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organizing the docs - put the content in a content folder. put the SimPEG core api docs in core_api
This commit is contained in:
Lindsey Heagy
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.. _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
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from resource exploration, including for hydrocarbons and minerals, to
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environmental applications, such as groundwater monitoring. The primary
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physical property of interest in EM is electrical conductivity, which
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describes the ease with which electric current flows through a material.
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Background
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==========
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Electromagnetic phenomena are governed by Maxwell's equations. They describe
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the behavior of EM fields and fluxes. Electromagnetic theory for geophysical
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applications by Ward and Hohmann (1988) is a highly recommended resource on
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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
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define our choice of harmonic time-dependence by choosing a Fourier transform
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convention. We use the :math:`e^{i \omega t}` convention, so we define our
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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 :math:`\omega` is angular frequency, :math:`t` is time, :math:`F(\omega)` is the function defined in the frequency domain and :math:`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} = \vec{S_m}\\
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\curl \vec{H} - \vec{J} - i \omega \vec{D} = \vec{S_e} \\
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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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- :math:`\vec{E}` : electric field (:math:`V/m` )
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- :math:`\vec{H}` : magnetic field (:math:`A/m` )
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- :math:`\vec{B}` : magnetic flux density (:math:`Wb/m^2` )
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- :math:`\vec{D}` : electric displacement / electric flux density (:math:`C/m^2` )
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- :math:`\vec{J}` : electric current density (:math:`A/m^2` )
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- :math:`\vec{S_m}` : magnetic source term (:math:`V/m^2` )
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- :math:`\vec{S_e}` : electric source term (:math:`A/m^2` )
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- :math:`\rho_f` : free charge density (:math:`\Omega m` )
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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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- :math:`\sigma` : electrical conductivity (:math:`S/m`)
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- :math:`\mu` : magnetic permeability (:math:`H/m`)
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- :math:`\varepsilon` : dielectric permittivity (:math:`F/m`)
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:math:`\sigma`, :math:`\mu`, :math:`\varepsilon` are physical properties which depend on the material. :math:`\sigma` describes how easily electric current passes through a material, :math:`\mu` describes how easily a material is magnetized, and :math:`\varepsilon` describes how easily a material is electrically polarized. In most geophysical applications of EM, :math:`\sigma` is the the primary physical property of interest, and :math:`\mu`, :math:`\varepsilon` are assumed to have their free-space values :math:`\mu_0 = 4\pi \times 10^{-7} H/m` , :math:`\varepsilon_0 = 8.85 \times 10^{-12} F/m`
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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 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} = \vec{S_m} \\
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\nabla \times \vec{H} - \vec{J} = \vec{S_e}
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Implementation in SimPEG.EM
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===========================
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We consider two formulations in SimPEG.EM, both first-order and both in terms of one field and one flux. We allow for the definition of magnetic and electric sources (see for example: Ward and Hohmann, starting on page 144). The E-B formulation is in terms of the electric field and the magnetic flux:
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.. math ::
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\nabla \times \vec{E} + i \omega \vec{B} = \vec{S}_m \\
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\nabla \times \mu^{-1} \vec{B} - \sigma \vec{E} = \vec{S}_e
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The H-J formulation is in terms of the current density and the magnetic field:
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.. math ::
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\nabla \times \sigma^{-1} \vec{J} + i \omega \mu \vec{H} = \vec{S}_m \\
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\nabla \times \vec{H} - \vec{J} = \vec{S}_e
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Discretizing
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------------
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For both formulations, we use a finite volume discretization
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and discretize fields on cell edges, fluxes on cell faces and
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physical properties in cell centers. This is particularly
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important when using symmetry to reduce the dimensionality of a problem
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(for instance on a 2D CylMesh, there are :math:`r`, :math:`z` faces and :math:`\theta` edges)
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.. figure:: ../../images/finitevolrealestate.png
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:align: center
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:scale: 60 %
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For the two formulations, the discretization of the physical properties, fields and fluxes are summarized below.
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.. figure:: ../../images/ebjhdiscretizations.png
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:align: center
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:scale: 60 %
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Note that resistivity is the inverse of conductivity, :math:`\rho = \sigma^{-1}`.
