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16 \section{Fault System}
17 \label{Fault System}
18 The \class{FaultSystem} class provides an easy-to-use interface to handle 2D
19 and 3D fault systems\index{faults} as used for instance in simulating fault
20 ruptures. The main purpose of the class is to provide a parameterization of
21 an individual fault in the system of faults.
22 In case of a 2D fault the fault is parameterized by a single value $w_{0}$ and
23 in the case of a 3D fault two parameters $w_{0}$ and $w_{1}$ are used.
24 This parameterization can be used to impose data (e.g. a slip distribution)
25 onto the fault. It can also be a useful tool to visualize or analyze the
26 results on the fault if the fault is not straight.
28 \begin{figure}
29 \centering
30 \includegraphics{FaultSystem2D}
31 \caption{\label{FAULTSYSTEM2D}Two dimensional fault system with one fault
32 named `t` in the $(x_{0},x_{1})$ space and its parameterization in the
33 $w_{0}$ space. The fault has three segments.}
34 \end{figure}
36 A fault $t$ in the fault system is represented by a starting point $V^{t0}$
37 and series of directions, called strikes\index{strike}, and the lengths $(l^{ti})$.
38 The strike of segment $i$ is defined by the angle $\sigma^{ti}$ between the
39 $x_{0}$-axis and the direction of the fault, see Figure~\ref{FAULTSYSTEM2D}.
40 The length and strike defines the polyline $(V^{ti})$ of the fault by
41 \begin{equation}
42 V^{ti} = V^{t(i-1)} +
43 l^{ti} \cdot S^{ti}
44 \mbox{ with }
45 S^{ti} =
46 \left[
47 \begin{array}{c}
48 cos(\sigma^{ti}) \\
49 sin(\sigma^{ti}) \\
50 0
51 \end{array}
52 \right]
53 \label{eq:fault 00}
54 \end{equation}
55 In the 3D case each fault segment $i$ has an additional dip\index{dip}
56 $\theta^{ti}$ and at each vertex $i$ a depth $\delta^{ti}$ is given.
57 The fault segment normal $n^{ti}$ is given by
58 \begin{equation}
59 n^{ti} =
60 \left[
61 \begin{array}{c}
62 -sin(\theta^{ti}) \cdot S^{ti}_{1} \\
63 sin(\theta^{ti}) \cdot S^{ti}_{0} \\
64 cos(\theta^{ti})
65 \end{array}
66 \right]
67 \label{eq:fault 0}
68 \end{equation}
69 At each vertex we define a depth vector $d^{ti}$ defined as the intersect of
70 the fault planes of segment $(i-1)$ and $i$ where for the first segment and
71 last segment the vector orthogonal to strike vector $S^{ti}$\index{strike}
72 and the segment normal $n^{ti}$ is used. The direction $\tilde{d}^{ti}$ of the
73 depth vector is given as
74 \begin{equation}
75 \tilde{d}^{ti} = n^{ti} \times n^{t(i-1)}
76 \label{eq:fault b}
77 \end{equation}
78 If $\tilde{d}^{ti}$ is zero the strike vectors $L^{t(i-1)}$ and $L^{ti}$ are
79 collinear and we can set $\tilde{d}^{ti} = l^{ti} \times n^{ti}$.
