Aperture effects in Kirchhoff migration

Seismic images obtained by Kirchhoff time or depth migration are always accompanied by some artifacts known as “migration noise”, “migration boundary effects”, or “diffraction smiles”, which may severely affect the quality of the migration result. Most of these undesirable effects are caused by a limited aperture if the algorithms make no special disposition to avoid them. Likewise, strong amplitude variation along reflection events may also cause similar artifacts. All these effects can be explained mathematically by means of the Method of Stationary Phase. However, such a purely theoretical explication is not always easy to understand for applied geophysicists. By relating the terms of the stationary-phase approximation to simple geometrical situations, a more physical interpretation of the migration artifacts can be obtained. A simple numerical experiment for poststack (zero-offset) data indicates the problem and helps to develop an intuitive understanding of the effects and the methods to avoid them. INTRODUCTION Since the early work of Hagedoorn (1954), migration concepts have strongly improved and are now an important tool in the world of seismic imaging, either as prestack or poststack time and depth migration. A frequently used method is Kirchhoff migration (Schneider, 1978) that treats each depth point M on a sufficiently dense grid like a diffraction point. In an a-priori given macrovelocity model, the relevant part of the Green’s function of a point source at any single diffraction point M in the depth domain is calculated. The kinematic part of this Green’s function is the configuration-specific diffraction-traveltime surface, also called “Huygens surface”. The amplitudes of the input seismograms (or, to be more specific, of their derivatives) are stacked along the Huygens surface and assigned to the depth point M . This explains why the Kirchhoff migration scheme is also called a “diffraction stack”. If so desired, the effect of geometrical spreading can be removed from the output amplitudes by multiplying the data during the stack with a true-amplitude weight factor that is calculated from the dynamic part of the Green’s function. Annual WIT report 2001 87 Ideally, the extent of the Huygens surfaces, that is, the migration aperture, should be limitless so that no contributions due to the abrupt truncation of the sum occur. In practice, of course, the aperture is always limited by the region over which seismic data have been acquired. In other words, because of the finiteness of the survey area, Kirchhoff migration will always be a “limited aperture migration” (LAM) (Sun, 1998). This is, however, not the only reason why we have to deal with the effects of a finite migration aperture. In practical migration implementations, even ranges of source and receiver positions might be excluded where data actually have been acquired. Such a procedure can be advantageous because • less traces to sum leads to a speedup of the whole migration process, • a smaller operator excludes steeper dips, which helps to avoid operator aliasing (see, e.g., Abma et al., 1999), • less summation of data away from the signal reduces the stacking of unwanted noise. For the best possible reduction of aliasing and noise as well as the best computational efficiency, one would like to use a model-based aperture restriction, i.e., one would like to make use of the (projected) Fresnel zone (see, e.g., Schleicher et al., 1997; Sun and Bancroft, 2001). Unfortunately, it is difficult to determine the exact center and size of the Fresnel zone for each depth point prior to or during migration. A reasonable compromise between accuracy and practicability is to specify a common maximum migration aperture radius or a maximum stacking operator dip. These aperture reductions lead to dip-restricted migration operators as, for example, a 45◦ migration. With these kinds of operators, higher dips cannot be imaged. In regions where dips are known to be restricted, this is a very convenient way of reducing aliasing and improving computational efficiency at the same time. It should, however, be kept in mind that close to the maximum dip, these dip-restricted migration operators will achieve only kinematically correct images (see, e.g., Schleicher et al., 1997; Sun, 1998). For true-amplitude migration, the maximum operator dip must always be chosen somewhat larger than the maximum reflector dip to be imaged. The fact that the migration aperture is limited causes artifacts known as migration noise, boundary or aperture effects, or migration smiles. In this paper, we relate the mathematical explanation of the migration artifacts by means of the Method of Stationary Phase (see, e.g., Bleistein, 1984; Sun, 1998; Bleistein et al., 2001) to simple geometrical situations. This more physical interpretation leads to a more intuitive insight into these effects. Of course, since the stacking operations are the same in Kirchhoff time and depth migration, the corresponding artifacts are conceptually identical in both processes. Thus, we restrict our present discussion to Kirchhoff depth migration. It should, however, be kept in mind that everything said and shown in this paper with respect to an image in depth holds in the same way for an image in time. KIRCHHOFF MIGRATION Mathematically, the Kirchhoff migration process is expressed as an integration over the recorded wavefield and reads in 3D (Tygel et al., 1996)