Modern magnetic microscopy (MM) provides high-resolution, ultra-high-sensitivity moment magnetometry, with the ability to measure at spatial resolutions better than 10 - 4 m and to detect magnetic moments weaker than 10 - 15 Am 2 . These characteristics make modern MM devices capable of particularly high-resolution analysis of the magnetic properties of materials, but generate extremely large data sets. Many studies utilizing MM attempt to solve an inverse problem to determine the magnitude of the magnetic moments that produce the measured component of the magnetic field. Fast Fourier techniques in the frequency domain and non-negative least-squares (NNLS) methods in the spatial domain are the two most frequently used methods to solve this inverse problem. Although extremely fast, Fourier techniques can produce solutions that violate the non-negativity of moments constraint. Inversions in the spatial domain do not violate non-negativity constraints, but the execution times of standard NNLS solvers (the Lawson and Hanson method and Matlab’s lsqlin) prohibit spatial domain inversions from operating at the full spatial resolution of an MM. In this paper, we present the applicability of the TNT-NN algorithm, a newly developed NNLS active set method, as a means to directly address the NNLS routine hindering existing spatial domain inversion methods. The TNT-NN algorithm enhances the performance of spatial domain inversions by accelerating the core NNLS routine. Using a conventional computing system, we show that the TNT-NN algorithm produces solutions with residuals comparable to conventional methods while reducing execution time of spatial domain inversions from months to hours or less. Using isothermal remanent magnetization measurements of multiple synthetic and natural samples, we show that the capabilities of the TNT-NN algorithm allow scans with sizes that made them previously inaccesible to NNLS techniques to be inverted. Ultimately, the TNT-NN algorithm enables spatial domain inversions of MM data on an accelerated timescale that renders spatial domain analyses for modern MM studies practical. In particular, this new technique enables MM experiments that would have required an impractical amount of inversion time such as high-resolution stepwise magnetization and demagnetization and 3-dimensional inversions.[Figure not available: see fulltext.].
Bibliographical noteFunding Information:
JMM co-developed TNT-NN and TNT, ran all inversion analyses, and led manuscript writing. IL acquired the SVC982 MM data and contributed to the manuscript. EAL and BPW acquired the Hawaiian basalt MM data, developed the least-squares spatial inversion method, and contributed to the manuscript. MOS and JMF coordinated the study design and contributed to the manuscript. All authors read and approved the final manuscript. We would like to thank Hirokuni Oda for graciously sharing the ferromanganese crust MM data, John Ackerman, the owner of Spring Valley Caverns, for access to, and his continued support of research in Spring Valley Caverns. We would also like to thank Richard Harrison and an anonymous reviewer for their constructive reviews and guest editor Hirokuni Oda for providing positive comments contributing to the improvement of this manuscript. The authors also acknowledge the Minnesota Supercomputing Institute (MSI) at the University of Minnesota for providing resources that contributed to the research results reported within this paper (http://www.msi.umn.edu). JMM thanks Anna K. Lindquist for engaging conversations on experimental rock magnetism. MOS thanks the George and Orpha Gibson endowment for its generous support of the Hydrogeology and Geofluids research group at the University of Minnesota. None of the authors have any competing interests. None of the work presented here is duplicated or in conflict with any of the authors other work, either published or in review. The availability of the data sets used and/or analyzed in this study are outlined below: Synthetic UMN logo—Available from corresponding author upon reasonable request. Hawaiian basalt—See Weiss et al. (2007b). Ferromanganese crust—see Oda et al. (2016). SVC982 Speleothem—Available from corresponding author upon reasonable request. A Matlab implementation of the TNT-NN algorithm is available on GitHub (Myre et al. 2017b). The Matlab code used to perform the spatial inversions is available upon request. This work was supported by National Science Foundation (NSF) Grants EAR-0941666 to MOS, EAR-1316385 to JMF, DMS-1521765 to EAL and BPW, a University of Minnesota Grants-In-Aid award and a McKnight Land-Grant Professorship awarded to JMF. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported by National Science Foundation (NSF) Grants EAR-0941666 to MOS, EAR-1316385 to JMF, DMS-1521765 to EAL and BPW, a University of Minnesota Grants-In-Aid award and a McKnight Land-Grant Professorship awarded to JMF. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF.
- Magnetic microscopy
- Non-negative least-squares
- Rock magnetism