 # Contents of /trunk/doc/inversion/ForwardMagnetic.tex

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some fixes in the inversion set-up

 1 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 % Copyright (c) 2003-2012 by University of Queensland 4 5 % 6 % Primary Business: Queensland, Australia 7 % Licensed under the Open Software License version 3.0 8 9 % 10 % Development until 2012 by Earth Systems Science Computational Center (ESSCC) 11 % Development since 2012 by School of Earth Sciences 12 % 13 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 14 15 16 \section{Magnetic Inversion}\label{sec:forward magnetic} 17 For the magnetic inversion we use the anomaly of the magnetic flux 18 density~\index{magnetic flux density} of the Earth. 19 The controlling material parameter is the susceptibility~\index{susceptibility} 20 $k$ of the rock. 21 With magnetization $M$ and inducing magnetic field anomaly $H^s$, the magnetic 22 flux density anomaly $B^s$ is given as 23 \begin{equation}\label{ref:MAG:EQU:1} 24 B_i = \mu_0 \cdot ( H^s_i + M_i ) 25 \end{equation} 26 where $\mu_0 = 4 \pi \cdot 10^{-7} \frac{Vs}{Am}$. 27 In this forward model we make the simplifying assumption that the magnetization 28 is proportional to the known geomagnetic flux density $B^b$: 29 \begin{equation}\label{ref:MAG:EQU:4} 30 \mu_0 \cdot M_i = k \cdot B^b_i \;. 31 \end{equation} 32 Values for the magnetic flux density can be obtained by the International 33 Geomagnetic Reference Field (IGRF)~\cite{IGRF} 34 (or the Australian Geomagnetic Reference Field (AGRF)~\cite{AGRF}). 35 A rough approximation at latitude $\theta$ is given by 36 \begin{equation}\label{ref:MAG:EQU:5} 37 \begin{array}{rcl} 38 B^b_{\theta} & = & \displaystyle{ \frac{ \mu_0 \cdot m_{earth}}{4 \pi \cdot R_{earth}^3} sin(\theta) } \\ 39 B^b_r & = & \displaystyle{ \frac{\mu_0 \cdot m_{earth}}{1 \pi \cdot R_{earth}^3} cos(\theta) } 40 \end{array} 41 \end{equation} 42 with the vacuum permeability\index{vacuum permeability} $\mu_0 = 4 \pi \cdot 10^{-7} \frac{Vs}{Am}$, 43 the magnetic dipole moment of Earth $m_{earth}=8.22 \cdot 10^{22} Am^2$ and 44 earth radius $R_{earth}= 6378137m$. 45 $B^b_r$ and $b^b_{\theta}$ denote the radial and latitudinal component of the 46 geomagnetic flux density. 47 Notice that convention~(\ref{REF:EQU:INTRO 9}) applies if Cartesian 48 coordinates\index{Cartesian coordinates} are used. 49 In most cases it is reasonable to assume that that the background field is 50 constant across the domain. 51 52 The magnetic field anomaly $H^s$ can be represented by the gradient of a 53 magnetic scalar potential\index{scalar potential!magnetic} $\psi$. 54 We use the form 55 \begin{equation}\label{ref:MAG:EQU:6} 56 \mu_0 \cdot H^s_i = - \psi_{,i} 57 \end{equation} 58 With this notation one gets from Equations~(\ref{ref:MAG:EQU:1}) and~(\ref{ref:MAG:EQU:4}): 59 \begin{equation}\label{ref:MAG:EQU:7} 60 B_i = - \psi_{,i} + k \cdot B^b_i 61 \end{equation} 62 As the $B^s$ magnetic flux density anomaly we obtain the PDE 63 \begin{equation}\label{ref:MAG:EQU:8} 64 - \psi_{,ii} = - (k B^b_i)_{,i} 65 \end{equation} 66 which needs to be solved for a given susceptibility $k$. 