混合范数下的最优化反演方法

Optimized inversion method based on mixed norm

在求解地球物理反问题时,通常根据最小二乘准则构造目标函数进行反演,并在实践中得到了广泛的应用.为进一步增强反演的稳健性及减少多解性,不损失反演结果的分辨率,本文提出了混合范数下的最优化反演方法,它根据数据和模型可能服从不同的概率分布,对数据空间和模型空间采用不同的范数来构造目标函数.在给出目标函数的基础上,导出了混合范数下的线性反演方程.由于该线性反演方程的复杂性,我们采用混合范数下迭代再加权共轭梯度法进行求解.最后,通过对模拟的电阻率数据进行反演,验证了本文计算方法是可行的.

When solving the geophysical inverse problems, we usually construct the objective function for inversion according to the least-squares standard, and it is widely used in practice. In order to enhance the robustness and reduce non-uniqueness without loss of the resolution of inversion results, this paper puts forward the optimized inversion method using mixed norm. Because data and model may obey different statistical distributions, different norms are adopted for data space and model space to construct the objective function, then the linear inversion equation based on mixed norm is deduced. Then we apply the iteratively reweighted conjugate gradient method based on mixed norm to solve the linear inversion equation because of its complexity. At last, we verified that the algorithm of this paper is workable through inversing the resistivity data.