# Contents of /trunk/doc/user/faultsystem.tex

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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} 6 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. 11 12 A fault $t$ in the fault system is represented by a staring point $V^{t0}$ and series of 13 direction, called strikes~\index{strike}, and the lengths $(l^{ti})$. 14 The strike of segment $i$ is defined by the angle $\sigma^{ti}$ between the $x\hackscore{0}$-axis and the 15 direction of the fault, see Figure~\ref{FAULTSYSTEM2D}. The length and strike defines the 16 polyline $(V^{ti})$ of the fault by 17 \begin{equation} 18 V^{ti} = V^{t(i-1)} + 19 l^{ti} \cdot S^{ti} 20 \mbox{ with } 21 S^{ti} = 22 \left[ 23 \begin{array}{c} 24 cos(\sigma^{ti}) \\ 25 sin(\sigma^{ti}) \\ 26 0 27 \end{array} 28 \right] 29 \label{eq:fault 00} 30 \end{equation} 31 In the 3D case each fault segment $i$ has an additional dip \index{dip} $\theta^{ti}$ and 32 and at each vertex $i$ a depth $\delta^{ti}$ is given. The fault segment normal $n^{ti}$ is given by 33 \begin{equation} 34 n^{ti} = 35 \left[ 36 \begin{array}{c} 37 -sin(\theta^{ti}) \cdot S^{ti}\hackscore{1} \\ 38 sin(\theta^{ti}) \cdot S^{ti}\hackscore{0} \\ 39 cos(\theta^{ti}) 40 \end{array} 41 \right] 42 \label{eq:fault 0} 43 \end{equation} 44 At each vertex we define a depth vector $d^{ti}$ defined as the 45 intersect of the fault planes of segment $(i-1)$ and $i$ where 46 for the first segment and last segment the vector orthogonal to strike vector $S^{ti}$\index{strike} 47 and the segment normal $n^{ti}$ is used. The direction $\tilde{d}^{ti}$ of the depth vector is 48 given as 49 \begin{equation} 50 \tilde{d}^{ti} = n^{ti} \times n^{t(i-1)} 51 \label{eq:fault b} 52 \end{equation} 53 If $\tilde{d}^{ti}$ is zero the strike vectors $L^{t(i-1)}$ and $L^{ti}$ are collinear and 54 we can set $\tilde{d}^{ti} = l^{ti} \times n^{ti}$. If the two fault segments are almost 55 orthogonal $\tilde{d}^{ti}$ is pointing in the direction of $L^{t(i-1)}$ and $L^{ti}$. In this case no 56 depth can be defined. So we will reject a fault system if 57 \begin{equation} 58 min(\| \tilde{d}^{ti} \times L^{t(i-1)} \|,\| \tilde{d}^{ti} \times L^{ti} \|) 59 \le 0.1 \cdot \| \tilde{d}^{ti} | 60 \label{eq:fault c} 61 \end{equation} 62 which corresponds to an angle of less then $10^o$ between the depth vector and the strike. We then set 63 \begin{equation} 64 d^{ti}=\delta^{ti} \cdot \frac{\tilde{d}^{ti}}{\|\tilde{d}^{ti}\|} 65 \label{eq:fault d} 66 \end{equation} 67 We can then define the polyline $(v^{ti})$ for the bottom of the fault as 68 \begin{equation} 69 v^{ti}= V^{ti}+d^{ti} 70 \label{eq:fault e} 71 \end{equation} 72 In order to simplify working on 73 a fault $t$ 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 74 \begin{equation} 75 0\le w\hackscore{0} \le w^t\hackscore{0 max} \mbox{ and } -w^t\hackscore{1max}\le w\hackscore{1} \le 0 76 \label{eq:fault 1} 77 \end{equation} 78 with positive if numbers $w^t\hackscore{0 max}$ and $w^t\hackscore{1 max}$. Typically one chooses 79 $w^t\hackscore{0 max}$ to be the unrolled length of the fault 80 $w^t\hackscore{1 max}$ to be the mean value of segment depth. Moreover we have 81 \begin{equation} 82 P^t(W^{ti})=V^{ti}\mbox{ and } P^t(w^{ti})=v^{ti}\ 83 \label{eq:fault 2} 84 \end{equation} 85 where 86 \begin{equation} 87 W^{ti}=(\Omega^{ti},0) \mbox{ and } w^{ti}=(\Omega^{ti},-w^t\hackscore{1 max}) 88 \label{eq:fault 3} 89 \end{equation} 90 and $\Omega^{ti}$ is the unrolled distance of $W^{ti}$ from $W^{t0}$, ie $l^{ti}=\Omega^{t(i+1)}-\Omega^{ti}$. In the 2D case $w^t\hackscore{1 max}$ is set to zero and therefore the second component is dropped, see Figure~\ref{FAULTSYSTEM2D}. 