Migration of a subsurface wavefield in reflection seismics: A mathematical study

TL;DR

This paper provides a rigorous mathematical analysis of seismic migration, deriving expressions for wavefield amplitude and phase, and demonstrating the equivalence of migration algorithms under certain conditions.

Contribution

It introduces a wave-equation-based approach to seismic migration, accounting for non-uniform velocity variations and establishing the connection between different migration methods.

Findings

Inclined reflectors cause elliptic moveout curves.

Wave equation approach captures amplitude and phase variations.

Stolt Migration and stationary phase are equivalent for planar wavefields.

Abstract

In this pedagogically motivated work, the process of migration in reflection seismics has been considered from a rigorously mathematical viewpoint. An inclined subsurface reflector with a constant dipping angle has been shown to cause a shift in the normal moveout equation, with the peak of the moveout curve tracing an elliptic locus. Since any subsurface reflector actually has a non-uniform spatial variation, the use of a more comprehensive principle of migration, by adopting the wave equation, has been argued to be necessary. By this approach an expression has been derived for both the amplitude and the phase of a subsurface wavefield with vertical velocity variation. This treatment has entailed the application of the WKB approximation, whose self-consistency has been established by the fact that the logarithmic variation of the velocity is very slow in the vertical direction, a feature that is much more strongly upheld at increasingly greater subsurface depths. Finally, it has been demonstrated that for a planar subsurface wavefield, there is an equivalence between the constant velocity Stolt Migration algorithm and the stationary phase approximation method (by which the origin of the reflected subsurface signals is determined).