Next: Flat waterbottom
Up: Alvarez: Multiples in image
Previous: Introduction
mul_sktch1
Figure 1 Waterbottom multiple. The
subscript s refers to the source and the subscript r to the receiver.

 
The propagation path of a waterbottom multiple, as shown
in Figure , consists of four
segments, such that the total traveltime for the multiple is given by

t_{m}=t_{s1}+t_{s2}+t_{r2}+t_{r1},

(1) 
where the subscript s refers to the sourceside rays and the subscript
r refers to the receiverside rays. The data space coordinates are
(m_{D},h_{D},t_{m}) where m_{D} is the horizontal position of the CMP gather
and h_{D} is the halfoffset between source and receiver.
Waveequation migration maps
the CMP gathers to SODCIGs with coordinates where
is the horizontal position of the image gather, and
and are the half subsurfaceoffset and the depth of the image,
respectively.
mul_sktch2
Figure 2 Imaging of waterbottom multiple
in SODCIG. The subscript refers to the image point.

 
As illustrated in the sketch of
Figure , at any given depth the image space coordinates
of the migrated multiple are given by:
 

 
 (2) 
 (3) 
 (4) 
where V_{1} is the water velocity, with V_{2} the sediment
velocity, and , are the acute takeoff angles of the source
and receiver rays with respect to the vertical. The
traveltime of the refracted ray segments and can be computed from two 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 extrapolated time
equals the traveltime of the multiple. As shown in Appendix A, the
traveltimes of the refracted rays are given by
 
(5) 
 (6) 
The refracted angles are related to the takeoff angles by Snell's law:
and
, from which we get
 
(7) 
 (8) 
 (9) 
 (10) 
Equations 210 are valid for any
waterbottom multiple, whether from
a flat or dipping waterbottom. They even describe the migration of
source or receiverside diffraction multiples, since no assumption has been
made relating and or the individual traveltime segments.
mul_sktch3
Figure 3 Imaging of waterbottom multiple
in ADCIG. The subscript refers to the image point. The line AB represents
the apparent reflector at the image point.

 
In ADCIGs, the mapping of the multiples can be directly related to the
previous equations by the geometry shown in Figure .
The halfaperture angle is given by
 
(11) 
which is the same equation used for converted waves
Rosales and Biondi (2005). The depth of the image point
() is given by (Appendix B)
 
(12) 
Equations 212 formally describe the image
coordinates in terms of
the data coordinates. They are, however, of little practical use unless
we can relate the individual traveltime segments (t_{s1}, t_{s2},
, t_{r2}, , t_{r1}), and the
angles and (which in turn determine and
) to the known data space
parameters (m_{D}, h_{D}, t_{m}, V_{1}, and ). This may not be easy or
even possible analytically for all situations, but it is for some simple
but important models that I will now examine.
Next: Flat waterbottom
Up: Alvarez: Multiples in image
Previous: Introduction
Stanford Exploration Project
11/1/2005