# The phase-shift-plus-correction method

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

The phase-shift method (Section G.3) is designed to accommodate vertically varying velocity field for migration, only. Nevertheless, it can be extended to accommodate lateral velocity variations [1] [2]. Consider a spatially varying 3-D migration velocity field v(x, y, z) and a laterally averaged, but vertically varying velocity function ${\displaystyle {\bar {v}}(z)}$. The basic idea is to first extrapolate in depth with the phase-shift extrapolator (equation G-22) using the vertically varying velocity function ${\displaystyle {\bar {v}}(z)}$. This is followed by the application of a convolutional operator as in the explicit schemes (Section G.2) to the extrapolated wavefield at the same depth. This second operation is fundamentally a residual extrapolation to account for the difference between the laterally varying velocity field v(x, y, z) and the vertically varying velocity function ${\displaystyle {\bar {v}}(z)}$. The migration algorithm that incorporates such a correction term into the phase-shift algorithm has been known as phase-shift-plus-correction (PSPC) method. Such a splitting of wave extrapolation within one depth step may appear to be computationally awkward. Nevertheless, numerically efficient schemes have been devised to apply the convolutional operator in the residual extrapolation step [1]. Depending on the degree of lateral velocity variations, the PSPC method can be used either as a time or depth migration algorithm.

Figure 7.3-17 shows selected crosslines from the 3-D migrated volumes of data associated with the 3-D survey of Figure 7.2-1 using three different migration algorithms — the implicit one-pass on top, the explicit McClellan transform in the middle, the PSPC at the bottom. Note that both of the explicit schemes based on the McClellan transform and the PSPC methods have produced comparable and better images of the subsurface in the vicinity of the overthrust between inline locations 200 and 300. The same region of the subsurface appears to be undermigrated by the implicit one-pass scheme. Incorrect positioning by the implicit one-pass scheme also is pronounced in the inline sections shown in Figure 7.3-18. Note, for instance, poor reflector continuity between 1.5-2 s and crossline locations 200 and 400 — events have not been moved completely into this section from the neighboring sections. Moreover, note the wobbly character of these events — this is caused by the circularly asymmetric impulse response of the implicit one-pass operator (Figure 7.3-12). Notice also the generally noisy character of the results from the implicit one-pass scheme — this may be attributed to the dispersive noise produced by the algorithm. Specifically, it treats evanescent energy as if it is propagating energy and generates a noisy impulse response at steep dips (Figure 7.3-12).

Figure 7.3-19 shows selected time slices from the results of migration using the three different migration algorithms — the implicit one-pass on the left, the explicit McClellan transform in the middle, the explicit phase-shift-plus-correction on the right. Note the wobbly behavior of the contours on the time slices from the implicit one-pass scheme — this is related to the azimuthal asymmetry of its impulse response (Figure 7.3-12). The differences are exhibited more clearly on the enlarged view in Figure 7.3-20. The two explicit schemes — the McClellan transform and the phase-shift-plus-correction, produce comparable images.

Study the crosslines (Figure 7.3-17), the inlines (Figure 7.3-18), and the time slices in Figure 7.3-19, and note the differences in imaging by the implicit and explicit schemes. It is almost certain that in the future, explicit schemes, because of their ease of design and implementation, will become standard. Increased computer power will further encourage use of the explicit schemes.

## References

1. Kosloff and Kessler, 1987, Kosloff, D. and Kessler, D., 1987, Accurate depth migration by a generalized phase-shift method: Geophysics, 52, 1074–1084.
2. Pai, 1988, Pai, D. M., 1988, Generalized f − k (frequency-wavenumber) migration in arbitrarily varying media: Geophysics, 53, 1547–1555.