Holographic approach for 3D DC resistivity modeling and CPU-GPU Parallel Architecture

The conventional numerical simulation methods have some problems, such as large computation, approximate boundary conditions and difficulty in accurately simulating the physical information characterized by 3D partial differential equations, we have developed an efficient and highly accurate holographic numerical simulation method. The core is to accurately obtain the space-wavenumber spectrum of the anomalous potential based on a realistic DC model using an appropriate approach, and then transform all wavenumber spectral information back into the space domain to obtain the true distribution of the electric field. Specifically, the approach converts the 3D partial differential equation governing the anomalous potential in the space domain into a set of 1D ordinary differential equations at different wavenumbers through a 2D Fourier transform in the horizontal direction. A holographic Fourier transform is applied in the horizontal direction to ensure the completeness of both space and wavenumber domain information. In the vertical direction, plane wave decomposition is introduced to mitigate the boundary effects caused by the upper and lower boundaries. Therefore, this method is called holographic approach. Additionally, the grid size can be freely adjusted in both the horizontal and vertical directions. Leveraging the high parallelism of the algorithm, thereby implementing a CPU-GPU parallel architecture. An abnormal sphere model is designed to analyze the wavenumber spectral distribution characteristics of the anomalous field and to validate the logarithmic sampling rule in the wavenumber domain. Constructing a random model and verifying the accuracy of the algorithm. Design boundary anomaly bodies and compare the algorithm proposed in this paper with the finite element algorithm. The results indicate that the algorithm proposed in this paper significantly improves computational efficiency while maintaining accuracy. Parallel experiments demonstrate that the CPU-GPU architecture is highly suitable for the proposed algorithm, enabling per iterations of models with tens of millions of nodes to be completed in just a few seconds. Additionally, an undulating terrain model was designed to test the algorithm's adaptability to complex topographies.

传统数值模拟方法存在计算量大、边界条件近似、难以准确模拟三维偏微分方程所表征的物理信息等问题，为此我们开发了一种高效且高精度的全息数值模拟方法。该方法的核心在于，通过合理的算法基于真实直流（DC）模型精准获取异常电位的空间-波数谱，随后将所有波数谱信息逆变换至空间域，以得到电场的真实分布。具体而言，该算法通过水平方向的二维傅里叶变换，将空间域中控制异常电位的三维偏微分方程转化为不同波数下的一维常微分方程集合。算法在水平方向引入全息傅里叶变换，以保障空间域与波数域信息的完备性；在垂直方向，则通过平面波分解削弱上下边界引发的边界效应，故此该方法被命名为全息算法。此外，该算法的水平与垂直方向网格尺寸均可自由调整。依托算法的高并行性，本研究实现了CPU-GPU并行架构。我们设计了异常球体模型，用于分析异常场的波数谱分布特征，并验证波数域的对数采样规则；通过构建随机模型，验证了算法的准确性；设计边界异常体，将本文提出的算法与有限元算法进行对比。实验结果表明，本文提出的算法在保证计算精度的同时，显著提升了计算效率。并行实验结果显示，CPU-GPU架构非常适配本文提出的算法，可使数千万节点规模的模型单次迭代仅需数秒即可完成。此外，我们还设计了起伏地形模型，以测试算法对复杂地形的适应性。