Analytical fundamentals of migration in reflection seismics

Abstract

We consider migration in reflection seismics from a completely analytical perspective. We review the basic geometrical ray-path approach to understanding the subject of migration, and discuss the limitations of this method. We stress the importance of the linear differential wave equation in migration. We also review briefly how a wavefield, travelling with a constant velocity, is extrapolated from the differential wave equation, with the aid of Fourier transforms. Then we present a non-numerical treatment by which we derive an asymptotic solution for both the amplitude and the phase of a planar subsurface wavefield that has a vertical velocity variation. This treatment entails the application of the Wentzel-Kramers-Brillouin approximation, whose self-consistency can be established due to a very slow logarithmic variation of the velocity in the vertical direction, a feature that holds more firmly at increasingly greater subsurface depths. For a planar subsurface wavefield, we also demonstrate an equivalence between two apparently different migration algorithms, namely, the constant-velocity Stolt Migration algorithm and the stationary-phase approximation method.

Keywords

• seismic migration
• body wave propagation
• fourier analysis
• integral transforms