基于乘积型目标函数的电阻率法三维反演

为了减小地球物理反演的多解性，通常采用累加型目标函数，即数据拟合差项加上正则化项。基于累加型目标函数的反演存在如何优选正则化因子的问题，这个过程通常需要做大量的反演计算。本研究系统分析和实现了基于乘积型目标函数(数据拟合差项和正则化项相乘)的电阻率法三维反演，乘积型目标函数的反演不存在优选正则化因子的问题。三维正演采用非结构化有限单元法，三维反演采用有限内存拟牛顿方法。使用理论模型合成数据进行了三维反演，检验了基于乘积型目标函数的电阻率法三维反演的可行性与有效性。反演算例结果表明：基于乘积型目标函数的反演方法能够可靠恢复异常体的电阻率值、形态和位置。

Three-dimensional Inversion for Resistivity Method Based on Multiplicative Regularization

To reduce the multiplicity of geophysical inversion solution, additive objective functions are usually used, i.e., the data term and regularization term. Inversion based on additive objective function has the problem of optimizing the regularization factor, which usually requires much inversion calculations. This study systematically analyzed and implemented the 3D resistivity inversion method, based on multiplicative objective function (i.e., multiplication of the data term and regularization term). The problem of optimizing regularization factor is absent in the inversion of multiplicative objective function. We used the unstructured finite element method for resolving the 3D forward problem, and the L-BFGS (Limited-memory Broyden-Flecher-Goldfarb-Shanno) method for 3D inversion. The feasibility and effectiveness of our approach is verified by using the synthetic data of the theoretical model. The results show that the inversion method (based on multiplicative objective function) can reliably recover the resistivity value, shape and position of anomalies.