next up previous print clean
Next: Tangent to the impulse Up: Rosales and Biondi: PS-ADCIG Previous: Conclusions and Future work


Biondi, B., and Symes, W., 2004, Angle-domain common-image gathers for migration velocity analysis by wavefield-continuation imaging: Geophysics, 69, no. 5, 1283-1298.

Biondi, B., 2005, Angle-domain common image gathers for anisotropic migration: SEP-120, 77-104.

Brandsberg-Dahl, S., de Hoop, M. V., and Ursin, B., 1999, The sensitivity transform in the common scattering-angle/azimuth domain: 61st Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1538-1541.

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.

Fomel, S., and Prucha, M., 1999, Angle-gather time migration: SEP-100, 141-150.

Fomel, S., 1996, Migration and velocity analysis by velocity continuation: SEP-92, 159-188.

Prucha, M., Biondi, B., and Symes, W., 1999, Angle-domain common-image gathers by wave-equation migration: 69th Ann. Internat. Meeting, Soc. of Expl. Geophys., Expanded Abstracts, 824-827.

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

Rosales, D., and Rickett, J., 2001a, ps-wave polarity reversal in angle domain common-image
gathers: SEP-108, 35-44.

Rosales, D., and Rickett, J., 2001b, PS-wave polarity reversal in angle domain common-image gathers: 71st Annual Internat. Mtg., Expanded Abstracts, 1843-1846.

Sava, P., and Fomel, S., 2000, Angle-gathers by Fourier Transform: SEP-103, 119-130.

Sava, P., and Fomel, S., 2003, Angle-domain common-image gathers by wavefield continuation methods: Geophysics, 68, no. 3, 1065-1074.

A In this part, we obtain the relation to transform subsurface offset-domain common-image gathers into angle-domain common-image gathers for the case of PS data. To perform this derivation, we use the geometry in Figure [*] in order to obtain the parametric equations for migration on a constant velocity medium.

Following the derivation of Fomel (1996) and Fomel and Prucha (1999), and applying simple trigonometry and geometry to Figure [*], we obtain parametric equations for migrating an impulse recorded at time tD, midpoint mD and surface offset hD as follows:

Figure 9
Parametric formulation of the impulse response.

z_\xi&=& (L_s+L_r)\frac{\cos{\beta_r} \cos{\beta_s}}{\cos{\beta...
where the total path length is:
t_D &=& S_sL_s+S_rL_r, \nonumber\\ z_s - z_r &=& L_s\cos{\beta_s}-L_r\cos{\beta_r}.\end{eqnarray}
From that system of equations, Biondi (2005) shows that the total path length is
L=\frac{t_D}{2}\frac{\cos{\beta_r}+\cos{\beta_s}}{S_s\cos{\beta_r}+S_r\cos{\beta_s}}.\end{displaymath} (16)
Appendix A shows that we can rewrite system (14) as:
 z_\xi&=& \frac{(L_s+L_r)}{2} \frac{\cos^2{\alpha}-\sin^2{\gamm...
 ...\xi&=& m_D - \frac{(L_s+L_r)}{2}\frac{\sin{\alpha}}{\cos{\gamma}}.\end{eqnarray}
where $\alpha$ and $\gamma$ follow the same definition as in equation (3). where, L in terms of the angles $\alpha$ and $\beta$ is:
L(\alpha,\beta)=\frac{t_D}{(S_r+S_s)+(S_r-S_s)\tan{\alpha}\tan{\gamma}}\end{displaymath} (18)