Explicit schemes combined with the McClellan transform
One-pass and two-pass 3-D poststack migration algorithms based on implicit finite-difference schemes are now scarcely used. The primary reason is the azimuth-dependent positioning errors they incur (Figure 7.3-8). Improvements to the one-pass scheme to achieve circular symmetry in the impulse response   are computationally involved. Instead, explicit schemes have gained a wide acceptance since the mid-90s. They provide circular symmetry and efficiency in the design and application of the extrapolation operators      . As with the implicit methods, explicit methods can be formulated to perform both time and depth migration of 3-D CMP stacked data.
There are three aspects to the design and application of 3-D explicit schemes.
- Design of a 2-D explicit operator in the frequency-space domain,
- Transformation of this 2-D operator to a 3-D wave extrapolator, and
- Application of the 3-D wave extrpolator to migrate 3-D CMP-stacked data.
Here, we shall review an explicit method based on the McClellan transform . An alternative design of 2-D explicit operators based on the Remez exchange algorithm (Section D.5) and the Laplace transform to convert the 2-D operator into a 3-D wave extrapolator is given by Soubaras . Table 7-2 provides a list of combinations of schemes used in the design of 2-D explicit operators, transforming them into 3-D wave extrapolators and the application of the 3-D extrapolators to migrate 3-D CMP-stacked data. Mathematical details of the design and application of 3-D explicit extrapolation operators are left to Section G.2.
As for the one-pass implicit schemes, explicit schemes that use the McClellan transform for 3-D poststack migration are implemented in the frequency-space (ω, x, y) domain. This enables application of the extrapolation filter at each depth level to each frequency component of the wavefield, independently.
Wave extrapolation in 3-D is described in the frequency-wavenumber domain by
where, kx and ky are inline and crossline wavenumbers, respectively, kz is the vertical wavenumber, and ω is frequency in units of radians per unit time, and P is the wavefield that is being extrapolated in depth z using a depth step size Δz. The desired extrapolation operator D(k) = exp(−ikzΔz) for a specific ratio of 2ω/v is given by
|2-D design||Least-squares||Remez exchange|
|3-D design||McClellan transform||Laplace transform|
|3-D application||Chebychev recursion||Chebychev recursion|
We want to compute an extrapolation filter hn in the frequency-space (ω, x, y) domain with a Fourier transform H(k, ω, v)
that best approximates the desired transform given by equation (16) (Mathematical foundation of migration#D.5 Stable explicit extrapolation|Section D.5]]). Since D(k) is symmetric with respect to k = 0, the complex filter coefficients hn are even — h−n = hn, and the number of filter coefficients 2N + 1 is odd. As a result, we only need to compute N coefficients.
Methods to obtain hn such that the actual Fourier transform H(k) given by equation (18) closely approximates the actual Fourier transform D(k) given by equation (16) is described by Holberg  and Hale . The former is based on least-squares and the latter is based on a modified Taylor series. The Hale method is described in Section D.5. Since the desired amplitude spectrum is |D(k)| = 1 for all k, then we require the actual amplitude spectrum to be |H(k)| ≤ 1 for all k. This will ensure that the operator is stable — that is, amplitudes of the extrapolated wavefield will not grow from one depth level to another.
Following the determination of the 1-D extrapolation filter for specific ω and v values, they are convolved with the McClellan transform filter template (Tables G-1 and G-2) to translate it to a 2-D filter in the (x, y) domain for each frequency ω. The filter is then applied to the appropriate frequency component of the wavefield P(x, y, ω, z) in the (ω, x, y) domain at depth z to extrapolate it to depth z + Δz by way of an efficient recursive scheme that is based on Chebychev polynomials. At each depth, the image P(x, y, z + Δz, t = 0) is obtained by invoking the imaging principle that is equivalent to summing the extrapolated wave components over frequency.
In practice, the usual implementation of the McClellan transform method requires equal spatial sampling intervals, Δx and Δy, in inline and crossline directions, respectively. When the two sampling intervals are not equal, as in most marine 3-D surveys, then trace interpolation needs to be done prior to migration to obtain data with equal sampling intervals in inline and crossline directions.
Figure 7.3-11 shows the desired impulse response of a 3-D poststack migration operator. There are 101 inlines and 101 crosslines, with line 51 being the center line. This impulse response was generated using the 3-D Stolt migration with constant velocity (Section G.4). Hence, it is accurate for all dips and frequencies. Note from the time slices the perfect circular symmetry. We also achieved the circular symmetry with the implicit 15-degree split operator (Figure 7.3-2), but failed with the implicit 45-degree split operator (Figure 7.3-3). In fact, using a steeper-dip implicit split operator also fails to respond with circular symmetry (Figure 7.3-12). In the case of Figure 7.3-12, the steep-dip implicit scheme was designed by the cascaded application (twice) of the 65-degree operator (Section D.5). Note from the vertical cross-sections the dispersive noise inherent to implicit schemes. Specifically, implicit schemes cause evanescent energy that behaves like propagating energy, rather than letting it vanish with depth. The time slices exhibit a distinctive diamond shape meaning that positioning errors in the form of undermigration with the splitting methods vary with azimuth. The largest error occurs along the two diagonals, and the least error occurs along the inline and crossline directions. Both dispersive noise and azimuthal asymmetry make the impulse response in Figure 7.3-12 undesirable.
