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1 \begin{figure}
2 \includegraphics[width=\textwidth]{figures/FaultSystem2D}
3 \caption{\label{FAULTSYSTEM2D}Two dimensional fault system with one fault named `t` in the $(x\hackscore{0},x\hackscore{1})$ and its parameterization in the
4 $w\hackscore{0}$ space. The fault has three segments.}
5 \end{figure}
7 \section{Fault System}
8 \label{Fault System}
9 The \class{FaultSystem} is an easy to use interface to handle 2D and 3D fault systems \index{faults} as used for instance in simulating fault ruptures. The main purpose of the class is to provide a parameterization of an individual fault in the system of fault. In case of a 2D fault the fault is parameterized by a single value $w\hackscore{0}$ and in the case of a 3D fault two parameters $w\hackscore{0}$ and $w\hackscore{1}$ are used. This parameterization can be used
10 to impose data (e.g. a slip distribution) onto the fault. It can also be a useful tool to visualize or analyze the results on the fault if the fault is not straight.
12 A fault $t$ in the fault system is represented by two polygons $(V^{ti})$ and $(v^{ti})$
13 defining the top and bottom line of the fault $t$.
14 $V^{ti}$ defines the $i$-th fault vertex at the top of the fault (typically at the surface of the Earth) and
15 $v^{ti}$ defines the $i$-th fault vertex at the bottom of the fault. Both polygons need to contain the same number of
16 vertices. For the 2D case the polygon $(v^{ti})$ for the bottom edge of the fault is dropped.
17 The patch with the vertices
18 $V^{t(i-1)}$, $V^{ti}$
19 $v^{t(i-1)}$, and $v^{ti}$
20 is called the $i$-th segment of the fault `t`. In 2D the the line with the start point $V^{t(i-1)}$
21 and the end point $V^{ti}$ is called the $i$-th segment, see Figure~\ref{FAULTSYSTEM2D}.
23 In general a fault does not define a plane surface (or a straight line in 2D). In order to simplify working on
24 a fault in a fault system a parameterization $P^t: (w\hackscore{0},w\hackscore{1}) \rightarrow (x\hackscore{0},x\hackscore{1},x\hackscore{2})$ over a rectangular domain is introduced such that
25 \begin{equation}
26 0\le w\hackscore{0} \le w^t\hackscore{0 max} \mbox{ and } -w^t\hackscore{1max}\le w\hackscore{1} \le 0
27 \label{eq:fault 1}
28 \end{equation}
29 with positive if numbers $w^t\hackscore{0 max}$ and $w^t\hackscore{1 max}$. Typically one chooses
30 $w^t\hackscore{0 max}$ to be the unrolled length of the fault
31 $w^t\hackscore{1 max}$ to be the mean value of segment depth. Moreover we have
32 \begin{equation}
33 P^t(W^{ti})=V^{ti}\mbox{ and } P^t(w^{ti})=v^{ti}\
34 \label{eq:fault 2}
35 \end{equation}
36 where
37 \begin{equation}
38 W^{ti}=(d^{ti},0) \mbox{ and } w^{ti}=(d^{ti},-w^t\hackscore{1 max})
39 \label{eq:fault 3}
40 \end{equation}
41 and $d^{ti}$ is the unrolled distance of $W^{ti}$ from $W^{t0}$. In the 2D case $w^t\hackscore{1 max}$ is set to zero and therefore the second component is dropped, see Figure~\ref{FAULTSYSTEM2D}.
43 In the case of 2D the parameterization $P^t$ is constructed as follows:
44 The line connecting $V^{t(i-1)}$ and $V^{ti}$ is given by
45 \begin{equation}
46 x=V^{t(i-1)} + s \cdot (V^{ti}-V^{t(i-1)})
47 \label{eq:2D line 1}
48 \end{equation}
49 where $s$ is between $0$ and $1$. The point $x$ is on $i$-th fault segment if and only if
50 such an $s$ exists. If assume $x$ is on the fault one can calculate $s$ as
51 \begin{equation}
52 s = \frac{ (x- V^{t(i-1)})^t \cdot (V^{ti}-V^{t(i-1)}) }{\|V^{ti}-V^{t(i-1)}\|^2}
53 \label{eq:2D line 1b}
54 \end{equation}
55 We then can set
56 \begin{equation}
57 w\hackscore{0}=d^{ti}+s \cdot (d^{ti}-d^{t(i-1)})
58 \label{eq:2D line 2}
59 \end{equation}
60 to get $P^t(w\hackscore{0})=x$.
61 It remains the question if the given $x$ is actual on the segment $i$ of fault $t$. To test this $s$ is restricted
62 between $0$ and $1$ (so if $s<0$ $s$ is set to $0$ and if $s>1$ $s$ is set to $1$) and the we check the
63 residual of equation~\ref{eq:2D line 1}, ie. $x$ is been accepted to be in the segment if
64 \begin{equation}
65 \|x-V^{t(i-1)} - s \cdot (V^{ti}-V^{t(i-1)}) \| \le tol \cdot max(\|V^{ti}-V^{t(i-1)}\|, \|x-V^{t(i-1)} \|)
66 \label{eq:2D line 3}
67 \end{equation}
68 where $tol$ is a given tolerance.
72 \subsection{Functions}
74 \begin{classdesc}{FaultSystem}{
75 \optional{dim =3}}
76 Creates a fault system in the \var{dim} dimensional space.
77 \end{classdesc}
81 \begin{methoddesc}[FaultSystem]{getDim}{}
82 returns the spatial dimension
83 \end{methoddesc}
84 \begin{methoddesc}[FaultSystem]{getLength}{tag}
85 returns the unrolled length of fault \var{tag}.
