Numerical modeling of 3D DC resistivity method in the mixed space-wavenumber domain

Forward modeling is the basis of inversion imaging and quantitative interpretation for DC resistivity exploration. Currently, a numerical model of the DC resistivity method must be finely divided to obtain a highly accurate solution under complex conditions, resulting in a long calculation time and large storage. Therefore, we propose a 3D numerical simulation method in a mixed space-wavenumber domain to overcome this challenge. The partial differential equation about abnormal potential is transformed into many independent ordinary differential equations with different wavenumbers using a 2D Fourier transform along the x axis and y axis direction. In this way, a large-scale 3D numerical simulation problem is decomposed into several 1D numerical simulation problems, which significantly reduces the computational and storage requirements. In addition, these ordinary 1D differential equations with different wavenumbers are independent of each other and high parallelelism of the algorithm. They are solved using a finite-element algorithm combined with a chasing method, and the obtained solution is modified using a contraction operator. In this method, the vertical direction is reserved as the spatial domain, then grid size can be determined flexibly based on the underground current density distribution, which considers the solution accuracy and calculation efficiency. In addition, for the first time, we use the contraction operator in the integral equation method to iterate the algorithm. The algorithm takes advantage of the high efficiency of the standard Fourier transform and chasing method, as well as the fast convergence of the contraction operator. We verified the accuracy of the algorithm and the convergence of the contraction operator. Compared with a volume integral method and goal-oriented adaptive finite-element method, the proposed algorithm has lower memory requirements and high computational efficiency, making it suitable for calculating a model with large-scale nodes. Moreover, different examples are used to verify the high adaptability and parallelism of the proposed algorithm. The findings show that the 3D numerical simulation method of DC resistivity method in a mixed space-wavenumber domain is highly efficient, precise, and parallel.