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

The least-squares method based on smooth constraint is one of the main methods 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.