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Claerbout (2004); Tarantola (1987) discuss the use of geophysical inversion as an
imaging tool. In recent applications, inverted seismic images are
computed by weighting
the migration result with the inverse of the Hessian matrix. The
associated large computational cost and complexity makes the explicit
computation of
the Hessian matrix and its inverse impracticable. Valenciano et al. (2006)
demonstrate that by taking the sparsity of the Hessian into account,
the inverse of the Hessian matrix may be computed cheaply and applied
in a target-oriented manner. This approach appears to yield better
results than simple
migration in subsalt reservoirs. For the time-lapse problem, one
approach would be to to compute the time-lapse image as a difference
between
inverse images computed as described above. Another approach would be
to solve for the time-lapse image through inversion rather a
difference between images.
Given a linear modeling operator , the synthetic data **d**
is computed using , where **m** is a
reflectivity model.
Two different surveys (say a baseline and monitor) may be represented as follows:

| |
(1) |

where and are the reflectivity models at the time we acquire the datasets, ( and ) respectively.
Taking and to be the modeling operators
for two different surveys, the quadratic cost functions are defined as
| |
(2) |

The least-squares solutions to the problems are given as
| |
(3) |

where and are the migrated
images, and are the
least-squares inverse images,
and are the migration operators, while
and
are the Hessian matrices.
In the first approach, the inverse time-lapse image () is given by

| |
(4) |

In the second approach, we express the modeling of operation for the two surveys as follows:

| |
(5) |

where . In matrix form, we can write
| |
(6) |

least-squares solution to equation 10 is given as
| |
(7) |

| |
(8) |

This, may be re-arranged as follows:

| |
(9) |

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Stanford Exploration Project

5/6/2007