80 If the two fault segments are almost orthogonal $\tilde{d}^{ti}$ is pointing
81 in the direction of $L^{t(i-1)}$ and $L^{ti}$. In this case no depth can be
82 defined. So we will reject a fault system if
83 \begin{equation}
84 min(\| \tilde{d}^{ti} \times L^{t(i-1)} \|,\| \tilde{d}^{ti} \times L^{ti} \|)
85 \le 0.1 \cdot \| \tilde{d}^{ti} |
86 \label{eq:fault c}
87 \end{equation}
88 which corresponds to an angle of less than $10^o$ between the depth vector and
89 the strike. We then set
90 \begin{equation}
91 d^{ti}=\delta^{ti} \cdot \frac{\tilde{d}^{ti}}{\|\tilde{d}^{ti}\|}
92 \label{eq:fault d}
93 \end{equation}
94 We can then define the polyline $(v^{ti})$ for the bottom of the fault as
95 \begin{equation}
96 v^{ti}= V^{ti}+d^{ti}
97 \label{eq:fault e}
98 \end{equation}
99 In order to simplify working on a fault $t$ in a fault system a
100 parameterization $P^t: (w_{0},w_{1}) \rightarrow (x_{0},x_{1},x_{2})$ over a
101 rectangular domain is introduced such that
102 \begin{equation}
103 0\le w_{0} \le w^t_{0 max} \mbox{ and } -w^t_{1max}\le w_{1} \le 0
104 \label{eq:fault 1}
105 \end{equation}
106 with positive numbers $w^t_{0 max}$ and $w^t_{1 max}$. Typically one chooses
107 $w^t_{0 max}$ to be the unrolled length of the fault and $w^t_{1 max}$ to be
108 the mean value of segment depth. Moreover we have
109 \begin{equation}
110 P^t(W^{ti})=V^{ti}\mbox{ and } P^t(w^{ti})=v^{ti}\
111 \label{eq:fault 2}
112 \end{equation}
113 where
114 \begin{equation}
115 W^{ti}=(\Omega^{ti},0) \mbox{ and } w^{ti}=(\Omega^{ti},-w^t_{1 max})
116 \label{eq:fault 3}
117 \end{equation}
118 and $\Omega^{ti}$ is the unrolled distance of $W^{ti}$ from $W^{t0}$, i.e.
119 $l^{ti}=\Omega^{t(i+1)}-\Omega^{ti}$. In the 2D case $w^t_{1 max}$ is set to
120 zero and therefore the second component is dropped, see Figure~\ref{FAULTSYSTEM2D}.
122 In the 2D case the parameterization $P^t$ is constructed as follows:
123 The line connecting $V^{t(i-1)}$ and $V^{ti}$ is given by
124 \begin{equation}
125 x=V^{ti} + s \cdot ( V^{t(i+1)}- V^{ti} )
126 \label{eq:2D line 1}
127 \end{equation}
128 where $s$ is between $0$ and $1$. The point $x$ is on $i$-th fault segment if
129 and only if such an $s$ exists. Assuming $x$ is on the fault it can be
130 calculated as
131 \begin{equation}
132 s = \frac{ (x- V^{ti})^t \cdot (V^{t(i+1)}- V^{ti}) }{ \|V^{t(i+1)}- V^{ti}\|^2}
133 \label{eq:2D line 1b}
134 \end{equation}
135 We then can set
136 \begin{equation}
137 w_{0}=\Omega^{ti}+s \cdot (\Omega^{ti}-\Omega^{t(i-1)})
138 \label{eq:2D line 2}
139 \end{equation}
140 to get $P^t(w_{0})=x$.
141 It remains the question if the given $x$ is actually on the segment $i$ of
142 fault $t$. To test this $s$ is restricted between $0$ and $1$ (so if $s<0$, $s$
143 is set to $0$ and if $s>1$, $s$ is set to $1$) and then we check the residual
144 of \eqn{eq:2D line 1}, i.e. $x$ has been accepted to be in the segment if
145 \begin{equation}
146 \|x-V^{ti} - s \cdot (V^{t(i+1)}- V^{ti}) \| \le tol \cdot
147 max(l^{ti}, \|x-V^{ti} \|)
148 \label{eq:2D line 3}
149 \end{equation}
150 where $tol$ is a given tolerance.
152 In the 3D case the situation is a bit more complicated: we split the fault
153 segment across the diagonal $V^{ti}$-$v^{t(i+1)}$ to produce two triangles.
