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Focusing inversion of potential field data for the recovery of sparsesubsurface structures from surface measurement data on a uniform grid isdiscussed. For the uniform grid the model sensitivity matrices exhibit blockToeplitz Toeplitz block structure, by blocks for each depth layer of thesubsurface. Then, through embedding in circulant matrices, all forwardoperations with the sensitivity matrix, or its transpose, are realized usingthe fast two dimensional Fourier transform. Simulations demonstrate that thisfast inversion algorithm can be implemented on standard desktop computers withsufficient memory for storage of volumes up to size $n \approx 1M$. The linearsystems of equations arising in the focusing inversion algorithm are solvedusing either Golub Kahan bidiagonalization or randomized singular valuedecomposition algorithms in which all matrix operations with the sensitivitymatrix are implemented using the fast Fourier transform. These two algorithmsare contrasted for efficiency for large-scale problems with respect to thesizes of the projected subspaces adopted for the solutions of the linearsystems. The presented results confirm earlier studies that the randomizedalgorithms are to be preferred for the inversion of gravity data, and that itis sufficient to use projected spaces of size approximately $m/8$, for datasets of size $m$. In contrast, the Golub Kahan bidiagonalization leads to moreefficient implementations for the inversion of magnetic data sets, and it isagain 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 supportthe presented conclusions and are verified on the inversion of a practicalmagnetic data set that is obtained over the Wuskwatim Lake region in Manitoba,Canada.

geophysical journal international

A fast methodology for large-scale focusing inversion of gravity and  magnetic data using the structured model matrix and the $2D$ fast Fourier  transform