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.

Figure 15

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.

Figure 16

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.

Figure 17

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.

Figure 18

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.

Figure 19

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.

Figure 20

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