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Cascading DMO and inverse DMO allowed us to evaluate the AMO
operator's summation path and the corresponding weighting
function.
However, this procedure is not sufficient for evaluating the third major
component of the integral operator, that is, its aperture (range of
integration). To solve this problem, we apply an alternative approach,
defining AMO as the cascade of the 3-D common-offset common-azimuth
migration and the 3-D modeling for a different azimuth and offset.

The impulse response of the common-offset common-azimuth migration is a
symmetric ellipsoid with the center in the input midpoint and axis of
symmetry along the input-offset direction. Such an ellipsoid is
described by the general formula

| |
(51) |

where *z* stands for the depth coordinate, is the surface
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 impulse in the data:
| |
(52) |

where *v* is the propagation velocity.
The impulse response of the AMO operator corresponds kinematically
to the reflections from the ellipsoid defined by
equation (52) to a different azimuth and different
offset.
To constrain the AMO aperture, we should look 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, it should be excluded from the AMO
aperture.

Fermat's principle provides a general method of solving the kinematic
reflection problem Goldin (1986). The formal expression for the
two-point reflection traveltime is given by

| |
(53) |

where is the vertical projection of the reflection point to the
surface, is the source location, and is the receiver
location for the output trace. According to Fermat's principle, the
reflection raypath between two fixed points must correspond to the
extremum value of the traveltime. Hence, in the vicinity of a
reflected ray,
| |
(54) |

Solving equation (55) for allows us to find the
reflection raypath for a given source-receiver pair on the surface.
To find the solution of (55), it is convenient to
decompose the reflection-point projection into three components:
, where is parallel to the
input offset vector , and is perpendicular to
. The plane, drawn through the reflection point and the
central line of ellipsoid (52), must contain the
zero-offset (normally reflected) ray because of the cylindrical
symmetry of the reflector. The fact that the zero-offset ray is normal
to the reflector gives us the following connection between the
zero-offset midpoint and the component of the
reflection point :

| |
(55) |

Equation (56) evaluates in terms of , as
follows:
| |
(56) |

where the length of the vector can be
determined from equation (44) for any given input and
output midpoints and and azimuths and
.
To find the third component of the reflection point projection
, we substitute expression (57)
into (54). Choosing a convenient parameterization
, , where
, and
,
we can rewrite the two-point
traveltime function from (54) in the form

| |
(57) |

Fermat's principle (55) leads to a simple linear equation
for the length of , which has the explicit solution
| |
(58) |

where is defined by (57), and satisfies relationship (44).
Because the reflection point is contained inside the ellipsoid,
its projection obeys the evident inequality

| |
(59) |

It is inequality (60) that defines the aperture of the
AMO operator.
After transformation (5) and algebraic
simplifications, it takes the form of inequality (12),
which is convenient for an efficient implementation of AMO.

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Stanford Exploration Project

11/11/1997