


 Angle domain common image gathers for steep reflectors  

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Reversetime migration solves the two issues in downward continuation migration in generating CIGs for steep reflectors,
but it is well known that it is expensive to apply reversetime routinely.
Planewave migration in tilted coordinates has been demonstrated useful imaging technology for steep reflectors (Shan and Biondi, 2004; Shan et al., 2007).
In planewave migration in tilted coordinates, the propagation direction of the waves illuminating steeply dipping reflectors is usually close to the extrapolation direction and thus they can be imaged correctly.
In this section, we discuss how to generate angle domain CIGs by planewave migration in tilted coordinates and show that
it can also produce reliable CIGs for steep reflectors. We start with CIGs in the conventional planewave migration.
As with shotprofile migration, offset domain CIGs in planewave migration are formed as follows:

(9) 
where is the horizontal subsurface offset,
and
are the source and receiver wavefields corresponding to the ray parameter , respectively.
Notice that the imaging condition in equation 9 is the crosscorrelation between the source and receiver wavefields weighted with the angular frequency , which is also called filter in Radon transform literature.
As with the conventional zerosubsurfaceoffset image, offset domain CIGs defined in equation 9 are equivalent to those obtained by shotprofile migration.
Offset domain CIGs are transformed to angle domain CIGs by local slantstacking (equation 2).


bpvel
Figure 1. Velocity model of the BP velocity Benchmark.






imagetilt
Figure 2. Image obtained by planewave migration in tilted coordinates. Both steep salt flank and nearflat sediments are present in this area.




Given a planewave source corresponding to the ray parameter , we use the tilted coordinates
with a tilting angle .
The subsurface offset domain CIGs for this planewave source are formed by:

(10) 
where the subsurface offset parallels the axis.
In planewave migration in tilted coordinates, the subsurface offset direction is not necessary the geologic dip direction,
but is usually closer to the dip direction for steeply dipping reflectors, than the conventional horizontal subsurface offset.
As for the transformation in the conventional planewave migration, we can transform offset domain CIGs
of planewave source corresponding to
to angle domain CIGs
in tilted coordinates by applying

(11) 
where
and are wavenumbers corresponding to and , respectively.
For each angle , we rotate the image
back to vertical Cartesian coordinates.
The angle domain CIGs of all possible planewave sources are then stacked in vertical Cartesian coordinates.
We can also transform the subsurface offset CIGs obtained by planewave migration in tilted coordinates into
horizontal offset and vertical offset CIGs, and merge them using equation 8 after transforming
them into angle domain CIGs, similarly to reversetime migration.
Equations 6 and 7 are the relationships linking the geologic offset ,
horizontal offset and vertical offset .
The horizontal and vertical offsets are two special cases and the relationship can be generalized to a generaldirection offset.
If the angle between the generaldirection offset and geologic offset is ,
the relationship between them is

(12) 
The angle in equation 12 for is and for is
.
From equation 12, the geologic offset is the optimal offset to generate angle domain CIGs and
the further the offset direction is from the dip direction, the larger the subsurface offset we need
given the same opening angle.
For the tilted coordinate system
, the angle between the subsurface offset and geologic offset is .
Therefore, the subsurface offset in tilted coordinates and the geologic offset can be linked by
the following relationship:

(13) 
From equations 13, 6 and 7, the subsurface offset in tilted coordinates , vertical offset and horizontal offset are linked by the following relationship:



(14) 



(15) 
By equations 14 and 15, the offset domain CIGs in tilted coordinates
can be decomposed into horizontal offset CIGs and vertical offset CIGs.
Vertical offset domain CIGs and horizontal offset domain CIGs of all possible planewave sources are stacked
after being rotated back to vertical Cartesian coordinates.
Being transformed to angle domain CIGs, they are merged using equation 8,
as with reversetime migration.
hxgathers
Figure 3. Horizontal offset domain CIGs with the true velocity obtained by reversetime migration. (a) For relatively flat
sediments, the energy focuses well at zero offset; (b) For steep salt flanks, the energy leaks to far offsets and the frequency is low.






hzgather
Figure 4. Vertical offset domain CIGs with the true velocity obtained by reversetime migration.
For the steeply dipping salt flank at km, the energy focuses well at zero offset.
For the nearflat sediments at km, the energy leaks to far offsets.







 Angle domain common image gathers for steep reflectors  

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Up: Shan and Biondi: Angle
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20070918