154 In the upper triangle we use the parameterization
155 \begin{equation}
156 x= V^{ti} + s \cdot (V^{t(i+1)}-V^{ti}) + r \cdot (v^{t(i+1)}-V^{t(i+1)})
157 \mbox{ with } r \le s;
158 \label{eq:2D line 4}
159 \end{equation}
160 while in the lower triangle we use
161 \begin{equation}
162 x= V^{ti} + s \cdot (v^{t(i+1)}-v^{ti}) + r \cdot (v^{ti}-V^{ti})
163 \mbox{ with } s \le r;
164 \label{eq:2D line 4b}
165 \end{equation}
166 where $0\le s,r \le 1$. Both equations are solved in the least-squares sense
167 e.g. using the Moore-Penrose pseudo-inverse for the coefficient matrices.
168 The resulting $s$ and $r$ are then restricted to the unit square. Similar to
169 the 2D case (see \eqn{eq:2D line 3}) we identify $x$ to be in the upper
170 triangle of the segment if
171 \begin{equation}
172 \|x- V^{ti} - s \cdot (V^{t(i+1)}-V^{ti}) - r \cdot (v^{t(i+1)}-V^{t(i+1)}) \|
173 \le tol \cdot max(\|x-V^{ti} \|,\|v^{t(i+1)}-V^{t(i)})\|)
174 \label{eq:2D line 4c}
175 \end{equation}
176 and in the lower part
177 \begin{equation}
178 \|x-V^{ti} - s \cdot (v^{t(i+1)}-v^{ti}) - r \cdot (v^{ti}-V^{ti}) \|
179 \le tol \cdot max(\|x-V^{ti} \|,\|v^{t(i+1)}-V^{t(i)})\|)
180 \label{eq:2D line 4d}
181 \end{equation}
182 after the restriction of $(s,t)$ to the unit square.
183 Note that $\|v^{t(i+1)}-V^{t(i)})\|$ is the length of the diagonal of the
184 fault segment. For those $x$ which have been located in the $i$-th segment we
185 then set
186 \begin{equation}
187 w_{0}=\Omega^{ti}+s \cdot (\Omega^{ti}-\Omega^{t(i-1)})
188 \mbox{ and }
189 w_{1}=w^t_{1max} (r-1)
190 \label{eq:2D line 5}
191 \end{equation}
193 \subsection{Functions}
195 \begin{classdesc}{FaultSystem}{\optional{dim =3}}
196 creates a fault system in the \var{dim} dimensional space.
197 \end{classdesc}
199 \begin{methoddesc}[FaultSystem]{getMediumDepth}{tag}
200 returns the medium depth of fault \var{tag}.
201 \end{methoddesc}
203 \begin{methoddesc}[FaultSystem]{getTags}{}
204 returns a list of the tags used by the fault system.
205 \end{methoddesc}
207 \begin{methoddesc}[FaultSystem]{getStart}{tag}
208 returns the starting point of fault \var{tag} as a \numpyNDA object.
209 \end{methoddesc}
211 \begin{methoddesc}[FaultSystem]{getDim}{}
212 returns the spatial dimension.
213 \end{methoddesc}
215 \begin{methoddesc}[FaultSystem]{getDepths}{tag}
216 returns the list of the depths of the segments in fault \var{tag}.
217 \end{methoddesc}
219 \begin{methoddesc}[FaultSystem]{getTopPolyline}{tag}
220 returns the polyline used to describe the fault tagged by \var{tag}.
221 \end{methoddesc}
223 \begin{methoddesc}[FaultSystem]{getStrikes}{tag}
224 returns the list of strikes $\sigma^{ti}$ of the segments in fault
225 $t=$\var{tag}.
226 \end{methoddesc}
228 \begin{methoddesc}[FaultSystem]{getStrikeVectors}{tag}
229 returns the strike vectors $S^{ti}$ of fault $t=$\var{tag}.
230 \end{methoddesc}
232 \begin{methoddesc}[FaultSystem]{getLengths}{tag}
233 returns the lengths $l^{ti}$ of the segments in fault $t=$\var{tag}.
234 \end{methoddesc}
236 \begin{methoddesc}[FaultSystem]{getTotalLength}{tag}
237 returns the total unrolled length of fault \var{tag}.
