Joint wave-equation inversion of time-lapse seismic data

Regularized joint-inversion for image differences: RJID

The data modeling process for three seismic datasets over an evolving earth model can be written as

$$\left [ \begin{array}{ccc} {\bf L}_{0} & {\bf0} & {\bf0} {\bf ... ...gin{array}{ccc} {\bf d}_{0} {\bf d}_{1} {\bf d}_{2} \end{array} \right ],$$ (A-18)

where ${\bf d}_{0}$, ${\bf d}_{1}$ and ${\bf d}_{2}$ are respectively datasets for the baseline, first and second monitor, ${\bf m}_{0}$ is the baseline reflectivity and the time-lapse reflectivities $\Delta {\bf m}_{1}$ and $\Delta {\bf m}_{2}$ are defined as

$$\begin{array}{ll} \Delta {\bf m}_{1}={\bf m}_{1}-{\bf m}_{0}, \Delta {\bf m}_{2}={\bf m}_{2}-{\bf m}_{1}, \end{array}$$ (A-19)

where ${\bf m}_{1}$ and ${\bf m}_{2}$ are respectively the monitor reflectivities at the times data ${\bf d}_{1}$ and ${\bf d}_{2}$ were acquired (with survey geometries defined by the linear ${\bf L}_{1}$ and ${\bf L}_{2}$).

The least-squares solution to equation A-18 is given as

$$\left [ \begin{array}{ccc} {\bf {L'}_0 L_0+{L'}_1 L_1+{L'}_2 L_2}... ...begin{array}{c} {\bf d}_{0} {\bf d}_{1} {\bf d}_{2} \end{array} \right ],$$ (A-20)

where the symbol $'$ denotes transpose complex conjugate. We rewrite equation A-20 as

$$\left [ \begin{array}{ccc} {\bf H_0+H_1+H_2}&{\bf H_1+H_2}&{\bf H... ... \tilde {\bf m_1} + \tilde {\bf m_2} \tilde {\bf m_2} \end{array} \right ],$$ (A-21)

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 regularization goals that incorporates prior knowledge of the reservoir geometry and location as well as constraints on the inverted time-lapse images into equation A-21 we obtain

$$\begin{array}{ll} \left ( \left [ \begin{array}{ccc} {\bf H_0... ... {\bf m_2} \tilde {\bf m_2} \end{array} \right ], \end{array}$$ (A-22)

where,

$$\begin{array}{ccc} {\bf R_{ij} = { {\epsilon_i R'}_i \epsilon... ...j} = {{ \zeta_i \Lambda'}_i \zeta_j \Lambda_j}} \end{array} ,$$ (A-23)

with ${\bf R_i}$ being the spatial regularization terms for the baseline and time-lapse images respectively while ${\bf {\Lambda}_i}$ is the temporal regularization between the surveys. Note that ${\bf R_{ij}}$ and ${\bf\Lambda_{ij}}$ are not explicitly computed, but instead, the regularization operators ${\bf R_i}$ and ${\bf\Lambda_i}$ (and their adjoints) are applied at each step of the inversion. Parameters ${ \epsilon_i}$ and ${\zeta_i}$ determine the relative strengths of the spatial and temporal regularization respectively. Equation A-22 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-24)

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

$${\bf\Xi }= \left [ \begin{array}{cccccc} {\bf H_0+..+H_N}&{\bf H_... ...bf H_N}&{\bf H_N}&{\bf H_N}&{\bf ...}&{\bf N_N}&{\bf H_N} \end{array} \right ].$$ (A-25)

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

$$\begin{array}{c} {\bf\Re = R' R}, {\bf\Gamma = \Lambda' \Lambda}, \end{array}$$ (A-26)

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-27)

and,

$${\bf\Lambda }= \left [ \begin{array}{cccccc} {\bf\Lambda_{0}}&{\b... ...f0 }&{\bf0 }&{\bf ...}&{\bf0 }&{\bf0 }&{\bf\Lambda_{N}} \end{array} \right ].$$ (A-28)

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

$$\tilde{{\bf M}} = \left [ \begin{array}{c} \tilde {\bf m_0}+{\bf ... ...{\bf m}_{N-1} + \tilde {\bf m}_{N} \tilde {\bf m}_{N} \end{array} \right ],$$ (A-29)

while the inversion targets are

$$\hat{{\bf M}} = \left [ \begin{array}{c} \hat{\bf m}_{0} \Delt... ...at{\bf m}_{2} { : } { : } \Delta \hat{\bf m}_{N} \end{array} \right ].$$ (A-30)

The temporal constraint on the baseline image, ${\bf\Lambda_{0}}$ may be set to zero, since it is assumed that the original geological structure is unchanged over time or that geomechanical changes are accounted for before/during inversion.

Joint wave-equation inversion of time-lapse seismic data