Wave-equation inversion of time-lapse seismic data sets |

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:

where is the temporal regularization operator and is the regularization parameter. Therefore the full regularized inversion requires a minimization of the norm:

which leads to the image-space problem

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

and the temporal regularization by minimizing

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

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

and for the baseline-aligned images, the temporal constraint in equation A-37 becomes

Wave-equation inversion of time-lapse seismic data sets |

2011-05-24