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In this chapter we will investigate the effects of a current flow and 
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resistivity in a medium. This type of problem is related to the DC resistivity method of 
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geophysical prospecting. Currents are injected into the ground at the surface 
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and measurements of the potential are taken at various potentialdipole 
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locations along or adjacent to the survey line. From these measurements of the 
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potential it is possible to infer an approximate apparent resistivity model of 
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the subsurface. 
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The following theory comes from a tutorial by \citet{Loke2004}. 
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We know from Ohm's law that the current flow in the ground is given in vector 
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form by 
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\begin{equation} 
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\vec{J}=\sigma\vec{E} 
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\end{equation} 
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where $\vec{J}$ is current density, $\vec{E}$ is the electric field intensity 
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and $\sigma$ is the conductivity. We can relate the potential to the electric 
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field intensity by 
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\begin{equation} 
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E=\nabla\Phi 
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\end{equation} 
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where $\Phi$ is the potential. We now note that the current density is related 
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to the potential via 
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\begin{equation} 
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\vec{J}=\sigma\nabla\Phi 
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\end{equation} 
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Geophysical surveys predominantly use current sources which individually act as 
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point poles. Considering our model will contain volumes, we can normalise the 
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input current and approximate the current density in a volume $\Delta V$ by 
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\begin{equation} 
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\nabla \vec{J} = 
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\left(\frac{I}{\Delta V} \right) \delta(xx_{s}) 
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\delta(yy_{s}) 
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\delta(zz_{s}) 
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\end{equation} 
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\begin{equation} 
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\nabla \cdot \left[ \sigma(x,y,z) \nabla \phi (x,y,z) \right] = 
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\left(\frac{I}{\Delta V} \right) \delta(xx_{s}) 
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\delta(yy_{s}) 
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\delta(zz_{s}) 
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\end{equation} 
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This form is quite simple to solve in \esc. 
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\section{3D CurrentDipole Potential} 
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\sslist{example11m.py; example11c.py} 
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\begin{figure}[ht] 
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\centering 
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\includegraphics[width=0.95\textwidth]{figures/ex11cstreamline.png} 
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\caption{Current Density Model for layered medium.} 
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\label{fig:ex11cstream} 
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\end{figure} 
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\section{Frequency Dependent Resistivity  Induced Polarisation} 
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With a more complicated resistivity model it is possible to calculate the 
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chargeability or IP effect in the model. A recent development has been the 
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Fractal model for complex resistivity \citep{Farias2010,Honig2007}. 
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The model is calculated over many frequencies and transformed to the time 
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domain using a discrete fourier transform. 
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