Figure 7.3-2 A 15-degree split 3-D migration operator impulse response. See Figure 4.2-1 for the 2-D equivalent response. Except for the amplitudes, the vertical cross-section at center line (51) is the same as the 2-D impulse response.
Figure 7.3-3 A 45-degree split 3-D migration operator impulse response. See Figure 4.2-1 for the 2-D equivalent response. Except for the amplitudes, the vertical cross-section at center line (51) is the same as the 2-D impulse response.
Figure 7.3-4 Two-pass versus one-pass 3-D migration. Refer to the text for details.
Figure 7.3-11 Desired impulse response of a 3-D migration operator is a hollow hemisphere. See Figure 7.3-3 for locations of the line numbers. Line 51 is the center line. Compare with Figures 7.3-12, 7.3-13 and 7.3-14.
Figure 7.3-12 An 80-degree split 3-D migration operator impulse response. See Figure 4.4-1 for the 2-D equivalent response. Line 51 is the center line, which also exhibits the 2-D impulse response except for the amplitudes. Compare with Figures 7.3-11, 7.3-13, and 7.3-14.
Now, examine the impulse response of the 3-D migration operator based on the explicit scheme discussed in Section D.5 and the McClellan transform discussed in Section G.2 (Figure 7.3-13). The 1-D explicit operator hn has 2N + 1 = 39 filter coefficients, and the McClellan filter template is 3 × 3 in size (Table G-1). The stable explicit operator hn provides cleaner vertical cross-sections, free of dispersive noise compared to the implicit scheme (Figure 7.3-12). The explicit scheme does not allow the evanescent energy to turn into propagating energy. The dip accuracy of the explicit scheme is governed by the length of the filter hn — the longer the filter the steeper the dip the algorithm can handle. The McClellan transform provides the more circular time slice compared to the splitting method (Figure 7.3-12).
Figure 7.3-14 shows the vertical cross-sections and time slices of the impulse response of another 3-D migration operator based on the McClellan transform. The 1-D explicit operator hn has 2N + 1 = 39 filter coefficients and the McClellan filter template is 5 × 5 in size (Table G-2). Because we have the same number of coefficients (39) in the explicit 1-D filter hn, the steep-dip accuracy is the same in both Figures 7.3-13 and 7.3-14. However, the McClellan filter template used in the impulse response of Figure 7.3-14 provides a more circular behavior at higher cost than that used in the impulse response of Figure 7.3-13.
We now perform 3-D migration of the 3-D zero-offset synthetic data set in Figure 7.3-4a. The velocity field is based on a horizontally layered earth model shown in Figure 5.1-17. Figure 7.3-15 shows selected vertical sections of the migrated output from the explicit scheme combined with the McClellan transform. The wave extrapolation in this case was performed in true depth z. The time slices from the input 3-D zero-offset wavefield and the depth slices from the 3-D image output of 3-D migration are shown in Figure 7.3-16. Note that energy associated with the three point scatterers has collapsed onto the apexes of the traveltime surfaces, which coincide with the depths where the scatterers are located.
Figure 7.3-13 Impulse response of a 3-D migration operator based on a 39-point explicit 1-D operator transformed into a 2-D operator using the 3 × 3 McClellan filter template in Table G-1. Compare with Figures 7.3-11, 7.3-12, and 7.3-14.
Figure 7.3-14 Impulse response of a 3-D migration operator based on a 39-point explicit 1-D operator transformed into a 2-D operator using the 5 × 5 McClellan filter template in Table G-2. Compare with Figures 7.3-11, 7.3-12, and 7.3-13.
Figure 7.3-15 3-D migration of the 3-D zero-offset wavefield associated with three point scatterers buried in a horizontally layered velocity-depth model (Figure 5.1-17). Top sections are the selected cross-sections of the wavefield as in Figure 7.3-4a, and the bottom sections are the cross-sections of the output from 3-D depth migration using an explicit scheme with the McClellan transform.
Figure 7.3-16 Time slices from the 3-D migration test of Figure 7.3-15. Shown on top are the time slices from the input 3-D zero-offset wavefield and bottom are the depth slices from 3-D poststack migration coincident with the depths of the point scatterers in the horizontally layered earth model of Figure 5.1-17.
Figure 7.3-17 Part 1: Selected crosslines from the volumes of migrated data using three different 3-D poststack migration algorithms — the implicit one-pass on top, the explicit McClellan transform in the middle, the explicit phase-shift-plus-correction at the bottom. The base map is shown in Figure 7.2-1, and the crosslines of the unmigrated CMP-stacked volume of data are shown in Figure 7.2-17. The annotation on top of the sections refer to the inline numbers.
Figure 7.3-17 Part 2: Selected crosslines from the volumes of migrated data using three different 3-D poststack migration algorithms — the implicit one-pass on top, the explicit McClellan transform in the middle, the explicit phase-shift-plus-correction at the bottom. The base map is shown in Figure 7.2-1, and the crosslines of the unmigrated CMP-stacked volume of data are shown in Figure 7.2-17. The annotation on top of the sections refer to the inline numbers.
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- 3-D poststack migration
- Separation versus splitting
- Impulse response of the one-pass implicit finite-difference 3-D migration
- Two-pass versus one-pass implicit finite-difference 3-D migration in practice
- The phase-shift-plus-correction method