Fundamentals of Frequency Domain Migration
J.H. Ghun, G.A. Jacewitz · Offshore Technology Conference · 1979
ABSTRACT Migration of seismic data has been important for some time. Graphical methods were employed first to accomplish correct positioning of seismic data. These procedures were succeeded by statistical methods using Kirchoff integral methods. Most of the recent work in migration has concentrated on finite difference techniques. Now, however, the approach to migration based on the frequency domain has become a viable alternative. This is founded on the wave equation, and so includes diffractions and other effects. This paper seeks to motivate and illuminate frequency domain migration using straightforward geometric techniques. INTRODUCTION Migration of seismic data is a process of mapping one time section onto a second time section, or a depth section in which events are repositioned under the appropriate surface location and at the correct time. That is, a migration output should be a time section of the geological depth section. No current migration technique perfectly handles all the difficulties of noise, rapidly varying velocities, steep dips, and other problems. Techniques vary greatly in performance relative to these problems. Three of the major techniques of migration are diffraction, finite difference, and frequency domain migration. Diffraction migration also is known as Kirchoff integral migration. The finite-difference approach commonly is known as time domain or wave equation migration. Frequency domain migration also may be referred to as FK migration, or Fourier transform migration. The diffraction stack process is a statistical approach. This procedure treats data that might have originated from certain subsurface locations. All such possible origins are treated as equally likely. The major advantage of diffraction migration is good performance with steep dip. One disadvantage is poor performance under low signal-to-noise ratio conditions. Finite-difference migration is a deterministic approach. The migration procedure is modeled by the wave equation. This partial differential equation then is approximated by a simpler type of equation appropriate for migration. This last equation then is approximated as a finite-difference scheme. An advantage of the finite-difference method is its good performance with a low signal-to-noise ratio. Disadvantages of this method include a relatively long running time and difficulty in handling steep dip data. Frequency domain migration is based also on a deterministic approach via the wave equation. Instead of utilizing finite-difference approximations, the two-dimensional Fourier transform is the fundamental technique of this method. The advantages of this method include fast running time, good performance under low signal-to-noise ratio conditions, and excellent performance for steep dip. Disadvantages include difficulties with widely varying velocity functions. This article is meant to serve as an overview to those working in the geophysical industry who wish to know more about migration. In particular, the authors hope to provide some new insights into the fundamental aspects of migration with a major emphasis on frequency migration. There will be much reliance on the geometry behind the usual physical and mathematical treatments of migration. The intimate geometric relations between the migration of a dipping event in time and the counterpart migration in the frequency domain will be explained.