With the help of the horizontal 2D Fourier transform, 3D convolutions can be transformed into vertical 1D integrations of different wavenumbers. It has already been proven that the Fourier ...

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.

This study presents an innovative 3D numerical simulation approach tailored for the direct current resistivity method under undulating terrain conditions. The method utilizes 2D Fourier transform technology to convert complex 3D problems into 1D problems in the space-wavenumber mixed domain, thereby simplifying the computational process. We assume a layered background medium and utilize the weighted residual method to solve this simplified 1D problem. Additionally, to precisely handle terrain boundary nodes, we introduce an isogeometric gradient method applied in the logarithmic domain of conductivity. By constructing 3D trapezoidal ridge and valley models, we have validated the effectiveness and accuracy of the proposed algorithm. Further experiments involving a spherical low-resistivity body and a prismatic high-resistivity body beneath ridge terrain demonstrate the significant impact of undulating terrain on the observed results. This highlights the importance of considering terrain undulations in field observations and emphasizes the necessity of making appropriate corrections when interpreting data to prevent erroneous interpretations or significant deviations in results.

The marine controlled-source electromagnetic (MCSEM) method is widely used in offshore oil and gas exploration. In this letter, we present a space-wavenumber domain integral equation (IE) algorithm for simulating marine CSEM responses in 3-D arbitrarily anisotropic media. The algorithm employs a space-wavenumber domain method and a contraction operator in 3-D IEs. The space-wavenumber domain method accelerates simulation speed and reduces memory requirements, while the contraction operator ensures solution convergence. Synthetic models are used to verify the accuracy and efficiency of the newly proposed method by comparing it with an edge-based finite element (FE) method, a space-wavenumber domain differential equation method, and a biconjugate gradient FFT (BCGS-FFT) method. The numerical results of the disk model demonstrate that the marine electromagnetic (EM) fields are significantly influenced by the anisotropic conductivity tensor.

In the realm of direct current (DC) resistivity exploration, traditional three-dimensional (3D) point-source forward calculations are often hindered by high computational complexity and substantial resource requirements. While Fourier transformation can reduce this complexity to a 2.5D model, it still does not match the efficiency of a two-dimensional (2D) line-source model. To address these challenges, we introduce an innovative algorithm for 2D line-source DC resistivity numerical simulation. Our approach leverages Fourier transformation to convert the 2D problem into a one-dimensional (1D) problem in the wavenumber domain, subsequently employing the finite difference method for resolution. We further incorporate the tight operator iterative technique to significantly enhance computational efficiency and convergence velocity. A suite of numerical experiments confirms that our algorithm not only maintains the accuracy of simulation results but also exhibits superior convergence characteristics. Additionally, by comparing the anomaly responses of point-source and line-source models in both high-resistivity and low-resistivity geophysical scenarios, we demonstrate the line-source model's heightened sensitivity to anomalies. This study not only equips the DC electrical exploration field with a efficient modeling tool but also contributes novel theoretical insights for the interpretation of geophysical data.