where the subscript refers to the source-side rays and the subscript refers to the receiver-side rays. The data space coordinates are where is the horizontal position of the common-midpoint (CMP) gather and is the half-offset between the source and the receiver.

mul-sktch1
Water-bottom multiple. The
subscript refers to the source and the subscript to the receiver.
Figure 1. | |
---|---|

Wave-equation migration maps
the CMP gathers to SODCIGs with coordinates
where
is the horizontal position of the image gather, and
and are the half subsurface-offset and the depth of the image,
respectively. As illustrated in the sketch of
Figure 2, at any given depth the spatial coordinates of the
downward-continued source and receiver rays are given by:

(2) | |||

(3) |

where is the water velocity, with the sediment velocity, , are the takeoff angles of the source and receiver rays with respect to the vertical and and are the angles of the refracted source and receiver rays, respectively. The coordinates of the migrated multiple in the image space are given by:

The traveltime of the refracted ray segments and can be computed from the two imaging conditions: (1) at the image point the depth of both rays has to be the same (since we are computing horizontal subsurface offset gathers) and (2) which follows immediately from equation 1 since at the image point the total extrapolated time equals the traveltime of the multiple. As shown in Appendix A, the traveltimes of the refracted rays are given by

The refracted angles are related to the takeoff angles by Snell's law: and , from which we get

mul-sktch3
Imaging of water-bottom multiple
in SODCIG and ADCIG. The subscript refers to the data space while the subscript refers to the image space.
The points
and
represent the end points of the source
and receiver ray after migration and must be at the same depth at the image point (for horizontal ADCIGs).
The coordinates
correspond to the image point in the angle domain. The coordinates
correspond to the image point in the subsurface offset gather. The line AB represents
the apparent reflector at the image point.
Figure 2. |
---|

In ADCIGs, the mapping of the multiples can be directly related to the
previous equations by the geometry shown in Figure 2.
The half-aperture angle is given by

Equations 4-6 describe the transformation performed by wave-equation migration between CMP gathers and SODCIGs . Equations 7-12 relate the traveltimes and angles of the refracted segments to parameters that can in principle be computed from the data (traveltimes, takeoff angles, reflector dips and velocities). Equations 13 and 14 provide the transformation from SODCIGs to ADCIGs. These equations are valid for any first-order water-bottom multiple, whether from a flat or dipping water-bottom. They even describe the migration of source- or receiver-side diffracted multiples with the diffractor at the water bottom, since no assumption has been made relating and or the individual traveltime segments. These equations, however, are of little practical use unless we can relate the individual traveltime segments (, , , ), and the angles and to the known data space coordinates (, , ) and the model parameters (, and ). This may not be easy or even possible analytically for all situations, but it is for the simple but important case of a specular multiple from a flat water-bottom.

2007-10-24