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

Lie group methods provide a valuable tool for examining invariance and non-uniqueness associated with geophysical inverse problems. The techniques are particularly well suited for the study of non-linear inverse problems. Using the infinitesimal generators of the group it is possible to move within the null space in an iterative fashion. The key computational step in determining the symmetry groups associated with an inverse problem is the singular value decomposition of a sparse matrix. I apply the methodology to the eikonal equation and examine the possible solutions associated with a crosswell tomographic experiment. Results from a synthetic test indicate that it is possible to vary the velocity model significantly and still fit the reference arrival times. The approach is also applied to data from crosswell surveys conducted before and after a CO2 injection at the Lost Hills field in California. The results highlight the fact that a fault cross-cutting the region between the wells may act as a conduit for the flow of water and CO2.