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REFERENCES

Baysal, E., Kosloff, D. D., and Sherwood, J. W. C., 1983, Reverse time migration: Geophysics, 48, no. 11, 1514-1524.

Biondi, B., and Sava, P., 1999, Wave-equation migration velocity analysis: 69th Ann. Internat. Meeting, Soc. of Expl. Geophys., Expanded Abstracts, 1723-1726.

Biondi, B., and Shan, G., 2002, Prestack imaging of overturned reflections by reverse time migration: 72nd Ann. Internat. Meeting, Soc. of Expl. Geophys., Expanded Abstracts, 1284-1287.

Biondi, B., 2003, Equivalence of source-receiver migration and shot-profile migration: Geophysics, accepted for publication.

Clapp, R., and Biondi, B., 2000, Tau domain migration velocity analysis using angle CRP gathers and geologic constraints: 70th Ann. Internat. Mtg., Soc. Expl. Geophys., 926-929.

de Bruin, C. G. M., Wapenaar, C. P. A., and Berkhout, A. J., 1990, Angle-dependent reflectivity by means of prestack migration: Geophysics, 55, no. 9, 1223-1234.

de Hoop, M., Le Rousseau, J., and Biondi, B., 2002, Symplectic structure of wave-equation imaging: A path-integral approach based on the double-square-root equation: Journal of Geophysical Research, accepted for publication.

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Etgen, J., 1990, Residual prestack migration and interval velocity estimation: Ph.D. thesis, Stanford University.

Liu, W., Popovici, A., Bevc, D., and Biondi, B., 2001, 3-D migration velocity analysis for common image gathers in the reflection angle domain: 69th Ann. Internat. Meeting, Soc. of Expl. Geophys., Expanded Abstracts, 885-888.

Mosher, C. C., Foster, D. J., and Hassanzadeh, S., 1997, Common angle imaging with offset plane waves: 67th Annual Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1379-1382.

Mosher, C., Jin, S., and Foster, D., 2001, Migration velocity analysis using common angle image gathers: 71st Ann. Internat. Mtg., Soc. of Expl. Geophys., 889-892.

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Rickett, J., and Sava, P., 2002, Offset and angle-domain common image-point gathers for shot-profile migration: Geophysics, 67, 883-889.

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A PROOF THAT THE TRANSFORMATION TO GOCIG CORRECTS FOR THE IMAGE-POINT SHIFT This appendix proves that by applying the offset transformations described in equations (9) and (10) we automatically remove the image-point shift characterized by equations (11) and (12). The demonstration for the VOCIG transformation is similar to the one for the HOCIG transformation, and thus we present only the demonstration for the HOCIGs. HOCIGs are transformed into GOCIGs by applying the following change of variables of the offset axis xh, in the vertical wavenumber kz and horizontal wavenumber kx domain:  
 \begin{displaymath}
x_h= \frac{\widetilde{h_0}}{\cos\alpha} = 
{\rm sign}\left(\...
 ...tilde{h_0}\left({1+\frac{k_{x}^2}{k_z^2}}\right)^{\frac{1}{2}}.\end{displaymath} (23)
For the sake of simplicity, in the rest of the appendix we will drop the ${\rm sign}$ in front of expression (23) and consider only the positive values of kx/kz.

We want to prove that by applying (23) we also automatically shift the image by  
 \begin{displaymath}
\Delta_z I_{x_h}= - \widetilde{h_0}\tan{\gamma}\tan{\alpha}\sin{\alpha}\end{displaymath} (24)
in the vertical direction, and  
 \begin{displaymath}
\Delta_x I_{x_h}= \widetilde{h_0}\tan{\gamma}\tan{\alpha}\cos{\alpha}\end{displaymath} (25)
in the horizontal direction.

The demonstration is carried out in two steps: 1) we compute the kinematics of the impulse response of transformation (23) by a stationary-phase approximation of the inverse Fourier transform along kz and kx, and 2) we evaluate the dips of the impulse response, relate them to the angles $\alpha$ and $\gamma$, and then demonstrate that relations (25) and (24) are satisfied.


 
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Next: Evaluation of the impulse Up: Biondi and Symes: ADCIGs Previous: Residual moveout in ADCIGs
Stanford Exploration Project
7/8/2003