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E-B Formulation
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---------------
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.. math ::
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\mathbf{C} \mathbf{e} + i \omega \mathbf{b} = \mathbf{s_m} \\
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\mathbf{C^T} \mathbf{M^f_{\mu^{-1}}} \mathbf{b} - \mathbf{M^e_\sigma} \mathbf{e} = \mathbf{M^e} \mathbf{s_e}
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H-J Formulation
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---------------
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.. math ::
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\mathbf{C^T} \mathbf{M^f_\rho} \mathbf{j} + i \omega \mathbf{M^e_\mu} \mathbf{h} = \mathbf{M^e} \mathbf{s_m} \\
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\mathbf{C} \mathbf{h} - \mathbf{j} = \mathbf{s_e}
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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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FDEM Problem
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------------
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.. automodule:: SimPEG.EM.FDEM.ProblemFDEM
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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:: SimPEG.EM.FDEM.SurveyFDEM
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:show-inheritance:
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:members:
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:undoc-members:
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.. automodule:: SimPEG.EM.FDEM.SrcFDEM
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:show-inheritance:
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:members:
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:undoc-members:
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.. automodule:: SimPEG.EM.FDEM.RxFDEM
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:show-inheritance:
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:members:
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:undoc-members:
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FDEM Fields
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-----------
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.. automodule:: SimPEG.EM.FDEM.FieldsFDEM
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:show-inheritance:
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:members:
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:undoc-members:
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@@ -0,0 +1,387 @@
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.. _api_TDEM:
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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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\newcommand {\J}{{\vec J}}
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\renewcommand{\H}{{\vec H}}
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\newcommand {\E}{{\vec E}}
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\newcommand{\dcurl}{{\mathbf C}}
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\newcommand{\dgrad}{{\mathbf G}}
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\newcommand{\Acf}{{\mathbf A_c^f}}
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\newcommand{\Ace}{{\mathbf A_c^e}}
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\renewcommand{\S}{{\mathbf \Sigma}}
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\newcommand{\St}{{\mathbf \Sigma_\tau}}
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\newcommand{\T}{{\mathbf T}}
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\newcommand{\Tt}{{\mathbf T_\tau}}
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\newcommand{\diag}[1]{\,{\sf diag}\left( #1 \right)}
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\newcommand{\M}{{\mathbf M}}
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\newcommand{\MfMui}{{\M^f_{\mu^{-1}}}}
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\newcommand{\MeSig}{{\M^e_\sigma}}
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\newcommand{\MeSigInf}{{\M^e_{\sigma_\infty}}}
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\newcommand{\MeSigO}{{\M^e_{\sigma_0}}}
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\newcommand{\Me}{{\M^e}}
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\newcommand{\Mes}[1]{{\M^e_{#1}}}
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\newcommand{\Mee}{{\M^e_e}}
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\newcommand{\Mej}{{\M^e_j}}
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\newcommand{\BigO}[1]{\mathcal{O}\bigl(#1\bigr)}
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\newcommand{\bE}{\mathbf{E}}
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\newcommand{\bH}{\mathbf{H}}
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\newcommand{\B}{\vec{B}}
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\newcommand{\D}{\vec{D}}
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\renewcommand{\H}{\vec{H}}
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\newcommand{\s}{\vec{s}}
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\newcommand{\bfJ}{\bf{J}}
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\newcommand{\vecm}{\vec m}
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\renewcommand{\Re}{\mathsf{Re}}
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\renewcommand{\Im}{\mathsf{Im}}
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\renewcommand {\j} { {\vec j} }
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\newcommand {\h} { {\vec h} }
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\renewcommand {\b} { {\vec b} }
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\newcommand {\e} { {\vec e} }
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\newcommand {\c} { {\vec c} }
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\renewcommand {\d} { {\vec d} }
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\renewcommand {\u} { {\vec u} }
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\newcommand{\I}{\vec{I}}
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Time Domain Electromagnetics
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****************************
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.. _api_TDEM_derivation:
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Time-Domain EM Derivation
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=========================
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The following shows the derivation for the TDEM problem. We use the b-formulation below.
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(More to come soon..!)
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Sensitivity Calculation
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-----------------------
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.. math::
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\begin{align}
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\dcurl \e^{(t+1)} + \frac{\b^{(t+1)} - \b^{(t)}}{\delta t} = 0 \\
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\dcurl^\top \MfMui \b^{(t+1)} - \MeSig \e^{(t+1)} = \Me \j_s^{(t+1)}
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\end{align}
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Using Gauss-Newton to solve the inverse problem requires the ability to calculate the product of the
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Jacobian and a vector, as well as the transpose of the Jacobian times a vector.
