A tutorial and open source software for the efficient evaluation of gravity and magnetic kernels
A tutorial and open source software for the efficient evaluation of gravity and magnetic kernels
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DOI：
10.1016/j.cageo.2020.104575
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发表时间：
2020-11-01
影响因子：
4.4
通讯作者：
Vatankhah, Saeed
中科院分区：
文献类型：
--
作者：
Hogue, Jarom D.;Renaut, Rosemary Anne;Vatankhah, Saeed
研究了三维重磁正演模型的快速计算。假设测量数据是在均匀网格上获得的，相对于参数体积的离散化是交错的，得到的核灵敏度矩阵呈现块- toeplitz - toeplitz块（BTTB）结构。这些矩阵对于重力问题是对称的，但是对于磁问题是不对称的。在每种情况下，该结构都便于使用二维快速傅里叶变换进行快速正演计算。对于每个问题，都详细描述了核矩阵的构造和快进乘法变换的应用。但是，为了与非变换方法进行比较，还解释了定义给定核矩阵的唯一项的生成。还演示了在引入填充时如何调整矩阵和变换。给出了一种不直接构造敏感矩阵的快速前向矩阵乘法及其转置变换算法。数值实验表明，使用转换实现可以显著减少计算时间和内存需求。因此，在减少内存需求和计算时间方面，实现大型三维体积的变换算法是可行的。所有提出的算法，包括变量填充，都是为了优化内存，存储和计算而编码的，作为一个开放源代码的MATLAB代码，可以适用于任何生成BTTB矩阵的卷积核，无论它是否对称。因此，这项工作为重力和磁场数据的有效模拟以及任何允许具有所需结构的灵敏度矩阵的公式提供了一个通用工具。
Fast computation of three-dimensional gravity and magnetic forward models is considered. When the measurement data is assumed to be obtained on a uniform grid which is staggered with respect to the discretization of the parameter volume, the resulting kernel sensitivity matrices exhibit block-Toeplitz-Toeplitzblock (BTTB) structure. These matrices are symmetric for the gravity problem but unsymmetric for the magnetic problem. In each case, the structure facilitates fast forward computation using two-dimensional fast Fourier transforms. The construction of the kernel matrices and the application of the transform for fast forward multiplication, for each problem, is carefully described. But, for purposes of comparison with the non-transform approach, the generation of the unique entries that define a given kernel matrix is also explained. It is also demonstrated how the matrices, and hence transforms, are adjusted when padding around the volume domain is introduced. The transform algorithms for fast forward matrix multiplication with the sensitivity matrix and its transpose, without the direct construction of the relevant matrices, are presented. Numerical experiments demonstrate the significant reduction in computation time and memory requirements that are achieved using the transform implementation. Thus, it becomes feasible, both in terms of reduced memory requirements and computational time, to implement the transform algorithms for large three-dimensional volumes. All presented algorithms, including with variable padding, are coded for optimal memory, storage and computation as an open source MATLAB code which can be adapted for any convolution kernel which generates a BTTB matrix, whether or not it is symmetric. This work, therefore, provides a general tool for the efficient simulation of gravity and magnetic field data, as well as any formulation which admits a sensitivity matrix with the required structure.