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

$$t_0=t_0(t_2,{\bf x}_2,w_0,{\bf k}_0) \hspace{.5in} \rm {and}... ....5in} {\bf x}_0={\bf x}_0({\bf x}_2)\; . \EQNLABEL{coord.relat}$$ (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

$$P_2(t_2,{\bf x}_2;{\bf h}_2)=\int d{\bf k}_0 \int d\omega_o ... ...t]} P_0(\omega_{0},{\bf k}_0;{\bf h}=0) \\ \EQNLABEL{dmoinv.eq}$$ (17)

where

$${J_2}={J_2}(t_2,{\bf x}_2,\omega_o,{\bf k}_0)$$ (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

$${J_2}=\frac {d\omega} {d\omega_o}\frac{d{\bf k}}{d{\bf k}_0}... ...rac {d{\bf x}_0}{d{\bf x}_2}\right]}^{-1}. \\ \EQNLABEL{jacob2}$$ (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

$${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]\$$ (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

$$H^{-1}=\frac {\partial{(\omega,{\bf k})}} {\partial{(\omega_... ...}} {d{\bf k}_0} \end{array} \right]\; . \EQNLABEL{Beylkin_inv}$$ (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

$$H^{-1}=\frac {d \omega} {d \omega_0} \frac {d{\bf k}} {d{\bf k}_0},\\$$ (22)

where $\omega$ and ${\bf k}$ are, respectively,

$$\omega=\omega_o \frac {d} {dt_2} \left[ t_0(t_2) \right] \EQNLABEL{omega}$$ (23)

$${\bf k}={\bf k}_0 \frac {d} {d{\bf x}_2} \left[{\bf x}_0 ({\bf x}_2) \right]\; . \EQNLABEL{wave.numb}$$ (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.