Focusing inversion of potential field data for the recovery of sparse subsurface structures from surface measurement data on a uniform grid is discussed. For the uniform grid the model sensitivity matrices exhibit block Toeplitz Toeplitz block structure, by blocks for each depth layer of the subsurface. Then, through embedding in circulant matrices, all forward operations with the sensitivity matrix, or its transpose, are realized using the fast two dimensional Fourier transform. Simulations demonstrate that this fast inversion algorithm can be implemented on standard desktop computers with sufficient memory for storage of volumes up to size $n \approx 1M$. The linear systems of equations arising in the focusing inversion algorithm are solved using either Golub Kahan bidiagonalization or randomized singular value decomposition algorithms in which all matrix operations with the sensitivity matrix are implemented using the fast Fourier transform. These two algorithms are contrasted for efficiency for large-scale problems with respect to the sizes of the projected subspaces adopted for the solutions of the linear systems. The presented results confirm earlier studies that the randomized algorithms are to be preferred for the inversion of gravity data, and that it is sufficient to use projected spaces of size approximately $m/8$, for data sets of size $m$. In contrast, the Golub Kahan bidiagonalization leads to more efficient implementations for the inversion of magnetic data sets, and it is again sufficient to use projected spaces of size approximately $m/8$. Moreover, it is sufficient to use projected spaces of size $m/20$ when $m$ is large, $m \approx 50000$, to reconstruct volumes with $n \approx 1M$. Simulations support the presented conclusions and are verified on the inversion of a practical magnetic data set that is obtained over the Wuskwatim Lake region in Manitoba, Canada.

中文翻译：

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一种使用结构化模型矩阵和二维快速傅立叶变换进行重磁数据大规模聚焦反演的快速方法

讨论了用于从均匀网格上的地表测量数据恢复稀疏地下结构的势场数据的聚焦反演。对于均匀网格，模型灵敏度矩阵表现出块 Toeplitz Toeplitz 块结构，每个块用于地下的每个深度层。然后，通过嵌入循环矩阵，使用快速二维傅立叶变换实现所有与灵敏度矩阵或其转置的前向运算。模拟表明，这种快速反演算法可以在标准台式计算机上实现，该计算机具有足够的内存来存储大小为 $n\大约 1M$ 的卷。聚焦反演算法中出现的线性方程组使用 Golub Kahan 双对角化或随机奇异值分解算法求解，其中使用快速傅立叶变换实现对灵敏度矩阵的所有矩阵运算。这两种算法在大规模问题的效率方面进行了对比，这与线性系统的解所采用的投影子空间的大小有关。所呈现的结果证实了早期的研究，即随机算法更适合用于重力数据的反演，并且对于大小为 $m$ 的数据集，使用大小约为 $m/8$ 的投影空间就足够了。相比之下，Golub Kahan 双对角化可以更有效地实现磁性数据集的反演，再次使用大小约为 $m/8$ 的投影空间就足够了。此外，当 $m$ 很大时，使用大小为 $m/20$ 的投影空间（$m \approx 50000$）就足以重建具有 $n \approx 1M$ 的体积。模拟支持了所提出的结论，并在加拿大马尼托巴省 Wuskwatim 湖地区获得的实用磁数据集的反演上得到了验证。