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# The residual AMO operator

The residual AMO operator includes a cascade of four 3-D v(z) DMO operations; two forward operations and two inverse ones. The difference between each of the pair of forward and inverse operations is the medium parameters. For example, a pair of forward and inverse DMO's, or AMO, is applied for a homogeneous medium followed by another pair corresponding to a v(z) medium. The result is a residual AMO operator that corrects for the velocity perturbation from a background homogeneous model to a v(z) one.

The size of the residual AMO operator is directly dependent on the amount of velocity perturbation from the homogeneous background model. The residual operator provides information on the impact of the perturbation in velocity on the AMO operator. The smaller the size of the residual operator, the lesser the velocity variations influenced the AMO operator, and thus the lesser the need to use it.

Figure 15 shows a side and a top view of a residual AMO operator that corrects a homogeneous AMO operator to a linear-velocity AMO operator. In other words, this residual AMO operator, when convolved with the homogeneous-medium AMO operator, provides us with the linear-velocity AMO operator. This AMO operator corresponds to a pure azimuth correction of 30 degrees. The resulting residual operator is about 10 times smaller than the corresponding full AMO operator shown in Figure 6 (upper-right). In fact, the maximum time correction exerted by this residual AMO operator is less than 10 ms, even for dips around 50 degrees. Such corrections are very much insignificant, and the homogeneous medium AMO operator is sufficient to correct for azimuth in such v(z) velocity variations.

Op2reslin30on
Figure 15
A side (left) and a top (right) view of a residual AMO operator responsible for the correction from the linear velocity model to a homogeneous medium for a pure azimuth correction of 30 degrees. The linear velocity model considered here, as previously stated, is v(z)=1.5+0.6z km/s.

Figure 16 includes residual AMO operators for corrections in offset, as well as azimuth, for the linear velocity model. However, the residual operator corresponding to a correction in azimuth only (middle), is smaller in size than the operators that include an offset correction as well (right), or has only an offset correction (left). The crossline component of the residual AMO operator that includes offset correction is important, because in homogeneous media the offset-correction operator does not include a crossline component. In fact, the size of the crossline component of the residual AMO operator corresponding to a purely offset correction should be about the same as the crossline component of the AMO operator for a similar correction, shown in Figure 6 (upper-left). In other words, the convolution of the residual DMO operator for a homogeneous medium, which is a 2-D operator, with the residual AMO operator in Figure 16 (left) should give us the AMO operator, shown in Figure 6 (upper-left). As expected, all residual AMO operators for the linear velocity case are smooth.

Op3vzreslin
Figure 16
Residual AMO operators corresponding to the difference between AMO operators in a homogeneous medium and AMO operators in the linear velocity medium. Both media have the same rms velocity at the NMO corrected time of 2 s. Left: corresponds to the AMO operator with a correction in offset from 2 to 1.5 km. Middle: corresponds to the AMO operator with a correction in azimuth only of 30 degrees. Right: corresponds to the AMO operator with a correction in offset and azimuth.

Not so, for the low-velocity-layer case, where the perturbation of the model from a homogeneous background caused, among other things, huge triplications. However, the residual operator, even for this case is generally small. Therefore, the correction needed to adjust for the low-velocity layer model, when a homogeneous AMO is applied, is generally small. In fact, it is as small as the linear velocity case model. Again, the residual operator corresponding to a correction in azimuth is the smallest.

Op3vzreslow
Figure 17
Residual AMO operators for the difference between AMO operators in a homogeneous medium and the low-velocity-layer medium. Both media have the same rms velocity at the NMO corrected time of 2 s. Left: corresponds to the AMO operator with a correction in offset from 2 to 1.5 km. Middle: corresponds to the AMO operator with a correction in azimuth of 30 degrees. Right: corresponds to the AMO operator with a correction in offset and azimuth.

For the case of the complicated high-velocity layer the observations are different. Even for the purely azimuth-correction operator, the residual operator, shown in Figure 18, is both complicated and large. In fact, the size of the residual AMO operator is almost the same as the size of the full AMO operator. The unequal distribution of ray parameters, as shown by the top view of Figure 18, suggests that steep angle dips are affected the most by applying a constant-velocity AMO operator. While reflections from small dip angles are generally helped the constant-velocity AMO operator.

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Figure 18
A side (left) and a top (right) view of a residual AMO operator responsible for the correction from the high-velocity-layer model in Figure 5 (left) to a homogeneous medium for a pure azimuth correction of 30 degrees.

Figure 19 shows the full range of residual AMO operators corresponding to correction in azimuth and offset. All operators have complicated shapes, however, now the size of the residual AMO operator corresponding to offset correction only is smaller than those that include azimuth correction. This reversal in size implies that such a velocity model impacts the azimuth correction more than the offset correction. This is a general statement, however a more accurate conclusion should include constant ray parameter comparisons, not shown here.

Op3vzreshigh
Figure 19
Residual AMO operators for the difference between AMO operators in a homogeneous medium and the high-velocity-layer medium. Both media have the same rms velocity at the NMO corrected time of 2 s. Left: corresponds to the AMO operator with a correction in offset from 2 to 1.5 km. Middle: corresponds to the AMO operator with a correction in azimuth of 30 degrees. Right: corresponds to the AMO operator with a correction in offset and azimuth.

The residual AMO operator in Figure 19 (left), that is responsible for offset correction, seems extremely complicated. The inline and crossline component of that operator, shown in Figure 20, displays the large number of triplications associated with the operator.

incrossvzhighres
Figure 20
The inline and crossline components of the residual AMO operator shown in Figure 19 (left), but with a wider aperture.

Next: cost issues Up: Alkhalifah & Biondi: AMO Previous: AMO operators in v(z)
Stanford Exploration Project
7/5/1998