Fractional Operators Applied to Geophysical Electromagnetics

Journal Article · Fri Nov 15 00:00:00 EST 2019 · Geophysical Journal International

DOI:https://doi.org/10.1093/gji/ggz516· OSTI ID:1575279

Weiss, C. J. ^[1]; van Bloemen Waanders, Bart G. ^[2]; Antil, H. ^[3]

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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

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.

• Research Organization: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
• Sponsoring Organization: National Science Foundation (NSF); USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: AC04-94AL85000; NA0003525
• OSTI ID: 1575279
• Report Number(s): SAND--2019-1547J; 672516
• Journal Information: Geophysical Journal International, Journal Name: Geophysical Journal International Journal Issue: 2 Vol. 220; ISSN 0956-540X
• Publisher: Oxford University Press
• Country of Publication: United States
• Language: English

Related Subjects

58 GEOSCIENCES
Fractional Helmholtz
electromagnetics
experimental geological data
finite element method for fractional Helmholtz
magnetotelluric method