Migration of seismic data by the WKBJ method

Migration is a term used in reflection seismology to describe the process of moving the recorded reflection events to their correct spatial positions by backward projection or depropagation. Although seismic records have been migrated since the very first reflection survey in 1921 in Oklahoma, it was the paper by Loewenthal, Lu, Robertson, and Sherwood [1] in 1976 that introduced the exploding-reflector model and numerical wave-equation methods that developed into the time migration techniques in use by the petroleum exploration industry today. Despite the fact that Hubral [2] introduced the more advanced topic of depth migration in 1977, the time migration methods have been the subject of so many research papers that an integrated treatment is warranted. The time migration methods can be described as various ways of implementing the classical WKBJ approximation (i.e., geometrical acoustics), well known in theoretical seismology and other areas of applied physics. The exploding reflectors hypothesis depends on an inherent (but, in fact, incidental) assumption that the amplitude of the seismic pulse is invariant as it is transmitted through the earth layers. Loewenthal et al. based this assumption on the requirement that the medium be weakly inhomogeneous as well as on the paper by Foster [3] in 1975, who had concluded that transmission effects in the continuous one-dimensional seismic model are unsubstantial, and at the surface (where the reflection seismograms are recorded) transmission effects are not present at all. Although Foster regarded this as a general result, the work of Gray [4] in 1984 shows that it is indeed a consequence of the WKBJ approximation, and so is valid only in those situations where the WKBJ approximation is applicable. The phase term which makes up the essential element of time migration (as well as various other types of migration methods) is the WKBJ phase correction factor, so time migration methods such as the conventional versions of Kirchhoff migration, finite-difference migration, and frequency-wavenumber (f-k) migration are not general wave-equation methods but are simply aspects of the WKBJ approximation. In fact, it can be said that most of conventional seismic processing in general is none other than WKBJ (i.e., geometrical acoustics).