Invariance, groups, and non-uniqueness; the discrete case

Lie group methods provide a valuable tool for examininginvariance and non-uniqueness associated with geophysical inverseproblems. The techniques are particularly well suited for the study ofnon-linear inverse problems. Using the infinitesimal generators of thegroup it is possible to move within the null space in an iterativefashion. The key computational step in determining the symmetry groupsassociated with an inverse problem is the singular value decomposition(SVD) of a sparse matrix. I apply the methodology to the eikonal equationand examine the possible solutions associated with a crosswelltomographic experiment. Results from a synthetic test indicate that it ispossible to vary the velocity model significantly and still fit thereference arrival times. the approach is also applied to data fromcorosswell surveys conducted before and after a CO2 injection at the LostHills field in California. The results highlight the fact that a faultcross-cutting the region between the wells may act as a conduit for theflow of water and CO2.

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