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| Wave-equation inversion of time-lapse seismic data sets | |
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Because in the JMI formulation, the models are completely decoupled, they can be regularized by minimizing the norm
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(A-32) |
where
is the spatial regularization operator and
the spatial regularization parameter for survey
.
To add any temporal regularization, we need to warp the inverted monitor images to the baseline and then apply temporal constraints
or we can regularize the time-lapse image directly by minimizing the norm:
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(A-33) |
where
is the temporal regularization operator and
is the regularization parameter.
Therefore the full regularized inversion requires a minimization of the norm:
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(A-34) |
which leads to the image-space problem
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(A-35) |
where
and
are the spatial and temporal constraints, respectively.
If the monitor has been aligned to the baseline, then we can impose the spatial regularization by minimizing
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(A-36) |
and the temporal regularization by minimizing
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(A-37) |
where
and
are defined with respect to the baseline-aligned monitor image.
If the time-lapse image at the baseline position, the regularized image-space inversion problem is given by
|
(A-38) |
where the superscript
denotes that the operators and images are referenced to the baseline position.
Note that in the simplest case, where the temporal regularization is a difference operator equation A-33 becomes
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(A-39) |
and for the baseline-aligned images, the temporal constraint in equation A-37 becomes
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(A-40) |
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| Wave-equation inversion of time-lapse seismic data sets | |
|
Next: Bibliography
Up: APPENDIX A
Previous: Joint Inversion of Multiple
2011-05-24