Hale DMO and its inverse

Starting from the following coordinate relationships between a finite-offset data and its equivalent zero-offset data

$$t_0=t_2 {\left[ 1+ {{\left(\frac{{\bf k}\cdot {\bf h}_2}{\om... ...{and} \hspace{.5in} {\bf x}_0={\bf x}_2 \EQNLABEL{weight.hale}$$ (25)

After differentiating weight.hale and taking into account a factor of $1/{2\pi}$ as scaling for the spatial Fourier transform we can write jacob2 as

$${J_2} =\frac {A_2} {2\pi} \frac {d\omega} {d\omega_o}\frac{d{\bf k}}{d{\bf k}_0} \; . \EQNLABEL{hale_inv1}$$ (26)

The remaining task reduces to performing the necessary derivatives, and, with some algebra, one can verify that H reduces to the simple expression Liner and Cohen (1988)

$$H=\frac {A_2^3} {2A_2^2-1} \EQNLABEL{hale_det}$$ (27)

and, therefore, we arrive at the inversion amplitude function

$${J_2}=\frac {1} {2\pi} \left[1+\frac{{\bf k}^2 {\bf h}^2}{\omega_o^2t_2^2A_2^2} \right] \; . \EQNLABEL{hale_inv}$$ (28)

For a detailed derivation, the reader should refer to the original work of Liner 1988.