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Appendix A

The impulse response of the DMO operator is the envelope to the family of hyperbolas obtained by considering each point on the migration ellipse, a point diffractor. This is a rather peculiar property of the DMO operator and we determine here the equation of the envelope of the hyperbolas, which coincides with the DMO operator. While the equation of the envelope is the equation of an ellipse with the horizontal semiaxis equal with the offset, the impulse response of the DMO is only a part of the total ellipse, the maximum x coordinate being

\begin{displaymath}
x_{max}=h^2
\over a\end{displaymath}

; where h is the offset and a is the semiaxis of the full constant offset migration ellipse

\begin{displaymath}
a= {\bf v}
t_h \over 2\end{displaymath}

.

The equation of the family of hyperbolas in Fig.A.1 is:

z2 = zm2 + (xm - x)2

(11)

The points of coordinates (xm,zm) are situated on the ellipse, representing a parametrized form of the ellipse equation.

\begin{displaymath}
x_m = -{a^2\tan{\alpha}\over \sqrt{a^2\tan^2{\alpha}+b^2} }\end{displaymath}

\begin{displaymath}
z_m = {b^2 \over \sqrt{a^2\tan^2{\alpha}+b^2}}\end{displaymath}

In expanded form the function F(x,z,t) where $t=\tan\alpha$ will be:

\begin{displaymath}
F(x,z,t)=z_m^2+(x_m-x)^2-z^2= {b^4\over {a^2t^2+b^2}} + ({-a^2t \over 
\sqrt{a^2t^2+b^2}} - x)^2 -z^2 = 0\end{displaymath} (12)
The envelope of the family of curves represented in equation ?? is obtained by eliminating the parameter $\bf t$ from the two equations:

F(x,z,t)=0

\begin{displaymath}
{\partial {F(x,z,t)} \over {t} } = 0\end{displaymath}

We have then the equations:

\begin{displaymath}
F(x,z,t)={ {b^4+a^4t^2} \over {a^2t^2+b^2} } + { {2a^2tx} \over 
\sqrt{a^2t^2+b^2} } + x^2 -z^2 = 0\end{displaymath} (13)

\begin{displaymath}
F'(x,z,t)={ {2a^2b^2t(a^2-b^2)} \over {(a^2t^2+b^2)^2} } +
{ {2a^2b^2x} \over {(a^2t^2+b^2)^{3 \over 2} } }\end{displaymath} (14)

\begin{displaymath}
F'(x,z,t)={ {2a^2b^2(t(a^2-b^2)+x \sqrt{a^2t^2+b^2})} \over 
{(a^2t^2+b^2)^2} } = 0 \end{displaymath}

Which gives for $\bf t$:

\begin{displaymath}
t(b^2-a^2)=x \sqrt{a^2t^2+b^2}\end{displaymath}

\begin{displaymath}
t^2={ {b^2x^2} \over {(b^2-a^2)^2-a^2x^2} }\end{displaymath} (15)

Introducing $\bf t$ in ?? we obtain:
\begin{displaymath}
z^2=b^2-{ {x^2b^2} \over {(b^2-a^2} }\end{displaymath} (16)

Because $\bf a$ and $\bf b$ represent the semiaxis of the ellipse, the offset $\bf h$ is given by h2 = a2 - b2. The equation ?? can be rewritten as:
\begin{displaymath}
z^2=b^2-{ {x^2b^2} \over {h^2} }\end{displaymath} (17)

\begin{displaymath}
{ {z^2} \over {b^2} } + {{x^2} \over {h^2} } = 1\end{displaymath}

The equation of the envelope is the equation of an ellipse, with the vertical semiaxis equal to $\bf b$ and the horizontal semiaxis equal to the offset $\bf h$.As the depth z is given by

\begin{displaymath}
z={\bf v} t_dmo\end{displaymath}

and the semiaxis b is given by

\begin{displaymath}
b={\bf v} t_n\end{displaymath}

where

tn

is the time impusle after the NMO correction, we can rewrite the equation as
\begin{displaymath}
{ {{t_dmo}^2} \over {{t_n}^2} } + {{x^2} \over {h^2} } = 1\end{displaymath} (18)
which is the well known equation of the full DMO ellipse.


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Stanford Exploration Project
1/13/1998