The traveltime of the first-order water-bottom multiple is given by

From the symmetry of the problem,
and
,
which in turn means
. Furthermore, from
Equations 7 and 8 it immediately follows that
and
which says that
the traveltimes of the refracted rays are equal to the traveltimes
of the corresponding segments of the multiple. Equation 4 thus simplifies
to

From Equation 5, the depth of the image point can be easily
computed as

This result shows that the multiple is mapped in the image space to the same horizontal position as the corresponding CMP even if migrated with sediment velocity. This result is a direct consequence of the symmetry of the raypaths of the multiple reflection in this case. For dipping water bottom or for diffracted multiples this is not the case (, ).

Equations 16-18 give the image
space coordinates in terms of the data space coordinates. An important
issue is the functional relationship between the subsurface
offset and the image depth, since it determines the moveout of the
multiples in the subsurface-offset-domain common-image-gathers (SODCIGs).
Replacing
and
in
Equation 17 we get

odcig1
Subsurface offset domain common
image gather of a water-bottom multiple from a flat water-bottom. Water
velocity is 1500 m/s, water depth 500 m, sediment velocity 2500 m/s and
surface offsets from 0 to 2000 m. Overlaid is the residual moveout curve
computed with Equation 19.
Figure 3. | |
---|---|

Figure 3 shows an SODCIG for a specular water-bottom multiple from a flat water-bottom 500 m deep. The data was migrated with a two-layer velocity model: the water layer of 1500 m/s and a sediment layer of velocity 2500 m/s. Larger subsurface offsets (which according to Equation 16 correspond to larger surface offsets) map to shallower depths for the usual situation of , as we should expect since the rays are refracted to increasingly larger angles until the critical reflection angle is reached. Also notice that the hyperbola is shifted down by a factor with respect to its image point when migrated with water velocity.

In angle-domain common-image-gathers (ADCIGs), the half-aperture angle
reduces to
, which in terms of the data
space coordinates is given by

If this expression simplifies to , which is the aperture angle of a primary at twice the water-bottom depth.

As I did with the SODCIG, it is important to find the functional
relationship between
and since it dictates the
residual moveout of the multiple in the ADCIG. Plugging
Equations 16 and 17 into equation 14,
using Equations 15, and 20 to eliminate and
simplifying we get

Once again, when the multiple is migrated with the water velocity () we get the expected result , that is, flat moveout (no angular dependence). The residual moveout in ADCIGs is therefore given by

This equation reduces to that of Biondi and Symes (2004) when is small (Appendix C), which is when we can neglect ray bending at the multiple-generating interface. Panel (a) of Figure 4 shows the ADCIG corresponding to the SODCIG shown in Figure 3. Notice that the migrated depth at zero aperture angle is the same as that for the zero sub-surface offset in Figure 3. For larger aperture angles, however, the migrated depth increases as indicated in equation 23. The continuous line corresponds to Equation 24 whereas the dotted line corresponds to the tangent-squared of Biondi and Symes (2004). For this model, the departure of the straight ray approximation can be more than 5% for large aperture angles as illustrated in panel (b). The relative error represents the different between the two approximations divided by that the more accurate of equation 24.

adcig1
Panel (a) is an ADCIG for a water-bottom multiple
from a two flat-layer model. The dotted curve corresponds to the straight
ray approximation whereas the solid curve corresponds to the ray-bending
approximation. Panel (b) is the relative error between the two approximations.
Figure 4. |
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2007-10-24