Amplitude and phase factors
Now consider several factors associated with the amplitude and phase behavior of the waveform along the diffraction hyperbola. From Figure 4.1-9, given the alternative of standing at location A or B, we intuitively think that it is safer to stand at location B. This is because the wave amplitude at location A, which is on the z-axis, is stronger than the wave amplitude at location B, which is at an oblique angle from the z-axis. As mentioned earlier, this is one difference between a point source with uniform amplitude response at all angles and the point aperture that produces a wavefront with angle-dependent amplitudes. This angle dependence of amplitudes, which is described by the obliquity factor, should be considered before summation. To correct for the obliquity factor, the amplitude at location B in Figure 4.1-15a is scaled by the cosine of the angle between BC and CA before it is placed at output location A.
Another factor is the spherical divergence of wave amplitudes. Again, from Figure 4.1-9, given the alternative of standing at location B or C, we prefer to stand at location C. The reason for this is that the wave amplitude along the wavefront at location C, which is farther from the point aperture source, is weaker than the wave amplitude at location B. Wave energy decays as (1/r2), where r is the distance from the source to the wavefront, while amplitudes decay as (1/r). Thus, amplitudes must be scaled by factor (1/r) before summation for wave propagation in three dimensions.
Finally, there is a third factor that involves the inherent property of Huygens’ secondary source waveform. This factor is difficult to explain from a physical viewpoint. Nevertheless, it is obvious from Figure 4.1-13 that Huygens’ secondary sources must respond as a wavelet along the hyperbolic paths with a unique phase and frequency characteristic. Otherwise, there would be no amplitude cancelation when they are close to one another. The waveform that results from the summation must be restored in both phase and amplitude.
Figure 4.1-9 The gap in the barrier acts as Huygens’ secondary source, causing the circular wavefronts that approach the beach line. Adapted from Claerbout.
Figure 4.1-15 Principles of migration based on diffraction summation. (a) Zero-offset section (trace interval, 25 m; constant velocity, 2500 m/s), (b) migration. The amplitude at input trace location B along the flank of the traveltime hyperbola is mapped onto output trace location A at the apex of the hyperbola by equation (4).
In summary, we must consider the following three factors before diffraction summation:
- The obliquity factor or the directivity factor, which describes the angle dependence of amplitudes and is given by the cosine of the angle between the direction of propagation and the vertical axis z (Figure 4.1-15).
- The spherical spreading factor, which is proportional to for 2-D wave propagation, and (1/vr) for 3-D wave propagation.
- The wavelet shaping factor, which is designed with a 45-degree constant phase spectrum and an amplitude spectrum proportional to the square root of the frequency for 2-D migration. For 3-D migration, the phase shift is 90 degrees and the amplitude is proportional to frequency.
- Claerbout, 1985, Claerbout, J.F., 1985, Imaging the earth’s interior: Blackwell Scientific Publications.
- Kirchhoff migration
- Diffraction summation
- 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
- Frequency-wavenumber migration
- Phase-shift migration
- Stolt migration
- Summary of domains of migration algorithms