Frequency-space explicit schemes
Start with the paraxial dispersion relation given by equation (13b) adapted to the exploding reflectors model. Operate on the pressure wavefield P and inverse Fourier transform in z to obtain the differential equation
whose solution can be used to extrapolate the wavefield at the surface down in depth
For a discrete depth step Δz, equation (21) takes the form
where, for convenience, the variables kx and ω have been omitted from P.
When designing extrapolation operators, whatever the differencing scheme, the objective must be to ensure that the phase and amplitude of the actual operator closely resembles those of the desired operator exp(−ikzΔz).
Discretize the one-way wave equation (20) and apply differencing approximation using an explicit scheme such as
and an implicit scheme (Section D.6):
The explicit extrapolation operator (1 − kzΔz) of equation (22b) actually is the first two terms of the Taylor expansion of the exact operator exp(−ikzΔz) of equation (22a). Table 4-5 provides the amplitude and phase of the exact, explicit and implicit operators used in equations (22a, 22b, 22c) for wavefield exptrapolation in depth.
A desired property of an extrapolation operator is that it must be stable — its amplitude should be less or equal to unity. The implicit operator defined by equation (22c) is stable, while the explicit operator defined by equation (22b) causes amplitudes of the extrapolated wavefield grow with depth (Section D.6). In fact, the larger the depth step Δz, the more unstable are the results of extrapolation. Another desired property of an extrapolation operator is that it should yield the least phase error. The inherently stable nature of implicit schemes has been the compelling reason for their use in practice. Recent developments in the design of stable explicit schemes, however, now have made them widely accepted   .
In principle, the exact extrapolation operator in equation (22a) can be inverse Fourier transformed to frequency-space (ω − x) domain and applied to P(z) in an explicit manner. Each output sample of P(z + Δz) at some x location for a frequency ω and velocity v is computed independently by convolving an explicit filter operator of a specified length centered at the output location x with the input data array P(z) in the x direction. In contrast, implicit schemes require solving a set of linear equations to obtain the output samples of P(z + Δz) — computationally more intensive than convolution. Efficiency is an advantage of the explicit schemes over the implicit schemes.
|Explicit 1 − ikzΔz||tan−1(kzΔz)|
Another attractive property of stable explicit schemes is their extension to 3-D extrapolation that preserves circular symmetry — a feature that is relatively more difficult to attain with implicit schemes (3-D poststack migration).
Whether an explicit filter is computed by inverse transforming the exact filter exp(−ikzΔz) of equation (22a) to the frequency-space domain or by Taylor expansion as in equation (22b), the problem is that neither approach yields a stable filter operator. A stable explicit extrapolation filter in the frequency-space domain can be designed using a constrained least-squares technique , or by a modified Taylor series expansion of the exact extrapolation filter exp(−ikzΔz) . Another method of explicit operator design based on on an alternative stability criterion is presented by Soubaras .
The objective is to find, for a specific frequency ω and velocity v, a symmetric explicit filter with complex coefficients h(x) in the frequency-space domain such that, when Fourier transformed to the frequency-wavenumber domain, the difference between the actual transform H(kx) and the desired transform D(kx) of equation (23) is minimum, subject to the stability constraint that the amplitude of H(kx) is never greater than unity within the propagation region kx ≤ (2ω/v). Details of the method of modified Taylor expansion based on this design criterion by Hale  are described in Section D.5.
As for the implicit schemes (Figure 4.1-23), a migration algorithm based on an explicit extrapolation filter design involves a loop over the depth step z and a loop over frequency ω. For each depth step:
- Convolve the explicit extrapolation filter h(x) with each of the frequency components of the wavefield.
- Sum over the frequencies to invoke the imaging principle which is equivalent to setting t = 0.
- Repeat the computation for all the depth steps to complete the imaging.
The length of the filter coefficients h(x) determines the dip accuracy of the explicit operator. The larger the number of filter coefficients 2N + 1, the steeper the dip accuracy. In practice, extrapolation filter lengths 7, 11, and 25 are often associated with 30-, 50-, and 70-degree dip accuracies. Phase error of the extrapolation operator at steep dips may be reduced by increasing the number of coefficients. Also, lateral velocity variations can be accommodated by varying the velocity at each x location of the filter coefficient.
- Holberg, 1988, Holberg, O., 1988, Towards optimum one-way wave propagation: Geophys. Prosp., 36, 99–114.
- Hale, 1991, Hale, I. D., 1991, Stable explicit depth extrapolation of seismic wavefields: Geophysics, 56, 1770–1777.
- Soubaras, 1992, Soubaras, R., 1992, Explicit 3-D migration using equiripple polynomial expansion and Laplacian synthesis: 62nd Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 905–908.
- 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-wavenumber migration
- Phase-shift migration
- Stolt migration
- Summary of domains of migration algorithms