True-amplitude DMO

The goal is to define an amplitude-preserving AMO from a true amplitude DMO and its true amplitude inverse. To select a true-amplitude DMO, I test the amplitude preservation by the DMO transformation applied to the dipping bed in the previous experiment. The input was corrected for normal moveout and spherical divergence. Figure dmo shows the peak amplitudes from the forward DMO operators compared to the theoretical amplitude for zero-offset equivalent data generated by Kirchhoff modeling. The theoretical curve almost coincides with the amplitudes of Zhang's DMO. The amplitudes given by Hale's algorithm fall below the theoretical curve whereas the peak-amplitudes from Bleistein's DMO overshoot the correct amplitudes.

The difference between Bleistein's DMO and Black/Zhang's DMO results from a philosophical difference about the definition of ``true-amplitude''. While the goal in the synthetic tests was to preserve the peak amplitude of each reflection event, Bleistein's algorithm is based on preserving the spectral density of the image wavelet. A second difference results from the sequence in the processing flow surrounding DMO. A divergence correction must be applied to the input prior to applying Black/Zhang's DMO, whereas both input and output of Bleistein's DMO decay with spherical divergence factors of $1 \over t_2$ and and $1 \over t_0$, respectively These two differences account for the A² factor between the two Jacobians leading to higher weights on Bleistein's DMO, which results in higher peak amplitudes than those on the predicted curve.

On the other hand, the difference between Black/Zhang's DMO and Hale's DMO results from the fact that the former algorithm accounts for the reflection point smear and, therefore, correctly positions input events at their true zero-offset locations. The two Jacobians differ by a factor of

$$\frac {B_Z} {B_H}= \frac {2A^2-1} {A^2}$$ (38)

Because this ratio being always larger than 1, it leads to lower weights on Hale's operator, which explains the lower peak amplitudes measured along the dipping event from Hale's DMO.

Consequently, to be consistent with the original definition of preserving the peak amplitudes of reflection events, I define the amplitude function of AMO in terms of Black/Zhang's DMO and its asymptotic inverse.