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The phase-shift method of migration
begins with a two-dimensional Fourier transform (2D-FT) of the dataset.
(See chapter .)
This transformed data is downward continued
with and subsequently evaluated
at (where the reflectors explode).
Of all migration methods,
the phase-shift method
most easily incorporates depth variation in velocity.
The phase angle and obliquity function are correctly included,
automatically.
Unlike Kirchhoff methods,
with the phase-shift method there is no danger of aliasing the operator.
(Aliasing the data, however, remains a danger.)
Equation (7.14) referred to upcoming waves.
However in the reflection experiment,
we also need to think about downgoing waves.
With the exploding-reflector concept of a zero-offset section,
the downgoing ray goes along the same path as the upgoing ray,
so both suffer the same delay.
The most straightforward way of converting one-way propagation
to two-way propagation is to multiply time everywhere by two.
Instead, it is customary to divide velocity everywhere by two.
Thus the Fourier transformed data values,
are downward continued to a depth by multiplying by
|
(15) |
Ordinarily the time-sample interval for the output-migrated
section is chosen equal to the time-sample rate
of the input data (often 4 milliseconds).
Thus, choosing the depth
,
the downward-extrapolation operator for a single time step is
|
(16) |
Data will be multiplied many times by , thereby downward
continuing it by many steps of .
Subsections
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2009-03-16