Obtaining vector magnetic field maps from single‐component measurements of geological samples

Lima, Eduardo A.; Weiss, B. P.

doi:10.1029/2008jb006006

Cited by 61 publications

(61 citation statements)

References 37 publications

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“…Maxwell's equations establish that the magnetic field in a region devoid of sources (i.e., outside the sample, where we take measurements of the field) can be completely represented by the gradient of a scalar function satisfying Laplace's equation. This scalar potential means that all three field components are tightly interconnected and that a single component essentially carries all the information about the full vector field [ Lima and Weiss , ]. This refutes recent unsubstantiated assertions that single‐component maps do not provide unique vector field constraints [ Cottrell et al ., ; Dare et al ., ].…”

Section: Description Of Net Moment Techniquementioning

confidence: 99%

Ultra-high sensitivity moment magnetometry of geological samples using magnetic microscopy

Lima

Weiss

2016

Geochem. Geophys. Geosyst.

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Useful paleomagnetic information is expected to be recorded by samples with moments up to three orders of magnitude below the detection limit of standard superconducting rock magnetometers. Such samples are now detectable using recently developed magnetic microscopes, which map the magnetic fields above room‐temperature samples with unprecedented spatial resolutions and field sensitivities. However, realizing this potential requires the development of techniques for retrieving sample moments from magnetic microscopy data. With this goal, we developed a technique for uniquely obtaining the net magnetic moment of geological samples from magnetic microscopy maps of unresolved or nearly unresolved magnetization. This technique is particularly powerful for analyzing small, weakly magnetized samples such as meteoritic chondrules and terrestrial silicate crystals like zircons. We validated this technique by applying it to field maps generated from synthetic sources and also to field maps measured using a superconducting quantum interference device (SQUID) microscope above geological samples with moments down to 10−15 Am2. For the most magnetic rock samples, the net moments estimated from the SQUID microscope data are within error of independent moment measurements acquired using lower sensitivity standard rock magnetometers. In addition to its superior moment sensitivity, SQUID microscope net moment magnetometry also enables the identification and isolation of magnetic contamination and background sources, which is critical for improving accuracy in paleomagnetic studies of weakly magnetic samples.

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Section: Description Of Net Moment Techniquementioning

confidence: 99%

Ultra-high sensitivity moment magnetometry of geological samples using magnetic microscopy

Lima

Weiss

2016

Geochem. Geophys. Geosyst.

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“…It is now straightforward if m ∈ BM O −∞ to obtain (13), as well as (14) in the distributional sense, following the steps we used when m ∈ W −∞,p , 1 < p < ∞.…”

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Characterizing kernels of operators related to thin-plate magnetizations via generalizations of Hodge decompositions

Baratchart¹,

Hardin²,

Lima³

et al. 2012

Inverse Problems

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Abstract. Recently developed scanning magnetic microscopes measure the magnetic field in a plane above a thin-plate magnetization distribution. These instruments have broad applications in geoscience and materials science, but are limited by the requirement that the sample magnetization must be retrieved from measured field data, which is a generically nonunique inverse problem. This problem leads to an analysis of the kernel of the related magnetization operators, which also has relevance to the "equivalent source problem" in the case of measurements taken from just one side of the magnetization. We characterize the kernel of the operator relating planar magnetization distributions to planar magnetic field maps in various function and distribution spaces (e.g., sums of derivatives of L p (Lebesgue spaces) or bounded mean oscillation (BMO) functions). For this purpose, we present a generalization of the Hodge decomposition in terms of Riesz transforms and utilize it to characterize sources that do not produce magnetic field either above or below the sample, or that are magnetically silent (i.e., no magnetic field anywhere outside the sample). For example, we show that a thin-plate magnetization is silent (i.e., in the kernel) when its normal component is zero and its tangential component is divergence-free. In addition, we show that compactly supported magnetizations (i.e., magnetizations that are zero outside of a bounded set in the source plane) that do not produce magnetic fields either above or below the sample are necessarily silent. In particular, neither a nontrivial planar magnetization with fixed direction (unidimensional) compact support nor a bidimensional planar magnetization (i.e., a sum of two unidimensional magnetizations) that is nontangential can be silent. We prove that any planar magnetization distribution is equivalent to a unidimensional one. We also discuss the advantages of mapping the field on both sides of a magnetization, whenever experimentally feasible. Examples of source recovery are given along with a brief discussion of the Fourier-based inversion techniques that are utilized.

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“…(In some cases, sophisticated spectral/spatial unmixing techniques can be employed to tackle nonuniqueness when attempting to invert multidirectional magnetization distributions [Baratchart et al, 2013], but such an approach is outside of the scope of this paper.) In addition, it shows that, for low or moderate noise levels, a single field component carries all the information about the magnetization distribution [Lima and Weiss, 2009]. For k = 0, g is the zero matrix, which implies that the uniform (constant) component of the magnetization distribution cannot be directly recovered from magnetic field measurements.…”

Section: The Inverse Problem For Scanning Magnetic Microscopymentioning

confidence: 99%

Fast inversion of magnetic field maps of unidirectional planar geological magnetization

et al. 2013

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[1] Scanning magnetic microscopes are being increasingly utilized in paleomagnetic studies of geological samples. These instruments typically map a single component of the sample's magnetic field at close proximity with submillimeter horizontal spatial resolution. However, in most applications, an image of the magnetization distribution within the sample is desired rather than its external magnetic field. This requires carefully solving an ill-posed inverse problem to obtain solutions that are nearly free of artifacts and consistent with both natural and laboratory magnetization processes. We present a new, fast inversion technique based on classic methods developed for the Fourier domain that retrieves planar unidirectional magnetization distributions from magnetic field maps. Whereas our approach considers the subtle peculiarities of scanning magnetic microscopy which otherwise can complicate this technique, much of the formalism and algorithms described in this work can also be directly applied to province-scale magnetic field data from aeromagnetic surveys and may be used as an initial step in the modeling of magnetic sources with complex three-dimensional geometries. We discuss sources of inaccuracy observed in practical implementations of the technique and present strategies to improve the quality of inversions. Numerous examples of inversion of both synthetic and experimental data demonstrate the performance of the technique under different conditions. In particular, we retrieve magnetization distributions of a Hawaiian basalt and compare it to inversions calculated in a previous work. We conclude by showing a reconstructed magnetization for the eucrite meteorite ALHA81001 that displays in high resolution the spatial distribution of high-coercivity grains within the sample.

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Obtaining vector magnetic field maps from single‐component measurements of geological samples

Cited by 61 publications

References 37 publications

Ultra-high sensitivity moment magnetometry of geological samples using magnetic microscopy

Ultra-high sensitivity moment magnetometry of geological samples using magnetic microscopy

Characterizing kernels of operators related to thin-plate magnetizations via generalizations of Hodge decompositions

Fast inversion of magnetic field maps of unidirectional planar geological magnetization

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