Fast computation of three-dimensional gravity and magnetic forward models is considered. Measurement data is assumed to be obtained on a uniform grid which is staggered with respect to the discretization of the parameter volume. Then, the resulting kernel sensitivity matrices exhibit block-Toeplitz Toeplitz-block (BTTB) structure. These matrices are symmetric for the gravity problem but non-symmetric 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 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 that is achieved using the transform implementation. Moreover, 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. 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.

中文翻译：

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用于有效评估重力和磁核的教程和开源软件

考虑了三维重力和磁正模型的快速计算。假设测量数据是在一个均匀网格上获得的，该网格相对于参数体积的离散化是交错的。然后，得到的内核灵敏度矩阵表现出块-托普利茨托普利茨块 (BTTB) 结构。这些矩阵对于重力问题是对称的，但对于磁问题是非对称的。在每种情况下，该结构都有助于使用二维快速傅立叶变换进行快速前向计算。对于每个问题，内核矩阵的构造和变换在快进乘法中的应用都被仔细描述。但是，为了与变换方法进行比较，还解释了定义给定内核矩阵的唯一条目的生成。还演示了在引入体积域周围的填充时如何调整矩阵，从而调整变换。介绍了快速前向矩阵乘法与灵敏度矩阵及其转置的变换算法，无需直接构造相关矩阵。数值实验表明，使用变换实现可以显着减少计算时间。此外，在减少内存需求和计算时间方面，为大型 3D 体积实现变换算法变得可行。所有提出的算法，包括可变填充，都被编码为最佳内存、存储和计算，作为开源 MATLAB 代码，可以适用于任何生成 BTTB 矩阵的卷积核。这项工作，