.. _api_FDEM:

.. math::
    \renewcommand{\div}{\nabla\cdot\,}
    \newcommand{\grad}{\vec \nabla}
    \newcommand{\curl}{{\vec \nabla}\times\,}


Frequency Domain Electromagnetics
*********************************

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.


Background
==========

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.

Fourier Transform Convention
----------------------------
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

.. math ::
	F(\omega) = \int_{-\infty}^{\infty} f(t) e^{- i \omega t} dt \\

	f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\omega) e^{i \omega t} d \omega

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.


Maxwell's Equations
===================
In the frequency domain, Maxwell's equations are given by

.. math ::
	\curl \vec{E} = - i \omega \vec{B} \\

	\curl \vec{H} = \vec{J} + i \omega \vec{D} + \vec{J}_s \\

	\div \vec{B} = 0 \\

	\div \vec{D} = \rho_f

where:
- \\(\\vec{E}\\) : electric field (\\(V/m\\))
- \\(\\vec{H}\\) : magnetic field (\\(A/m\\))
- \\(\\vec{B}\\) : magnetic flux density (\\(Wb/m^2\\))
- \\(\\vec{D}\\) : electric displacement / electric flux density (\\(C/m^2\\))
- \\(\\vec{J}\\) : electric current density (\\(A/m^2\\))
- \\(\\rho_f\\) : free charge density
The source term is \\(\\vec{J}_s\\)


Constitutive Relations
----------------------
The fields and fluxes are related through the constitutive relations. At each frequency, they are given by

.. math ::
	\vec{J} = \sigma \vec{E} \\

	\vec{B} = \mu \vec{H} \\

	\vec{D} = \varepsilon \vec{E}

where
- \\(\\sigma\\) : electrical conductivity \\(S/m\\)
- \\(\\mu\\) : magnetic permeability \\(H/m\\)
- \\(\\varepsilon\\) : dielectric permittivity \\(F/m\\)

\\(\\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\\)
For a more complete discussion of physical properties see `GPG <http://www.eos.ubc.ca/courses/eosc350/content/index.htm>`_


Quasi-static Approximation
--------------------------
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

.. math ::
	\nabla \times \vec{E} = -i \omega \vec{B} \\
	\nabla \times \vec{H} = \vec{J} + \vec{J}_s


Fields from a Dipole
--------------------

Forward Problem
===============

Inverse Problem
===============

API
===
.. automodule:: simpegEM.FDEM.FDEM
    :show-inheritance:
    :members:
    :undoc-members:


FDEM Survey
-----------

.. automodule:: simpegEM.FDEM.SurveyFDEM
    :show-inheritance:
    :members:
    :undoc-members: