Maximum dip to migrate
Kirchhoff migration in practice
During migration, we can specify the maximum dip we want migrated in the section. This may be useful when we need to suppress the steeply dipping coherent noise. Figure 4.2-8 shows migrations of the dipping events with four different maximum allowable dips. For a 4 ms/trace dip limit, events with dips greater than this value are suppressed. Similarly, for an 8 ms/trace dip value, events with dips greater than this value are suppressed. When the dip value is 12 ms/trace, no suppression occurs, since all events in the input section have dips less than this value. Limiting the dip parameter is a way to reduce computational cost, since it is related to aperture width (equation 1), which determines the cost.
From Figure 4.2-1, note that the Kirchhoff migration impulse response can be limited to various maximum dips. The smaller the maximum allowable dip, the smaller the aperture. This combination of maximum aperture width and maximum dip limit determines the actual effective aperture width used in migration. In particular, diffraction hyperbolas along which summation is done are truncated beyond the specified maximum dip limit.
A field data example of testing the maximum dip parameter is shown in Figure 4.2-9. Some steep dips are lost on the section that corresponds to the 2 ms/trace maximum allowable dip. The 8 ms/trace dip appears to be optimum. The maximum dip parameter must be chosen carefully so that the steep dips of interest in the input section are preserved. Finally, dip value can be changed spatially and in time; however, practical implementation can be cumbersome.
Frequency-wavenumber migration in practice
The phase-shift method of migration (migration principles and Section D.7) allows vertical variations in velocity and is accurate for up to dips of 90 degrees. Figure 4.5-1 shows the impulse response of the phase-shift algorithm. Clearly, for a constant-velocity medium, this response is equivalent to that of the Stolt migration. The impulse response shown in Figure 4.5-1 is considered to be the desired impulse response for 2-D zero-offset migration, and as such, responses of all migration algorithms discussed in this chapter are benchmarked against it.
Figure 4.5-1 Impulse response of phase-shift migration has a semicircular shape.
Figure 4.5-2 The impulse response of the f − k migration operator has a truncated semicircular shape when a maximum dip limit is imposed. For comparison, the desired response shape has been superimposed on the f − k responses.
As with the Kirchhoff summation method, migration with the phase-shift method can be limited to smaller dips by truncating the semicircular wavefront (Figure 4.5-2). This dip filtering capability is useful in rejecting coherent noise from the stacked section while migrating the data. If migration is constrained to small dip values, then the steeply dipping reflectors may be filtered out unintentionally. Edge effects also are pronounced when a very narrow range of dips is passed. Note the linear streaks on the impulse response with a dip limit of 2 ms/trace (Figure 4.5-2).
The dip-filtering action caused by imposing a dip limit on the impulse response also is visible on the results shown in Figure 4.5-3. Note that steep dips greater than the specified maximum dip to migrate have been annihilated. On the field data example shown in Figure 4.5-4, severe dip filtering action of the 2 ms/trace maximum dip has caused smearing and eliminated virtually all of the signal contained in the section.
- Aperture width
- Velocity errors
- Depth step size
- Velocity errors
- Stolt stretch factor
- Residual migration