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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).
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}
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.
78 \subsection{Usage}
79 The DC resistivity forward modelling classes are specified as follows:
81 \begin{classdesc}{WennerSurvey}{self, domain, primaryConductivity, secondaryConductivity,
82 current, a, midPoint, directionVector, numElectrodes}
83 \end{classdesc}
85 \begin{classdesc}{polepoleSurvey}{domain, primaryConductivity, secondaryConductivity,
86 current, a, midPoint, directionVector, numElectrodes}
87 \end{classdesc}
89 \begin{classdesc}{DipoleDipoleSurvey}{self, domain, primaryConductivity, secondaryConductivity,
90 current, a, n, midPoint, directionVector, numElectrodes}
91 \end{classdesc}
93 \begin{classdesc}{PoleDipoleSurvey}{self, domain, primaryConductivity, secondaryConductivity,
94 current, a, n, midPoint, directionVector, numElectrodes}
95 \end{classdesc}
97 \begin{classdesc}{SchlumbergerSurvey}{self, domain, primaryConductivity, secondaryConductivity,
98 current, a, n, midPoint, directionVector, numElectrodes}
99 \end{classdesc}
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}
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.
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}
133 \begin{methoddesc}[]{getElectrodes}{}
134 Returns a list containing the positions of the electrodes
135 \end{methoddesc}
137 \begin{methoddesc}[]{getApparentResistivityPrimary}{}
138 Returns a series of lists containing primary apparent resistivities one for each
139 value of $n$.
140 \end{methoddesc}
142 \begin{methoddesc}[]{getApparentResistivitySecondary}{}
143 Returns a series of lists containing secondary apparent resistivities one for each
144 value of $n$.
145 \end{methoddesc}
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}
152 The apparent resistivities are calculated by applying a geometric factor to the
153 measured potentials.

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