mirror of
https://github.com/wassname/simpeg.git
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Lindsey Heagy
158 lines
4.0 KiB
ReStructuredText
158 lines
4.0 KiB
ReStructuredText
.. 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{\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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\renewcommand{\Div}{{\mathbf {Div}}}
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\renewcommand{\Grad}{{\mathbf {Grad}}}
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\newcommand{\St}{{\mathbf \Sigma_\tau}}
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\newcommand{\diag}{\mathbf{diag}}
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\newcommand{\M}{{\mathbf M}}
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\newcommand{\Me}{{\M^e}}
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\newcommand{\Mes}[1]{{\M^e_{#1}}}
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\newcommand{\be}{\mathbf{e}}
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\newcommand{\bj}{\mathbf{j}}
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\newcommand{\bphi}{\mathbf{\phi}}
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\newcommand{\bq}{\mathbf{q}}
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\newcommand{\bJ}{\mathbf{J}}
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\newcommand{\bG}{\mathbf{G}}
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\newcommand{\bP}{\mathbf{P}}
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\newcommand{\bA}{\mathbf{A}}
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\newcommand{\bm}{\mathbf{m}}
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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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\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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Direct Current Resistivity
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**************************
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`SimPEG.DCIP` uses SimPEG as the framework for the forward and inverse
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direct current (DC) resistivity and induced polarization (IP) geophysical problems.
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DC resistivity survey
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=====================
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Electrical resistivity of subsurface materials is measured by causing an electrical current to flow in the earth between one pair of electrodes while the voltage across a second pair of electrodes is measured. The result is an "apparent" resistivity which is a value representing the weighted average resistivity over a volume of the earth. Variations in this measurement are caused by variations in the soil, rock, and pore fluid electrical resistivity. Surveys require contact with the ground, so they can be labour intensive. Results are sometimes interpreted directly, but more commonly, 1D, 2D or 3D models are estimated using inversion procedures (`GPG <http://www.eos.ubc.ca/courses/eosc350/content/>`_).
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Background
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==========
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As direct current (DC) implies, in DC resistivity survey, we assume steady-state. We consider Maxwell's equations in steady state as
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.. math::
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\curl \frac{1}{\mu} \vec{b} - \j = \j_s \\
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\curl \e = 0
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Then by taking \\(\\div\\) of the first equation, we have
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.. math::
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- \div\j = q \\
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where
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.. math::
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\div \j_s = q = I(\delta(\vec{r}-\vec{r}_{s+})-\delta(\vec{r}-\vec{r}_{s-}))
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Since \\(\\curl \\e = 0\\), we have
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.. math::
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\e = \grad \phi
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And by Ohm's law, we have
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.. math::
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\j = \sigma \grad \phi
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Finally, we can compute the solution of the system:
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.. math::
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- \div\j = q
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\j = \sigma \grad \phi
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\frac{\partial \phi}{\partial r}\Big|_{\partial \Omega_{BC}} = 0
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Discretization
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==============
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By using finite volume method (FVM), we discretize our system as
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.. math::
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-\Div \bj = \bq
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\diag(\Acf^{T}\sigma^{-1}) \bj = \Grad \bphi
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Here boundary condtions are embedded in the discrete differential operators. With some linear algebra we have
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.. math::
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\bA\bphi = -\bq
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where
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.. math::
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\bA = \Div (\diag(\Acf^{T}\sigma^{-1}))^{-1} \Grad
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By solving this linear equation, we can compute the solution of \\(\\phi\\). Based on this discretization, we derive sensitivity in discretized space. Sensitivity matrix can be in general can be written as
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.. math ::
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\bJ = -\bP\bA^{-1}\bG
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where
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.. math ::
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\bP: \text{Projection}
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\bJ = \bP\frac{\partial \phi}{\partial \bm}
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Here \\(\\bm\\) indicates model parameters in discretized space.
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Verification
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============
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Comparing to the analytic function:
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.. plot::
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from SimPEG import Examples
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Examples.DC_Analytic_Dipole.run(plotIt=True)
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API for DC codes
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================
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.. automodule:: SimPEG.DCIP.BaseDC
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:show-inheritance:
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:members:
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:undoc-members:
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:inherited-members:
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