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Title: 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 self-similar, 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 » 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.« less

Authors:
 [1];  [2];  [3]
+ Show Author Affiliations
  1. Sandia National Lab. (SNL-CA), Livermore, CA (United States). Geophysics Dept.
  2. Sandia National Lab. (SNL-CA), Livermore, CA (United States). Optimization and Uncertainty Quantification Dept.
  3. George Mason Univ., Fairfax, VA (United States). Dept. of Mathematical Sciences
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA); National Science Foundation (NSF)
OSTI Identifier:
1575279
Report Number(s):
SAND-2019-1547J
Journal ID: ISSN 0956-540X; 672516; TRN: US2001189
Grant/Contract Number:  
AC04-94AL85000; NA-0003525
Resource Type:
Accepted Manuscript
Journal Name:
Geophysical Journal International
Additional Journal Information:
Journal Volume: 220; Journal Issue: 2; Journal ID: ISSN 0956-540X
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. doi:10.1093/gji/ggz516.
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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
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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.
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@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 self-similar, 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 = {Fri Nov 15 00:00:00 EST 2019},
month = {Fri Nov 15 00:00:00 EST 2019}
}
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Publisher's Version of Record
https://doi.org/10.1093/gji/ggz516
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Works referenced in this record:

Corrigendum to the paper “Numerical approximation of fractional powers of regularly accretive operators”
journal, April 2017

  • Bonito, Andrea; Pasciak, Joseph E.
  • IMA Journal of Numerical Analysis, Vol. 37, Issue 4
  • DOI: 10.1093/imanum/drw067

Propagation of a chemical wave front in a quasi-two-dimensional superdiffusive flow
journal, June 2010

Note on fractional powers of linear operators
journal, January 1960

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    Works referencing / citing this record:

    External optimal control of fractional parabolic PDEs
    preprint, January 2019

    Convergence of solutions of discrete semi-linear space-time fractional evolution equations
    preprint, January 2019

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