Black/Zhang DMO and its inverse

Similar to the preceding discussion, I start the derivation for an asymptotic inverse for Black/Zhang's DMO by recognizing the coordinate relationships,

$$t_0=t_2 A_2^{-1} \hspace{.5in} \rm {and} \hspace{.5in} {\bf ... ...rac {{\bf k} {\bf h}^2}{\omega_ot_2A_2} \EQNLABEL{weight.zhang}$$ (29)

The Jacobian of the change of variables in the forward DMO is given by

$${J_1}=\frac {\partial{(t_0,{\bf x}_0)}} {\partial{(t_2,{\bf x}_2)}}=\frac {2A_2^2-1} {A_2^3}, \EQNLABEL{zhang.jacob}$$ (30)

which has the familiar form of Zhang's 1988 and Black's 1993 Jacobian. Zhang based his derivations on kinematic arguments that considered a fixed reflection point rather than a fixed midpoint. This derivation takes into account the reflection-point smear Black et al. (1993a); Deregowski and Rocca (1981), which means that the input event at location ${\bf x}_2$ will be positioned by DMO to the correct zero-offset location ${\bf x}_0$.

To compute the Beylkin determinant for Black/Zhang Jacobian I begin by writing the phase function in the DMO integral kernel as

$$\begin{eqnarray} \Phi&=&\omega t_0-{\bf k} \cdot {\bf x}_0\\ &=&\omega A_2t_2 - {\bf k} \cdot {\bf x}_2 \EQNLABEL{Hale.phase}\end{eqnarray}$$ (31) (32)

The phase in equation Hale.phase is identical to the phase function in Hale's DMO and, therefore, substituting for $\omega$ and ${\bf k}$ back in omega and wave.numb and differentiating with respect to $\omega_{0}$and ${\bf k}_0$, we end up with the following expression for H^-1:

$$H^{-1}=\frac {A_2^3} {2A_2^2 -1}\; .$$ (33)

Therefore, the Beylkin determinant for Black/Zhang's DMO becomes

$$H=\frac {2A_2^2-1}{A_2^3} \\ \EQNLABEL{zhang.beylkin}$$ (34)

which is the same as that for Hales's DMO.

Finally, by substituting back in jacob2 and accounting for the $1/{2\pi}$factor in the spatial Fourier transform, we obtain an expression for the weights of an asymptotic inverse for Black/Zhang's DMO:

$${J_2}=\frac {1} {2\pi} \\ \EQNLABEL{zhang_inv}$$ (35)

These weights have been also derived independently by Paul Fowler (personal communication).