Impulse response of the one-pass implicit finite-difference 3-D migration
The ideal impulse response of a 3-D migration operator is a hollow hemisphere, with semicircular cross-sections in the vertical planes and circular symmetry on time slices. Figures 7.3-2 and 7.3-3 show the 3-D 15-degree and 45-degree operator impulse responses for the one-pass approach. The corresponding 2-D operators are shown in Figure 4.2-1. Selected lines and time slices are shown to better illustrate the shape of the responses. There are 101 inlines and 101 crosslines.
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.
The center line 51 exhibits the elliptical shape of the 15-degree 2-D operator. As we go to lines away from the center line, the ellipse naturally gets smaller in size. The time slices of the 3-D 15-degree operator retain the circular symmetry. But the traveltimes are not quite correct — it causes undermigration in all directions.
Examine the time slices from the 15-degree 3-D impulse response (Figure 7.3-2) and note that the response is circularly symmetric. In contrast, the time slices of the 3-D 45-degree operator (Figure 7.3-3) do not have the circular symmetry; instead, they exhibit a diamond shape. The 15-degree operator causes the same amount of undermigration in all azimuthal directions. The 3-D 45-degree operator causes less undermigration, but the amount of migration error depends on the azimuth — the largest error being in the diagonal directions.
Although the one-pass algorithm is appropriate to handle lateral velocity variations — hence, suitable for depth migration as well as for time migration — it has problems of its own. The 15-degree 3-D operator is circularly symmetric (Figure 7.3-2), but when you ask for more steep-dip accuracy, as with the 45-degree 3-D operator, you end up with azimuthal variations in migration error that is manifested in the form of diamond-shaped time slices in the impulse response (Figure 7.3-3).
What is the remedy for the diamond-shaped behavior on the time slices of the one-pass 3-D 45-degree operator (Figure 7.3-3)? Ristow  discusses further splitting of the 3-D migration operator in the two diagonal directions, in addition to the inline and crossline directions, as a way to achieve a more circular time slice. Ristow  also gives a least-squares approximation (for a specified dip angle) to the 3-D migration operator by two 2-D operators. The problem with this technique is that data need to be sorted and operated upon in the two diagonal directions. This not only increases the computational cost, but also causes the extrapolation of smaller and smaller spatial data arrays in the proximity of the corners of the survey area.
Li  examined the difference between the exact 3-D phase-shift extrapolation operator and the one-pass extrapolation operator, and derived a residual extrapolation term to shape the diamond response of the 45-degree operator (Figure 7.3-3) to a circular response of a desired 3-D operator.
Certainly, the best way to achieve circular symmetry is by way of the 3-D phase-shift extrapolation operator (Section G.3). To accommodate lateral velocity variations, a residual correction can be applied at each depth level before extrapolating down to the next depth level. This algorithm, known as phase-shift-plus-correction scheme  , will be discussed later in the section.
Another way to circumvent the diamond shape is to Fourier transform the wavefield in the direction with the milder velocity variations, say in the crossline direction, and perform 2-D migration for each of the crossline wavenumbers . Again, this would require the constant-velocity assumption in the direction of Fourier transformation, something contrary to the reason for doing 3-D surveys and doing 3-D migration.
One other suggested solution   is to compute the exact 3-D phase shift operator in the Fourier transform domain, inverse Fourier transform in the inline and crossline directions, and then truncate to get a finite-size complex operator in the frequency-space domain. This would require the constant-velocity assumption over the spatial extent of the operator in the inline and crossline directions. Nevertheless, you would not have the errors in difference approximations to differential operators as with the finite-difference algorithms, since you are computing the derivatives of the wavefield in the inline and crossline directions in the Fourier transform domain. For the 3-D wave extrapolation from one depth step to another, the 2-D operator in the inline-crossline domain is then convolved with the complex wavefield for each frequency component.
While this scheme based on direct convolution of the extrapolation operator with the wavefield at one depth level is more accurate than the one-pass scheme based on operator splitting, it is computationally expensive. It can only be made practical if truncation errors are minimized, the 3-D extrapolation operator can be made spatially compact, and is applied, efficiently. Indeed, 2-D stable explicit extrapolation operators can be designed with minimal truncation errors within a specified dip range   (Section D.5). Blacquiere  extended the explicit method by Holberg  to 3-D poststack depth migration. This extension includes computing and tabulating the coefficients of 2-D explicit extrapolation operators in the frequency-space domain. Again, the operator is applied to the wavefield at each depth level by direct convolution. Finally, Hale >, devised an algorithm for 3-D poststack migration that circumvents the cost of computing and tabulating 2-D operators, and applying them by direct convolution. As will be shown later in the section, a 2-D explicit operator can be used to create a steep-dip 3-D extrapolation operator that preserves near-circular symmetry by way of the McClellan transform . This technique only requires computing 1-D explicit extrapolation filter coefficients and uses an efficient recursive filtering scheme based on Chebychev polynomials to perform the convolution of the extrapolation operator with the input wavefield.
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- Kitchenside, 1988, Kitchenside, P., 1988, Steep-dip 3-D migration: Some issues and examples: 58th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 976–978.
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- Hale, 1991b, Hale, D., 1991b, 3-D depth migration via McClellan transforms: Geophysics, 56, 1778–1785.
- 3-D poststack migration
- Separation versus splitting
- 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