Joint inversion of simultaneous source time-lapse seismic data sets

Linear least-squares migration/inversion

We re-write the linear modeling operation in equation (1) in matrix-vector form as follows:

$${\bf d}={\bf L}{\bf m},$$ (4)

where ${\bf L}$ is the modeling operator and ${\bf m}$ is the earth reflectivity. The encoding (or blending) operation in equation (2) is then defined as:

$${\bf\widetilde{d}}={\bf B{L}}{\bf m} = {\bf\widetilde{L}}{\bf m},$$ (5)

where ${\bf\widetilde{d}}$ is the encoded data, ${\bf B}$ is the encoding (or blending) operator, and ${\bf\widetilde{L}}$ is the combined modeling and encoding operator.

Given two surveys (baseline and monitor), acquired over an evolving earth model at times ${\bf t=0}$ and ${\bf t=1}$ respectively, we can write

$$\begin{array}{cc} {\bf\widetilde{d}}_{0}={\bf\widetilde{L}}_{... ...idetilde{d}}_{1}={\bf\widetilde{L}}_{1}{\bf m}_{1}, \end{array}$$ (6)

where ${\bf m}_{0}$ and ${\bf m}_{1}$ are the baseline and monitor reflectivities, and ${\bf\widetilde d}_{0}$ and ${\bf\widetilde d}_{1}$ are the encoded seismic data sets. Note that the modeling operators ${\bf\widetilde L}_{0}$ and ${\bf\widetilde L}_{1}$ in equation (6) can define both different acquisition geometries and different relative shot-timings.

By applying the adjoint operators to the data sets, we obtain the migrated images:

$$\begin{array}{cc} {\bf\acute m}_{0}={\bf\widetilde{L}}^{^{^{*... ...idetilde{L}}^{^{^{*}}}_{1}{\bf\widetilde d}_{1}, \\ \end{array}$$ (7)

where ${\bf\acute m}_{0}$ and ${\bf\acute m}_{1}$ are the migrated baseline and monitor images respectively, and the symbol $^{^{^{*}}}$ denotes the conjugate transpose of the modeling operators. The raw time-lapse seismic image $\Delta \tilde{{\bf m}}$ is the difference between the migrated images:

$$\Delta \acute{{\bf m}}= \acute{{\bf m}}_{1} - \acute{{\bf m}}_{0}.$$ (8)

Because of differences in relative shot-timings, cross-term artifacts (Romero et al., 2000; Tang and Biondi, 2009) will be different for each migrated data set. Conventional equalization methods (Rickett and Lumley, 2001; Calvert, 2005) will be inadequate to remove these artifacts.

The quadratic cost functions for equation (6) are

$$\begin{array}{cc} S({\bf m_0})=\Vert {\bf\widetilde L}_{0}{\b... ... L}_{1}{\bf m}_{1} - {\bf\widetilde d}_{1} \Vert^2, \end{array}$$ (9)

which when minimized gives the inverted baseline $\hat{\bf m}_{0}$ and monitor $\hat{{\bf m}}_{1}$ images:

$$\begin{array}{cc} \hat{{\bf m}}_{0}=({\bf\widetilde L^{^{^{*}... ...1})^{-1}{\bf\widetilde L^{^{^{*}}}}_{1}{\bf d}_{1}, \end{array}$$ (10)

This is the so-called data-space least-squares migration/inversion method.

Joint inversion of simultaneous source time-lapse seismic data sets