Joint wave-equation inversion of time-lapse seismic data

Linear inversion

Given a linear modeling operator ${\bf L}$, the seismic data ${\bf d}$ can be computed as

$${\bf L}{\bf m}={\bf d},$$ (A-1)

where ${\bf m}$ is the reflectivity model. The modeling operator, ${\bf L}$, in this study, represents the seismic acquisition process. Two different surveys -- say a baseline and monitor -- acquired at different times (${\bf t=0}$ and ${\bf t=1}$ respectively) over the same earth model can be represented as follows:

$$\begin{array}{cc} {\bf L}_{0}{\bf m}_{0}={\bf d}_{0}, {\bf L}_{1}{\bf m}_{1}={\bf d}_{1}, \end{array}$$ (A-2)

where ${\bf m}_{0}$ and ${\bf m}_{1}$ are respectively the reflectivity models at the times when the datasets ${\bf d}_{0}$ and ${\bf d}_{1}$ were acquired, and ${\bf L}_{0}$ and ${\bf L}_{1}$ are the modeling operators defining the acquisition process for the two surveys (baseline and monitor).

The quadratic cost functions for equation 2 are given by

$$\begin{array}{cc} S({\bf m_0})=\Vert {\bf L}_{0}{\bf m}_{0} -... ...\Vert {\bf L}_{1}{\bf m}_{1} - {\bf d}_{1} \Vert^2, \end{array}$$ (A-3)

and the least-squares solutions are

$$\begin{array}{cc} \hat{{\bf m}}_{0}=({\bf L'}_{0}{\bf L}_{0})... ...e {\bf m}_{1}= {\bf H}_{1}^{-1} \tilde {\bf m}_{1}, \end{array}$$ (A-4)

where $\tilde {\bf m}_{0}$ and $\tilde {\bf m}_{1}$ are the migrated baseline and monitor images, $\hat{{\bf m}}_{0}$ and $\hat{{\bf m}}_{1}$ are the inverted images, ${\bf L'}_{0}$ and ${\bf L'}_{1}$ are the migration operators (adjoints to the modeling operators ${\bf L}_{0}$ and ${\bf L}_{1}$ respectively), and ${\bf H}_{0}\equiv{\bf L'}_{0}{\bf L}_{0}$ and ${\bf H}_{1}\equiv{\bf L'}_{1}{\bf L}_{1}$, are the Hessian matrices. Here, and in other parts of this paper, the symbol $'$ denotes transposed complex conjugate. These formulations are based on (but not limited to) one-way wave-equation extrapolation methods.

The Hessian matrices are the second derivatives of the cost functions (equation 3) with respect to all model points in the image. Because the Hessian matrices are generally not invertible for almost any practical scenario, equation 4 is solved iteratively as follows:

$$\begin{array}{cc} {\bf H}_{0} \hat{{\bf m}}_{0}= \tilde {\bf ... ... {\bf H}_{1} \hat{{\bf m}}_{1}= \tilde {\bf m}_{1}. \end{array}$$ (A-5)

An inverted time-lapse image, $\Delta \hat{{\bf m}}$, can be obtained as the difference between the two images, $\hat{{\bf m}}_{1}$ and $\hat{{\bf m}}_{0}$, obtained from equation 5:

$$\Delta \hat{{\bf m}}= \hat{{\bf m}}_{1} - \hat{{\bf m}}_{0}.$$ (A-6)

We will refer to the method of computing the time-lapse image using equation 6 as $separate$ $inversion$ throughout the rest of this paper.

Joint wave-equation inversion of time-lapse seismic data