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). Subject headings: reflection seismics — migration — mathematical methods 1. Reflection seismology: A basic introduction In seismic processing, migration of seismic data is an exercise of paramount importance. Unless this aspect of processing is addressed satisfactorily, the possibility of making misleading estimates of subsurface depths remains wide open. Even by a simple quantitative estimate it can be shown that in many cases the error in the spatial measurement of the true reflector position could be of the order of a kilometre (Yilmaz 2001). Naturally, when it comes to exploring hydrocarbon reserves, this lack of precision will translate into a huge and wasted monetary expense. So an accurate representation of the depth and the local dip of subsurface reflectors is imperative. Migration of seismic data goes a long way in resolving this most difficult question (Lowrie 1997; Yilmaz 2001). It should be useful for any study on the relevance of migration to start with a simple and pedagogic understanding of reflection seismology (Lowrie 1997; Mari et al. 1999), whose primary purpose is to find the depths to reflecting interfaces and the corresponding seismic velocities of subsurface rock layers. This process first necessitates the artificial generation of mechanical waves at predetermined times and spatial positions on the surface of the earth. These waves pass through various earth layers and are reflected from the subsurface interfaces back to the surface of the earth, where they are recorded by receivers. For immediate purposes, the waves of interest are compressional (longitudinal) waves, which are also referred to as primary waves (Lowrie 1997; Mari et al. 1999). Considering the simplest possible case of this system, one might suppose the surface of the earth to be absolutely flat. From this surface a single source is creating waves which travel through the subsurface and are subsequently