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Lindsey Heagy
109 lines
3.8 KiB
ReStructuredText
109 lines
3.8 KiB
ReStructuredText
.. _api_FDEM:
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.. math::
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\renewcommand{\div}{\nabla\cdot\,}
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\newcommand{\grad}{\vec \nabla}
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\newcommand{\curl}{{\vec \nabla}\times\,}
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Frequency Domain Electromagnetics
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*********************************
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Electromagnetic (EM) geophysical methods are used in a variety of applications from resource exploration, including for hydrocarbons and minerals, to environmental applications, such as groundwater monitoring.
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Background
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==========
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Electromagnetic phenomena are governed by Maxwell's equations. They describe the behavior of EM fields and fluxes. Electromagnetic theory for geophysical applications by Ward and Hohmann (1988) is a highly recommended resource on this topic.
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Fourier Transform Convention
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----------------------------
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In order to examine Maxwell's equations in the frequency domain, we must first define our choice of harmonic time-dependence by choosing a Fourier transform convention. We use the \\(\\ e^{i \omega t} \\)\\ convention, so we define our Fourier Transform pair as
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.. math ::
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F(\omega) = \int_{-\infty}^{\infty} f(t) e^{- i \omega t} dt \\
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f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega) e^{i \omega t} d \omega
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where \\(\\omega\\) is angular frequency, \\(t\\) is time, \\(F(\omega)\\) is the function defined in the frequency domain and \\(f(t)\\) is the function defined in the time domain.
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Maxwell's Equations
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===================
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In the frequency domain, Maxwell's equations are given by
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.. math ::
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\curl \vec{E} = - i \omega \vec{B} \\
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\curl \vec{H} = \vec{J} + i \omega \vec{D} + \vec{J}_s \\
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\div \vec{B} = 0 \\
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\div \vec{D} = \rho_f
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where:
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- \\(\\vec{E}\\) : electric field (\\(V/m\\))
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- \\(\\vec{H}\\) : magnetic field (\\(A/m\\))
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- \\(\\vec{B}\\) : magnetic flux density (\\(Wb/m^2\\))
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- \\(\\vec{D}\\) : electric displacement / electric flux density (\\(C/m^2\\))
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- \\(\\vec{J}\\) : electric current density (\\(A/m^2\\))
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- \\(\\rho_f\\) : free charge density
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The source term is \\(\\vec{J}_s\\)
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Constitutive Relations
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----------------------
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The fields and fluxes are related through the constitutive relations. At each frequency, they are given by
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.. math ::
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\vec{J} = \sigma \vec{E} \\
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\vec{B} = \mu \vec{H} \\
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\vec{D} = \varepsilon \vec{E}
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where
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- \\(\\sigma\\) : electrical conductivity \\(S/m\\)
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- \\(\\mu\\) : magnetic permeability \\(H/m\\)
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- \\(\\varepsilon\\) : dielectric permittivity \\(F/m\\)
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\\(\\sigma\\), \\(\\mu\\), \\(\\varepsilon\\) are physical properties which depend on the material. \\(\\sigma\\) describes how easily electric current passes through a material, \\(\\mu\\) describes how easily a material is magnetized, and \\(\\varepsilon\\) describes how easily a material is electrically polarized. In most geophysical applications of EM, \\(\\sigma\\) is the the primary physical property of interest, and \\(\\mu\\), \\(\\varepsilon\\) are assumed to have their free-space values \\(\\mu_0 = 4\pi \times 10^{-7} H/m \\), \\(\\varepsilon_0 = 8.85 \times 10^{-12} F/m\\)
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For a more complete discussion of physical properties see `GPG <http://www.eos.ubc.ca/courses/eosc350/content/index.htm>`_
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Quasi-static Approximation
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--------------------------
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For the frequency range typical of most geophysical surveys, the contribution of the electric displacement is negligible compared to the electric current density. In this case, we use the \\(\\emph{Quasi-static approximation}\\) and assume that this term can be neglected, giving
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.. math ::
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\nabla \times \vec{E} = -i \omega \vec{B} \\
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\nabla \times \vec{H} = \vec{J} + \vec{J}_s
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Fields from a Dipole
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--------------------
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Forward Problem
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===============
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Inverse Problem
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===============
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API
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===
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.. automodule:: simpegEM.FDEM.FDEM
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:show-inheritance:
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:members:
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:undoc-members:
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FDEM Survey
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-----------
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.. automodule:: simpegEM.FDEM.SurveyFDEM
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:show-inheritance:
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:members:
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:undoc-members:
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