Phase shift approximation for the post-critical seismic wave

Abstract

Post-critical seismic waves are widely used in crustal exploration of the seismic velocity structure, and are gaining interest in the oil/gas seismic community to image the deeper structure beneath the high velocity basalt layer. They are featured with their phase shifts and strength changes, which should be taken into account in seismic data processing, such as velocity analysis and true amplitude migration, etc. In order to simplify the exact but complicated formula of reflection and transmission coefficients, numerous approximate expressions for reflection and transmission coefficients for pre-critical incidence are obtained. In the post-critical case, there is Downton's approximation with acceptable accuracy approximation when the velocity changes smoothly. However if the velocity model changes rapidly, the error will be relatively very large, limiting the use of the approach. In order to improve the post-critical approximation, we utilize Taylor expansion of ray parameters with angle increment (compared to critical angle) in wide-angle seismic reflection and transmission coefficients. The explicit expressions for amplitude and phase shift (time shift) for the post-critical incident angle are obtained. Our results confirm that the wide-angle seismic reflection/transmission phase shifts are strongly frequency dependent; phase shifts of low frequency wide-angle seismic waves are more predominant and their correction should be considered in seismic processing and imaging. Numerical examples demonstrate that (1) the accuracies of these approximations are high compared to the classic Aki's formula and Downton's approximation, and (2) the wide-angle effect can be effectively reduced with phase-shift correction by utilizing our time-shift approximation to the seismic traveltimes.