Prestack time migration of 2D seismic records by the Kirchhoff method



In exploration seismology, migration refers to a multi-channel processing step that attempts to spatially re-position events and improve focusing. Before migration, seismic data is usually displayed with traces plotted at the surface location of the receivers and with a vertical time axis. This means that dipping reflections are systematically mispositioned in the lateral coordinate and the vertical time axis needs a transformation to depth. Also problematic is the unfocused nature of seismic data before migration. Migration moves dipping reflections to their true subsurface positions and collapses diffractions, thus increasing spatial resolution and yielding a seismic image of the subsurface.
Time migration which produces a migrated time section is appropriate as long as lateral velocity variations are mild to moderate. Dipping events on a stacked section call for time migration. Conflicting dips with different stacking velocities is one case in which a conventional stacked section differs from a zero-offset section. Thus, poststack migration which assumes that the stacked section is equivalent to a zero-offset is not valid to handle the case of conflicting dips. Instead, one needs to do prestack time migration.
Kirchhoff migration methods are based on the diffraction summation technique, which sums the seismic amplitudes along a diffraction hyperbola whose curvature is governed by the medium velocity, and maps the result to apex of the hyperbola. The Kirchhoff summation technique applies amplitude and phase corrections to the data before summation. These corrections make the summation consistent with the wave equation in that they account for spherical spreading, the obliquity factor (angle-dependency of amplitudes), and the phase shift inherent in Huygens' secondary sources. Since the Kirchhoff migration method is based on summing the amplitudes along the hyperbolic trajectory, as long as the diffraction curve is known, it can be adapted for any domain. The velocity function used in the diffraction curve equation is vrms for prestack migration. Poststack migration uses zero-offset data while prestack time migration applies to unstacked data, so uses shot record, common-offset, and equivalent–offset data. The Kirchhoff prestack time migration sums through the input space along hyperbolic paths to compute each point in the output space. For variable velocity the hyperbola is replaced by a more general shape. Amplitudes change under migration. Velocity model is a matrix of interval velocity at each sampling point. Velocity model is generated using reflector velocities. An important factor in the Kirchhoff migration is migration aperture. Reducing the aperture reduces the maximum dip to migrate. The effect of migration aperture is illustrated using different apertures. Small apertures eliminate steeply dipping events from the migrated section.
A software in MATLAB is written which is capable of migrating common shot records. Traveltime from source to a scatterpoint (i.e. the image point) is approximated by a Dix equation using the rms velocity from the model at the lateral position halfway between the source and the receiver and at the vertical traveltime of the scatterpoint. Similarly, from the scatterpoint to a receiver, a Dix equation using the rms velocity halfway between the scatterpoint and the receiver is used. The source and all receivers are assumed to be on the same horizontal plane.
In this paper, two models consisting of a 2 layered trapezoid model and a 3 layered model are synthesized and inputted to the migration algorithm. The traveltimes are calculated via ray tracing with respect to a shot in the center of the model. Shot records were migrated with the interval velocity model. Since prestack migration is very sensitive to the velocity model, an rms velocity model was used. Using rms velocity field improved the migrated section. The critical parameter of the Kirchhoff migration is a migration aperture width whose effect is more evident in steep dips and depths.