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The above system can be rewritten as:
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.. math::
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\begin{align}
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\mathbf{A} \u^{(t+1)} + \mathbf{B} \u^{(t)}= \s^{(t+1)}
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\end{align}
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where
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.. math::
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\begin{align}
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\mathbf{A} =
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\left[
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\begin{array}{cc}
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\frac{1}{\delta t} \MfMui & \MfMui\dcurl \\
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\dcurl^\top \MfMui & -\MeSig
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\end{array}
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\right] \\
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\mathbf{B} =
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\left[
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\begin{array}{cc}
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-\frac{1}{\delta t} \MfMui & 0 \\
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0 & 0
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\end{array}
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\right] \\
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\u^{(k)} = \left[
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\begin{array}{c}
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\b^{(k)}\\
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\e^{(k)}
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\end{array}
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\right] \\
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\s^{(k)} = \left[
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\begin{array}{c}
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0\\
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\Me \j^{(k)}_s
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\end{array}
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\right]
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\end{align}
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.. note::
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Here we have multiplied through by \\(\\MfMui\\) to make A and B symmetric!
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|
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The entire time dependent system can be written in a single matrix expression
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||||
|
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.. math::
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|
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\begin{align}
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\hat{\mathbf{A}} \hat{u} = \hat{s}
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\end{align}
|
||||
|
||||
where
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|
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.. math::
|
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|
||||
\begin{align}
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||||
\mathbf{\hat{A}} = \left[
|
||||
\begin{array}{cccc}
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A & 0 & & \\
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||||
B & A & & \\
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||||
& \ddots & \ddots & \\
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||||
& & B & A
|
||||
\end{array}
|
||||
\right] \\
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\hat{u} = \left[
|
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\begin{array}{c}
|
||||
\u^{(1)} \\
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||||
\u^{(2)} \\
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||||
\vdots \\
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||||
\u^{(N)}
|
||||
\end{array} \right]\\
|
||||
\hat{s} = \left[
|
||||
\begin{array}{c}
|
||||
\s^{(1)} - \mathbf{B} \u^{(0)} \\
|
||||
\s^{(2)} \\
|
||||
\vdots \\
|
||||
\s^{(N)}
|
||||
\end{array}
|
||||
\right]
|
||||
\end{align}
|
||||
|
||||
For the fields \\(\\u\\), the measured data is given by
|
||||
|
||||
.. math::
|
||||
|
||||
\begin{align}
|
||||
\vec{d} = \mathbf{Q} \u
|
||||
\end{align}
|
||||
|
||||
The sensitivity matrix **J** is then defined as
|
||||
|
||||
.. math::
|
||||
|
||||
\begin{align}
|
||||
\mathbf{J} = \mathbf{Q} \frac{\partial \u}{\partial \sigma}
