The Modeling of 3-D DC Resistivity Based on Integral Equation of Potential

We perform a 2-D Fourier transform along the horizontal direction on the 3-D integration problem for the dc anomalous potential in the spatial domain, transforming it into a 1-D integration problem with independent solutions at different wavenumbers. The resulting 1-D integral equation in the wavenumber domain is expressed as a sum of element integrals. We then employ the shape function method using a quadratic shape function to characterize the scattering current density for each element and compute analytical expressions for the element integrals. Finally, we performed a 2-D inverse Fourier transform of the wavenumber-domain anomalous potential and electric field to obtain their values in the spatial domain, which are modified using iterative operators. This approach fully leverages the efficiency of the Fourier transform method and the high accuracy of the shape function integration method to achieve faster and more accurate solutions to the 3-D dc resistivity numerical simulation problem. Two examples demonstrate the accuracy and efficiency of our proposed algorithm. Numerical examples were used to analyze the relationship between the number of iterations and the anomaly conductivity difference during the convergence of the algorithm. It is shown that the algorithm converges faster for high-resistance anomalies and can be applied to simulate models with significantly different resistivities between the background medium and the anomaly.