大规模复杂重磁问题的高阶紧致有限差分快速正演

The fast and accurate three-dimensional gravity and magnetic modeling is the core engine of the interpretation and inversion of gravity and magnetic data. It is also one of the research hotspots and frontiers in the field of exploration geophysics. At present, the traditional gravity and magnetic forward modeling methods for solving partial differential equation (PDE) have only second-order accuracy. When computing potential field and gradient tensor from the obtained gravitational or magnetic potential, the traditional numerical differentiation methods inevitably lose accuracy. To overcome this problem, the project plans to present an accurate and fast forward modeling method for large-scale complex gravity and magnetic problems based on high-order compact difference schemes. Firstly, we intend to construct a high-order compact difference schemes on a non-uniform structured grid, and exploit the Extrapolation Cascadic Multigrid (EXCMG) method to accelerate the solution of large-scale linear equation systems. It can greatly improve the accuracy and efficiency of gravity or magnetic field and gradient modeling. Then, to accurately approximate the underground complex geological structure and arbitrary undulating terrain, we apply the immersion interface method to modify the above high-order compact difference schemes. So that we can realize high-accurate forward modeling for complex gravity and magnetic problems with terrain. Finally, some international standard models and actual models are used to verify the accuracy and efficiency of the new method. The expected results can not only realize the high-accurate and fast forward modeling for potential field and gradient tensor of complex gravity and magnetic models, but also provide dynamic support for inversion of actual measured data. It has important scientific significance and practical application value.

高精度快速三维正演是重磁数据解释和反演成像的核心引擎，是当前勘探地球物理领域的研究热点和前沿之一。针对基于偏微分方程的传统正演方法不能同时高精度地获得重磁场和梯度的关键难题，本项目拟提出一种基于高阶紧致差分算法的大规模复杂三维重磁快速高精度正演新方法。该方法首先构建适用于非均匀结构化网格的高阶紧致差分格式，结合外推瀑布多网格算法求解大规模线性方程组，大幅度提高重磁场及梯度的求解精度和效率。然后，利用浸入界面法修改高阶紧致差分格式，精确逼近地下复杂地质构造和任意起伏地形，实现带地形复杂重磁问题的高精度计算。最后，通过国际标准测试模型和野外实际模型验证算法的计算精度和效率。预期的研究成果不仅可实现复杂重磁模型的场和梯度的高精度快速计算，而且能为野外数据反演提供动力支持，具有重要的科学意义和实际应用价值。

高精度快速三维正演是重磁数据解释和反演成像的核心引擎，是当前勘探地球物理领域的研究热点和前沿之一。针对基于偏微分方程的传统正演方法不能同时高精度地获得重磁场和梯度的关键难题，本项目开展了基于高阶紧致（High-Order Compact, HOC）有限差分算法和外推瀑布式多网格（Extrapolation Cascadic Multigrid Method, EXCMG）算法的大规模复杂模型三维重磁快速高精度正演新方法研究。主要开展了以下工作：1）结合非均匀结构化网格上的HOC有限差分格式，应用五次拉格朗日插值和完整的Richardson外推技术，开发了一种改进的EXCMG算法，用于解决具有边界/内层的三维对角各向异性对流扩散问题；2）针对基于PDE的传统正演方法不能同时高精度地获得重磁场和梯度地关键难题，实现了基于非均匀六面体网格高阶紧致差分离散格和求导算子实现了3D重磁场及其梯度张量的高精度快速正演；3）为了适应复杂的地质结构，同时考虑到计算效率，在非均匀网格高阶紧致差分离散的基础上，基于四次插值和完全的Richardson外推在非均匀直线网格上构造新的广义外推瀑布式多重网格方法，实现了大规模重磁场的快速计算；4）针对复杂结构和地形模型，项目基于浸入界面法修改了高阶紧致差分格式，初步实现了复杂重磁问题的高精度正演计算。结果表明：1）HOC方案可以高精度地产生重磁场及其梯度张量，相比传统的有限差分方法而言，提高了至少1阶的收敛精度；2）改进的EXCMG算法比预处理的BiCGStab方法和最先进的AMG求解器收敛速度要快得多，极大地加速了大规模三维重磁问题的正演速度。研究成果在理论上具有创新性，不仅可实现复杂重磁模型的场和梯度的高精度快速计算，而且能为野外数据反演提供动力支持，具有重要的科学意义和实际应用价值。

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