From the previous section, we know that Kirchhoff migration involves a summation of amplitudes along diffraction hyperbolas. Given the rms velocity at a particular time sample of a particular input trace, a hyperbolic traveltime trajectory associated with a fictitious diffractor is overlaid on the input section with its apex at that time sample. In theory, a diffraction hyperbola extends to infinite time and distance. In practice, we have to deal with a truncated hyperbolic summation path. The spatial extent that the actual summation path spans, called the migration aperture, is measured in terms of the number of traces the hyperbolic path spans.
The curvature of the diffraction hyperbola is governed by the velocity function. Figure 4.2-2a shows a number of low-velocity diffraction hyperbolas, while Figure 4.2-2b shows a number of high-velocity hyperbolas. A low-velocity hyperbola has a narrower aperture when compared to a high-velocity hyperbola. This agrees with our intuition — high velocity means more migration. In practice, we deal with a velocity function that at least varies with depth. The diffraction hyperbolas can have different curvatures depending on the velocity value at a given time sample (Figure 4.2-2c). Because of the vertical variation in velocity, aperture width generally is time variant. For the usual case in which velocity increases with depth, migration of the shallow part of the section requires a narrow aperture, while migration of the deep portion requires a wider aperture (Figure 4.2-2c). This implies that, given the same dip, deep events migrate farther than shallow events.
Figure 4.2-3 shows a zero-offset diffraction hyperbola (8 ms/trace dip along the asymptotes) and migrations using four different aperture widths. The smaller the aperture, the less capable the migration is in collapsing the diffraction hyperbola. In this case, use of an aperture width that is equal to the width of the input section (half aperture, 256 traces) yields the best result.
Figure 4.2-4 shows a synthetic zero-offset section that consists of a number of dipping events ranging from 0 to 45 degrees in increments of 5 degrees. Aperture width is related closely to the horizontal displacement dx that takes place in migration as defined by equation (1). The number of traces an event migrates is nx = dx/Δx, where Δx is the CMP interval. Therefore, the aperture width that is required is 2nx+1. Figure 4.2-4 also shows Kirchhoff migrations of the dipping events using four different aperture widths. Small-aperture migration eliminates steeply dipping events on the output section. Increasing the aperture width allows proper migration of the steeply dipping events. From this we see that using too small an aperture width causes a dip filtering action during migration, because a small aperture excludes the steeper flanks of the diffraction hyperbola from the summation.
For any given event position in time t before migration, the optimal value for the aperture width is defined by twice the maximum horizontal displacement in migration for the steepest dip of interest in the input section. In this case, the horizontal displacement associated with the 45-degree dipping event is computed by substituting the values for v = 3500 m/s, Δx = 25 m, Δt/Δx = 12 ms/trace, where t = 2 s in equation (1). The value for the horizontal displacement is 118 traces, giving an aperture width of 237 traces. Typically, we consider somewhat larger values to allow for velocity errors.
A good way to determine aperture width is to generate diffraction hyperbolas as shown in Figure 4.2-2c using the regionally averaged, vertically varying velocity. Clearly, the larger the aperture width, the more traces are used in the summation. For the dipping events in Figure 4.2-4, the optimal value of the half-aperture width is 150 traces; increasing the width to 300 traces resulted in no further improvement.
Figure 4.2-3 Tests for aperture width in Kirchhoff migration: (a) a zero-offset section that contains a diffraction hyperbola with 2500-m/s velocity, (b) desired migration using the phase-shift method; Kirchhoff migration using (c) 35-trace, (d) 70-trace, (e) 150-trace, and (f) 256-trace half-aperture width.
Figure 4.2-4 Tests for aperture width in Kirchhoff migration: (a) a zero-offset section that contains dipping events with 3500-m/s velocity, (b) desired migration using the phase-shift method; Kirchhoff migration using (c) 35-trace, (d) 70-trace, (e) 150-trace, and (f) 256-trace half-aperture width.
Figure 4.2-5 Tests for aperture width in Kirchhoff migration: Insufficient aperture width causes removal of steeply dipping events.
Figure 4.2-6 Tests for aperture width in Kirchhoff migration: Insufficient aperture width causes spurious horizontal events in deep, noisy part of a stacked section.
Figure 4.2-7 Tests for aperture width in Kirchhoff migration: Input to migration is a section (a) that contains band-limited random noise uncorrelated from trace to trace. Note the spurious horizontal events in the deeper part of the section after migration using small aperture (60 traces); these gradually disappear at increasingly larger apertures.
A test of aperture width on the CMP stacked data example is shown in Figure 4.2-5. The small-aperture migration causes smearing in the deeper part of the section. This smearing effect destroys the dipping events and produces spurious horizontally dominant events. Smearing is reduced gradually with increasing aperture.
Figure 4.2-6 is the deeper portion of a stacked section with migrations using different aperture sizes. The smearing effect is much more noticeable at small aperture. The main difference between the stacked sections in Figures 4.2-5 and 4.2-6 is that the latter, being deeper in time, contains a large amount of noise. This smearing phenomenon was not noticed in the noise-free synthetic model in Figure 4.2-4. We now see that choice of aperture width is more critical than we originally thought. In particular, a small aperture changes the noise characteristics of the section.
Why do we see horizontally dominant smearing with small-aperture migration? To answer this, we must do a simple experiment with a section that contains only random noise and the velocity function used in migration increases in time (Figure 4.2-7a). We see two interesting phenomena on the migrated sections using three different apertures. First, in all three cases there is more smearing of noise in the deeper part of the data, where the velocities generally are higher than in the shallower part. Second, there is relatively more smearing in the small-aperture migration compared with others at a given time in the section. Moreover, this smearing is characterized by horizontally dominant spurious events, especially in the deeper part of the section. Note that even with a large aperture, some smearing still is present in the deepest part of the section in Figure 4.2-7d. As indicated in Figure 4.2-2b, because summation stops at the bottom of the section, the effective aperture (CD) at late times is much smaller than that used in other parts of the section (AB). Remember that summation using very small aperture includes only the apex portion of the diffraction hyperbola, where dips are nearly flat. Therefore, the small aperture with a dip filtering action passes flat or nearly flat events — those horizontal wavenumber components that are zero or nearly zero.
In conclusion, the following assessments are made concerning the choice of aperture width.
- Excessively small aperture width causes destruction of steeply dipping events and rapidly varying amplitude changes.
- Excessively small aperture width organizes random noise, especially in the deeper part of the section, as horizontally dominant spurious events.
- Excessively large aperture means more computer time. More importantly, large apertures can degrade the migration quality in poor signal-to-noise ratio conditions. Use of large aperture will cause random noise at late times to creep into the good shallow data. Aperture width always is a compromise with noise.
- Sometimes it is better to use a smaller aperture than would theoretically be required to avoid the adverse effect of noise on the migrated event. Noise considerations may even require a time-dependent aperture width.
- It is recommended that the aperture width be kept constant in migrating all lines from a particular survey so that an overall uniformity in amplitude characteristics on the migrated sections is maintained.
In practice, a regional velocity function and the steepest dip in a survey area are used to compute the optimal aperture width that can be used over the entire set of data from the area (equation 1).