Migration Of Seismic Data From Inhomogeneous Media
Authors: Hatton L. , Larner K. , Gibson B. 
Published: October 2008

Abstract
Because conventional time-migration algorithms include the implicit assumption of local homogeneity, they leave events mispositioned where overburden velocity varies laterally. Conventional time-migrated sections can be approximately corrected by a ray-theoretical "depth migration," a mapping from time into depth along image rays. Wavefront distortion due to complex geologic structure can however, render time-migrated sections barely interpretable and thus not readily correctable by ray-theoretical depth migration.
For complex media, accurate migration requires an algorithm that honors essential details of the wave equation for inhomogeneous media. That algorithm migrates a time section directly into depth in one processing step.
Unfortunately, where geologic structure is complicated we usually lack details of the velocity model. It is important, therefore, to understand the sensitivity of migration results to errors in the available velocity information. Migrations for synthetic models of differing complexity distinguish the types of media for which ray-theoretical depth migration is adequate from those which require the more accurate wave theoretical direct migration into depth. By perturbing migration velocities from their correct model values we also demonstrate the accuracy requirements of these two approaches. We find that the ability of the depth migration process to place reflections at correct lateral positions is less sensitive to variations of layer velocity than to orientations of interpreted horizons.
The depth migrations of data from offshore California demonstrates that the wave-theoretical approach yields a treatment of amplitudes superior to that of the image ray technique.
Frequently, where a medium is structurally complex, arguments are made in favor of migrating offset sections before stacking. When the overburden is laterally variable, however, conventional time migration of unstacked offset sections can be as destructive to steeply dipping reflections as is CDP stacking prior to migration. Thus, migration before stack should often be performed as depth migration before stack.
Once we squarely confront the problems of inhomogeneous media, we recognize that we must address three-dimensional complexities as well. The time may come when 3-D depth migration before stack becomes the new "conventional" migration.

Introduction
In many areas of exploration interest, seismic data are collected over geologic structures which have substantial lateral velocity variation. Most often, this lateral inhomogeneity is directly related to steeply dipping beds and, in processing the seismic data, migration will almost always be performed.
Hubral (1977) showed that the Kirchhoff-summation migration of data from laterally inhomogeneous media fails to position reflected events properly. Larner et al (1978) showed that the other conventional techniques yield similar results. Errors in position result from approximations made to the scalar wave equation - the foundation of all migration techniques commonly used today, regardless of their implementation (finite-difference, Kirchhoff summation, or frequency domain). The offending approximations can be identified in each technique, explaining why all three approaches similarly misplace reflected events.