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Appendix D: REFLECTION FROM A CIRCULAR REFLECTOR IN A HOMOGENEOUS VELOCITY MODEL

In the case of a circular (cylindrical or spherical) reflector in a homogeneous velocity model, there is no closed-form analytical solution. However, the moveout can be described analytically by parametric relationships (Glaeser, 1999).

crefl
crefl
Figure 10.
Reflection from a circular reflector in a homogeneous velocity model (a scheme).
[pdf] [png] [xfig]

Consider the geometry of the reflection shown in Figure D-1. According to the trigonometry of the reflection triangles, the source and receiver positions can be expressed as

$\displaystyle x_s$ $\textstyle =$ $\displaystyle R\,\sin{\alpha} + (H+R - R\,\cos{\alpha})\,\tan{(\alpha-\theta)}\;,$ (67)
$\displaystyle x_r$ $\textstyle =$ $\displaystyle R\,\sin{\alpha} + (H+R - R\,\cos{\alpha})\,\tan{(\alpha+\theta)}\;,$ (68)

where $R$ is the reflector radius, $H$ is the minimum reflector depth, $\alpha$ is the reflector dip angle at the reflection point, and $\theta$ is the reflection angle. Correspondingly, the midpoint and offset coordinates can be expressed as
$\displaystyle m$ $\textstyle =$ $\displaystyle \frac{x_s+x_r}{2} = R\,\sin{\alpha} + (H+R - R\,\cos{\alpha})\,\frac{\cos{\alpha}\,\sin{\alpha}}{\cos^2{\theta} - \sin^2{\alpha}}\;,$ (69)
$\displaystyle x$ $\textstyle =$ $\displaystyle x_r - x_s = 2\,(H+R - R\,\cos{\alpha})\,\frac{\cos{\theta}\,\sin{\theta}}{\cos^2{\theta} - \sin^2{\alpha}}\;,$ (70)

and the reflection traveltime can be expressed as
$\displaystyle t$ $\textstyle =$ $\displaystyle \frac{H+R - R\,\cos{\alpha}}{V}\,\left[\frac{1}{\cos{(\alpha-\theta)}} + \frac{1}{\cos{(\alpha+\theta)}}\right]$  
  $\textstyle =$ $\displaystyle 2\,\frac{H+R - R\,\cos{\alpha}}{V}\,\frac{\cos{\alpha}\,\cos{\theta}}{\cos^2{\theta} - \sin^2{\alpha}}\;,$ (71)

where $V$ is the medium velocity. Expressing the reflection angle $\theta$ from equation D-3 and substituting it into equations D-4 and D-5, we obtain a pair of parametric equations
$\displaystyle x^2(\alpha)$ $\textstyle =$ $\displaystyle 4\,\frac{[m\,\cos{\alpha} - (H+R)\,\sin{\alpha}]\,[m\,\sin{\alpha} + (H+R)\,\cos{\alpha} - R]}{\cos{\alpha}\,\sin{\alpha}}\;,$ (72)
$\displaystyle t^2(\alpha)$ $\textstyle =$ $\displaystyle \frac{4}{V^2}\,\frac{(m - R\,\sin{\alpha})\,[m\,\sin{\alpha} + (H+R)\,\cos{\alpha} - R]}{\sin{\alpha}}\;,$ (73)

which define the exact reflection moveout for the case of a circular reflector in a homogeneous medium.

The connection with parameters of equations 27-29 is given by

$\displaystyle L$ $\textstyle =$ $\displaystyle \sqrt{m^2 + (H+R)^2} - R\;,$ (74)
$\displaystyle \cos{\beta}$ $\textstyle =$ $\displaystyle \frac{H+R}{\sqrt{m^2 + (H+R)^2}}\;,$ (75)
$\displaystyle G$ $\textstyle =$ $\displaystyle \frac{L}{L+R} = 1 - \frac{R}{\sqrt{m^2 + (H+R)^2}}\;.$ (76)

The behavior of the moveout at infinitely large offsets is controlled by $P_{\infty} = 1/V$ and
\begin{displaymath}
T_{\infty} = \frac{2\,H}{V} = t_0\,\frac{G+\cos{\beta} - 1}{G\,\cos{\beta}}\;.
\end{displaymath} (77)

After substitution in equations 25-26, we obtain somewhat complicated but analytical expressions for parameters $B$ and $C$.


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Next: Appendix E: HOMOGENEOUS VTI Up: Fomel & Stovas: Generalized Previous: Appendix C: REFLECTION FROM

2013-07-26