Three examples of vertical velocity variations with depth
are considered here. All three examples are
plausible, and can be found in the subsurface, however, they do not represent
all possible vertical velocity
variations in the subsurface. These examples will, however, provide us
with a reasonable understanding on how the AMO operator
is sensitive to vertical inhomogeneity.
We will show three different AMO operators;
the first corresponding to
a correction in offset only, called typically the residual DMO operator.
The second corresponding to a correction in offset
and azimuth, with an azimuth correction of 30 degrees. The third
has no offset correction, but an azimuth
correction of 30 degrees. The offset correction, used in most of the examples,
is from 2.0 km to 1.5 km. For size comparison, we, also, display the
full *v*(*z*) DMO operator for an offset of 2 km. The NMO time for all
operators in this paper is 2 s.

All the 3-D graphs of AMO operators include an aperture
that covers half the maximum possible zero-offset ray parameter.
Since the
surface velocity for all three models is the same at 1.5 km/s,
this range includes ray emergence angle up to 30 degrees.
The corresponding reflector dip angle, however, should be much higher since velocity
increases with depth, and it will depend on the velocity model.
The 2-D operator cross-sections, on the other hand,
will include emerging angles up to the critical angle.
Figure 5 shows two of the three velocity models
considered in this paper. The left one
will be referred to as the low-velocity-layer example, while the right one will be
referred to as
the high-velocity-layer example. The third velocity model, not shown here, is a simple
linear velocity increase
with depth at a gradient of 0.6*s ^{-1}*.
All velocity models have a surface velocity of 1.5 km/s.

Figure 5

Figure 6 shows the AMO operators for the first example, which is a linear velocity increase will depth.

Figure 6

The AMO operator corresponding to a pure offset correction,
shown upper left, has a similar shape to the
full 3-D DMO operator, shown lower-right, which is generally a saddle,
but much smaller in size. The corresponding residual DMO operator
for homogeneous media is a purely 2-D operator.
The azimuth-correction-only operator, shown upper right, is very similar
to the homogeneous-medium
one shown in Figure 2,
with an overall skewed saddle shape.
When the offset and azimuth corrections
are combined in a single operator,
it is given by the one shown in the lower
left of Figure 6.
The full DMO operator, shown in the lower-right, is clearly the largest in size.
AMO operators that include offset correction alters the position of horizontal,
as well as dipping reflections.
This alteration is necessary to correct for the
non-hyperbolic moveout associated with *v*(*z*) media for horizontal and dipping
events.

Figure 7

Figure 7 shows the inline and crossline components of the AMO operator shown in Figure 6 (upper-left), which corrects for offset only from 2 km to 1.5 km. The operator here includes the full aperture of the AMO operator, and thus includes the triplication at high angles. Surprisingly, the size of the operator in the crossline component is larger than that in the inline component. This fact stresses the importance of the crossline component of the residual DMO operator. Figure 8 shows the inline and crossline components of the AMO operator corresponding to azimuth correction of 30 degrees. Again, we include the full possible aperture and conveniently no triplications exist. The absence of triplications simplify the application of such an operator in a Kirchhoff type of implementation.

Figure 8

Figure 9 shows the inline and crossline components of the AMO operator that includes both the offset and azimuth corrections. This operator includes triplications that are associated with the offset correction portion of the operator. This operator is simply the convolution of the two previous operators, with its overall shape resembling both operators.

Figure 9

An AMO (or residual DMO) correction from offset 1.5 to 2.0 km will provide us with an operator that is inverse (or adjoint) to the operator shown in Figure 7, which corresponds to an offset correction from 2.0 to 1.5 km. Figure 10 shows the inline and crossline components of such an AMO operator with the full aperture included. Triplications similar but opposite to the ones shown in Figure 7 appear here. The convolution of the operators in Figure 7 and Figure 10 should result in an impulse, which confirms the dot-product rule.

Figure 10

The second example has a low velocity zone as shown in Figure 5 (left). Figure 11 shows AMO operators for such a velocity model: corresponding to a pure offset correction (upper left), corresponding to a pure azimuth correction (upper right), corresponding to the combination of offset and azimuth correction (lower left), and corresponding to a full DMO operator (plotted at a larger scale, lower right).

Figure 11

The operators that include offset corrections are much more complicated then the ones corresponding to the linear velocity model example, while the operator that includes only azimuth corrections are very similar to the linear velocity model ones, as well as to the homogeneous model ones. This observation implies that vertical inhomogeneity has a greater impact on the offset correction part of the operator than the azimuth correction part.

A closer look given by the inline and crossline components shown in Figure 12 reveals the complications added to the operator by the offset correction. Specifically, the crossline component includes triplications at low reflector angles. These triplications will make any Kirchhoff-type application of this operator difficult.

Figure 12

The AMO operator corresponding to only azimuth correction, on the other hand, does not include triplications at any angle, as shown in Figure 13. The absence of triplications, despite the presence of a low velocity zone, is encouraging.

Figure 13

Figure 14, also, shows the four AMO operators, however, now for the complicated high-velocity layer model. Again the AMO operators are smaller in general than the full DMO operator shown at lower right. Interestingly, the full DMO operator and the residual DMO operator (upper left) have small crossline components, and in this aspect, they are similar to the constant-velocity operator. The azimuth correction gives the AMO operator a more 3-D shape as shown in Figure 14 (upper-right and lower-left). Again, the AMO operator that includes only azimuth correction (of 30 degrees) does not include triplication.

Figure 14

In summary, AMO operators correcting only the azimuth are much simpler than those that correct also the offset. These azimuth-only correction operators are overall triplication free, even for the case of the high velocity layer. Therefore, using such operators in Kirchhoff-type implementation should be straightforward. These operators are also, for the smooth velocity examples, very similar to the constant-velocity AMO operators.

7/5/1998