Discussion

The results of the previous sections illustrate that non-diffracted water-bottom multiples (whether from flat or dipping water-bottom) map to negative subsurface offsets (since $h_D\ge 0$ in this case), whereas primaries migrated with slower velocities would map to positive subsurface offsets. This suggests an easy strategy to attenuate these multiples. Migrate the data with a constant velocity that is faster than water velocity but slower than sediment velocity. Keep only the positive subsurface offsets and demigrate with the same velocity. In principle, the primaries would be restored (at least kinematically) whereas the multiples would be attenuated. Although not shown here, the same conclusion can be reached for higher-order non-diffracted water-bottom multiples. This strategy, however, would not work for diffracted multiples since they may map to positive subsurface offsets even when migrated with a velocity faster than water velocity as illustrated schematically in Figure . We can still separate these multiples from the primaries, but that requires the application of an appropriate Radon transform. An apex-shifted tangent-squared Radon transform was applied by Alvarez et. al. 2004 to a real 2D section with good results, but the basic assumption there was that of no ray-bending at the water-bottom interface. It is expected that the more accurate equations derived here will allow the design of a better Radon transform and therefore a better degree of separation between primaries and diffracted multiples. This is the subject of continuing research.

 

mul_sktch7 Figure 18 Sketch illustrating that diffracted multiples may map to positive subsurface offsets.

For the non-diffracted multiple from a flat water-bottom the mapping between the image-space coordinates and the data-space coordinates is essentially 2D since $m_D=m\xi$, which allowed the computation of closed-form expressions for the residual moveout of the multiples in both SODCIGs and ADCIGs. For diffracted multiples in particular, it is not easy to compute equivalent closed-form expressions, but we can compute numerically the residual moveout curves given the expression for $(h_\xi,z_\xi,m_\xi)$ in terms of the data-space coordinates (t_m,h_D,m_D), $\varphi$ and X_diff. In principle, the dip of the water-bottom can be estimated from the data and the position of the diffractor corresponds to the lateral position of the apex of the multiple diffraction in a shot gather as illustrated in the sketch of Figure .

 

mul_sktch8 Figure 19 Sketch illustrating the raypaths of a diffracted multiple in a shot gather. The lateral position of the diffractor corresponds to the apex of the moveout curve.