# Contents of /release/4.0/doc/inversion/ForwardDCRES.tex

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 1 %!TEX root = inversion.tex 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 % Copyright (c) 2003-2014 by University of Queensland 4 5 % 6 % Primary Business: Queensland, Australia 7 % Licensed under the Open Software License version 3.0 8 9 % 10 % Development until 2012 by Earth Systems Science Computational Center (ESSCC) 11 % Development 2012-2013 by School of Earth Sciences 12 % Development from 2014 by Centre for Geoscience Computing (GeoComp) 13 % 14 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 15 16 \section{DC resistivity inversion: 3D}\label{sec:forward DCRES} 17 This section will discuss DC resistivity\index{DC forward} forward modelling, as well as an \escript 18 class which allows for solutions of these forward problems. The DC resistivity 19 forward problem is modelled via the application of Ohm's Law to the flow of current 20 through the ground. When sources are treated as a point sources and Ohm's Law 21 is written in terms of the potential field, the equation becomes: 22 \begin{equation} \label{ref:dcres:eq1} 23 \nabla \cdot (\sigma \nabla \phi) = -I \delta(x-x_s) \delta(y-y_s) \delta(z-z_s) 24 \end{equation} 25 Where $(x,y,z)$ and $(x_s, y_s, z_s)$ are the coordinates of the observation and source 26 points respectively. The total potential, $\phi$, is split into primary and secondary 27 potentials $\phi = \phi_p + \phi_s$, where the primary potential is analytically calculated 28 as a flat half-space background model with conductivity of $\sigma_p$. 29 The secondary potential is due to conductivity deviations 30 from the background model and has its conductivity denoted as $\sigma_s$. 31 This approach effectively removes the singularities of the Dirac delta 32 source and provides more accurate results \cite{rucker2006three}. 33 An analytical solution is available for the primary potential of a uniform half-space due to a single pole source and is given by: 34 \begin{equation} \label{ref:dcres:eq2} 35 \phi_p = \frac{I}{2 \pi \sigma_1 R} 36 \end{equation} 37 Where $I$ is the current and $R$ is the distance from the observation points to the source. 38 In \escript the observation points are the nodes of the domain and $R$ is given by 39 \begin{equation} \label{ref:dcres:eq3} 40 R = \sqrt{(x-x_s)^2+(y-y_s)^2 + z^2} 41 \end{equation} 42 The secondary potential, $\phi_s$, is given by 43 \begin{equation}\label{ref:dcres:eq4} 44 -\mathbf{\nabla}\cdot\left(\sigma\,\nabla \phi_s \right) = 45 \mathbf{\nabla}\cdot\left( \left(\sigma_p-\sigma\right)\,\nabla \phi_p \right) 46 \end{equation} 47 where $\sigma_p$ is the conductivity of the background half-space. 48 The weak form of above PDE is given by multiplication of a suitable test function, $w$, and integrating over the domain $\Omega$: 49 \begin{multline}\label{ref:dcres:eq5} 50 -\int_{\partial\Omega} \sigma\,\nabla \phi_s \cdot \hat{n} w\,ds + 51 \int_{\Omega} \sigma\,\nabla \phi_s \cdot \nabla w\,d\Omega =\\ 52 -\int_{\partial\Omega} \left(\sigma_p-\sigma\right)\,\nabla \phi_p 53 \cdot \hat{n} w\,ds + \int_{\Omega} \left(\sigma_p-\sigma\right)\,\nabla \phi_p \cdot \nabla w\,d\Omega 54 \end{multline} 55 The integrals over the domain boundary provide the boundary conditions which are 56 implemented as Dirichlet conditions (i.e. zero potential) at all interfaces except the 57 top, where Neumann conditions apply (i.e. no current flux through the air-earth interface). 58 From the integrals over the domain, the \escript coefficients can be deduced: the 59 left-hand-side conforms to \escript coefficient $A$, whereas the right-hand-side agrees 60 with the coefficient $X$ (see User Guide). 61 62 A number a of different configurations for electrode set-up are available \cite[pg 5]{LOKE2014}. 63 An \escript class is provided for each of the following survey types: 64 \begin{itemize} 65 \item Wenner alpha 66 \item Pole-Pole 67 \item Dipole-Dipole 68 \item Pole-Dipole 69 \item Schlumberger 70 \end{itemize} 71 72 These configurations are comprised of at least one pair of current and potential 73 electrodes separated by a distance $a$. In those configurations which use $n$, 74 electrodes in the currently active set may be separated by $na$. In the classes 75 that follow, the specified value of $n$ is an upper limit. That is $n$ will 76 start at 1 and iterate up to the value specified. 