Joint wave-equation inversion of time-lapse seismic data

Regularized joint-inversion of multiple images: RJMI

The data modeling process for three seismic datasets (a baseline and two monitors) over an evolving earth model can be written as

$$\left [ \begin{array}{ccc} {\bf L}_{0} & {\bf0} & {\bf0} {\bf0... ...gin{array}{ccc} {\bf d}_{0} {\bf d}_{1} {\bf d}_{2} \end{array} \right ],$$ (A-7)

where ${\bf d}_{0}$, ${\bf d}_{1}$ and ${\bf d}_{2}$ are respectively datasets for the baseline, first and second monitor, ${\bf m}_{0}$, ${\bf m}_{1}$, and ${\bf m}_{2}$ are the baseline and monitor reflectivity models. The linear operators ( ${\bf L}_{0}$, ${\bf L}_{1}$ and ${\bf L}_{2}$) define the modeling/acquisition experiments for datasets ${\bf d}_{0}$, ${\bf d}_{1}$ and ${\bf d}_{2}$ respectively. The least-squares solution to equation A-7 is given as

$$\left [ \begin{array}{ccc} {\bf {L'}_0 L_0}&{\bf0 }&{\bf0} {\bf... ...begin{array}{c} {\bf d}_{0} {\bf d}_{1} {\bf d}_{2} \end{array} \right ],$$ (A-8)

where the symbol $'$ denotes transposed complex conjugate.

We rewrite equation A-8 as

$$\left [ \begin{array}{ccc} {\bf H_0}&{\bf0}&{\bf0} {\bf0}&{\bf ... ...c} \tilde {\bf m_0} \tilde {\bf m_1} \tilde {\bf m_2} \end{array} \right ],$$ (A-9)

where $\tilde {\bf m_i}$ is the migrated image from the ${{\bf i}^{th}}$ survey, and ${\bf H_i}$ is the corresponding Hessian matrix. Introducing spatial and temporal constraints into equation A-9 we obtain

$$\left ( \left [ \begin{array}{ccc} {\bf H_0}&{\bf0}&{\bf0} {\bf... ... \tilde {\bf m_0} \tilde {\bf m_1} \tilde {\bf m_2} \end{array} \right ].$$ (A-10)

Equation A-10 can be generalized to an arbitrary number of surveys as follows

$$\left [ {\bf\Xi } + {\bf\Re } + {\bf\Gamma} \right ] \left [ \hat{{\bf M}} \right ] = \left [ \tilde {\bf M} \right ],$$ (A-11)

where, ${\bf\Xi }$ is the Hessian operator, defined as

$${\bf\Xi }= \left [ \begin{array}{cccccc} {\bf H_0}&{\bf0}&{\bf0}&... ...&{\bf0} {\bf0}&{\bf0}&{\bf0}&{\bf ...}&{\bf0}&{\bf H_N} \end{array} \right ].$$ (A-12)

The spatial and temporal regularization operators, ${\bf\Re}$ and ${\bf\Gamma}$ are defined as

$$\begin{array}{c} {\bf\Re = R' R}, {\bf\Gamma = \Lambda' \Lambda}, \end{array}$$ (A-13)

where,

$${\bf R }= \left [ \begin{array}{cccccc} {\bf R_{0}}&{\bf0}&{\bf0}... ...}&{\bf0} {\bf0}&{\bf0}&{\bf0}&{\bf0}&{\bf0}&{\bf R_N} \end{array} \right ],$$ (A-14)

and,

$${\bf\Lambda }= \left [ \begin{array}{cccccc} {\bf\Lambda_{0}}&{\b... ...\bf0 }&{\bf0 }&{\bf ...}&{\bf0 }&{\bf0 }&{\bf\Lambda_{N}} \end{array} \right ].$$ (A-15)

The input vector into the RJMI formulation, ${\tilde{\bf M}}$ is given as

$$\tilde{{\bf M}} = \left [ \begin{array}{cc} \tilde{\bf m}_{0} ... ... {\bf :} \tilde{\bf m}_{N-1} \tilde{\bf m}_{N} \end{array} \right ],$$ (A-16)

while the inversion targets are in the vector:

$$\hat{{\bf M}} = \left [ \begin{array}{cc} \hat{\bf m}_{0} \hat... ...{2} {\bf :} \hat{\bf m}_{N-1} \hat{\bf m}_{N} \end{array} \right ].$$ (A-17)

Joint wave-equation inversion of time-lapse seismic data