 # Contents of /trunk/doc/cookbook/example11.tex

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Fri Oct 1 02:08:38 2010 UTC (10 years, 8 months ago) by ahallam
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Updates to cookbook, new chapter DC Resis and IP. Using new packages hyperref, natbib

 1 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 % 4 % Copyright (c) 2003-2010 by University of Queensland 5 % Earth Systems Science Computational Center (ESSCC) 6 7 % 8 % Primary Business: Queensland, Australia 9 % Licensed under the Open Software License version 3.0 10 11 % 12 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 13 14 In this chapter we will investigate the effects of a current flow and 15 resistivity in a medium. This type of problem is related to the DC resistivity method of 16 geophysical prospecting. Currents are injected into the ground at the surface 17 and measurements of the potential are taken at various potential-dipole 18 locations along or adjacent to the survey line. From these measurements of the 19 potential it is possible to infer an approximate apparent resistivity model of 20 the subsurface. 21 22 The following theory comes from a tutorial by \citet{Loke2004}. 23 We know from Ohm's law that the current flow in the ground is given in vector 24 form by 25 \begin{equation} 26 \vec{J}=\sigma\vec{E} 27 \end{equation} 28 where $\vec{J}$ is current density, $\vec{E}$ is the electric field intensity 29 and $\sigma$ is the conductivity. We can relate the potential to the electric 30 field intensity by 31 \begin{equation} 32 E=-\nabla\Phi 33 \end{equation} 34 where $\Phi$ is the potential. We now note that the current density is related 35 to the potential via 36 \begin{equation} 37 \vec{J}=-\sigma\nabla\Phi 38 \end{equation} 39 Geophysical surveys predominantly use current sources which individually act as 40 point poles. Considering our model will contain volumes, we can normalise the 41 input current and approximate the current density in a volume $\Delta V$ by 42 \begin{equation} 43 \nabla \vec{J} = 44 \left(\frac{I}{\Delta V} \right) \delta(x-x\hackscore{s}) 45 \delta(y-y\hackscore{s}) 46 \delta(z-z\hackscore{s}) 47 \end{equation} 48 49 \begin{equation} 50 -\nabla \cdot \left[ \sigma(x,y,z) \nabla \phi (x,y,z) \right] = 51 \left(\frac{I}{\Delta V} \right) \delta(x-x\hackscore{s}) 52 \delta(y-y\hackscore{s}) 53 \delta(z-z\hackscore{s}) 54 \end{equation} 55 56 This form is quite simple to solve in \esc. 57 58 \section{3D Current-Dipole Potential} 59 \sslist{example11m.py; example11c.py} 60 61 \begin{figure}[ht] 62 \centering 63 \includegraphics[width=0.95\textwidth]{figures/ex11cstreamline.png} 64 \caption{Current Density Model for layered medium.} 65 \label{fig:ex11cstream} 66 \end{figure} 67 68 \section{Frequency Dependent Resistivity - Induced Polarisation} 69 With a more complicated resistivity model it is possible to calculate the 70 chargeability or IP effect in the model. A recent development has been the 71 Fractal model for complex resistivity \citep{Farias2010,Honig2007}. 72 73 74 The model is calculated over many frequencyies and transformed to the time 75 domain using a discrete fourier transform. 76

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