If the medium velocity is constant, migration can be expressed as a direct mapping  from temporal frequency ω to vertical wavenumber kz (Figure 4.1-25). Figure 4.1-30 is a flowchart of the Stolt algorithm; the mathematical details are left to Section D.7. The equation for Stolt mapping is
where P(kx, z = 0, ω) is the zero-offset section and P(kx, kz, t = 0)is the migrated section in the frequency-wavenumber domain.
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.
Figure 4.1-30 Flowchart for Stolt’s constant-velocity migration method in the f − k domain.
Note that Stolt migration involves, first, mapping from ω to kz for a specific kx by using the dispersion relation of equation (13a) recast as
The output of mapping is then scaled by the quantity S
Stolt’s algorithm for constant velocity thus involves the following steps:
- Start with the input wavefield P(x, z = 0, t) approximated by the CMP stack, and apply 2-D Fourier transform to get P(kx, z = 0, ω).
- Map the wavefield from ω to kz using the dispersion relation given by equation (24b).
- Apply the scaling factor S of equation (24c) as part of the mapping procedure (Section D.7).
- Invoke the imaging principle by setting t = 0 and obtain P(kx, kz, t = 0).
- Finally, apply 2-D inverse transform to get the migrated section P(x, z, t = 0).
It may be questionable as to whether the constant-velocity Stolt method has value on its own as a practical migration algorithm. Nevertheless, Stolt’s method can be used efficiently to perform a constant-velocity migration as the first step in a residual migration scheme (frequency-wavenumber migration in practice). Additionally, the method constitutes an essential procedural step for migration velocity analysis as described in migration velocity analysis.
Stolt extended his method to handle velocity variations (Section D.7). For the variable-velocity case, Stolt’s extension consists of
- modifying the input wavefield to make it appear as if it were the response of a constant-velocity earth,
- applying the constant-velocity algorithm outlined in Figure 4.1-30, and
- reversing the original modification of the input wavefield.
This modification essentially is a type of stretching of the time axis (Section D.7) to make the reflection times approximately equivalent to those recorded for a constant-velocity earth. The nature of stretching is described by the stretch factor W. The constant-velocity case is equivalent to W = 1.
Note that the phase-shift and Stolt migration outputs normally are displayed in two-way vertical zero-offset time τ = 2z/v, as are the outputs from the finite-difference and Kirchhoff migrations. In practice, mapping in the f − k domain really is from ω − kx to ωτ − kx rather than to kz − kx, where ωτ is the Fourier dual of τ and is simply kz of equation (13b) scaled by v/2 (Section D.3):
One important concept must be pointed out from equation (25). Note that for a constant kx, ωτ < ω; thus, migration shifts the bandwidth to lower frequencies. This is analogous to the conclusion derived in relation to the NMO correction, since the latter also causes data stretching to lower frequencies (normal moveout). The implication from equation (25) is demonstrated by the dipping events model in Figure 4.1-24. While the bandwidth of the zero-dip event is retained after migration, the bandwidth of the event with steepest dip has shifted from approximately 40 Hz to 36 Hz at the high-frequency end of the spectrum. In fact, the shift in bandwidth is dip-dependent; events with different dips which have the same bandwidth before migration will have different bandwidths after migration.
- Stolt (1978), Stolt, R.H., 1978, Migration by Fourier transform: Geophysics, 43, 23–48.
- 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
- Frequency-wavenumber migration
- Phase-shift migration
- Summary of domains of migration algorithms