238 \end{methoddesc}
240 \begin{methoddesc}[FaultSystem]{getDips}{tag}
241 returns the list of the dips of the segments in fault \var{tag}.
242 \end{methoddesc}
244 \begin{methoddesc}[FaultSystem]{getBottomPolyline}{tag}
245 returns the list of the vertices defining the bottom of the fault \var{tag}.
246 \end{methoddesc}
248 \begin{methoddesc}[FaultSystem]{getSegmentNormals}{tag}
249 returns the list of the normals of the segments in fault \var{tag}.
250 \end{methoddesc}
252 \begin{methoddesc}[FaultSystem]{getDepthVectors}{tag}
253 returns the list of the depth vectors $d^{ti}$ for fault $t=$\var{tag}.
254 \end{methoddesc}
256 \begin{methoddesc}[FaultSystem]{getDepths}{tag}
257 returns the list of the depths of the segments in fault \var{tag}.
258 \end{methoddesc}
260 \begin{methoddesc}[FaultSystem]{getW0Range}{tag}
261 returns the range of the parameterization in $w_{0}$.
262 For tag $t$ this is the pair $(\Omega^{t0},\Omega^{tn})$ where $n$ is the
263 number of segments in the fault.
264 In most cases one has $(\Omega^{t0},\Omega^{tn})=(0,w^t_{0 max})$.
265 \end{methoddesc}
267 \begin{methoddesc}[FaultSystem]{getW1Range}{tag}
268 returns the range of the parameterization in $w_{1}$.
269 For tag $t$ this is the pair $(-w^t_{1max},0)$.
270 \end{methoddesc}
272 \begin{methoddesc}[FaultSystem]{getW0Offsets}{tag}
273 returns the offsets for the parameterization of fault \var{tag}.
274 For tag \var{tag}=$t$ this is the list $[\Omega^{ti}]$.
275 \end{methoddesc}
277 \begin{methoddesc}[FaultSystem]{getCenterOnSurface}{}
278 returns the center point of the fault system at the surfaces.
279 In 3D the calculation of the center is considering the top edge of the faults
280 and projects the edge to the surface (the $x_{2}$ component is assumed to be
281 0). An \numpyNDA object is returned.
282 \end{methoddesc}
284 \begin{methoddesc}[FaultSystem]{getOrientationOnSurface}{}
285 returns the orientation of the fault system in RAD on the surface
286 ($x_{2}=0$ plane) around the fault system center.
287 \end{methoddesc}
289 \begin{methoddesc}[FaultSystem]{transform}{\optional{rot=0, \optional{shift=numpy.zeros((3,)}}}
290 applies a shift \var{shift} and a consecutive rotation in the $x_{2}=0$ plane.
291 \var{rot} is a float number and \var{shift} an \numpyNDA object.
292 \end{methoddesc}
294 \begin{methoddesc}[FaultSystem]{getMaxValue}{f\optional{, tol=1.e-8}}
295 returns the tag of the fault where \var{f} takes the maximum value and a
296 \class{Locator} object which can be used to collect values from \Data objects
297 at the location where the maximum is taken, e.g.
298 \begin{python}
299 fs=FaultSystem()
300 f=Scalar(..)
301 t, loc=fs.getMaxValue(f)
302 print("maximum value of f on the fault %s is %s at location %s."%(t, \
303 loc(f), loc.getX()))
304 \end{python}
305 \var{f} must be a \Scalar. When the maximum is calculated only
306 \DataSamplePoints are considered which are on a fault in the fault system in
307 the sense of condition~\ref{eq:2D line 3} or \ref{eq:2D line 4d}, respectively.
308 In the case no \DataSamplePoints are found the returned tag is \var{None} and
309 the maximum value as well as the location of the maximum value are undefined.
310 \end{methoddesc}
312 \begin{methoddesc}[FaultSystem]{getMinValue}{f\optional{, tol=1.e-8}}
313 returns the tag of the fault where \var{f} takes the minimum value and a
314 \class{Locator} object which can be used to collect values from \Data objects
315 at the location where the minimum is taken, e.g.
