三维大地电磁自适应L1范数正则化反演

常规三维大地电磁反演的正则项为L₂范数，它以电阻率空间分布函数处处光滑为模型期望，弱化了算法对电性突变界面的分辨能力.本文实现了正则项为L₁范数的三维大地电磁反演算法，让模型空间梯度向量更有机会取得稀疏解，在充分正则的迭代下能够有效突出模型真实电性界面.为避免L₁范数零点不可导带来的求解困难，使用迭代重加权最小二乘法把原问题转换为一系列L₂正则子问题迭代求解.每个子问题的极小方法使用改进型拟牛顿法，其下降方向既能保证正则项海塞矩阵的精确性，又能允许反演过程随迭代灵活更新正则因子.使用比值法或分段衰减法自适应更新正则因子以避免迭代早期陷入奇异解，从而提升反演收敛的稳定性并降低初始模型依赖度.合成的无噪数据反演表明L₁正则算法的模型恢复效果优于L₂正则；不同噪声水平的合成数据反演表明本文的算法具有稳健性；实测数据反演对比表明在合理的正则因子调整策略下，L₁正则反演结果的模型分辨率优于L₂正则.另外，不同初始模型的反演测试还表明，正则因子选取不合理时L₁正则可能造成方块状假异常.

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.