Conical-wave source migration

For narrow azimuth data, conical-wave source migration has been demonstrated as an efficient way to image the subsurface Duquet et al. (2001); Whitmore (1995); Zhang et al. (2005). The conical-wave source data are generated as follows

$$R_c(p_x,s_y;r_x,r_y,z=0,\omega)=\int R(s_x,s_y;r_x,r_y,z=0,\omega)e^{i\omega (s_xp_x)}ds_x.$$ (9)

And the corresponding conical source at the surface is

$$S_c(p_x,s_y;r_x,r_y,z=0,\omega)=\int e^{i\omega(s_xp_x)} ds_x.$$ (10)

Similar to the plane-wave source migration, the image of a conical-wave source can be obtained by cross-correlating the source and receiver wavefields weighted with the frequency $\omega$:

$$I_{p_x,s_y}(x,y,z)=\int \omega \bar{S_c}(p_x,s_y;x,y,z,\omega)R_c(p_x,s_y;x,y,z,\omega)d\omega,$$ (11)

where $S_c(p_x,s_y;x,y,z,\omega)$ and $R_c(p_x,s_y;x,y,z,\omega)$ are the conical-wave source and receiver wavefields extrapolated from the surface using equation 1 and 2. The final image is generated by stacking images of all possible conical-wave sources of all sail lines:

$$I_c=\int\int I_{p_x,s_y}(x,y,z) dp_xds_y.$$ (12)

Similar to the 3D plane-wave migration, the image of conical-wave migration is equivalent to the shot-profile migration.