Seismic time migration is a cost and time effective process for imaging of the subsurface. However, the approach is traditionally limited to velocity models with mildly varying lateral variations.The objective of the present work is to study the potential for extending the prestack Kirchhoff time migration method to cases where previously depth migration was required, because of lateral velocity variations and/or anisotropy. New additional higher-order terms have been introduced to the conventional double square root (DSR) diffraction-time function. The results of their approximations to diffraction-time surfaces obtained using the NORSAR-3D software have been discussed. Tests have been performed for both isotropic and anisotropic media. For the isotropic media, tests were performed for both vertically varying velocity fields, and also velocity fields dominated by lateral velocity variations. For the anisotropic media, tests have been performed for both vertical transverse isotropic (VTI) and tilted transverse isotropic (TTI) media with varying anisotropy parameter η. Here, the tests were performed on vertically varying velocity fields and when the velocity gradient was tilted 15 degrees with respect to the vertical.
The accuracy of the various diffraction-time surface approximations was determined by the root mean square (RMS) error in seconds. Our results show that additional terms can provide better approximations of the diffraction-time surface. Diffraction-time surfaces corresponding to a vertically varying velocity field are well approximated by symmetric functions of higher-order. This is also true for tests in anisotropic geomodels with a vertical symmetry axis. On the other hand, approximations of simple asymmetric diffraction-time surfaces caused by lateral velocity variation or TTI can benefit from introducing odd terms of higher-order to the DSR function. The results also underscores that additional terms are not a guarantee to achieve good approximations, as the possibility of a good result depends largely on the character of the diffraction-time surface to be approximated.
Our results also show that the diffraction-time surface approximation error is more sensitive to the lateral velocity gradient than to the vertical velocity gradient in isotropic media. Lastly, the effect of changing the axis of symmetry from vertical, to 15 degrees with respect to the vertical for anisotropic media had less impact on the approximation error, than when the velocity gradient was changed similarly.