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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)

$\displaystyle t_0$ $\textstyle =$ $\displaystyle \frac{2\,L}{V}\;,$ (27)
$\displaystyle v$ $\textstyle =$ $\displaystyle \frac{V}{\cos{\beta}}\;,$ (28)
$\displaystyle A$ $\textstyle =$ $\displaystyle 2\,\tan^2{\beta}\,G\;,$ (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,

$\displaystyle T_{\infty}^2$ $\textstyle =$ $\displaystyle t_0^2\,\frac{G}{G+\tan^2{\beta}}\;,$ (30)
$\displaystyle P_{\infty}$ $\textstyle =$ $\displaystyle \frac{1}{V} = \frac{1}{v\,\cos{\beta}}\;,$ (31)

which, after substitution in equations 25-26, produce
$\displaystyle B$ $\textstyle =$ $\displaystyle G - \tan^2{\beta}\;,$ (32)
$\displaystyle C$ $\textstyle =$ $\displaystyle \left(G + \tan^2{\beta}\right)^2\;.$ (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
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.
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Next: Homogeneous VTI layer Up: Analytical examples Previous: Linear velocity and linear

2013-07-26