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Next: Appendix D: REFLECTION FROM
Up: Fomel & Stovas: Generalized
Previous: Appendix B: LINEAR SLOTH
In this appendix, we derive an analytical expression for reflection
traveltime from a hyperbolic reflector in a homogeneous velocity
model (Figure C-1). Similar derivations apply to an elliptic reflector and were
used previously in the theory of dip moveout, offset continuation, and
non-hyperbolic common-reflection surface
(Fomel and Kazinnik, 2009; Stovas and Fomel, 1996; Fomel, 2003).
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hyper
Figure 9. Reflection from a hyperbolic reflector in a
homogeneous velocity model (a scheme).
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Consider the source point
and the receiver point
at the
surface
above a 2-D constant-velocity medium and a hyperbolic
reflector defined by the equation
![\begin{displaymath}
z(x) = \sqrt{h^2 +
x^2\,\tan^2{\alpha}}\;.
\end{displaymath}](img134.png) |
(54) |
The reflection traveltime as a function of the reflection point
location
is
![\begin{displaymath}
t = \frac{\sqrt{(x_s-y)^2 + z^2(y)} + \sqrt{(x_r-y)^2+z^2(y)}}{V}\;.
\end{displaymath}](img136.png) |
(55) |
According to Fermat's principle, the traveltime should be stationary
with respect to the reflection point
:
![\begin{displaymath}
0 = \frac{\partial t}{\partial y} =
\frac{y-x_s + y\,\tan^2{...
...2{\alpha}}{V\,\sqrt{(x_r-y)^2 + h^2 + y^2\,\tan^2{\alpha}}}\;.
\end{displaymath}](img137.png) |
(56) |
Putting two terms in equation C-3 on different sides of
the equation, squaring them, and reducing their difference to a common
denominator, we arrive at the equation
which simplifies to the following quadratic equation with respect to
:
![\begin{displaymath}
y^2\,(x_s+x_r)\,\tan^2{\alpha} - 2\,y\,\left(x_s\,x_r\,\s...
...{\alpha} - h^2\right) -
h^2\,(x_s+x_r)\,\cos^2{\alpha} = 0\;.
\end{displaymath}](img142.png) |
(58) |
The discriminant is
![\begin{displaymath}
D = \left(x_s\,x_r\,\sin^2{\alpha}-h^2\right)^2 + h^2\,(x...
... = (h^2+x_s^2\,\sin^2{\alpha})\,(h^2+x_r^2\,\sin^2{\alpha})\;.
\end{displaymath}](img143.png) |
(59) |
Only one of the two branches of the solution
has physical meaning. Substituting equation C-7 into
equation C-2, we obtain, after a number of algebraic
simplifications,
![\begin{displaymath}
t = \frac{\sqrt{2 h^2 + x_s^2 + x_r^2 - 2\,x_s\,x_r\,\cos...
...2+x_s^2\,\sin^2{\alpha})\,(h^2+x_r^2\,\sin^2{\alpha})}}}{V}\;.
\end{displaymath}](img147.png) |
(61) |
Making the variable change in equation C-8 from
and
to the midpoint and offset coordinates
and
according to
,
, we notice that this equation is exactly
equivalent to equation 1 with the following definition of
parameters:
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Next: Appendix D: REFLECTION FROM
Up: Fomel & Stovas: Generalized
Previous: Appendix B: LINEAR SLOTH
2013-07-26