77 78 \subsection{Usage} 79 The DC resistivity forward modelling classes are specified as follows: 80 81 \begin{classdesc}{WennerSurvey}{self, domain, primaryConductivity, secondaryConductivity, 82 current, a, midPoint, directionVector, numElectrodes} 83 \end{classdesc} 84 85 \begin{classdesc}{polepoleSurvey}{domain, primaryConductivity, secondaryConductivity, 86 current, a, midPoint, directionVector, numElectrodes} 87 \end{classdesc} 88 89 \begin{classdesc}{DipoleDipoleSurvey}{self, domain, primaryConductivity, secondaryConductivity, 90 current, a, n, midPoint, directionVector, numElectrodes} 91 \end{classdesc} 92 93 \begin{classdesc}{PoleDipoleSurvey}{self, domain, primaryConductivity, secondaryConductivity, 94 current, a, n, midPoint, directionVector, numElectrodes} 95 \end{classdesc} 96 97 \begin{classdesc}{SchlumbergerSurvey}{self, domain, primaryConductivity, secondaryConductivity, 98 current, a, n, midPoint, directionVector, numElectrodes} 99 \end{classdesc} 100 101 \noindent Where: 102 \begin{itemize} 103 \item \texttt{domain} is the domain which represent the half-space of interest. 104 it is important that a node exists at the points where the electrodes will be placed. 105 \item \texttt{primaryConductivity} is a data object which defines the primary conductivity 106 it should be defined on the ContinuousFunction function space. 107 \item \texttt{secondaryConductivity} is a data object which defines the secondary conductivity 108 it should be defined on the ContinuousFunction function space. 109 \item \texttt{current} is the value of the injection current to be used in amps this is a currently a 110 constant. 111 \item \texttt{a} is the electrode separation distance. 112 \item \texttt{n} is the electrode separation distance multiplier. 113 \item \texttt{midpoint} is the centre of the survey. Electrodes will spread from this point 114 in the direction defined by the direction vector and in the opposite direction, placing 115 half of the electrodes on either side. 116 \item \texttt{directionVector} defines as the direction in which electrodes are spread. 117 \item \texttt{numElectrodes} is the number of electrodes to be used in the survey. 118 \end{itemize} 119 120 When calculating the potentials the survey is moved along the set of electrodes. 121 The process of moving the electrodes along is repeated for each consecutive value of $n$. 122 As $n$ increases less potentials are calculated, this is because a greater spacing is 123 required and hence some electrodes are skipped. The process of building up these 124 pseudo-sections is covered in greater depth by Loke (2014)\cite[pg 19]{LOKE2014}. 125 These classes all share common member functions described below. For the surveys 126 where $n$ is not specified only one list will be returned. 127 128 \begin{methoddesc}[]{getPotential}{} 129 Returns 3 lists, each made up of a number of lists containing primary, secondary and total 130 potential differences. Each of the lists contains $n$ sublists. 131 \end{methoddesc} 132 133 \begin{methoddesc}[]{getElectrodes}{} 134 Returns a list containing the positions of the electrodes 135 \end{methoddesc} 136 137 \begin{methoddesc}[]{getApparentResistivityPrimary}{} 138 Returns a series of lists containing primary apparent resistivities one for each 139 value of $n$. 140 \end{methoddesc} 141 142 \begin{methoddesc}[]{getApparentResistivitySecondary}{} 143 Returns a series of lists containing secondary apparent resistivities one for each 144 value of $n$. 145 \end{methoddesc} 146 147 \begin{methoddesc}[]{getApparentResistivityTotal}{} 148 Returns a series of lists containing total apparent resistivities, one for each 149 value of $n$. This is generally the result of interest. 150 \end{methoddesc} 151 152 The apparent resistivities are calculated by applying a geometric factor to the 153 measured potentials. 154