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The recursion in equation (6) can
be also written in matrix form as
| |
(16) |
| |
(17) |
| |
where
- W^{-} is a upper bidiagonal matrix containing the upward continuation
operator for all depth levels,
- P^{-} is a column vector containing the receiver wavefield at all depth levels,
- P^{+} is a diagonal square matrix containing the source wavefield at all depth
levels, and
- is the reflectivity at all depth levels.
Equation (16) represents the upward continuation
recursion written for a given frequency. We can write a similar
relationship for each of the frequencies in the data, and
group them all in a matrix relationship:
| |
(18) |
where
- is a upper bidiagonal matrix containing the upward continuation
operators for all the frequencies in the data,
- is a column vector containing the receiver wavefield for all
the frequencies,
- is a diagonal square matrix containing the source wavefield for all
the frequencies,
- and is the sum over frequency matrix with dimensions (the transpose of the spreading over frequencies),
Equation (18) represents the upward continuation
recursion written for a given shot position. We can write a similar
relationship for each of the shot positions in the data, and
group them all in a matrix relationship:
| |
(19) |
where
- is a upper bidiagonal matrix containing the upward continuation
operators for all the shots in the data,
- is a column vector containing the receiver wavefield for all the shots,
- is a diagonal square matrix containing the source wavefield for all
the shots,
- and is the sum over the frequency matrix with dimensions (the transpose of the spreading over frequencies),
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Previous: Source wavefield downward extrapolation
Stanford Exploration Project
5/23/2004