Wave-equation inversion of time-lapse seismic data sets

Joint Inversion for Image Differences (JID)

We can formulate baseline and monitor data modeling as follows:

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

which can be divided into the following two parts:

$${\bf L}_{0} {\bf m}_{0} = {\bf d}_{0},$$ (A-2)

$${\bf L}_{1} {\bf m}_{0} +{\bf L}_{1} {\bf\Delta m} = {\bf d}_{1},$$ (A-3)

where the time-lapse reflectivity image ${\bf\Delta m}$ is given by

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

Note that Equation A-1 assumes that both ${\bf m}_{0}$ and ${\bf m}_{1}$ are collocated. This means that there is no physical movement of the reflector between the baseline and the monitor images. In addition, equation A-1 assumes that there are no overburden velocity changes. If stress changes cause any physical movement of a point from baseline position $(x_{0},y_{0},z_{0})$ in ${\bf m}_{0}$ to monitor position $(x_{1},y_{1},z_{1})$ in ${\bf m}_{1}$ , we can update equation A-3 such that the point in ${\bf m}_{0}$ is repositioned at $(x_{1},y_{1},z_{1})$ . The updated modeling equation for the monitor data then becomes

$${\bf L}_{1} {\bf S}^{^{m-}}{\bf m}_{0} +({\bf L}_{1} {\bf m}_{1}-... ... L}_{1} {\bf S}^{^{m-}}{\bf m}_{0} +{\bf L}_{1} {\bf\Delta m}^{m}= {\bf d}_{1},$$ (A-5)

where ${\bf S}^{m-}$ is an orthogonal warping operator that aligns ${\bf m}_{0}$ to ${\bf m}_{1}$ , and

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

is the time-lapse image estimated at the monitor position $(x_{1},y_{1},z_{1})$ . And the combined modeling equation becomes

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

However, note that equation A-7 requires that we know the true reflector position in the monitor which we may obtain from a geomechanical model. Furthermore, any regularization on the time-lapse image must be applied at the monitor position, or by first repositioning the time-lapse image to the baseline position as follows:

$${\bf\Delta m}={\bf\Delta m}^{b}={\bf S}^{^{m+}}{\bf\Delta m}^{m},$$ (A-8)

where ${\bf\Delta m}^{b}$ is the time-lapse image at the baseline position and ${\bf S}^{^{m+}}=({\bf S}^{^{m-}})^{-1}$ is an operator that repositions events from the monitor position to the baseline position.

Assuming we migrate the monitor data with the true monitor velocity, we arrive at the image-space inversion problem by minimizing the quadratic-norm

$$\begin{array}{ccc} \left \vert\left\vert \left [ \begin{array... ...array} \right ] \right \vert \right \vert \approx 0 \end{array}$$ (A-9)

for which the solutions ${\bf\hat m}_{0}$ and ${\bf\Delta \hat m}^{m}$ satisfy the solution

$$\begin{array}{ccc} \left [ \begin{array}{cc} {\bf L}_{0}^{T}{... ... {\bf d}_{0} \\ {\bf d}_{1}\\ \end{array} \right ] \end{array},$$ (A-10)

or simply

$$\begin{array}{ccc} \left [ \begin{array}{cc} {\bf H}_{0}+{\bf... ...de m}_{1}\\ {\bf\tilde m}_{1} \end{array} \right ] \end{array}.$$ (A-11)

Note that the time-lapse image we obtain is ${\bf\Delta \hat m}^{m}$ at the monitor position and not ${\bf\Delta \hat m}={\bf\Delta \hat m}^{b}$ at the baseline position. Although what is most interesting is ${\bf\Delta \hat m}^{m}$ , as shown later in this section, we may choose to re-write the formulation as a function of ${\bf\Delta \hat m}^{b}$ .

Assuming we migrate the monitor data with the wrong (e.g. baseline) velocity, then equation A-10 becomes

$$\begin{array}{ccc} \left [ \begin{array}{cc} {\bf L}_{0}^{T}{... ...{\bf d}_{0} \\ {\bf d}_{1} \\ \end{array} \right ] \end{array},$$ (A-12)

where ${\bf\check{L}}_{1}^{T}$ , the migration operator with the monitor geometry but with baseline velocity migrates the monitor data to apparent position $(x_{1}',y_{1}',z_{1}')$ , and ${\bf S}^{^{\alpha+}}$ repositions the migrated data from $(x_{1}',y_{1}',z_{1}')$ to $(x_{0},y_{0},z_{0})$ . However, because the operator ${\bf L}_{1}$ is a function of the true monitor velocity, if the true monitor velocity is known, we should solve equation A-11 instead of equation A-12. Note that in the case where the monitor migration velocity is the correct one, equation A-12 becomes equation A-11. If we have neither the true monitor velocity nor a geomechanical model, we may modify the Hessian in equation A-12 using the apparent displacements between $(x_{1}',y_{1}',z_{1}')$ and $(x_{0},y_{0},z_{0})$ so we can approximate equation A-12 as

$$\begin{array}{ccc} \left [ \begin{array}{cc} {\bf L}_{0}^{T}{... ...{\bf d}_{0} \\ {\bf d}_{1} \\ \end{array} \right ] \end{array},$$ (A-13)

