/[escript]/release/4.0/doc/inversion/CookDcRes.tex
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16  \chapter{DC Resistivity Forward modelling}\label{Chp:cook:Dc Resistivity inversion}  \chapter{DC Resistivity Forward modelling}\label{Chp:cook:Dc Resistivity inversion}
17  \section{Introduction}  \section{Introduction}
18  Dc resistivity involves placing electrodes into the ground and injecting a current  DC resistivity surveys involve placing electrodes into the ground and injecting a current
19  into them. The current propagates through the ground and generates a potential field.  into them. The current propagates through the ground and generates a potential field.
20  The change in potential between a set of electrodes can then be measured. There are  The change in potential between a set of electrodes can then be measured.
21  a number of different ways to set up dc resistivity survey these can be found in  
22  \cite[pg 5]{LOKE2014}. The aim of the different survey set-ups is to gather more  As the separation between the electrodes increases, the current has a longer
 spacial information. Some surveys include a separation multiplier n this is because  
 as the separation between the electrodes increases, the current has a longer  
23  distance to travel and will consequently travel deeper. This is not always true  distance to travel and will consequently travel deeper. This is not always true
24  as a highly conductive layer will keep the current close to the surface.  as a highly conductive layer will keep the current close to the surface.
25    
26  The final objective is to perform an inversion and develop a resistivity profile of the subsurface.  The final objective is to perform an inversion and develop a resistivity profile of the subsurface.
27  This profile can then be matched up to known resistivities of material and used  This profile can then be matched up to known resistivities of material and used
28  to make inferences about the material contained in the subsurface.  to make inferences about the material contained in the subsurface.
29    There are
30    a number of different ways to set up a DC resistivity survey, each providing
31    different spatial information\cite[pg 5]{LOKE2014}.
32    
33  Escript currently supports the forward modelling of dc resistivity. Forward modelling  Escript currently supports the forward modelling of DC resistivity surveys. Forward modelling
34  involves performing a survey artificially by solving the pde's which describe the underlying  involves performing a survey artificially by solving the PDEs which describe the underlying
35  physics. Escript provides a number of classes for solving forward modelling problems, these are  physics. Escript provides a number of classes for solving forward modelling problems, these are
36  detailed in section \ref{sec:forward DCRES}.  detailed in section \ref{sec:forward DCRES}.
37    
38  \section{Example}  \section{Example}
39  In this section we will look at an example forward problem. The domain consists of  In this section we will look at an example forward problem. The domain consists of
40  a homogenous half space with a half sphere embedded see figure \ref{fig:HalfSphere}.  a homogenous half-space with a half-sphere embedded within it (Figure~\ref{fig:HalfSphere}).
41  In following script~\ref{code: dc1}\footnote{The script is similar to  In this example\footnote{The script is similar to
42  \examplefile{dc_forward.py} within the \escript example file directory.} run a  \examplefile{dc_forward.py} within the \escript example file directory.}
43  Schlumberger survey on this domain.   a Schlumberger survey is used.
44    
45    
46  \begin{figure}  \begin{figure}
47  \centering  \centering
# Line 62  pi = math.pi Line 65  pi = math.pi
65    
66  #Setup Input  #Setup Input
67  mesh_file = "data/HalfSphere_v1.4.msh"  mesh_file = "data/HalfSphere_v1.4.msh"
68  # Tag volume names and conductivity values (Sm/m) for primary and secondary potential:  # Tag volume names and conductivity values (S/m) for primary and secondary potential:
69  tag_p = {"domain" : 1/10.0, "sphere" :  1/10.0} # Primary (homogeneous).  tag_p = {"domain" : 1/10.0, "sphere" :  1/10.0} # Primary (homogeneous).
70  tag_s = {"domain" : 1/10.0, "sphere" :  1/1.0 } # Secondary.  tag_s = {"domain" : 1/10.0, "sphere" :  1/1.0 } # Secondary.
