Wave-equation inversion of time-lapse seismic data sets

Regularized Joint inversion of Multiple Images (RJMI)

Because in the JMI formulation, the models are completely decoupled, they can be regularized by minimizing the norm

$$\begin{array}{ccc} \left \vert\left\vert \left [ \begin{array... ...rray} \right ] \right \vert \right \vert \approx 0 \end{array},$$ (A-32)

where ${\bf R}_{i}$ is the spatial regularization operator and ${\bf\epsilon}_{i}$ the spatial regularization parameter for survey $i$ . 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:

$$\begin{array}{ccc} \left \vert\left\vert \left [ \begin{array... ...rray} \right ] \right \vert \right \vert \approx 0 \end{array},$$ (A-33)

where $\Lambda_{i}$ is the temporal regularization operator and $\zeta_{i}$ is the regularization parameter. Therefore the full regularized inversion requires a minimization of the norm:

$$\begin{array}{ccc} \left \vert\left\vert \left [ \begin{array... ...rray} \right ] \right \vert \right \vert \approx 0 \end{array},$$ (A-34)

which leads to the image-space problem

$$\begin{array}{c} \left [ \begin{array}{ccc} {\bf H }_{0} & {\... ...\\ \hline {\bf0} \\ {\bf0} \\ \end{array} \right ], \end{array}$$ (A-35)

where ${\bf R}_{ii}={\bf\epsilon}^{2}_{ i}{\bf R}_{ i}^{^T}{\bf R}_{ i}$ and ${{\bf\Lambda}_{ij} = {\bf\zeta}_i {\bf\Lambda}_{ i}^{^T} {\bf\zeta}_j {\bf\Lambda}_j}$ 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

$$\begin{array}{ccc} \left \vert\left\vert \left [ \begin{array... ...rray} \right ] \right \vert \right \vert \approx 0 \end{array},$$ (A-36)

and the temporal regularization by minimizing

$$\begin{array}{ccc} \left \vert\left\vert \left [ \begin{array... ...rray} \right ] \right \vert \right \vert \approx 0 \end{array},$$ (A-37)

where ${\bf R}^{b}_{1}$ and ${\bf\Lambda}^{b}_{1}$ 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

$$\begin{array}{c} \left [ \begin{array}{ccc} {\bf H }_{0} & {\... ...\\ \hline {\bf0} \\ {\bf0} \\ \end{array} \right ], \end{array}$$ (A-38)

where the superscript $(^{b})$ 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

$$\begin{array}{ccc} \left \vert\left\vert \zeta \left [ \begin... ...rray} \right ] \right \vert \right \vert \approx 0 \end{array},$$ (A-39)

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

$$\begin{array}{ccc} \left \vert\left\vert \zeta \left [ \begin... ...rray} \right ] \right \vert \right \vert \approx 0 \end{array}.$$ (A-40)

Wave-equation inversion of time-lapse seismic data sets