Fictitious Point Technique Based on Finite-Difference Method for 2.5D Direct-Current Resistivity Forward Problem

Abstract

With the widespread application of the direct-current resistivity method, searching for accurate and fast-forward algorithms has become the focus of research for geophysicists and engineers. Three-dimensional forward modeling can be the best way to identify geo-electrical anomalies but are hampered by computational limitations because of the large amount of data. A practical compromise, or even alternative, is represented by 2.5D modeling characterized using a 3D source in a 2D medium. Thus, we develop a 2.5D direct-current resistivity forward modeling algorithm. The algorithm incorporates the finite-difference approximation and fictitious point technique that can improve the efficiency and accuracy of numerical simulation. Firstly, from the boundary value problem of the electric potential generated by the point source, the discrete expressions of the governing equation are derived from the finite-difference approach. The numerical solutions of the discrete electric potential are calculated after the approximate treatment of the boundary conditions with a finite-difference method based on a fictitious point scheme. Secondly, through the simulation of a homogeneous half-space model and a one-dimensional model, and compared with the analytical results, the correctness and stability of the finite-difference forward algorithm are verified. Lastly, through the numerical simulation for a two-dimensional model, 2.5D direct-current sounding responses are summarized, which can provide a qualitative interpretation of field data.

A Legendre Spectral-Element Method to Incorporate Topography for 2.5D Direct-Current-Resistivity Forward Modeling

“…Numerical modeling approaches, such as the finite-difference method (FDM) [12][13][14][15] and finite-element method (FEM) [16][17][18], have been applied for high-dimensional directcurrent-resistivity forward modeling and inversion. The FDM utilizes difference operators to approximate the first or second partial derivatives in the partial differential equation (PDE), but its effectiveness is limited to a relatively flat topography.…”