基于不等式约束的最小二乘法三维电阻率反演及其算法优化  

3D electrical resistivity inversion with least-squares method based on inequality constraint and its computation efficiency optimization

作　　者：刘斌[1] 李术才[1] 李树忱[1] 聂利超[1] 钟世航[1] 李利平[1] 宋杰[1] 刘征宇[1] 

机构地区：[1]山东大学岩土与结构工程研究中心,济南250061

出　　处：《地球物理学报》2012年第1期260-268,共9页Chinese Journal of Geophysics

基　　金：国家973计划项目(2010CB732002);国家自然科学基金重点项目(51139004);国家自然科学基金青年项目(41102183);山东省自然科学基金项目(ZR2011EEQ013);中国博士后科学基金(20110491596);山东大学自主创新基金(2009JC005;2010GN059;2010TS038)资助

摘　　要：基于光滑约束的最小二乘法是三维电阻率反演的主要方法,但该方法在某些情况下存在着多解性较强的问题,且普遍耗时较长,严重制约了三维反演方法的推广与发展.为改善上述问题,将表征模型参数变化范围的不等式约束作为先验信息引入最小二乘线性反演方法中,有效地改善了反演结果的精度,降低了反演的多解性问题.为了解决耗时较长的问题,基于预条件共轭梯度(PCG)算法和Cholesky分解法的特点提出了一套优化三维电阻率反演计算效率的计算方案.在该方案中,Cholesky分解法被用来求解敏感度矩阵计算中的多个点源场的正演问题,Cholesky分解法只需对总体系数矩阵进行一次分解,然后对不同的右端向量进行回代即可.将预条件共轭梯度法引入到三维电阻率反演方程的求解中,将雅可比迭代中的对角阵作为预处理矩阵,其具有求逆方便、无需内存空间的特点,有效地加快了收敛速度.对合成数据以及实测数据的反演算例表明,借助不等式约束和反演效率优化方案,最小二乘反演方法可得到较为精确的反演结果,有效地提高了反演计算效率,具有良好的推广前景.The least-squares method based on smooth constraint is one of the mare metnoos for 3D electrical resistivity inversion. However, multiplicity of solution of this method is serious in some cases, and it is usually time-consuming and needs large computer memory, which restricts the application of 3D resistivity inversion in practical engineering. For solving above problems, inequality constraint as a priori information representing the parameter variation range is introduced into 3D resistivity inversion method, by which the accuracy and the solution multiplicity of inversion result is efficiently improved. An optimization scheme is presented for 3D resistivity inversion based on Preconditioned Conjugate Gradient （PCG） algorithm and Cholesky factorization algorithm. In this scheme, Cholesky factorization algorithm is used to solve the forward modeling problem of multiple point-source electrical field for sensitivity matrix. PCG algorithm is used for solving the 3D resistivity inversion equation. For improving the convergence speed efficiently, diagonal block matrix in Jacobi Iteration is used as the preconditioning matrix in PCG algorithm, the inversion of which is convenient to solve and doesn＇t occupy memory space. The inversion examples for synthetic data and measured data show that the least-squares method makes the accuracy and computation efficiency of 3D resistivity inversion greatly improved by means of inequality constraint and computation efficiency optimization scheme. Therefore, this inversion method has a bright prospect of application.

关 键 词：三维电阻率反演 最小二乘反演 不等式约束 预条件共轭梯度算法 Cholesky分解算法 电阻率层析成像 

分 类 号：P631[天文地球—地质矿产勘探]