Three-dimensional magnetotelluric inversion based on adaptive L1-norm regularization

The conventional regularization term in three-dimensional magnetotelluric (MT) inversion takes the form of L2-norm, which requires the underground resistivity model be smooth enough everywhere, thus weakening the resolution of inversion algorithm for the resolving electrical interface when the true model is very complex. This work applied L1-norm regularization to solve three-dimensional MT inversion, which allowed more probability for sparse solution of model space gradient vectors, and can effectively highlight the true electrical interface of the resistivity model if every inversion iteration was sufficiently regularized. In order to avoid the non-differentiability on zero points of the L1-norm, we transformed the new inversion problem into a series of L2-norm sub-inversion problems using the iterative re-weighted least squares method. Every sub-inversion problem was solved by an improved quasi-Newton method, which retained the exact form of regularization term's Hessian matrix and meanwhile allowed us to flexibly update the regularization parameter on every inversion iteration. For the purpose of preventing singular solutions due to insufficient regularization in the early stage of inversion, we introduced gradient norm ratio strategy or piecewise attenuation strategy to adaptively update the regularization parameter, so that inversion convergence could be improved and initial model dependence could be reduced. Tests on synthetic data show that L1-norm regularization inversion recovers the electrical boundary better than the L2-nrom, and inversion tests under different artificial noise levels indicate that our algorithm is quite robust. We also compared the results of inversion results on real data in L1-norm and conventional L2-norm regularization, which demonstrate that L1-norm regularization could give better results than L2-norm if an appropriate regularization parameter updating strategy is used, otherwise, it might yield many cube-shaped false anomalies in the result due to weaker constraint of L1-norm regularization. © 2020, Science Press. All right reserved.