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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% Copyright (c) 2003-2018 by The University of Queensland |
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% http://www.uq.edu.au |
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% |
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% Primary Business: Queensland, Australia |
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% Licensed under the Apache License, version 2.0 |
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% http://www.apache.org/licenses/LICENSE-2.0 |
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% |
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% Development until 2012 by Earth Systems Science Computational Center (ESSCC) |
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% Development 2012-2013 by School of Earth Sciences |
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% Development from 2014 by Centre for Geoscience Computing (GeoComp) |
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% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Fault System} |
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\label{Fault System} |
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The \class{FaultSystem} class provides an easy-to-use interface to handle 2D |
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and 3D fault systems\index{faults} as used for instance in simulating fault |
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ruptures. The main purpose of the class is to provide a parameterization of |
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an individual fault in the system of faults. |
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In case of a 2D fault the fault is parameterized by a single value $w_{0}$ and |
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in the case of a 3D fault two parameters $w_{0}$ and $w_{1}$ are used. |
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This parameterization can be used to impose data (e.g. a slip distribution) |
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onto the fault. It can also be a useful tool to visualize or analyze the |
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results on the fault if the fault is not straight. |
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|
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\begin{figure} |
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\centering |
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\includegraphics{FaultSystem2D} |
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\caption{\label{FAULTSYSTEM2D}Two dimensional fault system with one fault |
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named `t` in the $(x_{0},x_{1})$ space and its parameterization in the |
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$w_{0}$ space. The fault has three segments.} |
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\end{figure} |
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|
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A fault $t$ in the fault system is represented by a starting point $V^{t0}$ |
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and series of directions, called strikes\index{strike}, and the lengths $(l^{ti})$. |
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The strike of segment $i$ is defined by the angle $\sigma^{ti}$ between the |
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$x_{0}$-axis and the direction of the fault, see Figure~\ref{FAULTSYSTEM2D}. |
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The length and strike defines the polyline $(V^{ti})$ of the fault by |
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\begin{equation} |
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V^{ti} = V^{t(i-1)} + |
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l^{ti} \cdot S^{ti} |
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\mbox{ with } |
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S^{ti} = |
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\left[ |
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\begin{array}{c} |
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cos(\sigma^{ti}) \\ |
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sin(\sigma^{ti}) \\ |
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0 |
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\end{array} |
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\right] |
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\label{eq:fault 00} |
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\end{equation} |
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In the 3D case each fault segment $i$ has an additional dip\index{dip} |
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$\theta^{ti}$ and at each vertex $i$ a depth $\delta^{ti}$ is given. |
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The fault segment normal $n^{ti}$ is given by |
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\begin{equation} |
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n^{ti} = |
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\left[ |
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\begin{array}{c} |
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-sin(\theta^{ti}) \cdot S^{ti}_{1} \\ |
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sin(\theta^{ti}) \cdot S^{ti}_{0} \\ |
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cos(\theta^{ti}) |
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\end{array} |
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\right] |
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\label{eq:fault 0} |
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\end{equation} |
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At each vertex we define a depth vector $d^{ti}$ defined as the intersect of |
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the fault planes of segment $(i-1)$ and $i$ where for the first segment and |
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last segment the vector orthogonal to strike vector $S^{ti}$\index{strike} |
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and the segment normal $n^{ti}$ is used. The direction $\tilde{d}^{ti}$ of the |
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depth vector is given as |
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\begin{equation} |
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\tilde{d}^{ti} = n^{ti} \times n^{t(i-1)} |
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\label{eq:fault b} |
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\end{equation} |
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If $\tilde{d}^{ti}$ is zero the strike vectors $L^{t(i-1)}$ and $L^{ti}$ are |
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collinear and we can set $\tilde{d}^{ti} = l^{ti} \times n^{ti}$. |
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If the two fault segments are almost orthogonal $\tilde{d}^{ti}$ is pointing |
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in the direction of $L^{t(i-1)}$ and $L^{ti}$. In this case no depth can be |
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defined. So we will reject a fault system if |
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\begin{equation} |
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min(\| \tilde{d}^{ti} \times L^{t(i-1)} \|,\| \tilde{d}^{ti} \times L^{ti} \|) |
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\le 0.1 \cdot \| \tilde{d}^{ti} | |
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\label{eq:fault c} |
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\end{equation} |
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which corresponds to an angle of less than $10^o$ between the depth vector and |
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the strike. We then set |
