2020 **Abstract:** Summary
We discuss the focusing inversion of potential field data for the recovery of sparse subsurface structures from surface measurement data on a uniform grid. For the uniform grid, the model sensitivity matrices have a block Toeplitz Toeplitz block structure for each block of columns related to a fixed depth layer of the subsurface. Then, all forward operations with the sensitivity matrix, or its transpose, are performed using the two dimensional fast Fourier transform. Simulations are prov…

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“…In general, regardless of the choice of λ (i) = 0 or > 0, we observe that the algorithm requires many more iterations for convergence, as compared to the focusing inversion. On the other hand, consistent with smoothing inversion, the data misfits gradually and stably decrease with the iterations, and, as has been seen in other situations, Renaut et al (2020); Vatankhah et al (2022), the data misfit term for the magnetic problem satisfies the noise level much faster than is the case for the gravity data misfit. Consequently, we generally need both λ (2) ≫ λ (1) and α…”

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“…In general, regardless of the choice of λ (i) = 0 or > 0, we observe that the algorithm requires many more iterations for convergence, as compared to the focusing inversion. On the other hand, consistent with smoothing inversion, the data misfits gradually and stably decrease with the iterations, and, as has been seen in other situations, Renaut et al (2020); Vatankhah et al (2022), the data misfit term for the magnetic problem satisfies the noise level much faster than is the case for the gravity data misfit. Consequently, we generally need both λ (2) ≫ λ (1) and α…”

“…Multiple approaches have been adopted to mitigate the computational challenges, including (i) the use of wavelet and compression techniques to significantly reduce the size of the sensitivity matrices (Portniaguine & Zhdanov 2002;Li & Oldenburg 2003;Voronin et al 2015) and (ii) the use of iterative and/or randomization algorithms that project the problem onto a smaller subspace, (Oldenburg et al 1993;Oldenburg & Li 1994;Vatankhah et al 2017) and (Vatankhah et al 2018;Vatankhah et al 2020b), respectively. By designing a discretization of the volume domain such that the resulting matrices exhibit a structure that is amenable to reduced storage, and also to efficient forward and transpose computations, it becomes feasible to apply standard iterative and/or randomization algorithms for the solution of large-scale problems (Renaut et al 2020). Specifically, for a uniform volume discretization, the sensitivity matrices exhibit a block Toeplitz Toeplitz block (BTTB) structure for each depth layer of the model, (Zhang & Wong 2015;Chen & Liu 2019), and both forward and transpose matrix operations can be implemented using the 2−D fast Fourier transform (2DFFT), as described by Vogel (2002) and as explicitly explained for the gravity and magnetic forward models by Hogue et al (2020).…”

“…Second, gravity inversion suffers from severe ambiguity in that the same external gravity field can be explained by different models of density distribution. To improve computational efficiency, previous studies mainly worked on speeding up forward modeling, such as using wavelet transforms (Li & Oldenburg, 2003), adaptive discretization of study regions based on features of observed data (Davis & Li, 2013; Yang et al., 2019), downsampling of observed data (Foks et al., 2014), adaptive multilevel fast multipole (AMFM) method (Ren et al., 2017), block Toeplitz Toeplitz Block (BTTB) matrices combined with 2D fast Fourier transform (Hogue et al., 2020; Renaut et al., 2020), and the equivalence of the kernel matrices (Zhao et al., 2019). To alleviate the ambiguity of gravity inversion, many efforts have been made to incorporate different constraints.…”