Frequency-wavenumber (f − k) migration is not as easily explained as the Kirchhoff or finite-difference migration from a physical point of view. Chun and Jacewitz  provide practical insight into the principles of f − k migration.
In the 2-D Fourier transform, we learned that dipping events in the t − x domain map onto the f − k domain along radial lines. The steeper the dip, the closer the radial line is to the wavenumber axis. Figure 4.1-24 shows dipping events before and after migration in the t − x and f − k domains. The Nyquist wavenumber is 20 cycles/km and the bandwidth is given by the corner frequencies 6, 12 - 36, 48 Hz for the passband region of the spectrum. (See Figure 1.1-26 for the definition of corner frequencies.) The red is associated with the flat part of the passband region and the blue is associated with the taper zone. Note that migration rotates the radial lines in the 2-D amplitude spectrum outward and away from the frequency axis. The steepest event represented by radial line A maps onto radial line B after migration. The feather-like energy especially prominent in the left quadrant of the f − k spectrum is associated with the flanks of the diffraction hyperbolas in the t − x domain. The energy associated with the left flanks which are dipping opposite to the dipping reflections maps onto the left quadrant of the f − k plane. And the energy associated with the right flanks of the diffraction hyperbolas that are dipping in the same direction as that of the dipping reflections maps onto the right quadrant of the f − k plane and is superimposed on the energy associated with the dipping reflections themselves.
Figure 4.1-24 (a) A zero-offset section that contains a set of dipping reflectors, (b) migration of (a), (c) the f − k spectrum of (a), and (d) the f − k spectrum of (b). Red and blue represent high and low amplitudes, respectively. Energy along the radial line A, which is associated with the steepest dip, maps onto the radial line B after migration. Medium velocity is constant, 3500 m/s.
Figure 4.1-25 Migration in the f − k domain. (Migration in the t − x domain is illustrated in Figure 4.1-1.) (a) A dipping reflector is represented by a radial line OB in the f − k plane. (b) After migration, the radial line OB maps onto another radial line OB′, while B maps onto B′. The horizontal wavenumber is invariant under migration. For comparison, the f − k response of the dipping event before migration (a) has been superimposed on the f − k response after migration (b). Adapted from Chun and Jacewitz .
Figure 4.1-26 (a) A zero-offset section that contains a diffraction hyperbola, (b) migration of (a), (c) the f − k spectrum of (a), and (d) the f − k spectrum of (b). Red and blue represent high and low amplitudes, respectively. Energy within the triangular area in (c) maps onto the semicircular area in (d) after migration. Medium velocity is constant, 2500 m/s.
Migration of a dipping event in the f − k domain is sketched in Figure 4.1-25. Note that this figure is the f − k equivalent of Figure 4.1-1. In both figures, we assume velocity equal to 1. The vertical axis in Figure 4.1-25 represents the temporal frequency ω for the event in its unmigrated position B, and the vertical wavenumber kz for the event in its migrated position B′.
Migration in the frequency-wavenumber domain involves mapping the lines of constant frequency AB in the ω − kx plane to circles AB′ in the kz − kx plane. Therefore, migration maps point B vertically onto point B′. Note that in this process, the horizontal wavenumber kx does not change as a result of mapping. When this mapping is completed, the dipping event OB is mapped along OB′ after migration; thus, the dip angle after migration is greater than the dip angle θ before migration. For comparison, these two radial lines are shown on the same plane kz − kx.
We now examine the diffraction hyperbola and its collapse to the apex after migration in the f − k domain. A diffraction hyperbola is represented by an inverted triangular area in the frequency domain as shown in Figure 4.1-26. The Nyquist wavenumber is 40 cycles/km and the bandwidth is given by the corner frequencies 6, 12 - 36, 48 Hz for the passband region of the spectrum. As for the dipping events model in Figure 4.1-24, the red is associated with the flat part of the passband region and the blue is associated with the taper zone. The two edges in the right and left quadrant of the f − k plane correspond to the asymptotes of the flanks of the diffraction hyperbola, the base of the inverted triangle corresponds to the high-frequency end of the passband, and the tip of the triangle in the proximity of the origin of the f − k plane corresponds to the low-frequency end of the passband. Migration turns the triangular area into a circular shape as shown in Figure 4.1-26.
The f − k analysis of the diffraction hyperbola shown in Figure 4.1-26 is based on the representation of the hyperbola as a series of discrete dipping segments. Figure 4.1-27 depicts a diffraction hyperbola in the t − x and f − k domains. We imagine that the hyperbola is made up of a series of dipping segments, such as A, B, C, D and E. The zero-dip segment A is at the apex, while the steepest dip segment E is along the asymptotes. In the f − k domain, the zero-dip segment A maps along the frequency axis, while the dipping segments B, C and D map along the radial lines, increasingly further away from the frequency axis. Finally, the asymptotic tail E maps along the radial line that represents the boundary between the propagation and the evanescent region. The evanescent region corresponds to the energy that is located at or greater than 90 degrees from the vertical. The opposite side of the hyperbola maps onto the second quadrant (negative kx) in the f − k domain. In the continuous case, a diffraction hyperbola is represented by a series of continuous radial lines that constitute an inverted triangular area in the f − k domain (Figure 4.1-26).
A curious fact emerges from the f − k spectrum of the migrated section in Figure 4.1-26. We expect migration to collapse the diffraction hyperbola to a point at the apex. The spectrum of this migrated section really should be more like that in Figure 4.1-28 — a rectangle. Why is there a difference between this spectrum and the spectrum after migration in Figure 4.1-26?
If you start with a point and model it, you get the diffraction hyperbola in Figure 4.1-28. However, in reality we deal with a diffraction hyperbola as shown in Figure 4.1-26. The hyperbolas do not look different in the t − x domain, but note the difference in their f − k spectra. The f − k spectrum of the real-life diffraction, which is always subjected to bandpass filtering (Figure 4.1-26), is missing the energy above the 48-Hz line that is present in the f − k spectrum of the modeled diffraction curve (Figure 4.1-28). These missing high frequencies cause the difference between the spectra after migration.
- Chun and Jacewitz, 1981, Chun, J.H. and Jacewitz, C., 1981, Fundamentals of frequency-domain migration: Geophysics, 46, 717–732.
- Kirchhoff migration
- Diffraction summation
- Amplitude and phase factors
- Kirchhoff summation
- Finite-difference migration
- Downward continuation
- Differencing schemes
- Rational approximations for implicit schemes
- Reverse time migration
- Frequency-space implicit schemes
- Frequency-space explicit schemes
- Phase-shift migration
- Stolt migration
- Summary of domains of migration algorithms