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\begin{equation} |
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d^{ti}=\delta^{ti} \cdot \frac{\tilde{d}^{ti}}{\|\tilde{d}^{ti}\|} |
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\label{eq:fault d} |
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\end{equation} |
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We can then define the polyline $(v^{ti})$ for the bottom of the fault as |
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\begin{equation} |
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v^{ti}= V^{ti}+d^{ti} |
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\label{eq:fault e} |
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\end{equation} |
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In order to simplify working on a fault $t$ in a fault system a |
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parameterization $P^t: (w_{0},w_{1}) \rightarrow (x_{0},x_{1},x_{2})$ over a |
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rectangular domain is introduced such that |
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\begin{equation} |
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0\le w_{0} \le w^t_{0 max} \mbox{ and } -w^t_{1max}\le w_{1} \le 0 |
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\label{eq:fault 1} |
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\end{equation} |
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with positive numbers $w^t_{0 max}$ and $w^t_{1 max}$. Typically one chooses |
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$w^t_{0 max}$ to be the unrolled length of the fault and $w^t_{1 max}$ to be |
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the mean value of segment depth. Moreover we have |
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\begin{equation} |
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P^t(W^{ti})=V^{ti}\mbox{ and } P^t(w^{ti})=v^{ti}\ |
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\label{eq:fault 2} |
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\end{equation} |
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where |
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\begin{equation} |
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W^{ti}=(\Omega^{ti},0) \mbox{ and } w^{ti}=(\Omega^{ti},-w^t_{1 max}) |
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\label{eq:fault 3} |
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\end{equation} |
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and $\Omega^{ti}$ is the unrolled distance of $W^{ti}$ from $W^{t0}$, i.e. |
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$l^{ti}=\Omega^{t(i+1)}-\Omega^{ti}$. In the 2D case $w^t_{1 max}$ is set to |
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zero and therefore the second component is dropped, see Figure~\ref{FAULTSYSTEM2D}. |
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|
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In the 2D case the parameterization $P^t$ is constructed as follows: |
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The line connecting $V^{t(i-1)}$ and $V^{ti}$ is given by |
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\begin{equation} |
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x=V^{ti} + s \cdot ( V^{t(i+1)}- V^{ti} ) |
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\label{eq:2D line 1} |
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\end{equation} |
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where $s$ is between $0$ and $1$. The point $x$ is on $i$-th fault segment if |
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and only if such an $s$ exists. Assuming $x$ is on the fault it can be |
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calculated as |
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\begin{equation} |
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s = \frac{ (x- V^{ti})^t \cdot (V^{t(i+1)}- V^{ti}) }{ \|V^{t(i+1)}- V^{ti}\|^2} |
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\label{eq:2D line 1b} |
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\end{equation} |
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We then can set |
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\begin{equation} |
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w_{0}=\Omega^{ti}+s \cdot (\Omega^{ti}-\Omega^{t(i-1)}) |
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\label{eq:2D line 2} |
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\end{equation} |
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to get $P^t(w_{0})=x$. |
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It remains the question if the given $x$ is actually on the segment $i$ of |
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fault $t$. To test this $s$ is restricted between $0$ and $1$ (so if $s<0$, $s$ |
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is set to $0$ and if $s>1$, $s$ is set to $1$) and then we check the residual |
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of \eqn{eq:2D line 1}, i.e. $x$ has been accepted to be in the segment if |
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\begin{equation} |
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\|x-V^{ti} - s \cdot (V^{t(i+1)}- V^{ti}) \| \le tol \cdot |
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max(l^{ti}, \|x-V^{ti} \|) |
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\label{eq:2D line 3} |
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\end{equation} |
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where $tol$ is a given tolerance. |
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|
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In the 3D case the situation is a bit more complicated: we split the fault |
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segment across the diagonal $V^{ti}$-$v^{t(i+1)}$ to produce two triangles. |
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In the upper triangle we use the parameterization |
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\begin{equation} |
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x= V^{ti} + s \cdot (V^{t(i+1)}-V^{ti}) + r \cdot (v^{t(i+1)}-V^{t(i+1)}) |
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\mbox{ with } r \le s; |
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\label{eq:2D line 4} |
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\end{equation} |
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while in the lower triangle we use |
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\begin{equation} |
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x= V^{ti} + s \cdot (v^{t(i+1)}-v^{ti}) + r \cdot (v^{ti}-V^{ti}) |
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\mbox{ with } s \le r; |
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\label{eq:2D line 4b} |
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\end{equation} |
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where $0\le s,r \le 1$. Both equations are solved in the least-squares sense |
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e.g. using the Moore-Penrose pseudo-inverse for the coefficient matrices. |
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The resulting $s$ and $r$ are then restricted to the unit square. Similar to |
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the 2D case (see \eqn{eq:2D line 3}) we identify $x$ to be in the upper |
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triangle of the segment if |
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\begin{equation} |
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\|x- V^{ti} - s \cdot (V^{t(i+1)}-V^{ti}) - r \cdot (v^{t(i+1)}-V^{t(i+1)}) \| |
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\le tol \cdot max(\|x-V^{ti} \|,\|v^{t(i+1)}-V^{t(i)})\|) |
