Joint wave-equation inversion of time-lapse seismic data

Joint-inversion of multiple images (JMI)

The data modeling operations for two surveys can be written as follows

$$\left [ \begin{array}{cc} {\bf L}_{0} & {\bf0 } {\bf0 } & {\bf... ... ] = \left [ \begin{array}{cc} {\bf d}_{0} {\bf d}_{1} \end{array} \right ].$$ (A-13)

In principle, it is possible to solve for a least-squares solution to equation 13 by minimizing the cost 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-14)

As discussed in the JID formulation, this would be too expensive to be practical since the cost of one iteration is at least the cost of four migrations. Ajo-Franklin et al. (2005) have shown a tomographic example of this formulation, but since each migration is orders of magnitudes more expensive than ray-based tomography, this approach would be too expensive for wave-equation inversion. Therefore, we reformulate equation 14 as

$$\left [ \begin{array}{cc} {\bf L'}_{0} {\bf L}_{0} & {\bf0} {\... ...begin{array}{cc} \tilde {\bf m}_{0} \tilde {\bf m}_{1} \end{array} \right ],$$ (A-15)

or

$$\left [ \begin{array}{ccc} {\bf H}_{0} & {\bf0} {\bf0} & {\bf ... ...begin{array}{cc} \tilde {\bf m}_{0} \tilde {\bf m}_{1} \end{array} \right ],$$ (A-16)

which can be written as

$$\left [ \begin{array}{cc} \hat{{\bf m}}_{0} \hat{{\bf m}}_{1} ... ...begin{array}{cc} \tilde {\bf m}_{0} \tilde {\bf m}_{1} \end{array} \right ].$$ (A-17)

Thus, the inverted baseline and monitor images ( ${\hat{{\bf m}}}_{0}$ and ${\hat{{\bf m}}}_{1}$ respectively) can be obtained from equation 17 and the time-lapse image as a difference between the two images as done in equation 5. Also, note that without coupling, as done in the next section, equation 16 is equivalent to equation 5. Since the Hessian matrices ${\bf H}_{0}$ and ${\bf H}_{1}$ (and hence the joint Hessian operator) are not invertible, equation 16 is solved iteratively. An extension of equation 16 to multiple surveys is given in Appendix A.

Joint wave-equation inversion of time-lapse seismic data