Seismic reservoir monitoring with simultaneous sources

Regularized joint inversion

For an arbitrary survey $i$ , we can simplify the modeling equation into the form

$${\bf\widetilde{d}}_{i}={\bf B}_{i}{\bf L}_{i}{\bf m}_{i} = {\bf\widetilde{L}}_{i}{\bf m}_{i},$$ (4)

where ${\bf\widetilde{d}}$ is the recorded data, ${\bf B}$ is the encoding operator, ${\bf L}$ is the modeling operator, ${\bf m}$ is the earth reflectivity, and ${\bf\widetilde{L}}={\bf B}{\bf L}$ . The migrated image, computed by applying the adjoint operator ${\bf\widetilde{L}}^{T}$ to ${\bf\widetilde{d}}$ , will contain cross-term artifacts generated by cross-correlation between incongruous source and receiver wavefields (Romero et al., 2000; Tang and Biondi, 2009). In addition, because of the associated geometry and relative shot-time non-repeatability, different surveys have unique cross-term artifacts. To attenuate these artifacts, for $N$ surveys, we minimize a joint (global) cost function $S$ given by

$$S({\bf m}_{0},..,{\bf m}_{N})= \sum_{i=0}^{N}\left \vert\left\vert {\bf\widetilde L}_i{\bf m}_i-... ...\left \vert\left\vert \epsilon_i{\bf R}_i{\bf m}_i\right \vert \right \vert ^2+$$

$$\sum_{i=1}^{N}\left \vert\left\vert \zeta_{i}{\bf\Lambda}_{i}\lef... ...a_{i}{\bf\Gamma}_{i}\left(\Delta m_{i} \right) \right \vert \right \vert _{hb},$$ (5)

where the parameters $\epsilon_{i}$ and $\zeta_{i}$ determine the strengths of the spatial and temporal regularization operators, ${\bf R}_{i}$ and ${\bf\Lambda}_{i}$ respectively. Because only a small region in the model space contain desired in time-lapse signal, a sparseness requirement is desirable. Parameter $\beta_{i}$ determines the strength of the sparseness operator ${\bf\Gamma}_{i}$ . Related formulations have been applied to other time-lapse imaging problems (Ajo-Franklin et al., 2005). We compute the time-lapse image as the difference between the migrated or inverted image at time $t$ and that at time 0 . Because several shots are encoded and directly imaged, the computational cost of this approach is considerably reduced compared to non-encoded data sets.

In this paper, the spatial regularization operator is a system of non-stationary dip-filters, whereas the temporal regularization operator is a gradient between surveys. We compute dips using the plane-wave destruction method (Fomel, 2002), and we compute dip-filters using factorized directional Laplacians (Hale, 2007). To ensure stable transitions at sharp boundaries, the filter corresponding to any image point is scaled according to a dip-contrast-dependent variance. We estimate the spatial and temporal regularization parameters by scaling the maximum amplitude in each data set. Finally, we minimize the objective function using an iterative hybrid conjugate direction algorithm (Li et al., 2010) which enforces desired spareness on the time-lapse images.

Seismic reservoir monitoring with simultaneous sources