起伏地形下大地电磁L-BFGS三维反演方法

为推进大地电磁三维反演的实用化，本文实现了基于L-BFGS算法的带地形大地电磁三维反演.首先推导了大地电磁法三维反演的Tikhonov正则化目标函数以及Hessian矩阵逆矩阵近似表达式和计算方法，然后设计了一种既能保证空气电阻率固定不变又能保证模型平滑约束的协方差矩阵统一表达式，解决带地形反演问题.在反演算法中采用正则化因子冷却法以及基于Wolf条件的步长搜索策略，提升了反演的稳定性.利用开发的算法对多个带地形地电模型（山峰地形下的单个异常模型、峰-谷地形下的棋盘模型）的合成数据进行了三维反演，并与已有大地电磁三维反演程序（ModEM）进行对比，验证了本文开发的三维反演算法的正确性和可靠性.最后，利用该算法反演了华南某山区大地电磁实测数据，得到该区三维电性结构，揭示了研究区以高阻介质为基底，中间以低阻不整合面和相对低阻介质连续分布，浅部覆盖高阻介质的电性结构特征，进一步验证了本文算法的实用性.

Three-dimensional magnetotelluric inversion under topographic relief based on the limited-memory quasi-Newton algorithm (L-BFGS)

We present an algorithm of three-dimensional (3D) magnetotelluric (MT) inversion under topographic relief using a limited-memory quasi-Newton optimization based on Broyden-Fletcher-Goldfarb-Shanno (BFGS) formula in this study. Firstly, we establish the approximate expression and calculation of the inverse of Hessian matrix for 3D MT inversion based on iterative minimization of a classical Tikhonov regularized penalty function. A covariance matrix is designed to keep the resistivity of air unchanged and to impose the smoothness of the model while implementing the inversion with topographic relief. In addition, the cooling method for the regularization factor and a strategy to search for the step length based on the Wolf condition are adopted in this algorithm to improve the stability of the inversion. Secondly, we make a 3D inversion of synthetic data which are generated utilizing a series of conductivity structure models with topographic relief and compare the results with that of the well-known 3D MT inversion program (ModEM) to verify the correctness and reliability of the algorithm in this paper. Finally, we use this 3D algorithm to invert observed MT data with topographic relief from a mountainous survey area in South China, yielding an electrical structure model. It can be roughly divided into three layers, i.e. a high-resistivity basement, a conductive layer and the unconformity surface in the middle and the sedimentary cover with high-resistivity. The applicability of this algorithm is further verified.