Generalized nonhyperbolic moveout approximation

Curved reflector in a constant-velocity medium

Our next analytical example is a curved reflector under a constant-velocity overburden. The reflector curvature is one of the possible causes of non-hyperbolic moveout (Fomel and Grechka, 2001). The Taylor expansion around zero offset for the case of a curved reflector has the form of equation 18 with the following set of parameters (Fomel, 1994)

$$t_0 = \frac{2\,L}{V}\;,$$ (27)

$$v = \frac{V}{\cos{\beta}}\;,$$ (28)

$$A = 2\,\tan^2{\beta}\,G\;,$$ (29)

where $L$ is the length of the normal (zero-offset) ray, $V$ is the true velocity, $\beta$ is the reflector dip angle at the normal reflection point, $G=K\,L/(1+K\,L)$, and $K$ is the reflector curvature at the normal reflection point.

The two additional parameters $B$ and $C$ depend on the particular shape of the reflector. In the case of a hyperbolic reflector, analyzed in Appendix C, equation 2 happens to be exact. In this case,

$$T_{\infty}^2 = t_0^2\,\frac{G}{G+\tan^2{\beta}}\;,$$ (30)

$$P_{\infty} = \frac{1}{V} = \frac{1}{v\,\cos{\beta}}\;,$$ (31)

which, after substitution in equations 25-26, produce

$$B = G - \tan^2{\beta}\;,$$ (32)

$$C = \left(G + \tan^2{\beta}\right)^2\;.$$ (33)

In the special case of a plane (zero curvature) reflector, $G=0$, and the generalized approximation reduces to a hyperbolic form. In the special case of a diffraction point (infinite curvature), $G=1$, and the generalized approximation reduces to the double-square-root equation 17. In both of those cases, as well as in the case of a hyperbolic reflector, the generalized approximation is simply exact.

Figure 3 shows a comparison between different approximations for the case of a circular reflector, analyzed in Appendix D. As in the other examples, the proposed five-parameter generalized approximation brings an improvement in accuracy in several orders of magnitude in comparison with the three-parameter approximations.

circle
Figure 3. Relative absolute error of different traveltime approximations for the case of a circular reflector as a function of the radius/depth ratio and the offset/depth ratio. The midpoint location with respect to the center of the circle is equal to the depth of the reflector. (a) Hyperbolic approximation, (b) Shifted hyperbola approximation, (c) Alkhalifah-Tsvankin approximation, (d) Generalized nonhyperbolic approximation. The proposed generalized approximation reduces the maximum approximation error by several orders of magnitude.

Generalized nonhyperbolic moveout approximation