86 \end{methoddesc}
88 \begin{methoddesc}[FaultSystem]{getDepth}{tag}
89 returns the medium depth of fault \var{tag}.
90 \end{methoddesc}
92 \begin{methoddesc}[FaultSystem]{getTags}{}
93 returns a list of the tags used by the fault system
94 \end{methoddesc}
96 \begin{methoddesc}[FaultSystem]{getW0Range}{tag}
97 returns the range of the parameterization in $w\hackscore{0}$.
98 For tag $t$ this is the pair $(d^{t0},d^{tn})$ where $n$ is the number of segments in the fault.
99 In most cases one has $(d^{t0},d^{tn})=(0,w^t\hackscore{0 max})$.
100 \end{methoddesc}
102 \begin{methoddesc}[FaultSystem]{getW1Range}{tag}
103 returns the range of the parameterization in $w\hackscore{1}$.
104 For tag $t$ this is the pair $(-w^t\hackscore{1max},0)$.
105 \end{methoddesc}
107 \begin{methoddesc}[FaultSystem]{getW0Offsets}{tag}
108 returns the offsets for the parameterization of fault \var{tag}.
109 For tag \var{tag}=$t$ this is the list $[d^{ti}]$.
110 \end{methoddesc}
113 \begin{methoddesc}[FaultSystem]{getFaultSegments}{tag}
114 returns the polygons used to describe fault \var{tag}. For \var{tag}=$t$ this is the list of the vertices
115 $[V^{ti}]$ for the 2D and the pair of lists of the top vertices $[V^{ti}]$ and the bottom vertices $[v^{ti}]$ in 3D.
116 Note that the coordinates are represented as \numpyNDA objects.
117 \end{methoddesc}
119 \begin{methoddesc}[FaultSystem]{getCenterOnSurface}{}
120 returns the center point of the fault system at the surfaces. In 3D the calculation of the center is
121 considering the top edge of the faults and projects the edge to the surface (the $x\hackscore{2}$ component is assumed to be 0). An \numpyNDA object is returned.
122 \end{methoddesc}
124 \begin{methoddesc}[FaultSystem]{getOrientationOnSurface}{}
125 returns the orientation of the fault system in RAD on the surface ($x\hackscore{2}=0$ plane) around the fault system center.
126 \end{methoddesc}
127 \begin{methoddesc}[FaultSystem]{transform}{\optional{rot=0, \optional{shift=numpy.zeros((3,)}}}
128 applies a shift \var{shift} and a consecutive rotation in the $x\hackscore{2}=0$ plane.
129 \var{rot} is a float number and \var{shift} an \numpyNDA object.
130 \end{methoddesc}
132 \begin{methoddesc}[FaultSystem]{addFault}{top, tag \optional{, bottom=None \optional{, w0_offsets=None\optional{, w1_max=None}}}}
133 adds the fault \var{tag} to the fault system.
135 \var{top} is the list of the vertices defing the top of the fault
136 while \var{bottom} is the list of the vertices defing the bottom of the fault.
137 In the case of 2D \var{bottom} must not be present. Both list, if present, must have the same length.
138 \var{w1_max} defines the range of the $w\hackscore{1}$. If not present the mean value over the depth of
139 all segment edges in the fault is used.
140 \var{w0_offsets} sets the offsets $d^{ti}$. If not present it i chosen such that $d^{ti}-d^{t(i-1)}$ is the length of the $i$-th segment. In some cases, eg. when kinks in the fault are relevant, it can be useful
141 to explicitly specify the offsets in order to simplify the assignment of values.
142 \end{methoddesc}
144 \begin{methoddesc}[FaultSystem]{getMaxValue}{f\optional{, tol=1.e-8}}
145 returns the maximum value of \var{f}, the fault where the maximum is found and the location on the fault in fault coordinates. \var{f} must be a \Scalar. When the maximum is calculated only \DataSamplePoints are considered
146 which are on a fault in the fault system in the sense of condition~\ref{eq:2D line 3} or \ref{eq:3D line 3}, respectively. In the case no \DataSamplePoints is found the returned tag is \var{None} and
147 the maximum value as well as the location of the maximum value are undefined.
148 \end{methoddesc}
150 \begin{methoddesc}[FaultSystem]{getParametrization}{x,tag \optional{\optional{, tol=1.e-8}, outsider=None}}
151 returns the argument $w$ of the parameterization $P^t$ for \var{tag}=$t$ to provide \var{x}
152 together with a mask indicating where the given location if on a fault in the fault system by the value $1$ (otherwise the value is set to $0$). \var{x} needs to be \Vector or \numpyNDA object. \var{tol} defines the tolerance to decide if a given \DataSamplePoints is on fault \var{tag}. The value
153 \var{outside} is the value used as a replacement value for $w$ where the corresponding value in \var{x} is not
154 on a fault. If not \var{outside} is not present an appropriate value is used.
155 \end{methoddesc}
157 \begin{methoddesc}[FaultSystem]{getSideAndDistance}{x,tag}
158 returns the side and the distance at locations \var{x} from the fault \var{tag}.
159 \var{x} needs to be \Vector or \numpyNDA object. Positive values for side means that the corresponding location is
160 to the right of the fault, a negative value means that the corresponding location is
161 to the left of the fault. The value zero means that the side is undefined.
162 \end{methoddesc}
164 \subsection{Example}
165 See section~\ref{Slip CHAP}

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