67 The magnetic scalar potential is set to zero at the top of the domain 68 $\Gamma_{0}$. 69 On all other faces the normal component of the magnetic flux density anomaly 70 $B_i$ is set to zero, i.e. $n_i \psi_{,i} = k \cdot n_i B^b_i$ with outer 71 normal field $n_i$. 72 73 From the magnetic scalar potential we can calculate the magnetic flux density 74 anomaly via Equation~(\ref{ref:MAG:EQU:8}) to calculate the defect to the given 75 data. 76 If $B^{(s)}_i$ is a measurement of the magnetic flux density anomaly for 77 survey $s$ and $\omega^{(s)}_i$ is a weighting factor the data defect 78 $J^{mag}(k)$ in the notation of Chapter~\ref{Chp:ref:introduction} is given as 79 \begin{equation}\label{ref:MAG:EQU:9} 80 J^{mag}(k) = \frac{1}{2}\sum_{s} \int_{\Omega} ( \omega^{(s)}_i \cdot (B_{i}- B^{(s)}_i) ) ^2 dx 81 \end{equation} 82 Summation over $i$ is performed. 83 The cost function kernel\index{cost function!kernel} is given as 84 \begin{equation}\label{ref:MAG:EQU:10} 85 K^{mag}(\psi_{,i},k) = \frac{1}{2}\sum_{s} ( \omega^{(s)}_i \cdot (k \cdot B^b_i - \psi_{,i} - B^{(s)}_i) ) ^2 86 \end{equation} 87 Notice that if magnetic flux density is measured in air one can ignore the 88 $k\cdot B^b_i$ as the susceptibility is zero. 89 90 In practice the magnetic flux density $b^{(s)}$ is measured along a certain 91 direction $d^{(s)}_i$ with a standard error deviation $\sigma^{(s)}$ at 92 certain locations in the domain. 93 In this case one sets $B^{(s)}_i=b^{(s)} \cdot d^{(s)}_i$ and the weighting 94 factors $\omega^{(s)}$ as 95 \begin{equation}\label{ref:MAG:EQU:11} 96 \omega^{(s)}_i 97 = \left\{ 98 \begin{array}{lcl} 99 f \cdot \frac{d^{(s)}_i}{\sigma^{(s)}} & & \mbox{data are available} \\ 100 & \mbox{ where } & \\ 101 0 & & \mbox{ otherwise } \\ 102 \end{array} 103 \right. 104 \end{equation} 105 where it is assumed that $d^{(s)}_i \cdot d^{(s)}_i =1$. With the objective to control the 106 gradient of the cost function the scaling factor $f$ is chosen in the way that 107 \begin{equation}\label{ref:MAG:EQU:12} 108 \sum_{s} \int_{\Omega} ( \omega^{(s)}_i B^{(s)}_i ) 109 \cdot ( \omega^{(s)}_j \frac{1}{L_j} ) \cdot L^2 \cdot 110 ( B^b_n \frac{1}{L_n} ) 111 \cdot k' \; 112 dx =\alpha 113 \end{equation} 114 where $\alpha$ defines a scaling factor which is typically set to one and $L$ is defined by equation~(\ref{ref:EQU:REG:6b}). 115 $k'$ is considering the 116 derivative of the density with respect to the level set function. 117 118 \subsection{Usage} 119 120 \LG{Add example} 121 122 \begin{classdesc}{MagneticModel}{domain, w, B, background_field, 123 \optional{, useSphericalCoordinates=False} 124 \optional{, tol=1e-8}} 125 opens a magnetic forward model over the \Domain \member{domain} with 126 weighting factors \member{w} ($=\omega^{(s)}$) and measured magnetic flux 127 density anomalies \member{B} ($=B^{(s)}$). 128 The weighting factors and the measured magnetic flux density anomalies must be vectors. 129 \member{background_field} defines the background magnetic flux density $B^b$ 130 as a vector. 131 If \member{useSphericalCoordinates} is \True spherical coordinates are used. 132 \member{tol} sets the tolerance for the solution of the PDE~(\ref{ref:MAG:EQU:8}). 