Reversed time migration in spatial frequency domain

Abstract

During the past decade, finite‐difference methods have become important tools for direct modeling of seismic data as well as for certain interpretation processes. One of the earliest applications of these methods to seismics is the pioneering contribution of Alterman who, in a series of papers (Alterman and Karal, 1968; Alterman and Aboudi, 1968; Alterman and Rotenberg, 1969; Alterman and Loewenthal, 1972) demonstrated the usefulness of such numerical computations for the propagation of seismic waves in elastic media. A clear exposition of these techniques, as well as a comparison of results obtained from them with the corresponding analytical solutions, can be found in Alterman and Karal (1968). This subject was further developed and extended to more complicated models by Boore (1970), Ottaviani (1971), and Kelly et al (1976). Claerbout introduced a somewhat different finite‐difference approach (Claerbout, 1970; Claerbout and Johnson, 1971) for modeling the acoustic waves which often dominate the reflection seismogram. In his approach, the original wave equation, which governs the propagation of the acoustic waves, is modified in such a way so as to allow the propagation of either only upcoming or only downgoing waves. By moving the coordinate frame with the downgoing waves, Claerbout showed that one could greatly reduce computation time. Using the same concepts, he showed (Claerbout and Doherty, 1972) how to use a similar scheme for migrating a seismic section by downward continuation of the upcoming waves. This migration method is an interesting extension of the ideas of Hagedoorn (1954) and was found to be extremely useful with real data (Larner and Hatton, 1976; Loewenthal et al, 1976).