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\label{eq:2D line 4c} |
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\end{equation} |
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and in the lower part |
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\begin{equation} |
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\|x-V^{ti} - s \cdot (v^{t(i+1)}-v^{ti}) - r \cdot (v^{ti}-V^{ti}) \| |
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\le tol \cdot max(\|x-V^{ti} \|,\|v^{t(i+1)}-V^{t(i)})\|) |
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\label{eq:2D line 4d} |
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\end{equation} |
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after the restriction of $(s,t)$ to the unit square. |
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Note that $\|v^{t(i+1)}-V^{t(i)})\|$ is the length of the diagonal of the |
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fault segment. For those $x$ which have been located in the $i$-th segment we |
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then set |
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\begin{equation} |
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w_{0}=\Omega^{ti}+s \cdot (\Omega^{ti}-\Omega^{t(i-1)}) |
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\mbox{ and } |
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w_{1}=w^t_{1max} (r-1) |
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\label{eq:2D line 5} |
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\end{equation} |
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|
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\subsection{Functions} |
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|
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\begin{classdesc}{FaultSystem}{\optional{dim =3}} |
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creates a fault system in the \var{dim} dimensional space. |
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\end{classdesc} |
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|
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\begin{methoddesc}[FaultSystem]{getMediumDepth}{tag} |
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returns the medium depth of fault \var{tag}. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getTags}{} |
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returns a list of the tags used by the fault system. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getStart}{tag} |
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returns the starting point of fault \var{tag} as a \numpyNDA object. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getDim}{} |
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returns the spatial dimension. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getDepths}{tag} |
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returns the list of the depths of the segments in fault \var{tag}. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getTopPolyline}{tag} |
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returns the polyline used to describe the fault tagged by \var{tag}. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getStrikes}{tag} |
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returns the list of strikes $\sigma^{ti}$ of the segments in fault |
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$t=$\var{tag}. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getStrikeVectors}{tag} |
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returns the strike vectors $S^{ti}$ of fault $t=$\var{tag}. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getLengths}{tag} |
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returns the lengths $l^{ti}$ of the segments in fault $t=$\var{tag}. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getTotalLength}{tag} |
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returns the total unrolled length of fault \var{tag}. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getDips}{tag} |
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returns the list of the dips of the segments in fault \var{tag}. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getBottomPolyline}{tag} |
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returns the list of the vertices defining the bottom of the fault \var{tag}. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getSegmentNormals}{tag} |
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returns the list of the normals of the segments in fault \var{tag}. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getDepthVectors}{tag} |
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returns the list of the depth vectors $d^{ti}$ for fault $t=$\var{tag}. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getDepths}{tag} |
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returns the list of the depths of the segments in fault \var{tag}. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getW0Range}{tag} |
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returns the range of the parameterization in $w_{0}$. |
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For tag $t$ this is the pair $(\Omega^{t0},\Omega^{tn})$ where $n$ is the |
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number of segments in the fault. |
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In most cases one has $(\Omega^{t0},\Omega^{tn})=(0,w^t_{0 max})$. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getW1Range}{tag} |
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returns the range of the parameterization in $w_{1}$. |
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For tag $t$ this is the pair $(-w^t_{1max},0)$. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getW0Offsets}{tag} |
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returns the offsets for the parameterization of fault \var{tag}. |
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For tag \var{tag}=$t$ this is the list $[\Omega^{ti}]$. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getCenterOnSurface}{} |
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returns the center point of the fault system at the surfaces. |
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In 3D the calculation of the center is considering the top edge of the faults |
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and projects the edge to the surface (the $x_{2}$ component is assumed to be |
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0). An \numpyNDA object is returned. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getOrientationOnSurface}{} |
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returns the orientation of the fault system in RAD on the surface |
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($x_{2}=0$ plane) around the fault system center. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{transform}{\optional{rot=0, \optional{shift=numpy.zeros((3,)}}} |
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applies a shift \var{shift} and a consecutive rotation in the $x_{2}=0$ plane. |
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\var{rot} is a float number and \var{shift} an \numpyNDA object. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getMaxValue}{f\optional{, tol=1.e-8}} |
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returns the tag of the fault where \var{f} takes the maximum value and a |
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\class{Locator} object which can be used to collect values from \Data objects |
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at the location where the maximum is taken, e.g. |
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\begin{python} |
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fs=FaultSystem() |
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f=Scalar(..) |
