Claerbout's method has been implemented for the migration of stacked seismic data. A simplified description of the method is given together with an account of some of the practical programming problems and the types of inaccuracy encountered. Routine production results are considered to be comparable or superior to the results derived from alternative migration techniques. Particular advantages are 1) the possibility of using a detailed velocity model for the migration and 2) the preservation of the amplitude and character of the seismic events on the migrated time section.
The complete motion of an elastic quarter plane and of a three-quarter plane with free boundaries caused by an explosive point source, is obtained by finite difference methods.Varying ratio /?/a of the shear to compressional wave velocity shows that in the quarter plane the amplitude of motion at the corner increases with increasing /?/a, in the three-quarter plane it decreases. The motion in the quarter plane differs from the sum of reflections at perpendicular half planes. The amplitude of diffracted P waves varies mainly with distance from the corner. The amplitude of diffracted S waves varies mainly in angular direction. Comer-generated surface waves and elliptical particle motion in the waves are analysed. At the corner of a quarter plane, the amplitude of the Rayleigh wave is three to five times as large as on a half plane, the particle motion is elliptic and the major axes of the ellipses are inclined at 45" to the free surface.
The manner in which boundary conditions are approximated and introduced into finite difference schemes may have an important influence on the stability and accuracy of the results. The standard von Neumann condition for stability applies only for points which are not in the vicinity of the boundaries. This stability condition does not take into consideration the effects caused by introducing the boundary conditions to the scheme. Working on elastic media with free stress boundary conditions we found that the boundary approximation gives rise to serious stability problems especially for regions with high Poisson's ratio. In order to detect these effects apriori and to analyse them, we have used a more elaborate procedure for checking the stability of the scheme which takes into consideration the boundary conditions. It is based on studying a locally spaced time propagating matrix which governs the time‐space behavior of a small region of the grid which includes free surface points. By using this procedure a better insight into the nature of instability caused by the approximations to the boundary conditions was gained which led us to a new stable approximation for the free surface boundary conditions.
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).
In many instances in exploration geophysics we are interested in the so‐called one‐way wave equation. This equation allows the wave fields to propagate in the positive depth direction, but not in the reverse (−Z) direction. Some modeling and migration methods, such as the f-k method (Stolt, 1978) and the phase‐shift method (Gazdag, 1978), produce in a natural way the one‐way wave equation. The main advantage of the one‐way wave equation is that it does not give rise to multiples or interlayer reverberations and enables the observation of primary events only.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.