<commit_before>.. _TKC_DC_story:

DC resistivity
==============

.. topic:: Purpose

    Understand basic setup and physics of a direct current (DC) resistivity
    survey within the context of a kimberlite exploration. Run DC forward
    modelling and inversion using SimPEG-Static package.

.. image:: images/tkc/dc.png
    :width: 80%
    :align: center

Set-up
------

The physical behavior of DC resistivity survey is governed by the steady-state
Maxwell's equations:

.. math::
    \vec{j} = \sigma \vec{e}

.. math::
    \vec{e} = -\nabla \phi

.. math::
    \nabla \cdot \vec{j} = -\vec{j}_s = I_0 (\delta(\vec{r}-\vec{r}_+)-\delta(\vec{r}-\vec{r}_-))

.. math::
    \vec{j} \cdot \hat{n} \ \Big|_{\partial \Omega} = 0


where:
- :math:`\vec{j}`: Current density (A/m :math:`^2`)

- :math:`\vec{e}`: Electric field (V/m)

- :math:`I_0`: Current (A)

- :math:`\delta`: Volumetric delta function (m :math:`^{-3}`)


Consider a simple gradient array having a pair of A (+) and B (+) current
electrodes (Tx) with multiple M (+) and N (-) potential electrodes (Rx). Using
giant battery (?), we setup a significant potential difference allowing
electrical currents to flow between the A to B electrodes. If the earth
includes conductors or resistors, these will distorts current flow, and
measured potential differences on the surface electrodes (MN) will be
reflective of those distortions. Typically kimberlitic pipes (including those
containing diamonds!) will be more conductive than the background rock
(granitic), hence, the measured potential difference will be low. That is,
contrasts in electrical conductivity between different rocks induce anomalous
voltages. From the observed voltages, we want to estimate conductivity<commit_msg>fix typos in Maxwell's equations<commit_after>.. _TKC_DC_story:

DC resistivity
==============

.. topic:: Purpose

    Understand basic setup and physics of a direct current (DC) resistivity
    survey within the context of a kimberlite exploration. Run DC forward
    modelling and inversion using SimPEG-Static package.

.. image:: images/tkc/dc.png
    :width: 80%
    :align: center

Set-up
------

The physical behavior of DC resistivity survey is governed by the steady-state
Maxwell's equations:

.. math::
    \vec{j} = \sigma \vec{e}

.. math::
    \vec{e} = -\nabla \phi

.. math::
    \nabla \cdot \vec{j} = -\nabla \cdot \vec{j}_s = I_0 (\delta(\vec{r}-\vec{r}_+)-\delta(\vec{r}-\vec{r}_-))

.. math::
    \vec{j} \cdot \hat{n} \ \Big|_{\partial \Omega} = 0


where:
- :math:`\vec{j}`: Current density (A/m :math:`^2`)

- :math:`\vec{e}`: Electric field (V/m)

- :math:`I_0`: Current (A)

- :math:`\delta`: Volumetric delta function (m :math:`^{-3}`)


Consider a simple gradient array having a pair of A (+) and B (+) current
electrodes (Tx) with multiple M (+) and N (-) potential electrodes (Rx). Using
giant battery (?), we setup a significant potential difference allowing
electrical currents to flow between the A to B electrodes. If the earth
includes conductors or resistors, these will distorts current flow, and
measured potential differences on the surface electrodes (MN) will be
reflective of those distortions. Typically kimberlitic pipes (including those
containing diamonds!) will be more conductive than the background rock
(granitic), hence, the measured potential difference will be low. That is,
contrasts in electrical conductivity between different rocks induce anomalous
voltages. From the observed voltages, we want to estimate conductivity
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