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

Abstract

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.

arXiv comment39 pages. 16 figures

References in corpus (7)

• Hybrid and iteratively reweighted regularization by unbiased predictive risk and weighted GCV for projected systems
• 3-D Projected $L_1$ inversion of gravity data
• Application of the ${\chi }^2$ principle and unbiased predictive risk estimator for determining the regularization parameter in 3D focusing gravity inversion
• A fast algorithm for regularized focused 3-D inversion of gravity data using the randomized SVD
• A fast methodology for large-scale focusing inversion of gravity and magnetic data using the structured model matrix and the $2D$ fast Fourier transform
• IGUG: A MATLAB package for $3$D inversion of gravity data using graph theory
• Compression Approaches for the Regularized Solutions of Linear Systems from Large-Scale Inverse Problems

Cited by in corpus (3)

• Generalized L${}_p$-norm joint inversion of gravity and magnetic data using cross-gradient constraint
• A fast methodology for large-scale focusing inversion of gravity and magnetic data using the structured model matrix and the $2D$ fast Fourier transform
• Large-scale focusing joint inversion of gravity and magnetic data with Gramian constraint