Wave-equation inversion of time-lapse seismic data sets

Joint Inversion of Multiple Images (JMI)

We can pose the problem as an inversion for the individual (baseline and monitor) images. Then, the modeling equation becomes

$$\begin{array}{ccc} \left [ \begin{array}{cc} {\bf L}_{0} & {\... ... {\bf d}_{0} \\ {\bf d}_{1}\\ \end{array} \right ] \end{array}.$$ (A-29)

Using the same assumptions and procedure as in the previous section, we arrive at the following formulation:

$$\begin{array}{ccc} \left [ \begin{array}{cc} {\bf H}_{0} & {\... ...e m}_{0} \\ {\bf\tilde m}_{1} \end{array} \right ] \end{array},$$ (A-30)

where the baseline and monitor velocities are known. Where the monitor has been aligned with baseline, we have

$$\begin{array}{ccc} \left [ \begin{array}{cc} {\bf H}_{0} & {\... ..._{0} \\ {\bf\tilde m}^{b}_{1} \end{array} \right ] \end{array},$$ (A-31)

which holds approximately whether or not we migrate the monitor data with the baseline or monitor velocity.