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The impulse response of the AMO operators corresponds to a spike on
the initial constant-offset constant-azimuth gather. Such a spike
can physically occur in the case of a focusing ellipsoidal
reflector whose focuses are coincident with the initial source and
receiver locations (the impulse response of prestack common-offset
migration). Therefore, the impulse response of AMO corresponds
kinematically to a reflection from this ellipsoid. These
considerations allow us to define AMO as the cascade of the 3-D
common-offset common-azimuth migration and the 3-D modeling for a different
azimuth and offset. An analogous point of
view was
developed for the 2-D case
by Deregowski and Rocca 1981.
Let's consider the general symmetric ellipsoid equation
| ![\begin{displaymath}
z(x,y)=\sqrt{R^2-\beta\,\left(x-x_1\right)^2-\left(y-y_1\right)^2}\;,\end{displaymath}](img43.gif) |
(15) |
where z stands for the depth coordinate, R is the small semi-axis of the
ellipsoid, and
is a nondimensional parameter describing the
stretching of the ellipse
. Deregowski and
Rocca 1981 derived the following connections
between the geometric properties of the reflector and the coordinates
of the corresponding spike in the data:
| ![\begin{displaymath}
R={{v\,t_1}\over 2}\;;\;
\beta={t_1^2 \over t_1^2+{{4\,{\bf h_1}^2}\over v^2}}\;,\end{displaymath}](img46.gif) |
(16) |
where v is the wave velocity.
The center of the ellipsoid is at the initial midpoint
.
This section addresses the kinematic problem of reflection
from the ellipsoid defined by (15). In particular, we are looking for
the answer to the following question: For a given elliptic
reflector defined by the input midpoint, offset, and time coordinates,
what points on the surface can form a source-receiver pair valid for a
reflection? If a point in the output midpoint-offset space
cannot be related to a reflection pattern, we should exclude it from the AMO
impulse response defined in (1).
Fermat's principle provides a general method of solving the kinematic reflection problems. Consider a formal expression for the two-point
reflection traveltime
| ![\begin{displaymath}
t={\sqrt{
(\mbox{\boldmath{$s - \xi$}})^2+z^2(\xi_x,\xi_y)}
...
...{
(\mbox{\boldmath{$r - \xi$}})^2+z^2(\xi_x,\xi_y)}
\over v}\;,\end{displaymath}](img47.gif) |
(17) |
where ![$\xi=$](img48.gif)
is the vertical
projection of the reflection
point to the surface,
is the source location, and
is the receiver location.
According to Fermat's principle, the reflection ray path between two
fixed points must correspond to the
extremum value of the traveltime. Hence, in the vicinity of a
reflected ray,
| ![\begin{displaymath}
{\partial t \over \partial \xi_x}=0\;;\;
{\partial t \over \partial \xi_y}=0\;.\end{displaymath}](img52.gif) |
(18) |
Solving the system of equations (18) for
and
allows us to find the reflection ray path for a given source-receiver
pair on the surface. The solution is derived in Appendix B to be
| ![\begin{displaymath}
\xi_x={{x_0-\beta\,x_1}\over{1-\beta}}\;,\end{displaymath}](img55.gif) |
(19) |
| ![\begin{displaymath}
\xi_y=y_1+\left(x_0-\xi_x\right)\,\cot{\varphi}-
{{\left(y_2...
...]}\over
{{\bf h_2^2}\sin^2{\varphi}-\left(y_2-y_1\right)^2}}\;,\end{displaymath}](img56.gif) |
(20) |
where x0 has the same meaning as in the preceding section
and is defined by (7).
Since the reflection point is contained inside the ellipsoid,
its projection obeys the evident inequality
| ![\begin{displaymath}
\left(\xi_y-y_1\right)^2\leq R^2-\beta\,\left(\xi_x-x_1\right)^2\;.\end{displaymath}](img57.gif) |
(21) |
It is inequality (21) that defines the aperture of the AMO
operator.
amoapp
Figure 2 The AMO impulse
response traveltime. Parameters:
m,
m, t1=1 sec. The top plots illustrate
the case of an unrealistically low velocity (v=10 m/s); on the bottom,
v=2000 m/s. On the left side the azimuth rotation
; on the right,
The AMO operator's contours for
different azimuth rotation angles are shown in
Figure 2.
Comparing the results for the case of an unrealistically low velocity
(the top two plots in Figure 2) and the case of a realistic
velocity (the bottom two plots) clearly demonstrates
the gain in the reduction of the aperture size
achieved by the aperture limitation.
The gain is
especially spectacular for small azimuths. When the azimuth rotation
approaches zero, the area of the 3-D aperture monotonously shrinks to a
line, and the limit of the traveltime of the AMO impulse response
(the inverse of (9)) approaches the offset continuation operator
(14) (Figures 3). This means that
taking into account
the aperture limitations of AMO provides a consistent description
valid for small azimuth rotations including zero (the offset
continuation case). Obviously, the cost of an integral operator is
proportional to its size. The size of the offset continuation
operator cannot extend the difference between the offsets
. If we applied DMO and
inverse DMO explicitly, the total size of the two operators would be
about
, which is
substantially greater. This fact proves that in the case of small
azimuth rotations the AMO price is less than those of not only 3-D prestack
migration, but also 3-D DMO and inverse DMO combined Canning and Gardner (1992).
Figure 4 shows the saddle shape of the AMO operator impulse
response in a 3-D AVS display.
amocom
Figure 3 Traveltime curves
of the impulse responses. The dashed lines indicate the AMO impulse
response with an azimuth rotation of 3 degrees (projection on the x
plane); the solid lines, the 2-D offset continuation impulse response.
|
| ![amocom](../Gif/amocom.gif) |
amoavs
Figure 4 AMO impulse
response traveltime in three dimensions (the AVS display). Parameters:
m,
m, t1=1 sec, v=2000 m/s,
.
Next: Conclusions
Up: Fomel & Biondi: t-x
Previous: CASCADING DMO AND INVERSE
Stanford Exploration Project
9/12/2000