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# Electromagnetic Wave Propagation based upon Spectral-Element Methodology in Dispersive and Attenuating Media

## Abstract

We build on mathematical equivalences between Maxwell’s wave equations for an electromagnetic medium and elastic seismic wave equations. This allows us to readily model Maxwell’s wave propagation in the spectral-element codes SPECFEM2D and SPECFEM3D, written for acoustic, viscoelastic and poroelastic seismic wave propagation, providing the ability to handle complex geometries, inherent to finite element methods, and retaining the strength of exponential convergence and accuracy due to the use of high-degree polynomials to interpolate field functions on the elements, characteristic to spectral-element methods. Attenuation and dispersion processes related to the frequency dependence of dielectric permittivity and conductivity are also included using a Zener model, similar to shear attenuation in viscoelastic media or viscous diffusion in poroelastic media, and a Kelvin-Voigt model, respectively. Ability to account for anisotropic media is also discussed. Here, we limit ourselves to certain dielectric permittivity tensor geometries, in order to conserve a diagonal mass matrix after discretization of the system of equations. Doing so, simulation of Maxwell’s wave equations in the radar frequency range based on spectral-element method can be solved using explicit time integration schemes well suited for parallel computation. We validate our formulation with analytical solutions. In 2D, our implementation allows for the modelling ofmore »

- Authors:

- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1580324

- Report Number(s):
- LLNL-JRNL-755165

Journal ID: ISSN 0956-540X; 941790

- Grant/Contract Number:
- AC52-07NA27344

- 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; 97 MATHEMATICS AND COMPUTING; electromagnetic theory; ground penetrating radar; numerical modelling; wave propagation

### Citation Formats

```
Morency, Christina. Electromagnetic Wave Propagation based upon Spectral-Element Methodology in Dispersive and Attenuating Media. United States: N. p., 2019.
Web. doi:10.1093/gji/ggz510.
```

```
Morency, Christina. Electromagnetic Wave Propagation based upon Spectral-Element Methodology in Dispersive and Attenuating Media. United States. doi:10.1093/gji/ggz510.
```

```
Morency, Christina. Wed .
"Electromagnetic Wave Propagation based upon Spectral-Element Methodology in Dispersive and Attenuating Media". United States. doi:10.1093/gji/ggz510.
```

```
@article{osti_1580324,
```

title = {Electromagnetic Wave Propagation based upon Spectral-Element Methodology in Dispersive and Attenuating Media},

author = {Morency, Christina},

abstractNote = {We build on mathematical equivalences between Maxwell’s wave equations for an electromagnetic medium and elastic seismic wave equations. This allows us to readily model Maxwell’s wave propagation in the spectral-element codes SPECFEM2D and SPECFEM3D, written for acoustic, viscoelastic and poroelastic seismic wave propagation, providing the ability to handle complex geometries, inherent to finite element methods, and retaining the strength of exponential convergence and accuracy due to the use of high-degree polynomials to interpolate field functions on the elements, characteristic to spectral-element methods. Attenuation and dispersion processes related to the frequency dependence of dielectric permittivity and conductivity are also included using a Zener model, similar to shear attenuation in viscoelastic media or viscous diffusion in poroelastic media, and a Kelvin-Voigt model, respectively. Ability to account for anisotropic media is also discussed. Here, we limit ourselves to certain dielectric permittivity tensor geometries, in order to conserve a diagonal mass matrix after discretization of the system of equations. Doing so, simulation of Maxwell’s wave equations in the radar frequency range based on spectral-element method can be solved using explicit time integration schemes well suited for parallel computation. We validate our formulation with analytical solutions. In 2D, our implementation allows for the modelling of both a transverse magnetic (TM) mode, suitable for surface based reflection ground penetration radar type of applications, and a transverse electric (TE) mode more suitable for crosshole and vertical radar profiling setups. Two 2D examples are designed to demonstrated the use of the TM and TE modes. In conclusion, a 3D example is also presented, which allows for the full TEM solution, different antenna orientations, and out-of-plane variations in material properties.},

doi = {10.1093/gji/ggz510},

journal = {Geophysical Journal International},

number = 2,

volume = 220,

place = {United States},

year = {2019},

month = {11}

}