316 \begin{python}
317 fs=FaultSystem()
318 f=Scalar(..)
319 t, loc=fs.getMinValue(f)
320 print("minimum value of f on the fault %s is %s at location."%\
321 (t,loc(f),loc.getX()))
322 \end{python}
323 \var{f} must be a \Scalar. When the minimum is calculated only
324 \DataSamplePoints are considered which are on a fault in the fault system in
325 the sense of condition~\ref{eq:2D line 3} or \ref{eq:2D line 4d}, respectively.
326 In the case no \DataSamplePoints are found the returned tag is \var{None} and
327 the minimum value as well as the location of the minimum value are undefined.
328 \end{methoddesc}
330 \begin{methoddesc}[FaultSystem]{getParametrization}{x,tag \optional{\optional{, tol=1.e-8}, outsider=None}}
331 returns the argument $w$ of the parameterization $P^t$ for \var{tag}=$t$ to
332 provide \var{x} together with a mask indicating where the given location if on
333 a fault in the fault system by the value $1$ (otherwise the value is set to $0$).
334 \var{x} needs to be a \Vector or \numpyNDA object.
335 \var{tol} defines the tolerance to decide if given \DataSamplePoints are on
336 fault \var{tag}. The value \var{outside} is the value used as a replacement
337 value for $w$ where the corresponding value in \var{x} is not on a fault.
338 If \var{outside} is not present an appropriate value is used.
339 \end{methoddesc}
341 \begin{methoddesc}[FaultSystem]{getSideAndDistance}{x,tag}
342 returns the side and the distance at locations \var{x} from the fault \var{tag}.
343 \var{x} needs to be a \Vector or \numpyNDA object.
344 Positive values for side means that the corresponding location is to the right
345 of the fault, a negative value means that the corresponding location is
346 to the left of the fault. The value zero means that the side is undefined.
347 \end{methoddesc}
349 \begin{methoddesc}[FaultSystem]{getFaultSegments}{tag}
350 returns the polylines used to describe fault \var{tag}. For \var{tag}=$t$ this
351 is the list of the vertices $[V^{ti}]$ for the 2D and the pair of lists of the
352 top vertices $[V^{ti}]$ and the bottom vertices $[v^{ti}]$ in 3D.
353 Note that the coordinates are represented as \numpyNDA objects.
354 \end{methoddesc}
356 \begin{methoddesc}[FaultSystem]{addFault}{
357 strikes\optional{,
358 ls\optional{,
359 V0=[0.,0.,0.]\optional{,
360 tag=None\optional{,
361 dips=None\optional{,
362 depths= None\optional{,
363 w0_offsets=None\optional{,
364 w1_max=None}}}}}}}}
365 adds the fault \var{tag} to the fault system.
366 \var{V0} defines the start point of fault named $t=$\var{tag}.
367 The polyline defining the fault segments on the surface are set by the strike
368 angles \var{strikes} (=$\sigma^{ti}$, north = $\pi/2$, the orientation is
369 counterclockwise.) and the length \var{ls} (=$l^{ti}$).
370 In the 3D case one also needs to define the dip angles \var{dips}
371 (=$\delta^{ti}$, vertical=$0$, right-hand rule applies.) and the depth
372 \var{depths} for each segment.
373 \var{w1_max} defines the range of $w_{1}$.
374 If not present the mean value over the depth of all segment edges in the fault
375 is used.
376 \var{w0_offsets} sets the offsets $\Omega^{ti}$. If not present it is chosen
377 such that $\Omega^{ti}-\Omega^{t(i-1)}$ is the length of the $i$-th segment.
378 In some cases, e.g. when kinks in the fault are relevant, it can be useful
379 to explicitly specify the offsets in order to simplify the assignment of values.
380 \end{methoddesc}
382 \subsection{Example}
383 See \Sec{Slip CHAP}.

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