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One way to reduce the computational cost of 3D prestack depth migration is to use commonazimuth data, which is only a 4D dataset. Commonazimuth migration is then derived as the solution of the oneway wave equation through a recursive downwardcontinuation operator. In the frequencywavenumber domain, this operator can be expressed as a simple phaseshift applied to the wavefield. We use the notation indicated below. The vertical wavenumber k_{z} is given by a 3D dispersion relation called the Double Square Root (DSR) equation Claerbout (1984):
 

 (1) 
where is the midpoint wavenumber (), the offset wavenumber () and k_{z} the vertical wavenumber (), with , , and unit vectors of the four midpoint and offset axes. The propagation velocities and correspond respectively to the receiver and source locations.
The first square root in equation (1) downwardcontinues the receiver wavefield, whereas the second one downwardcontinues the source wavefield. In the algorithm developed by Biondi and Palacharla 1996, the data at the new depth level D_{z+dz} are obtained from commonazimuth data D_{z} by the following integration:
 

 (2) 
 
In practice, we use the stationaryphase approximation to compute this integral (see Appendix A). Popovici 1995 finds a very similar expression for the kernel of migration to zerooffset (MZO) in 2D and asserts that the stationaryphase approximation avoids the very high computational cost of a numerical evaluation of the integral.
In our case, the stationary phase approximation of the commonazimuth downwardcontinuation operator for data where h_{y}=0 can be written as
 
(3) 
with the phase .
For data evaluated only at the origin of the crossline offset axis (h_{y}=0), Biondi and Palacharla 1996 derived an analytical expression for the stationary point:
 
(4) 
We have implemented the kernel of downwardcontinuation and phaseshift migration with the preceding theory. Simulations performed with the standard commonazimuth migration code have already proven its efficiency in imaging complex media Biondi (1999). As shown in Figures 1 and 2, the commonazimuth technique provides accurate subsalt images. Still, on the left side of Figure 1(b), imaging the major fault has obviously met with some difficulties, which can be attributed to rapid lateral variations of the velocity model. As seen in Figures 2(a) and 2(b), the same fault, now on the right side, is also dipping in the crossline direction. Extending the commonazimuth technique may help improve the imaging of these lateral velocity variations.
SEGEAGEin
Figure 1 Commonazimuth prestack depth migration of data from the SEGEAGE salt model. Inline section at CMPy=10400 (m). The top plot (a) represents the exact velocity model and the bottom plot (b) represents the migration result.
SEGEAGEcross
Figure 2 Commonazimuth prestack depth migration of data from the SEGEAGE salt model. Crossline section at CMPx=2100 (m). The top plot (a) represents the exact velocity model and the bottom plot (b) represents the migration result.
Next: Extending crossline offset range
Up: Vaillant & Biondi: Extending
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Stanford Exploration Project
4/20/1999