|
||||
\end{align}
|
||||
|
||||
|
||||
Defining the function \\(\\c(m,\\u)\\) to be
|
||||
|
||||
.. math::
|
||||
|
||||
\begin{align}
|
||||
\vec{c}(m,\u) = \hat{\mathbf{A}} \vec{u} - \vec{q} = \vec{0}
|
||||
\end{align}
|
||||
|
||||
then
|
||||
|
||||
.. math::
|
||||
|
||||
\begin{align}
|
||||
\frac{\partial \vec{c}}{\partial m} \partial m
|
||||
+ \frac{\partial \vec{c}}{\partial \u} \partial \vec{u} = 0
|
||||
\end{align}
|
||||
|
||||
or
|
||||
|
||||
.. math::
|
||||
|
||||
\begin{align}
|
||||
\frac{\partial \vec{u}}{\partial m} = -\left(\frac{\partial \vec{c}}{\partial \u} \right)^{-1} \frac{\partial \vec{c}}{\partial m}
|
||||
\end{align}
|
||||
|
||||
|
||||
Differentiating, we find that
|
||||
|
||||
.. math::
|
||||
|
||||
\begin{align}
|
||||
\frac{\partial \vec{c}}{\partial \hat{u}} = \hat{\mathbf{A}}
|
||||
\end{align}
|
||||
|
||||
and
|
||||
|
||||
.. math::
|
||||
|
||||
\begin{align}
|
||||
\frac{\partial \vec{c}}{\partial \sigma} = \mathbf{G}_\sigma =
|
||||
\left[
|
||||
\begin{array}{c}
|
||||
g_\sigma^{(1)}\\
|
||||
g_\sigma^{(2)}\\
|
||||
\vdots \\
|
||||
g_\sigma^{(N)}
|
||||
\end{array}
|
||||
\right]
|
||||
\end{align}
|
||||
|
||||
with
|
||||
|
||||
.. math::
|
||||
|
||||
\begin{align}
|
||||
g_\sigma^{(n)} =
|
||||
\left[
|
||||
\begin{array}{c}
|
||||
\mathbf{0} \\
|
||||
- \diag{\e^{(n)}} \Ace \diag{\vec{V}}
|
||||
\end{array}
|
||||
\right]
|
||||
\end{align}
|
||||
|
||||
|
||||
Implementing **J** times a vector
|
||||
---------------------------------
|
||||
|
||||
Multiplying **J** onto a vector can be broken into three steps
|
||||
|
||||
|
||||
* Compute \\(\\vec{p} = \\mathbf{G}m\\)
|
||||
* Solve \\(\\hat{\\mathbf{A}} \\vec{y} = \\vec{p}\\)
|
||||
* Compute \\(\\vec{w} = -\\mathbf{Q} \\vec{y}\\)
|
||||
|
||||
.. math::
|
||||
|
||||
\begin{align}
|
||||
\vec{p}^{(n)} = \left[
|
||||
\begin{array}{c}
|
||||
\vec{p}_b^{(n)} \\
|
||||
\vec{p}_e^{(n)}
|
||||
\end{array}
|
||||
\right] \\
|
||||
\vec{p}_b^{(n)} = 0 \\
|
||||
\vec{p}_e^{(n)} = - \diag{\e^{(n)}} \Ace \diag{V} m
|
||||
\end{align}
|
||||
|
||||
|
||||
For all time steps:
|
||||
|
||||
.. math::
|
||||
|
||||
\begin{align}
|
||||
\frac{1}{\delta t} \MfMui\vec{y}_{b}^{(t+1)} + \MfMui\dcurl \vec{y}_{e}^{(t+1)}
|
||||
- \frac{1}{\delta t} \MfMui \vec{y}_{b}^{(t)}
|
||||
= \vec{p}_b^{(t+1)} \\
|
||||
\dcurl^\top \MfMui \vec{y}_b^{(t+1)} - \MeSig \vec{y}_e^{(t+1)} = \vec{p}_e^{(t+1)}
|
||||
\end{align}
|
||||
|
||||
and
|
||||
|
||||
.. math::
|
||||
|
||||
\begin{align}
|
||||
\left( \MfMui \dcurl \MeSig^{-1} \dcurl^\top \MfMui + \frac{1}{\delta t} \MfMui \right) \vec{y}_{b}^{(t+1)} =
|
||||
\frac{1}{\delta t} \MfMui \vec{y}_b^{(t)}
|
||||
+ \MfMui \dcurl \MeSig^{-1} \vec{p}_e^{(t+1)} + \vec{p}_b^{(t+1)} \\
|
||||
\vec{y}_e^{(t+1)} = \MeSig^{-1} \dcurl^\top \MfMui \vec{y}_b^{(t+1)} - \MeSig^{-1} \vec{p}_e^{(t+1)}
|
||||
\end{align}
|
||||
|
||||
.. note::
|
||||
|
||||
For the first time step, \\\(t=0\\\), the term: \\\(\\frac{1}{\\delta t} \\MfMui \\vec{y}_b^{(0)}\\\) is zero.
|
||||
|
||||
|
||||
|
||||
|
||||
Implementing **J** transpose times a vector
|
||||
-------------------------------------------
|
||||
|
||||
Multiplying \\(\\mathbf{J}^\\top\\) onto a vector can be broken into three steps
|
||||
|
||||
|
||||
* Compute \\(\\vec{p} = \\mathbf{Q}^\\top \\vec{v}\\)
|
||||
* Solve \\(\\hat{\\mathbf{A}}^\\top \\vec{y} = \\vec{p}\\)
|
||||
* Compute \\(\\vec{w} = -\\mathbf{G}^\\top y\\)
|
||||
|
||||
|
||||
.. math::
|
||||
|
||||
\mathbf{\hat{A}}^\top = \left[
|
||||
\begin{array}{cccc}
|
||||
A & B & & \\
|
||||
& \ddots & \ddots & \\
|
||||
& & A & B \\
|
||||
& & 0 & A
|
||||
\end{array}
|
||||
\right]
|
||||
|
||||
For the all time-steps (going backwards in time):
|
||||
|
||||
|
||||
.. math::
|
||||
|
||||
A \vec{y}^{(t)} + B \vec{y}^{(t+1)} = \vec{p}^{(t)}
|
||||
|
||||
|
||||
.. math::
|
||||
|
||||
\begin{align}
|
||||
\frac{1}{\delta t} \MfMui\vec{y}_{b}^{(t)} + \MfMui\dcurl \vec{y}_{e}^{(t)}
|
||||
- \frac{1}{\delta t} \MfMui \vec{y}_{b}^{(t+1)}
|
||||
= \vec{p}_b^{(t)} \\
|
||||
\dcurl^\top \MfMui \vec{y}_b^{(t)} - \MeSig \vec{y}_e^{(t)} = \vec{p}_e^{(t)}
|
||||
\end{align}
|
||||
|
||||
and
|
||||
|
||||
.. math::
|
||||
|
||||
\begin{align}
|
||||
\left( \MfMui \dcurl \MeSig^{-1} \dcurl^\top \MfMui + \frac{1}{\delta t} \MfMui \right) \vec{y}_{b}^{(t)} =
|
||||
\frac{1}{\delta t} \MfMui \vec{y}_b^{(t+1)}
|
||||
+ \MfMui \dcurl \MeSig^{-1} \vec{p}_e^{(t)} + \vec{p}_b^{(t)} \\
|
||||
\vec{y}_e^{(t)} = \MeSig^{-1} \dcurl^\top \MfMui \vec{y}_b^{(t)} - \MeSig^{-1} \vec{p}_e^{(t)}
|
||||
\end{align}
|
||||
|
||||
|
||||
.. note::
|
||||
|
||||
For the last time step, \\\(t=N\\\), the term: \\\(\\frac{1}{\\delta t} \\MfMui \\vec{y}_b^{(N+1)}\\\) is zero.
