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## Water-bottom multiple

The water-bottom multiple from a dipping reflector has exactly the same kinematics as a primary from a reflector with twice the dip Alvarez (2005), that is,
 (31)
where is the dip of the reflector, is the perpendicular depth to the equivalent reflector with twice the dip (at the CMP location) and is the NMO velocity of the equivalent primary .

Following the same procedure as for the flat water-bottom, we compute the coordinates of the image point using equations 2-4 and noting that in this case ,
 (32) (33) (34)
where, according to equations 7-10,
 (35) (36) (37) (38)
The traveltimes of the individual ray segments are computed by repeated application of the law of sines as shown in Appendix C:
 (39) (40) (41) (42)
where is the perpendicular depth to the reflector at the CMP location and is given by (Appendix C):
 (43)
Notice that this is not the same as in equation 31, which corresponds to the perpendicular depth to the equivalent reflector whose primary has the same kinematics as the water-bottom multiple.

The traveltime of the refracted ray segments are given by equations 5 and 6 with
 (44)
In order for equation 32-34 to be useful in practice, we need to express them entirely in terms of the known data coordinates, which means that we need to find an expression for in terms of . In Appendix C it is shown that
 (45)
We now have all the pieces to compute the image space coordinates, since and can be computed from equations 5 and 6 using equations 35-45.

Appendix D shows that equations 32-34 reduce to the corresponding equations for the non-diffracted multiple from a flat water bottom when , as they should.

Figure  shows the zero subsurface offset section from a migrated non-diffracted multiple from a dipping water-bottom. The overlaid curve was computed with equations 32-34. The dip of the water-bottom is 15 degrees and intercepts the surface at CMP location zero. The CMP range of the data is from 2000 to 3000 m and the surface offsets from 0 to 2000 m. The multiple was migrated with the same two-layer model described before. Notice how the multiple was migrated as a primary. Since the migration velocity is faster than water-velocity, the multiple is over-migrated and appears much steeper and shallower than it should (recall that it would be migrated as a reflector with twice the dip is the migration velocity were that of the water.)

 image3 Figure 11 image section at zero subsurface offset for a non-diffracted multiple from a dipping water-bottom. The overlaid curve was computed with equation 34 and 33.

Figure  shows the SODCIG at CMP 1500 m in i Figure . Just as for the flat water-bottom, the multiple energy is mapped to negative subsurface offsets since . The overlaid curve is the moveout computed with equations 32-34.

 odcig3 Figure 12 SODCIG from a non-diffracted multiple from a dipping water-bottom. The overlaid residual moveout curve was computed with equation 32 and 33.

The aperture angle is given by equation 11 with
 (46)
and given by equation 45. The image depth in the ADCIG is given by equation 12 with and given by equation 46 and and given by equations 32 and 33. Figure  shows the ADCIG corresponding to the SODCIG in Figure . Notice that the apex is at zero aperture angle.

 adcig3 Figure 13 ADCIG corresponding to the SODCIG shown in Figure . The overlaid residual moveout curve was computed with equation 33,  11,  12, and 46.

Next: Diffracted multiple Up: Dipping water-bottom Previous: Dipping water-bottom
Stanford Exploration Project
11/1/2005