71    
72  xe_0 = -5.0 # start X-coordinate  xe_0 = -5.0 # start X-coordinate
73  numEle =  21  # number of electrodes  numEle =  21  # number of electrodes
74  estp =  0.5 # step size  a =  0.5 # step size
75  n=9 #  n = 9 #max electrode step
76  midPoint = [xe_0 + (((numEle-1)*estp)/2), 0, 0]  midPoint = [xe_0 + (((numEle-1)*estp)/2), 0, 0]
77  current = 1.0 # (Ampere)  current = 1.0 # (Ampere)
78  domain = finley.ReadGmsh(mesh_file, 3)  domain = finley.ReadGmsh(mesh_file, 3)
# Line 92  for tag in tag_p: Line 95  for tag in tag_p:
95  sig_p.expand()  sig_p.expand()
96  sig_s.expand()  sig_s.expand()
97  #solve for result  #solve for result
98  schs=SchlumbergerSurvey(domain, sig_p, sig_s, current, estp,n, midPoint, directionVector, nume)  schs=SchlumbergerSurvey(domain, sig_p, sig_s, current, a, n, midPoint, directionVector, numEle)
99  pot=schs.getPotential()  pot=schs.getPotential()
100  totalApparentRes=schs.getApparentResistivityTotal()  totalApparentRes=schs.getApparentResistivityTotal()
101  #print result  #print result
# Line 106  for i in totalApparentRes: Line 109  for i in totalApparentRes:
109  \end{pyc}  \end{pyc}
110    
111    
 The example starts with by loading in a preprepared gmsh model which describes  
 the domain. the gmsh script can be found in \examplefile{data/HalfSphere_v1.4.geo}  
 gmsh can be used to generate the msh file. The values for for primary and secondary  
 conductivity is specified in the different regions. these regions have been tagged  
 in the gmsh script. The survey is set-up to have 21 electrodes spanning from -5 to 5  
 in x. Finally the potentials and total apparent resistivity is calculated. The Schlumberger  
 survey starts with the first 4 electrodes, with the electrode 1 and 4 as current electrodes and  
 2 and 3 as potential electrodes. The electrodes used are all incremented by why before another  
 measurement is taken. This is repeated until the last 4 electrodes are used. The value of n is then increased  
 causing the  separation between the current and potential electrodes to become 2 $\times$ a rather than a  
 such that electrode 1 and 6 are current electrodes and electrodes 3 and 4 are potential electrodes.  
 this process is then repeated until the maximal value of n is achieved.  
   
112    The example begins with constructing the domain, loaded from a pre-prepared
113    \emph{gmsh} model. The \emph{gmsh} script used can be found in \examplefile{data/HalfSphere_v1.4.geo}.
114    \emph{gmsh} can be used to generate the \texttt{msh} file. The values for primary and secondary
115    conductivity, in siemens per meter, are specified for the different regions. These regions have been tagged
116    in the \emph{gmsh} script. The survey is constructed to have 21 electrodes spanning from -5m to 5m
117    in the $x$-axis, with a fixed interval between each electrode and the next in
118    line. These electrodes, once placed, are not moved for the remainder of the
119    survey. The potentials and total apparent resistivity is then calculated.
120    
121    The SchlumbergerSurvey class uses four electrodes at a time, beginning with the
122    electrode at location \texttt{xe_0}. In each set of four electrodes, electrodes
123    1 and 4 are used as current electrodes and
124    2 and 3 used as potential electrodes. The next set of electrodes are then used
125    for the next measurement, beginning with the previous electrode 2. This is
126    repeated until the last four electrodes are used.
127    
128    This process is itself repeated with a continually increasing step size between
129    the electrodes pairs at each end of the set. As an example, the first time the
130    process is repeated, the initial set will be made up of electrodes 1, 3, 4, and
131    6. During the second repeat the initial set will be electrodes 1, 4, 5, and 8.
132    The maximal step size in the above script is given as $n$.
133    

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