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In this pedagogically motivated work, the process of migration in reflectionseismics has been considered from a rigorously mathematical viewpoint. Aninclined subsurface reflector with a constant dipping angle has been shown tocause a shift in the normal moveout equation, with the peak of the moveoutcurve tracing an elliptic locus. Since any subsurface reflector actually has anon-uniform spatial variation, the use of a more comprehensive principle ofmigration, by adopting the wave equation, has been argued to be necessary. Bythis approach an expression has been derived for both the amplitude and thephase of a subsurface wavefield with vertical velocity variation. Thistreatment has entailed the application of the WKB approximation, whoseself-consistency has been established by the fact that the logarithmicvariation of the velocity is very slow in the vertical direction, a featurethat is much more strongly upheld at increasingly greater subsurface depths.Finally, it has been demonstrated that for a planar subsurface wavefield, thereis an equivalence between the constant velocity Stolt Migration algorithm andthe stationary phase approximation method (by which the origin of the reflectedsubsurface signals is determined).

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