A high-frequency, asymptotic solution for transient head, appropriate for a medium containing smoothly varying heterogeneity, provides a basis for efficient inverse modeling. The semi analytic solution is trajectory based, akin to ray methods used in modeling wave propagation, and may be constructed by post processing the output of a numerical simulator. For high frequencies, the amplitude sensitivities, the relationship between changes in flow properties and changes in head ampliude, are dominated by the phase term which may be computed directly from the output of the simulator. Thus, transient head waveforms may be inverted with little more computation than is required to invert arrival times. An applicatino to synthetic head values indicates that the technique can be used to improve the fit to waveforms. An application to transient head data from the Migration experiment in Switzerland reveals a narrow, high conductivity pathway within a 0.5 m thick zone of fracturing.

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 (SVD) 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 corosswell 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.