Target-oriented joint inversion of incomplete time-lapse seismic data sets

Least-squares inversion of time-lapse seismic data sets

Within limits of the Born approximation of the linearized acoustic wave equation, synthetic seismic data set ${\bf d}$ is obtained by the action of a modeling operator ${\bf L}$ on the earth reflectivity ${\bf m}$:

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

Given two data sets (baseline and monitor), acquired over an evolving earth model at times ${\bf0}$ and ${\bf 1}$ respectively, we can write

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

where ${\bf m}_{0}$ and ${\bf m}_{1}$ are the baseline and monitor reflectivities, and ${\bf d}_{0}$ and ${\bf d}_{1}$ are the data sets modeled by ${\bf L}_{0}$ and ${\bf L}_{1}$.

Applying the adjoint operators ${\bf\bar L}^{T}_{0}$ and ${\bf\bar L}^{T}_{1}$ to ${\bf d}_{0}$ and ${\bf d}_{1}$ respectively, we obtain the migrated baseline ${\bf\tilde m}_{0}$ and monitor ${\bf\tilde m}_{1}$ images:

$$\begin{array}{cc} {\bf\tilde m}_{0}={\bf\bar L}^{T}_{0}{\bf d... ...\bf\tilde m}_{1}={\bf\bar L}^{T}_{1}{\bf d}_{1}, \end{array}$$ (A-3)

where ${\bf\bar L}^{T}_{i}$ denotes conjugate transpose of ${\bf L}_{i}$. The raw time-lapse image $\Delta \tilde{{\bf m}}$ is the difference between the migrated images:

$$\Delta \tilde{{\bf m}}= \tilde{{\bf m}}_{1} - \tilde{{\bf m}}_{0}.$$ (A-4)

Because incomplete seismic data sets leads to high non-repeatability, ${\bf\tilde m_{0}}$ and ${\bf\tilde m_{1}}$ must be cross-equalized before $\Delta \tilde{\bf m}$ is computed. The high level of non-repeatability makes it difficult to adapt existing cross-equalization methods (Rickett and Lumley, 2001; Calvert, 2005; Hall, 2006) to randomly sampled time-lapse seismic data sets. The RJMI method takes the data acquisition geometry and sampling into account and hence can correct for the non-repeatability of the data sets.

We define two quadratic cost functions for the modeling experiments (equation 2):

$$\begin{array}{cc} S({\bf m_0})=\Vert {\bf L}_{0}{\bf m}_{0} -... ...t {\bf L}_{1}{\bf m}_{1} - {\bf d}_{1} \Vert^2_{2}, \end{array}$$ (A-5)

which, when minimized, give the least-squares solutions ${\bf\hat m_0}$ and ${\bf\hat m_1}$, where

$$\begin{array}{cc} \hat{{\bf m}}_{0}=({\bf\bar L}^{T}_{0}{\bf ... ...bf L}_{1})^{\dagger}{\bf\bar L}^{T}_{1}{\bf d}_{1}, \end{array}$$ (A-6)

and ${(\cdot)^{\dagger}}$ denotes approximate inverse.

Because seismic inversion is ill-posed, model regularization is often required to ensure stability and convergence to a geologically consistent solution. For many seismic monitoring objectives, the known geology and reservoir architecture provide useful regularization information. Including baseline and monitor regularization operators ( ${\bf R}_{0}$ and ${\bf R}_{1}$ respectively) in the cost functions gives

$$\begin{array}{cc} S({\bf m_0})=\Vert {\bf L}_{0}{\bf m}_{0} -... ...lon^{2}_{1} \Vert{\bf R}_{1} {\bf m}_{1} \Vert^{2}, \end{array}$$ (A-7)

which have the solutions

$$\begin{array}{cc} \hat{{\bf m}}_{0}=({\bf\bar L}^{T}_{0}{\bf ... ...bf R}_{1})^{\dagger}{\bf\bar L}^{T}_{1}{\bf d}_{1}. \end{array}$$ (A-8)

where $\epsilon_{i}$ is a regularization parameter that determines the strength of the regularization relative to the data fitting goal. Although there is a wide range of suggested methods for selecting $\epsilon_{i}$, in most practical applications, the final choice of the parameter is subjective. Unless otherwise stated, we use a fixed, heuristically determined, data-dependent regularization parameter given by

$$\epsilon_{i} = \frac{max \vert {\bf d}_{i} \vert }{50}.$$ (A-9)

Estimating ${\bf\hat m_0}$ or ${\bf\hat m_1}$ by minimizing equation 7 is the so-called data-space least-squares migration/inversion method (Clapp, 2005).

Substituting equation 3 into equation 8, and re-arranging the terms, we get

$$\begin{array}{cc} \left [ {\bf H}_{0}+{\bf R}_{00} \right ] \... ...11} \right ] \hat{{\bf m}}_{1}= \tilde {\bf m}_{1}, \end{array}$$ (A-10)

where ${\bf H}_{i}={\bf\bar L}^{T}_{i}{\bf L}_{i}$ is the Hessian, and ${\bf R}_{ii}=\epsilon^{2}_{i}{\bf R}_{i}^{T}{\bf R}_{i}$ is the regularization term. Equation 10 can be solved using iterative inverse filtering leading to the so-called model-space least-squares migration/inversion method (Valenciano, 2008). We summarize linearized (Born) wave-equation data modeling and the least-squares Hessian derivation in Appendix ${A}$. Throughout this paper, our discussion of the Hessian refers to its target-oriented approximation defined in equation A-5.

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

$$\Delta \hat{{\bf m}}= \hat{{\bf m}}_{1} - \hat{{\bf m}}_{0}.$$ (A-11)

In this paper, we refer to the method of computing the time-lapse image using equation 11 as separate inversion.

Target-oriented joint inversion of incomplete time-lapse seismic data sets