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When the input-offset vector is parallel
to the output-offset vector ,
the triangle --, formed
by the midpoints of the input trace, zero-offset trace, and output
trace, degenerates to a line. The location of the zero-offset midpoint
is not constrained by the input and output midpoints and can
take different values on the line. The cascade of DMO and inverse DMO
becomes a convolution on that line. To find the summation path of 2-D
AMO (offset continuation), one needs to consider the envelope of the
family of traveltime curves (where m0 is the parameter of a
curve in the family):
| |
(49) |
Solving the envelope condition
for the zero-offset midpoint m0 produces
| |
(50) |
where . Substituting (51)
into (50), we obtain the explicit
expression (3) of the offset continuation summation
path.
C
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Stanford Exploration Project
11/11/1997