio for the case of a linear sloth model. Compare
with Figure 2.
Figure 7. |
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linsloth1
Relative absolute error of
the generalized approximation
fitted to the zero offset as opposed to the critical offset. Compare
with Figure 2.
Figure 8. |
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Generalized nonhyperbolic moveout approximation |

2013-07-26