|
||||
|
||||
|
||||
|
||||
TDEM - B formulation
|
||||
====================
|
||||
|
||||
.. automodule:: SimPEG.EM.TDEM.TDEM_b
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
|
||||
Field Storage
|
||||
=============
|
||||
|
||||
.. autoclass:: SimPEG.EM.TDEM.BaseTDEM.FieldsTDEM
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
:inherited-members:
|
||||
|
||||
|
||||
TDEM Survey Classes
|
||||
===================
|
||||
|
||||
.. autoclass:: SimPEG.EM.TDEM.SurveyTDEM.SurveyTDEM
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
:inherited-members:
|
||||
|
||||
|
||||
Base Classes
|
||||
============
|
||||
|
||||
.. automodule:: SimPEG.EM.TDEM.BaseTDEM
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
:inherited-members:
|
||||
|
||||
@@ -0,0 +1,34 @@
|
||||
simpegEM Utilities
|
||||
******************
|
||||
|
||||
SimPEG for EM provides a few EM specific utility codes,
|
||||
sources, and analytic functions.
|
||||
|
||||
Utilities for Electromagnetics
|
||||
==============================
|
||||
|
||||
.. automodule:: SimPEG.EM.Utils
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
:inherited-members:
|
||||
|
||||
|
||||
Analytic Functions - Time
|
||||
=========================
|
||||
|
||||
.. automodule:: SimPEG.EM.Analytics.TDEM
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
:inherited-members:
|
||||
|
||||
|
||||
Analytic Functions - Frequency
|
||||
==============================
|
||||
|
||||
.. automodule:: SimPEG.EM.Analytics.FDEM
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
:inherited-members:
|
||||
@@ -0,0 +1,33 @@
|
||||
Overview of Electromagnetics in SimPEG
|
||||
**************************************
|
||||
|
||||
|
||||
The API
|
||||
=======
|
||||
|
||||
Physical Properties
|
||||
-------------------
|
||||
|
||||
.. autoclass:: SimPEG.EM.Base.EMPropMap
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
Problem
|
||||
-------
|
||||
|
||||
.. autoclass:: SimPEG.EM.Base.BaseEMProblem
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
|
||||
Survey
|
||||
------
|
||||
|
||||
.. autoclass:: SimPEG.EM.Base.BaseEMSurvey
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
|
||||
@@ -0,0 +1,27 @@
|
||||
|
||||
Electromagnetics
|
||||
================
|
||||
|
||||
`SimPEG.EM` uses SimPEG as the framework for the forward and inverse
|
||||
electromagnetics geophysical problems.
|
||||
|
||||
To solve for predicted data, we follow the framework shown below. The model is
|
||||
what we invert for. This is mapped to a physical property on the simulation
|
||||
mesh. A source which is used to excite the system is specified. Having a model
|
||||
and a source, we can solve Maxwell's equations for fields. We sample these
|
||||
fields with recievers to give us predicted data.
|
||||
|
||||
|
||||
.. image:: ../../images/simpegEM_noMath.png
|
||||
:scale: 50%
|
||||
|
||||
|
||||
.. toctree::
|
||||
:maxdepth: 2
|
||||
|
||||
api_basic
|
||||
api_FDEM
|
||||
api_TDEM
|
||||
api_Utils
|
||||
|
||||
|
||||
Reference in New Issue
Block a user