91 92 In the case of 2D the parameterization $P^t$ is constructed as follows: 93 The line connecting $V^{t(i-1)}$ and $V^{ti}$ is given by 94 \begin{equation} 95 x=V^{ti} + s \cdot ( V^{t(i+1)}- V^{ti} ) 96 \label{eq:2D line 1} 97 \end{equation} 98 where $s$ is between $0$ and $1$. The point $x$ is on $i$-th fault segment if and only if 99 such an $s$ exists. If assume $x$ is on the fault one can calculate $s$ as 100 \begin{equation} 101 s = \frac{ (x- V^{ti})^t \cdot (V^{t(i+1)}- V^{ti}) }{ \|V^{t(i+1)}- V^{ti}\|^2} 102 \label{eq:2D line 1b} 103 \end{equation} 104 We then can set 105 \begin{equation} 106 w\hackscore{0}=\Omega^{ti}+s \cdot (\Omega^{ti}-\Omega^{t(i-1)}) 107 \label{eq:2D line 2} 108 \end{equation} 109 to get $P^t(w\hackscore{0})=x$. 110 It remains the question if the given $x$ is actual on the segment $i$ of fault $t$. To test this $s$ is restricted 111 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 112 residual of equation~\ref{eq:2D line 1}, ie. $x$ is been accepted to be in the segment if 113 \begin{equation} 114 \|x-V^{ti} - s \cdot (V^{t(i+1)}- V^{ti}) \| \le tol \cdot 115 max(l^{ti}, \|x-V^{ti} \|) 116 \label{eq:2D line 3} 117 \end{equation} 118 where $tol$ is a given tolerance. 119 120 In the 3D case the situation is a bit more complicated: we split the fault segment across 121 the diagonal $V^{ti}$-$v^{t(i+1)}$ to produce two triangles. In the upper triangle we use the 122 parameterization 123 \begin{equation} 124 x= V^{ti} + s \cdot (V^{t(i+1)}-V^{ti}) + r \cdot (v^{t(i+1)}-V^{t(i+1)}) 125 \mbox{ with } r \le s; 126 \label{eq:2D line 4} 127 \end{equation} 128 while in the lower triangle we use 129 \begin{equation} 130 x= V^{ti} + s \cdot (v^{t(i+1)}-v^{ti}) + r \cdot (v^{ti}-V^{ti}) 131 \mbox{ with } s \le r; 132 \label{eq:2D line 4b} 133 \end{equation} 134 where $0\le s,r \le 1$. Both equations are solved in the least-squares sense e.g. using the Mooreâ€“Penrose pseudo-inverse for the coefficient matrices. The resulting $s$ and $r$ are then restricted to the unit square. Similar to the 2D case (see equation~\ref{eq:2D line 3}) we identify $x$ to be in the upper triangle of the segment if 135 \begin{equation} 136 \|x- V^{ti} - s \cdot (V^{t(i+1)}-V^{ti}) - r \cdot (v^{t(i+1)}-V^{t(i+1)}) \| 137 \le tol \cdot max(\|x-V^{ti} \|,\|v^{t(i+1)}-V^{t(i)})\|) 138 \label{eq:2D line 4c} 139 \end{equation} 140 and in the lower part 141 \begin{equation} 142 \|x-V^{ti} - s \cdot (v^{t(i+1)}-v^{ti}) - r \cdot (v^{ti}-V^{ti}) \| 143 \le tol \cdot max(\|x-V^{ti} \|,\|v^{t(i+1)}-V^{t(i)})\|) 144 \label{eq:2D line 4d} 145 \end{equation} 146 after the restriction of $(s,t)$ to the unit square. Note that $\|v^{t(i+1)}-V^{t(i)})\|$ is length of the 147 diagonal of the fault segment. For those $x$ which have been located in the $i$-th segment we then set 148 \begin{equation} 149 w\hackscore{0}=\Omega^{ti}+s \cdot (\Omega^{ti}-\Omega^{t(i-1)}) 150 \mbox{ and } 151 w\hackscore{1}=w^t\hackscore{1max} (r-1) 152 \label{eq:2D line 5} 153 \end{equation} 154 155 156 \subsection{Functions} 157 158 \begin{classdesc}{FaultSystem}{ 159 \optional{dim =3}} 160 Creates a fault system in the \var{dim} dimensional space. 161 \end{classdesc} 162 163 \begin{methoddesc}[FaultSystem]{getMediumDepth}{tag} 164 returns the medium depth of fault \var{tag}. 