Stereographic imaging condition for wave-equation migration |

stereo2
Comparison of
conventional imaging (a) and stereographic imaging (b).
Figure 4. |
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A simple way of decomposing the source and receiver wavefields
function of local slope at every position and time is by local
slant-stacks at coordinates and in the four-dimensional
source and receiver wavefields. Thus, we can write the total source
and receiver wavefields ( and ) as a sum of decomposed
wavefields ( and ):

Here, the three-dimensional vector represents the local slope function of position and time. Using the wavefields decomposed function of local slope, and , we can design a stereographic imaging condition which cross-correlates the wavefields in the decomposed domain, followed by summation over the decomposition variable:

Correspondence between the slopes p of the decomposed source and receiver wavefields occurs only in planes dipping with the slope of the imaged reflector at every location in space. Therefore, an approximate measure of the expected reflector slope is required for correct comparison of corresponding reflection data in the decomposed wavefields. The choice of the word ``stereographic'' for this imaging condition is analogous to that made for the velocity estimation method called stereotomography (Billette et al., 2003; Billette and Lambare, 1997) which employs two parameters (time and slope) to constrain traveltime seismic tomography.

For comparison with the stereographic imaging condition 7, the
conventional imaging condition can be reformulated using the wavefield
notation 5-6 as follows:

Figure 3(b) shows the image produced by stereographic imaging of the data generated for the model depicted in Figures 1(a)-1(b), and Figure 5(b) shows the similar image for the model depicted in Figures 2(a)-2(b). Images 3(b) and 5(b) use the same source receiver wavefields as images 3(a) and 5(a), respectively. In both cases, the cross-talk artifacts have been eliminated by the stereographic imaging condition.

ii,kk
Images obtained for the model in Figures 2(a)-2(c)
using the conventional imaging condition (a) and the stereographic
imaging condition (b).
Figure 5. |
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ttr1,ii1,ttr2,ii2,ttr0,ii0,vel,kk
Data corresponding to shots located at coordinates kft (a),
kft (c), and the sum of data corresponding to both shot
locations (e). Image obtained by conventional imaging condition for
the shots located at coordinates kft (b), kft (d) and
the sum of data for both shots (f). Velocity model extracted from the
Sigsbee 2A model (g) and image from the sum of the shots located at
kft and kft obtained using the stereographic imaging
condition (h).
Figure 6. |
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Stereographic imaging condition for wave-equation migration |

2013-08-29