133 \end{classdesc} 134 135 \begin{methoddesc}[MagneticModel]{rescaleWeights}{ 136 \optional{scale=1.} 137 \optional{k_scale=1.}} 138 rescale the weighting factors such condition~(\ref{ref:MAG:EQU:12}) holds where 139 \member{scale} sets the scale $\alpha$ 140 and \member{k_scale} sets $k'$. This method should be called before any inversion is started 141 in order to make sure that all components of the cost function are appropriately scaled. 142 \end{methoddesc} 143 144 145 \subsection{Gradient Calculation} 146 This section briefly explains how the gradient 147 $\frac{\partial J^{mag}}{\partial k}$ of the cost function $J^{mag}$ with 148 respect to the susceptibility $k$ is calculated. 149 The magnetic potential $\psi$ from PDE~(\ref{ref:MAG:EQU:8}) is solved in weak form: 150 \begin{equation}\label{ref:MAG:EQU:201} 151 \int_{\Omega} q_{,i} \psi_{,i} \; dx = \int_{\Omega} k \cdot q_{,i} B^b_i \; dx 152 \end{equation} 153 for all $q$ with $q=0$ on $\Gamma_{0}$. 154 In the following we set $\Psi[k]=\psi$ for a given susceptibility $k$ as 155 solution of the variational problem~(\ref{ref:MAG:EQU:201}). 156 If $\Gamma_{k}$ denotes the region of the domain where the susceptibility is 157 known and for a given direction $p$ with $p=0$ on $\Gamma_{k}$ one has 158 \begin{equation}\label{ref:MAG:EQU:201aa} 159 \int_{\Omega} \frac{\partial J^{mag}}{\partial k} \cdot p \; dx = \int_{\Omega} 160 \sum_{s} (\omega^{(s)}_j 161 ( B^{(s)}_j-B_{j})) \cdot ( \omega^{(s)}_i ( \Psi[p]_{,i} - p \cdot B^b_i ) ) \; dx 162 \end{equation} 163 with 164 \begin{equation}\label{ref:MAG:EQU:202c} 165 Y_i[\psi]= \sum_{s} (\omega^{(s)}_j 166 (B^{(s)}_j - B_{j}) ) \cdot \omega^{(s)}_i 167 \end{equation} 168 This is written as 169 \begin{equation}\label{ref:MAG:EQU:202cc} 170 \int_{\Omega} \frac{\partial J^{mag}}{\partial k} \cdot p \; dx = \int_{\Omega} 171 Y_i[\psi] \Psi[p]_{,i} - p \cdot Y_i[\psi]B^b_i \; dx 172 \end{equation} 173 We then set $Y^*[\psi]$ as the solution of the equation 174 \begin{equation}\label{ref:MAG:EQU:202d} 175 \int_{\Omega} r_{,i} Y^*[\psi]_{,i} \; dx = \int_{\Omega} r_{,i} ,Y_i[\psi] \; dx \mbox{ for all } p \mbox{ with } r=0 \mbox{ on } \Gamma_{0} 176 \end{equation} 177 with $Y^*[\psi]=0$ on $\Gamma_{0}$. With $r=\Psi[p]$ we get 178 \begin{equation}\label{ref:MAG:EQU:202dd} 179 \int_{\Omega} \Psi[p]_{,i} Y^*[\psi]_{,i} \; dx = \int_{\Omega} \Psi[p]_{,i} ,Y_i[\psi] \; dx 180 \end{equation} 181 and from Equation~(\ref{ref:MAG:EQU:201}) with $q=Y^*[\psi]$ we get 182 \begin{equation}\label{ref:MAG:EQU:20e} 183 \int_{\Omega} Y^*[\psi]_{,i} \Psi[p]_{,i} \; dx = \int_{\Omega} p \cdot Y^*[\psi]_{,i} B^b_i \; dx 184 \end{equation} 185 which leads to 186 \begin{equation}\label{ref:MAG:EQU:20ee} 187 \int_{\Omega} \Psi[p]_{,i} ,Y_i[\psi] \; dx = \int_{\Omega} p \cdot Y^*[\psi]_{,i} B^b_i \; dx 188 \end{equation} 189 and finally 190 \begin{equation}\label{ref:MAG:EQU:201a} 191 \int_{\Omega} \frac{\partial J^{mag}}{\partial k} \cdot p \; dx = \int_{\Omega} 192 p \cdot (Y_i[\psi] - Y^*[\psi]_{,i}) B^b_i \; dx 193 \end{equation} 194 or 195 \begin{equation}\label{ref:MAG:EQU:201b} 196 \frac{\partial J^{mag}}{\partial k} = (Y^*[\psi]_{,i}-Y_i[\psi]) B^b_i 197 \end{equation} 198

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