Generalized nonhyperbolic moveout approximation

Discussion

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

$$t^2(x) \approx t_0^2+\frac{x^2}{v^2} + \frac{A\,x^4} {\displaystyle v^4\,\left(2\,t_0^2+D\,\frac{x^2}{v^2}\right)}\;,$$ (37)

where $D=B+\sqrt{C}$. Equation 37 has the same behavior as equation 2 at small offsets and the same asymptote as $x$ approaches infinity. However, its accuracy is not nearly as spectacular (Figure 7).

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 $B$ and $C$ 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
Figure 7. 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.

linsloth1
Figure 8. Relative absolute error of the generalized approximation fitted to the zero offset as opposed to the critical offset. Compare with Figure 2.

Generalized nonhyperbolic moveout approximation