diff --git a/docs/api_FDEM.rst b/docs/api_FDEM.rst index 2a8c2268..ab6dc611 100644 --- a/docs/api_FDEM.rst +++ b/docs/api_FDEM.rst @@ -9,7 +9,7 @@ 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. +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. The primary physical property of interest in EM is electrical conductivity, which describes the ease with which electric current flows through a material. Background @@ -19,14 +19,14 @@ Electromagnetic phenomena are governed by Maxwell's equations. They describe the 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 +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. +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 @@ -43,6 +43,7 @@ In the frequency domain, Maxwell's equations are given by \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\\)) @@ -63,12 +64,13 @@ The fields and fluxes are related through the constitutive relations. At each fr \vec{D} = \varepsilon \vec{E} -where +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\\) +\\(\\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 `_