A new four-dimensional (4D) inversion algorithm is developed so that any of data misfits and model roughness in the space and time domains can be selectively minimized either in terms of L1 norm or L2 norm. This study is motivated by the realization that a particular criterion of either L1 or L2 norm cannot be universally optimal for accurately reconstructing the subsurface condition. To overcome difficulties of jointly choosing two optimal regularization parameters for the inverse model constraints in the space and time domains, we devise automatic determination methods of two different kinds of the Lagrangian multipliers. We conducted inversion experiments using synthetic and field monitoring data to test the proposed algorithms. Both of the synthetic and field data experiments proved that the automatic determination method developed in this study is effective for calculating the ground changes that are closer to the ground truth. Inversion experiments showed that L1 norm minimization of the time-domain roughness could cure the problem of unnecessary smooth model changes when the subsurface changes are locally confined, but at the same time, the L2 norm approach would be more reasonable when the changes are expected widespread.


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