Explicit expressions for map time-migration in weakly anisotropic VTI media

If the velocity of the medium is known and the medium does not incorporate different reflectors that have identical surface seismic measurements that remain identical under small perturbations of the medium, map migration achieves a one-to-one correspondence between the reflections recorded at the surface, and the reflectors in the subsurface, through explicit use of the slopes in the data. Because of this property, it has had many applications in velocity estimation. In this paper, I apply the weak anisotropy approximation to the 3D post-stack and pre-stack map time-migration equations for qP-waves in transversely isotropic homogeneous media with a vertical symmetry axis. In this way the dependence of qP-wave time imaging (in this approximation) on the anellipticity parameter η and the zero-dip NMO velocity VNMO(0) only, is analytically confirmed in the context of map time-migration. The accuracy of the equations is verified with numerical examples. The developed equations allow for the estimation of the anellipticity parameter η in the context of time-migration velocity analysis.