Joint wave-equation inversion of time-lapse seismic data

Regularized joint-inversion

The process of acquiring two seismic datasets over an evolving earth model can be represented as

$$\left [ \begin{array}{ccc} {\bf L}_{0} & {\bf0} {\bf0} &{\bf L... ... \left [ \begin{array}{ccc} {\bf d}_{0} {\bf d}_{1} \end{array} \right ],$$ (A-1)

where ${\bf d}_{0}$ and ${\bf d}_{1}$ are the baseline and monitor datasets, and ${\bf m}_{0}$ and ${\bf m}_{1}$ are the baseline and monitor reflectivity models respectively. The linear operators ( ${\bf L}_{0}$ and ${\bf L}_{1}$) define the modeling/acquisition experiments for datasets ${\bf d}_{0}$ and ${\bf d}_{1}$ respectively. We rewrite equation A-1 to include spatial and regularization operators ( ${\bf R}_{0}$ and ${\bf\Lambda}_{0}$ respectively), and we seek to minimize the objective function

$$\begin{array}{ccc} S({\bf m_0}, {\bf m_1})= \left \vert\left\... ...end{array} \right ] \right \vert \right \vert ^2. \end{array}$$ (A-2)

This cost function can be expanded as follows:

$$\left ( \left [ \begin{array}{ccc} {\bf {L'}_0 L_0 } & {\bf0 } \\... ...begin{array}{cc} \tilde {\bf m}_{0} \tilde {\bf m}_{1} \end{array} \right ],$$ (A-3)

which can be written as

$$\left ( \left [ \begin{array}{ccc} {\bf H_0 } & {\bf0 } {\bf0 ... ...begin{array}{cc} \tilde {\bf m}_{0} \tilde {\bf m}_{1} \end{array} \right ],$$ (A-4)

where

$$\begin{array}{ccc} {\bf R_{ij} = { {\epsilon_i R'}_i \epsilon... ...j} = {{ \zeta_i \Lambda'}_i \zeta_j \Lambda_j}}, \end{array}$$ (A-5)

while ${\bf R_0}$ and ${\bf R_1}$ are the spatial/imaging constraints for the baseline and monitor images respectively, and ${\bf {\Lambda}_0}$ and ${\bf {\Lambda}_1}$ the temporal constraints between the surveys. The parameters ${\bf {\epsilon}_0}$ and ${\bf {\epsilon}_1}$ determine the strength of the spatial regularization on the baseline and monitor images respectively, while ${\bf { \zeta }_0}$ and ${\bf {\zeta }_1}$ determine the coupling between surveys. Equation A-4 is the RJMI formulation. Using a similar procedure, the RJID formulation for two seismic datasets can be shown to be

$$\left ( \left [ \begin{array}{ccc} {\bf H_0+H_1} & {\bf H_1} {... ...ilde {\bf m}_{0}+\tilde {\bf m}_{1} \tilde {\bf m}_{1} \end{array} \right ].$$ (A-6)

In the next sections, we derive the RJID and RJMI formulations for multiple surveys.

Joint wave-equation inversion of time-lapse seismic data