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Integral inverse DMO

DMO is a method of transformation of finite-offset data to zero-offset data. Let the normal moveout corrected input data be denoted $P_2(t_2,{\bf x}_2;{\bf h}_2)$ and the zero-offset desired output denoted $P_0(t_0,{\bf x}_0;{\bf h}=0)$. Assume known relationships between the coordinates of the general form
\begin{displaymath}
t_0=t_0(t_2,{\bf x}_2,w_0,{\bf k}_0) \hspace{.5in} \rm {and}...
 ....5in} {\bf x}_0={\bf x}_0({\bf x}_2)\; .
\EQNLABEL{coord.relat}\end{displaymath} (14)
The DMO operator can be defined in the zero-offset frequency $\omega_{0}$ and midpoint wavenumber ${\bf k}$ as Liner (1988)
\begin{eqnarray}
P_0(\omega_{0},{\bf k}_0;{\bf h}=0)&=&\int d{\bf x}_2 \frac {d{...
 ...bf x}_2)\right]} P_2(t_2,{\bf x}_2;{\bf h}_2), \
\EQNLABEL{dmo.eq}\end{eqnarray} (15)
(16)
whereas its inverse can be defined as
\begin{displaymath}
P_2(t_2,{\bf x}_2;{\bf h}_2)=\int d{\bf k}_0 \int d\omega_o ...
 ...t]} P_0(\omega_{0},{\bf k}_0;{\bf h}=0) \\ \EQNLABEL{dmoinv.eq}\end{displaymath} (17)
where
\begin{displaymath}
{J_2}={J_2}(t_2,{\bf x}_2,\omega_o,{\bf k}_0)\end{displaymath} (18)

A detailed derivation of J2 is given by Liner 1988. The method is based on a general formalism Beylkin (1985); Cohen and Hagin (1985) for inverting integral equations such as dmo.eq. It involves inserting dmo.eq into dmoinv.eq and expanding the resulting amplitude and phase as a Taylor series and making a change of variables according to Beylkin 1985. The solution provides an asymptotic inverse for dmo.eq, where the weights are given by
\begin{displaymath}
{J_2}=\frac {d\omega} {d\omega_o}\frac{d{\bf k}}{d{\bf k}_0}...
 ...rac {d{\bf x}_0}{d{\bf x}_2}\right]}^{-1}. \\ \EQNLABEL{jacob2}\end{displaymath} (19)
In this expression, $\frac {dt_0}{dt_2} \frac {d{\bf x}_0}{d{\bf x}_2}$ is the Jacobian of the change of variables in the forward DMO given by
\begin{displaymath}
{J_1} =\frac {\partial{(t_0,{\bf x}_0)}} {\partial{(t_2,{\bf...
 ...{dt_2} & \frac {d{\bf x}_0} {d{\bf x}_2}, \end{array} \right]\ \end{displaymath} (20)
which reduces to ${J_1}=\frac {dt_0}{dt_2} \frac {d{\bf x}_0}{d{\bf x}_2}$, assuming the general coordinate relationships coord.relat where ${\bf x}_0$ is independent of t2, leading to a zero lower left element in the determinant matrix above.

The quantity $\frac {d \omega}{d \omega_0} \frac {d{\bf k}}{d{\bf k}_0}$ is the inverse of the Beylkin determinant, H, and is given by
\begin{displaymath}
H^{-1}=\frac {\partial{(\omega,{\bf k})}} {\partial{(\omega_...
 ...}} {d{\bf k}_0} \end{array} \right]\; . 
\EQNLABEL{Beylkin_inv}\end{displaymath} (21)
If we recognize that ${\bf k}$ is independent of $\omega_{0}$, then the lower element of H-1 is zero and Beylkin_inv reduces to
\begin{displaymath}
H^{-1}=\frac {d \omega} {d \omega_0} \frac {d{\bf k}} {d{\bf k}_0},\\ \end{displaymath} (22)
where $\omega$ and ${\bf k}$ are, respectively,
\begin{displaymath}
\omega=\omega_o \frac {d} {dt_2} \left[ t_0(t_2) \right]
\EQNLABEL{omega}\end{displaymath} (23)
\begin{displaymath}
{\bf k}={\bf k}_0 \frac {d} {d{\bf x}_2} \left[{\bf x}_0 ({\bf x}_2) \right]\; .
\EQNLABEL{wave.numb}\end{displaymath} (24)

Notice that $\omega$ and ${\bf k}$ depend on the coordinate relationships coord.relat. Therefore, the Beylkin determinant, H, varies according to the DMO operator but is constant for kinematically equivalent operators.



 
next up previous print clean
Next: Hale DMO and its Up: Amplitude-preserving AMO Previous: Chaining DMO and inverse
Stanford Exploration Project
1/18/2001