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ahallam 
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% Copyright (c) 20032010 by University of Queensland 
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% Licensed under the Open Software License version 3.0 
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\section{Newtonian Potential} 
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In this chapter we examine the gravitational potential field due to source 
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bodies and geological structure. The use of vector data and visualisation in 
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Mayavi is investigated. 
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The gravitational potential $U$ at a point $P$ due to a region with a mass 
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distribution $\rho(P)$ is given by Poisson's equation 
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\begin{equation} \label{eqn:poisson} 
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\nabla^2 U(P) = 4\pi\gamma\rho(P) 
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\end{equation} 
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where $\gamma$ is the gravitational constant. 
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If this is now related to the simplified general form 
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\begin{equation} 
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\left(A\hackscore{jl} u\hackscore{,l} \right)\hackscore{,j} = Y 
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\end{equation} 
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where the LHS side is equivalent to 
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\begin{equation} \label{eqn:ex10a} 
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\nabla A \nabla u 
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\end{equation} 
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If $A=\delta\hackscore{jl}$ then equation \ref{eqn:ex10a} is equivalent to 
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\begin{equation*} 
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\nabla^2 u 
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\end{equation*} 
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and thus Poisson's Equation \ref{eqn:poisson} satisfies the general form when 
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\begin{equation} 
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A=\delta\hackscore{jl} \text{ and } Y= 4\pi\gamma\rho 
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\end{equation} 
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At least one boundary point must be set for the problem to be solvable. For this 
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example we have set all of the boundaries to zero. The normal flux condition is 
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also zero by default. 
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Setting the boundary condition is relatively simple using the \verb!q! and 
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\verb!r! variables of the general form, where 
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\begin{python} 
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q=whereZero(x[1]my)+whereZero(x[1])+whereZero(x[0])+whereZero(x[0]mx) 
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\end{python} 
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identifies the points on the boundary and \verb!r=0.0! sets the dirichlet 
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condition. 
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\section{Gravity Profile} 
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\sslist{example10a.py} 
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\begin{figure}[ht] 
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\centering 
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\includegraphics[width=0.75\textwidth]{figures/ex10apot.png} 
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\caption{Newtonian potential with g field directionality.} 
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\label{fig:ex10pot} 
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\end{figure} 
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\section{Gravity Well} 
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\sslist{example10b.py} 
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\begin{figure}[htp] 
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\centering 
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\includegraphics[width=0.75\textwidth]{figures/ex10bpot.png} 
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\caption{Gravity well with iso surfaces and streamlines of the vector 
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gravitational potential.} 
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\label{fig:ex10bpot} 
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\end{figure} 
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\section{Gravity Surface over a fault model.} 
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\sslist{example10c.py,example10m.py} 
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\begin{figure}[htp] 
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\centering 
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\subfigure[The geometry of the fault model in example10c.py.] 
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{\label{fig:ex10cgeo} 
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\includegraphics[width=0.8\textwidth]{figures/ex10potfaultgeo.png}} \\ 
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\subfigure[The fault of interest from the fault model in 
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example10c.py.] 
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{\label{fig:ex10cmsh} 
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\includegraphics[width=0.8\textwidth]{figures/ex10potfaultmsh.png}} 
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\end{figure} 
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\begin{figure}[htp] 
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\centering 
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\includegraphics[width=0.8\textwidth]{figures/ex10cpot.png} 
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\caption{Gravitational potential of the fault model with primary layers and 
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faults identified as isosurfaces.} 
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\label{fig:ex10cpot} 
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\end{figure} 
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