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t, loc=fs.getMaxValue(f) |
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print("maximum value of f on the fault %s is %s at location %s."%(t, \ |
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loc(f), loc.getX())) |
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\end{python} |
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\var{f} must be a \Scalar. When the maximum is calculated only |
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\DataSamplePoints are considered which are on a fault in the fault system in |
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the sense of condition~\ref{eq:2D line 3} or \ref{eq:2D line 4d}, respectively. |
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In the case no \DataSamplePoints are found the returned tag is \var{None} and |
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the maximum value as well as the location of the maximum value are undefined. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getMinValue}{f\optional{, tol=1.e-8}} |
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returns the tag of the fault where \var{f} takes the minimum value and a |
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\class{Locator} object which can be used to collect values from \Data objects |
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at the location where the minimum is taken, e.g. |
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\begin{python} |
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fs=FaultSystem() |
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f=Scalar(..) |
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t, loc=fs.getMinValue(f) |
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print("minimum value of f on the fault %s is %s at location."%\ |
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(t,loc(f),loc.getX())) |
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\end{python} |
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\var{f} must be a \Scalar. When the minimum is calculated only |
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\DataSamplePoints are considered which are on a fault in the fault system in |
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the sense of condition~\ref{eq:2D line 3} or \ref{eq:2D line 4d}, respectively. |
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In the case no \DataSamplePoints are found the returned tag is \var{None} and |
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the minimum value as well as the location of the minimum value are undefined. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getParametrization}{x,tag \optional{\optional{, tol=1.e-8}, outsider=None}} |
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returns the argument $w$ of the parameterization $P^t$ for \var{tag}=$t$ to |
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provide \var{x} together with a mask indicating where the given location if on |
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a fault in the fault system by the value $1$ (otherwise the value is set to $0$). |
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\var{x} needs to be a \Vector or \numpyNDA object. |
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\var{tol} defines the tolerance to decide if given \DataSamplePoints are on |
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fault \var{tag}. The value \var{outside} is the value used as a replacement |
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value for $w$ where the corresponding value in \var{x} is not on a fault. |
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If \var{outside} is not present an appropriate value is used. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getSideAndDistance}{x,tag} |
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returns the side and the distance at locations \var{x} from the fault \var{tag}. |
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\var{x} needs to be a \Vector or \numpyNDA object. |
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Positive values for side means that the corresponding location is to the right |
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of the fault, a negative value means that the corresponding location is |
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to the left of the fault. The value zero means that the side is undefined. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{getFaultSegments}{tag} |
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returns the polylines used to describe fault \var{tag}. For \var{tag}=$t$ this |
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is the list of the vertices $[V^{ti}]$ for the 2D and the pair of lists of the |
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top vertices $[V^{ti}]$ and the bottom vertices $[v^{ti}]$ in 3D. |
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Note that the coordinates are represented as \numpyNDA objects. |
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\end{methoddesc} |
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|
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\begin{methoddesc}[FaultSystem]{addFault}{ |
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strikes\optional{, |
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ls\optional{, |
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V0=[0.,0.,0.]\optional{, |
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tag=None\optional{, |
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dips=None\optional{, |
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depths= None\optional{, |
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w0_offsets=None\optional{, |
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w1_max=None}}}}}}}} |
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adds the fault \var{tag} to the fault system. |
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\var{V0} defines the start point of fault named $t=$\var{tag}. |
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The polyline defining the fault segments on the surface are set by the strike |
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angles \var{strikes} (=$\sigma^{ti}$, north = $\pi/2$, the orientation is |
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counterclockwise.) and the length \var{ls} (=$l^{ti}$). |
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In the 3D case one also needs to define the dip angles \var{dips} |
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(=$\delta^{ti}$, vertical=$0$, right-hand rule applies.) and the depth |
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\var{depths} for each segment. |
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\var{w1_max} defines the range of $w_{1}$. |
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If not present the mean value over the depth of all segment edges in the fault |
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is used. |
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\var{w0_offsets} sets the offsets $\Omega^{ti}$. If not present it is chosen |
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such that $\Omega^{ti}-\Omega^{t(i-1)}$ is the length of the $i$-th segment. |
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In some cases, e.g. when kinks in the fault are relevant, it can be useful |
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to explicitly specify the offsets in order to simplify the assignment of values. |
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\end{methoddesc} |
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|
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\subsection{Example} |
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See \Sec{Slip CHAP}. |
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