Fractional Operators Applied to Geophysical Electromagnetics
Abstract
A growing body of applied mathematics literature in recent years has focused on the application of fractional calculus to problems of anomalous transport. In these analyses, the anomalous transport (of charge, tracers, fluid, etc.) is presumed attributable to long–range correlations of material properties within an inherently complex, and in some cases selfsimilar, conducting medium. Rather than considering an exquisitely discretized (and computationally intractable) representation of the medium, the complex and spatially correlated heterogeneity is represented through reformulation of the governing equation for the relevant transport physics such that its coefficients are, instead, smooth but paired with fractional–order space derivatives. Here we apply these concepts to the scalar Helmholtz equation and its use in electromagnetic interrogation of Earth’s interior through the magnetotelluric method. We outline a practical algorithm for solving the Helmholtz equation using spectral methods coupled with finite element discretizations. Execution of this algorithm for the magnetotelluric problem reveals several interesting features observable in field data: long–range correlation of the predicted electromagnetic fields; a power–law relationship between the squared impedance amplitude and squared wavenumber whose slope is a function of the fractional exponent within the governing Helmholtz equation; and, a non–constant apparent resistivity spectrum whose variability arises solely from themore »
 Authors:

 Sandia National Lab. (SNLCA), Livermore, CA (United States). Geophysics Dept.
 Sandia National Lab. (SNLCA), Livermore, CA (United States). Optimization and Uncertainty Quantification Dept.
 George Mason Univ., Fairfax, VA (United States). Dept. of Mathematical Sciences
 Publication Date:
 Research Org.:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Sponsoring Org.:
 USDOE National Nuclear Security Administration (NNSA); National Science Foundation (NSF)
 OSTI Identifier:
 1575279
 Report Number(s):
 SAND20191547J
Journal ID: ISSN 0956540X; 672516; TRN: US2001189
 Grant/Contract Number:
 AC0494AL85000; NA0003525
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Geophysical Journal International
 Additional Journal Information:
 Journal Volume: 220; Journal Issue: 2; Journal ID: ISSN 0956540X
 Publisher:
 Oxford University Press
 Country of Publication:
 United States
 Language:
 English
 Subject:
 58 GEOSCIENCES; Fractional Helmholtz; electromagnetics; magnetotelluric method; finite element method for fractional Helmholtz; experimental geological data
Citation Formats
Weiss, C. J., van Bloemen Waanders, Bart G., and Antil, H.. Fractional Operators Applied to Geophysical Electromagnetics. United States: N. p., 2019.
Web. https://doi.org/10.1093/gji/ggz516.
Weiss, C. J., van Bloemen Waanders, Bart G., & Antil, H.. Fractional Operators Applied to Geophysical Electromagnetics. United States. https://doi.org/10.1093/gji/ggz516
Weiss, C. J., van Bloemen Waanders, Bart G., and Antil, H.. Fri .
"Fractional Operators Applied to Geophysical Electromagnetics". United States. https://doi.org/10.1093/gji/ggz516. https://www.osti.gov/servlets/purl/1575279.
@article{osti_1575279,
title = {Fractional Operators Applied to Geophysical Electromagnetics},
author = {Weiss, C. J. and van Bloemen Waanders, Bart G. and Antil, H.},
abstractNote = {A growing body of applied mathematics literature in recent years has focused on the application of fractional calculus to problems of anomalous transport. In these analyses, the anomalous transport (of charge, tracers, fluid, etc.) is presumed attributable to long–range correlations of material properties within an inherently complex, and in some cases selfsimilar, conducting medium. Rather than considering an exquisitely discretized (and computationally intractable) representation of the medium, the complex and spatially correlated heterogeneity is represented through reformulation of the governing equation for the relevant transport physics such that its coefficients are, instead, smooth but paired with fractional–order space derivatives. Here we apply these concepts to the scalar Helmholtz equation and its use in electromagnetic interrogation of Earth’s interior through the magnetotelluric method. We outline a practical algorithm for solving the Helmholtz equation using spectral methods coupled with finite element discretizations. Execution of this algorithm for the magnetotelluric problem reveals several interesting features observable in field data: long–range correlation of the predicted electromagnetic fields; a power–law relationship between the squared impedance amplitude and squared wavenumber whose slope is a function of the fractional exponent within the governing Helmholtz equation; and, a non–constant apparent resistivity spectrum whose variability arises solely from the fractional exponent. In geologic settings characterized by self–similarity (e.g. fracture systems; thick and richly–textured sedimentary sequences, etc.) we posit that these diagnostics are useful for geologic characterization of features far below the typical resolution limit of electromagnetic methods in geophysics.},
doi = {10.1093/gji/ggz516},
journal = {Geophysical Journal International},
number = 2,
volume = 220,
place = {United States},
year = {2019},
month = {11}
}
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