Four-dimensional inversion of resistivity monitoring data through Lp norm minimizations

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A new 4-D inversion algorithm is developed so that any data misfits and model roughness in the space and time domains can be selectively minimized, in terms of either the L1 norm or the L2 norm. This study is motivated by the experience that a 4-D inversion adopting full L2 norm minimization may sometimes result in a model that is too smoothly varying with time. It is further encouraged by the realization that a particular criterion of either L1 or L2 norm cannot be universally optimal for accurately reconstructing the subsurface condition. In addition, we try to overcome difficulties of jointly choosing two optimal regularization parameters in space and time domains. To achieve this, we devise automatic determination methods, not only of the Lagrangian multipliers for the space-domain smoothness constraint, but also of the regularization parameter for penalizing the model roughness along the time axis. Both kinds of regularization parameters are actively updated at each iteration, according to variations in data misfit and model roughness. We conducted inversion experiments using synthetic and field monitoring data to test the proposed algorithms, and further to compare the performance of L1 and L2 norm minimizations. Both the synthetic and field data experiments demonstrated that the proposed automatic determination method produced ground changes that were more similar to the true changes than those of approaches using pre-determined parameter values. Inversion experiments showed that L1 norm minimization of the time-domain roughness could reduce the problem of overly smooth model changes when the subsurface changes are locally confined, but that the L2 norm approach would be more reasonable when the changes are expected to be widespread.