Then equation A-11 becomes

$$\begin{array}{ccc} \left [ \begin{array}{cc} {\bf H}_{0}+{\bf... ...}_{1}\\ {\bf\tilde m}^{a}_{1} \end{array} \right ] \end{array},$$ (A-14)

where ${\bf\check H}_{1}$ is the modified Hessian in which we account for the mis-positioning due to compaction and velocity change. To account for such mis-positioning, we compute the updated Hessian using perturbed Green's functions:

$${\check H} \left ({\bf y_{T}}, {\bf y_{T+a_{x}}} \right ) = \sum_... ...}, y_{T},\omega}) \bar G_{\alpha}({\bf y_{s},y_{T+a_y},\omega}) \hspace{-0.8cm}$$

$$\sum_{\bf y_{r}} G_{\alpha} ({\bf y_{T}, y_{r}, \omega}) \bar G_{\alpha} ({\bf y_{T+a_y}, y_{r}, \omega}) ,$$ (A-15)

where ${\bf y}$ denotes an apparent point in the monitor image that corresponds to baseline point ${\bf x}$ . The modified Green's function $G_{\alpha}$ is given by

$$G_{\alpha}= G({\bf x})\exp^{-i\omega \Delta t_{\alpha}}\approx G({\bf x})\exp^{-i\omega\frac{\left\vert \bf x_{\alpha}\right\vert}{v_{o}}},$$ (A-16)

where ${\Delta t_{\alpha}}$ is the time-delay corresponding to the absolute apparent displacement ${\left\vert \bf x_{\alpha} \right\vert}$ and $v_{o}$ is the baseline velocity.

Instead of inverting for the time-lapse image at the monitor position, another approach is to directly invert for ${\bf\Delta \hat m}^{b}={\bf\Delta \hat m}$ at the baseline position by making the substitution

$${\bf\Delta m}^{m}={\bf S}^{^{m-}}{\bf\Delta m}$$ (A-17)

into equation A-9 to obtain

$$\begin{array}{ccc} <tex2html_comment_mark>32 \left [ \begin{a... ...} {\bf d}_{0} \\ {\bf d}_{1}\\ \end{array} \right ] \end{array}$$ (A-18)

which leads to the image-space problem

$$\begin{array}{ccc} \left [ \begin{array}{cc} {\bf H}_{0}+{\bf... ...}_{1}\\ {\bf\tilde m}^{b}_{1} \end{array} \right ] \end{array}.$$ (A-19)

where ${\bf\tilde m}^{b}_{1}$ , the migrated monitor image repositioned to the baseline position $(x_{0},y_{0},z_{0})$ , is defined as

$${\bf\tilde m}^{b}_{1}={\bf S}^{^{m+}}{\bf\tilde m}_{1}.$$ (A-20)

If we migrate the monitor data with the baseline velocity, equation A-19 becomes

$$\begin{array}{ccc} \left [ \begin{array}{cc} {\bf H}_{0}+{\bf... ...}_{1}\\ {\bf\tilde m}^{b}_{1} \end{array} \right ] \end{array},$$ (A-21)

where,

$${\bf S}^{^{\alpha+}}{\bf\tilde m}^{a}_{1}\approx{\bf\tilde m}^{b}_{1}={\bf S}^{^{m+}}{\bf\tilde m}_{1}.$$ (A-22)

However, provided the velocity change is isotropic, compaction effects are small, differences in kinematics are small, and the velocity change is small, we can make the following approximation:

$${\bf L}_{1} {\bf S}^{^{m-}}{\bf m}^{b}_{1} \approx {\bf U}^{^{m-}}{\bf L}^{b}_{1} {\bf m}^{b}_{1} ={\bf d}_{1},$$ (A-23)

where the operator ${\bf L}_{1}$ is a function of both the monitor velocity and geometry, whereas ${\bf L}^{b}_{1}$ is a function of the baseline velocity but the monitor geometry. ${\bf U}^{^{m-}}$ is an orthogonal operator that translates a data due to a reflectivity spike at baseline position $(x_{0},y_{0},z_{0})$ and baseline background velocity $v_{0}$ , to data due to a spike at $(x_{1},y_{1},z_{1})$ and monitor background velocity $v_{1}$ . Note that in equation A-23, we have made the following approximation

$${\bf U}^{^{m-}}\approx {\bf L}_{1} {\bf S}^{^{m-}}\left[ \left[({\bf L}^{b}_{1})^{T}{\bf L}^{b}_{1}\right]^{-1}({\bf L}^{b}_{1})^{T}\right].$$ (A-24)

Provided equation A-23 holds, we can write

$$({\bf L}^{b}_{1})^{T}({\bf U}^{^{m-}})^{T}{\bf U}^{^{m-}}{\bf L}^... ...} {\bf m}^{b}_{1} \approx({\bf L}^{b}_{1})^{T}({\bf U}^{^{m-}})^{T}{\bf d}_{1},$$ (A-25)

where,

$$\begin{array}{cc} ({\bf U}^{^{m-}})^{T}{\bf U}^{^{m-}}=({\bf ... ...\bf U}^{^{m-}})^{T}={\bf S}^{^{m+}}{\bf L}^{T}_{1}. \end{array}$$ (A-26)

Therefore, we can write

$$({\bf L}^{b}_{1})^{T}{\bf L}^{b}_{1} {\bf m}^{b}_{1} \approx {\bf... ...\approx {\bf S}^{^{m+}}{\bf L}^{T}_{1}{\bf d}_{1}\approx {\bf\tilde m}^{b}_{1},$$ (A-27)

where, $({\bf L}^{b}_{1})^{T}{\bf L}^{b}_{1} ={\bf H}^{b}_{1}$ is the Hessian computed using the baseline velocity but with the monitor geometry. Making these substitutions into equation A-19, we have

$$\begin{array}{ccc} \left [ \begin{array}{cc} {\bf H}_{0}+{\bf... ...}_{1}\\ {\bf\tilde m}^{b}_{1} \end{array} \right ]. \end{array}$$ (A-28)

An important advantage of the formulation in equation A-28 is that it allows us to readily regularize the time-lapse image. However, it may be desirable to invert directly for the individual seismic images, as shown in the following sections.

Wave-equation inversion of time-lapse seismic data sets