Separation versus splitting
We now discuss some practical alternatives to summing over the surface of the hyperboloid. The diffraction hyperboloid, which is strictly associated with a constant-velocity medium, has a hyperbolic cross-section in any azimuthal direction. A two-stage summation over the surface of the hyperboloid can be performed as follows:
- Sum along the hyperbolic cross-sections in the inline direction and place the summed amplitudes at the local apexes of these hyperbolas. The hyperboloid now is collapsed to a hyperbola that is in the plane perpendicular to the direction of the first summation. This hyperbola comprises the first summed amplitudes at the local apexes and is contained in the plane of the crossline direction.
- Next, sum the energy along this hyperbola and place it to its apex, which is also the apex of the original hyperboloid and is where the image should be placed.
Gardner et al. (1978) proposed this two-pass approach to do 3-D migration that involves successive 2-D migrations of the inlines followed by 2-D migrations of the crosslines. A rigorous theoretical treatise of the two-stage Kirchhoff summation for 3-D zero-offset migration was later given by Jacubowicz and Levin .
The underlying mathematical notion for the two-pass approach is full separation of extrapolation operators in the inline and crossline directions . Another method of 3-D migration is based on the numerical procedure called splitting  . Here, extrapolation operators are independently applied to the data in both the inline and crossline directions at each downward-continuation step (Section G.1). In contrast, full separation calls for complete migration in one direction — for instance, the inline direction, before any operation is applied in the orthogonal direction.
The sequence of numerical operations in the splitting and separation methods is shown in Figure 7.3-1. Note that the number of operations in both schemes is the same, only that the order of the operations differ. It turns out that if you have strong dependency of velocities on spatial variables in the inline and crossline directions, then splitting is more accurate than separation. So splitting would be the required scheme for depth migration, whereas separation may be acceptable for time migration. As noted earlier, 3-D migration based on full separation is known as the two-pass approach. Three-dimensional migration based on splitting is considered a one-pass approach.
The two-pass or one-pass strategy for doing 3-D poststack migration should not be confused with the type of algorithm — finite-difference, Kirchhoff summation, or frequency-wavenumber used in migration. The two-pass approach based on separation is now outmoded by the one-pass approach based on the splitting of the 3-D extrapolation operator with various improvements in accuracy and efficiency. Li  introduced a correction term to the one-pass implicit scheme to compensate for the accumulated errors resulting from splitting. Li’s correction was adapted later to explicit schemes by Etgen and Nichols  (Section G.2). One-pass algorithms based on stable explicit extrapolation operators designed in frequency-space  ,     and frequency-wavenumber domains   have now become more widely used than those based on implicit schemes. Before we discuss the more modern one-pass explicit schemes, however, for their historical significance, we shall first discuss the two-pass finite-difference implicit scheme based on separation and one-pass implicit finite-difference scheme based on splitting of the 3-D extrapolation operator without any additional correction term.
As for 2-D migration (migration principles), a convenient domain to do 3-D time or depth migration is the frequency-space domain. Since the need for 3-D imaging becomes especially significant in areas with steep dips and lateral velocity variations, the 2-D frequency-space algorithm (Section D.4) adapted for one-pass 3-D migration is used in the examples shown in this section. Extrapolation operators based on rational approximations to the exact dispersion relation are applied to the 3-D stacked volume of data in the inline and crossline directions. A concise theoretical description of 3-D migration based on the one-pass approach is provided in Section G.1.
- Jacubowicz and Levin (1983), Jakubowicz, H. and Levin, S., 1983, A simple exact method of 3-D migration — theory: Geophys. Prosp., 31, 34–56.
- Claerbout, 1985, Claerbout, J. F., 1985, Imaging the earth’s interior: Blackwell Scientific Publications.
- Brown, 1983, Brown, D. L., 1983, Applications of operator separation in reflection seismology: Geophysics, 48, 288–294.
- Li (1991), Li., Z, 1991, Compensating finite-difference errors in 3-D migration and modeling: Geophysics, 56, 1650–1660.
- Etgen and Nichols (1999), Etgen, J. T. and Nichols, D., 1999, Application of the Li correction to explicit depth extrapolation methods: 69th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1366–1369.
- Holberg, 1988, Holberg, O., 1988, Towards optimum one-way wave propagation: Geophys. Prosp., 36, 99–114.
- Hale, 1991a, Hale, D., 1991a, Stable explicit depth extrapolation of seismic wavefields: Geophysics, 56, 1770-1777.
- Hale, 1991b, Hale, D., 1991b, 3-D depth migration via McClellan transforms: Geophysics, 56, 1778–1785.
- Blacquiere, 1991, Blacquiere, G., 1991, Optimized McClellan transformation filters applied to one-pass 3-D depth extrapolation filters: 61st Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1126-1129.
- Biondi and Palacharla, 1994, Biondi, B. and Palacharla, G., 1994, 3-D migration using rotated McClellan filters: 64th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1278–1281.
- Gaiser, 1994, Gaiser, J. E., 1994, Optimum transformation and filter structure for explicit finite-difference 3-D migration: Presented at the 56th Mtg. Eur. Assoc. Expl. Geophys.
- Notfors, 1995, Notfors, C. D. B., 1995, Accurate and efficient explicit 3-D migration: 65th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1224–1227.
- Kosloff and Kessler, 1987, Kosloff, D. and Kessler, D., 1987, Accurate depth migration by a generalized phase-shift method: Geophysics, 52, 1074–1084.
- Reshef and Kessler, 1989, Reshef, M. and Kessler, D., 1989, Practical implementation of three-dimensional poststack depth migration: Geophysics, 54, 309–318.
- 3-D poststack migration
- 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
- Explicit schemes combined with the McClellan transform
- The phase-shift-plus-correction method