Inverse DMO vs. Adjoint DMO

Figures Spike and inv-dmo compare results of different inverse DMO operators. The left panel of Figure Spike is an in-line section from a common offset cube consisting of five unit-amplitude spikes. The offset is 800m and the CMP spacing is 20 m in both directions. I compare the output of each true inverse to the output of the adjoint Ronen (1987). The ideal output would be five spikes with unit amplitudes. The table below summarizes the output of each inverse DMO for increasingly deep spikes.

 

Spike
Figure 1 From left to right: input data, cascade of several DMO operators with their inverse.

IDEAL Amp. Hale Black/Zhang Bleistein Ronen

1.00 .76 .76 .76 .35

1.00 .95 .95 .95 .66

1.00 .97 .97 .97 .77

1.00 .97 .97 .97 .82

1.00 .98 .98 .98 .86

Figure inv-dmo represents a similar test for a synthetic model consisting of a single bed dipping at 35 degrees angle. The input data is a constant-offset section recorded at an offset of 1600 m with CMP spacing of 25 m. Each DMO operator is applied to the input section to construct its zero-offset equivalent data. The zero-offset sections are then back projected to the original offset of 1600 m using the inverse DMO process. The plot of peak amplitudes from the cascade of each DMO with its asymptotic inverse coincides perfectly with the original amplitudes from the constant-offset input section. On the same graph I also display the peak amplitudes from the cascade of DMO with its adjoint. The results of the adjoint clearly fall below the theoretical values.

 

inv-dmo
Figure 2 Peak amplitudes along a dipping event from the cascade of several DMO operators and their inverse. The solid curve is the result of the adjoint inverse. The results of the asymptotic inverse for Hale, Zhang, and Bleistein operators coincide with original input amplitudes

 

dmo
Figure 3 Peak amplitudes along a dipping event from a zero-offset section obtained by different DMO operators. The continuous curve is the the predicted result, the dashed curve is Hale's result, the dotted curve is Black/Zhang's output and the large dashed curve (top curve) is Bleistein's output