165 \end{methoddesc} 166 167 \begin{methoddesc}[FaultSystem]{getTags}{} 168 returns a list of the tags used by the fault system 169 \end{methoddesc} 170 171 \begin{methoddesc}[FaultSystem]{getStart}{tag} 172 returns the starting point of fault \var{tag} as a \numpyNDA object. 173 \end{methoddesc} 174 175 \begin{methoddesc}[FaultSystem]{getDim}{} 176 returns the spatial dimension 177 \end{methoddesc} 178 179 \begin{methoddesc}[FaultSystem]{getDepths}{tag} 180 returns the list of the depths of the segements in fault \var{tag}. 181 \end{methoddesc} 182 183 184 \begin{methoddesc}[FaultSystem]{getTopPolyline}{tag} 185 returns the polyline used to describe fault tagged by \var{tag} 186 \end{methoddesc} 187 188 \begin{methoddesc}[FaultSystem]{getStrikes}{tag} 189 returns the list of strikes $\sigma^{ti}$ of the segements in fault $t=$\var{tag} 190 \end{methoddesc} 191 192 193 \begin{methoddesc}[FaultSystem]{getStrikeVectors}{tag} 194 returns the strike vectors $S^{ti}$ of fault $t=$\var{tag} 195 \end{methoddesc} 196 197 198 \begin{methoddesc}[FaultSystem]{getLengths}{tag} 199 returns the lengths $l^{ti}$ of segments in fault $t=$\var{tag} 200 \end{methoddesc} 201 202 203 \begin{methoddesc}[FaultSystem]{getTotalLength}{tag} 204 returns the total unrolled length of fault \var{tag} 205 \end{methoddesc} 206 207 \begin{methoddesc}[FaultSystem]{getDips}{tag} 208 returns the list of the dips of the segements in fault \var{tag}. 209 \end{methoddesc} 210 211 \begin{methoddesc}[FaultSystem]{getBottomPolyline}{tag} 212 returns the list of the vertices defining the bottom of the fault \var{tag}. 213 \end{methoddesc} 214 215 \begin{methoddesc}[FaultSystem]{getSegmentNormals}{tag} 216 returns the list of the normals of the segments in fault \var{tag}. 217 \end{methoddesc} 218 219 \begin{methoddesc}[FaultSystem]{getDepthVectors}{tag} 220 returns the list of the depth vectors $d^{ti}$ for fault $t=$\var{tag}. 221 \end{methoddesc} 222 223 \begin{methoddesc}[FaultSystem]{getDepths}{tag} 224 returns the list of the depths of the segements in fault \var{tag}. 225 \end{methoddesc} 226 227 \begin{methoddesc}[FaultSystem]{getW0Range}{tag} 228 returns the range of the parameterization in $w\hackscore{0}$. 229 For tag $t$ this is the pair $(\Omega^{t0},\Omega^{tn})$ where $n$ is the number of segments in the fault. 230 In most cases one has $(\Omega^{t0},\Omega^{tn})=(0,w^t\hackscore{0 max})$. 231 \end{methoddesc} 232 233 \begin{methoddesc}[FaultSystem]{getW1Range}{tag} 234 returns the range of the parameterization in $w\hackscore{1}$. 235 For tag $t$ this is the pair $(-w^t\hackscore{1max},0)$. 236 \end{methoddesc} 237 238 \begin{methoddesc}[FaultSystem]{getW0Offsets}{tag} 239 returns the offsets for the parameterization of fault \var{tag}. 240 For tag \var{tag}=$t$ this is the list $[\Omega^{ti}]$. 241 \end{methoddesc} 242 243 \begin{methoddesc}[FaultSystem]{getCenterOnSurface}{} 244 returns the center point of the fault system at the surfaces. In 3D the calculation of the center is 245 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. 246 \end{methoddesc} 247 248 \begin{methoddesc}[FaultSystem]{getOrientationOnSurface}{} 249 returns the orientation of the fault system in RAD on the surface ($x\hackscore{2}=0$ plane) around the fault system center. 250 \end{methoddesc} 251 \begin{methoddesc}[FaultSystem]{transform}{\optional{rot=0, \optional{shift=numpy.zeros((3,)}}} 252 applies a shift \var{shift} and a consecutive rotation in the $x\hackscore{2}=0$ plane. 