Discussion

Although it is not feasible to explicitly compute the inverse matrix in equation 13, an iterative least squares solution of equation 12 is expected to yield reasonably good results for $\hat{{\bf m}}_{0}$ and $\Delta \hat{{\bf m}}$. In the 2D case considered, for a given frequency ${\bf \omega}$ and an image-point ${\it i}(x,z)$, the measured data ${\bf d}\left ({\it i}_{s}, {\it i}_{r}\right)$ in equation 1 is given by

$${\bf d} \left ({\it i}_{s}, {\it i}_{r}; \omega \right )= {\... ...{i_{xz}}}, {\it i}_{r}; \omega \right )}{\bf m_{\it {i_{xz}}}},$$ (10)

where ${\bf G \left ({\it {i_{xz}}}, {\it i}_{s}; \omega \right )}$ and ${\bf G \left ({\it {i_{xz}}}, {\it i}_{r}; \omega \right )}$ are the Green's functions from the shot and receiver positions (${\it i}_{s} \left ({\it x}_{s},0 \right)$ and ${\it i}_{s} \left ({\it x}_{r},0 \right)$) respectively.

The Hessian sub-matrices (${\bf H}_{0}$ and ${\bf H}_{1}$), which are the second derivatives of the cost functions (equations 3 and 4) with respect to the model parameters ${\bf m_{\it {i_{xz}}}}$ and ${\bf m_{\it {j_{xz}}}}$ (where ${\it j_{xz}}$ is some other point in the model space), may be computed as follows:  

$${\bf H} \left ({\it {i_{xz}}}, {\it {j_{xz}}} \right )=\sum_... ...}{\bf G \left ({\it {j_{xz}}}, {\it {i_{r}}}; \omega \right )}.$$ (11)

Using a target-oriented approach and limiting the computation to near-diagonal elements Valenciano et al. (2006), equation 15 is reduced to  

$${\bf H} \left ({\it {i_{T}}}, {\it {i_{T}}} + {\it {a_{i}}} ... ...{\it {i_{T}}} + {\it {a_{i}}}, {\it {i_{r}}}; \omega \right )},$$ (12)

where ${\it {a_{i}}}$ is the offset from the target image-point ${\it {i_{T}}}$.

Computation of the target-oriented wave-equation Hessian is discussed in detail by Valenciano and Biondi (2004). Hence, it is possible to compute the matrix of Hessian terms in equation 12 for specific targets of interest (e.g. regions around a sub-salt reservoir) and therefore compute the least-squares time-lapse image, $\Delta \hat{{\bf m}}$. Since the geometry (and other unwanted) information are contained in the Hessian terms, it should be possible to such effects from the time-lapse image by solving the inverse problem in equation 13 (or practically by solving equation 12 in a least-squares sense). The inversion technique is itself limited by the amount of good quality data from the sub-salt reflectors, but we expect that it provides better results than presently obtainable with migration. We intend to compare results from the inversion schemes with those from standard cross-equalization and to determine whether our techniques could make seismic monitoring of sub-salt reservoirs a reality.