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

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).

💡 Research Summary

The paper presents a rigorous, mathematically‑driven treatment of migration in reflection seismics, beginning with the simplest 2‑D geometry and progressing to a full 3‑D wave‑field formulation. In the introductory section the authors revisit the classic normal‑move‑out (NMO) relationship for a flat reflector, deriving the hyperbolic travel‑time curve (t^{2}=t_{0}^{2}+x^{2}/v^{2}) where (t_{0}=2d/v) is the two‑way vertical travel time. They then consider a single dipping interface of constant dip θ. By shifting the horizontal coordinate to (\bar{x}=x-2d\sin\theta) and redefining the zero‑offset time to (\bar{t}{0}=2d\cos\theta/v), the same hyperbolic form is retained, but the apex of the hyperbola moves to ((-2d\sin\theta,\ \bar{t}{0})). Varying θ traces an ellipse in the x‑t plane, expressed as ((x_{m}^{2}/(2d)^{2})+(t_{m}^{2}v^{2}/(2d)^{2})=1). This geometric insight shows that dipping reflectors shift the NMO apex along an elliptic locus.

The authors argue that real subsurface structures are far more complex: multiple layers, non‑uniform dip, faults, and a velocity that varies with depth. Consequently, a simple NMO correction is insufficient, and migration must be defined as the extrapolation of the recorded surface wavefield (P(x,y,0,t)) into the earth’s interior using the full acoustic wave equation \