253 \var{rot} is a float number and \var{shift} an \numpyNDA object. 254 \end{methoddesc} 255 256 \begin{methoddesc}[FaultSystem]{getMaxValue}{f\optional{, tol=1.e-8}} 257 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 258 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 259 the maximum value as well as the location of the maximum value are undefined. 260 \end{methoddesc} 261 262 \begin{methoddesc}[FaultSystem]{getMinValue}{f\optional{, tol=1.e-8}} 263 returns the minimum value of \var{f}, the fault where the minimum is found and the location on the fault in fault coordinates. \var{f} must be a \Scalar. When the minimum is calculated only \DataSamplePoints are considered 264 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 265 the minimum value as well as the location of the mimimum value are undefined. 266 \end{methoddesc} 267 268 \begin{methoddesc}[FaultSystem]{getParametrization}{x,tag \optional{\optional{, tol=1.e-8}, outsider=None}} 269 returns the argument $w$ of the parameterization $P^t$ for \var{tag}=$t$ to provide \var{x} 270 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 271 \var{outside} is the value used as a replacement value for $w$ where the corresponding value in \var{x} is not 272 on a fault. If not \var{outside} is not present an appropriate value is used. 273 \end{methoddesc} 274 275 \begin{methoddesc}[FaultSystem]{getSideAndDistance}{x,tag} 276 returns the side and the distance at locations \var{x} from the fault \var{tag}. 277 \var{x} needs to be \Vector or \numpyNDA object. Positive values for side means that the corresponding location is 278 to the right of the fault, a negative value means that the corresponding location is 279 to the left of the fault. The value zero means that the side is undefined. 280 \end{methoddesc} 281 282 283 284 285 \begin{methoddesc}[FaultSystem]{getFaultSegments}{tag} 286 returns the polylines used to describe fault \var{tag}. For \var{tag}=$t$ this is the list of the vertices 287 $[V^{ti}]$ for the 2D and the pair of lists of the top vertices $[V^{ti}]$ and the bottom vertices $[v^{ti}]$ in 3D. 288 Note that the coordinates are represented as \numpyNDA objects. 289 \end{methoddesc} 290 291 \begin{methoddesc}[FaultSystem]{addFault}{ 292 strikes\optional{, 293 ls\optional{, 294 V0=[0.,0.,0.]\optional{, 295 tag=None\optional{, 296 dips=None\optional{, 297 depths= None\optional{, 298 w0_offsets=None\optional{, 299 w1_max=None}}}}}}}} 300 adds the fault \var{tag} to the fault system. 301 \var{V0} defines the start point of fault names $t=$\var{tag}. The polyline 302 defining the fault segments on the surface are set 303 by the strike angles \var{strikes} (=$\sigma^{ti}$, north = $\pi/2$, the orientation is counter clockwise.) and 304 the length \var{ls} (=$l^{ti}$). In the 3D one also needs to define the 305 dip angles \var{dips} (=$\delta^{ti}$, vertical=$0$, right-hand rule applies.) and 306 the depth \var{depths} for each segment. 307 \var{w1_max} defines the range of the $w\hackscore{1}$. If not present the mean value over the depth of 308 all segment edges in the fault is used. 309 \var{w0_offsets} sets the offsets $\Omega^{ti}$. If not present it i chosen such that $\Omega^{ti}-\Omega^{t(i-1)}$ is the length of the $i$-th segment. In some cases, eg. when kinks in the fault are relevant, it can be useful 310 to explicitly specify the offsets in order to simplify the assignment of values. 311 \end{methoddesc} 312 313 \subsection{Example} 314 See section~\ref{Slip CHAP} 315 316 317 318 319 320 321