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15
16 \section{Linear 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 In most cases it is reasonable to assume that that the background field is
36 constant across the domain.
37
38 The magnetic field anomaly $H^s$ can be represented by the gradient of a
39 magnetic scalar potential\index{scalar potential!magnetic} $\psi$.
40 We use the form
41 \begin{equation}\label{ref:MAG:EQU:6}
42 \mu_0 \cdot H^s_i = - \psi_{,i}
43 \end{equation}
44 With this notation one gets from Equations~(\ref{ref:MAG:EQU:1}) and~(\ref{ref:MAG:EQU:4}):
45 \begin{equation}\label{ref:MAG:EQU:7}
46 B_i = - \psi_{,i} + k \cdot B^b_i
47 \end{equation}
48 As the $B^s$ magnetic flux density anomaly we obtain the PDE
49 \begin{equation}\label{ref:MAG:EQU:8}
50 - \psi_{,ii} = - (k B^b_i)_{,i}
51 \end{equation}
52 which needs to be solved for a given susceptibility $k$.
53 The magnetic scalar potential is set to zero at the top of the domain
54 $\Gamma_{0}$.
55 On all other faces the normal component of the magnetic flux density anomaly
56 $B_i$ is set to zero, i.e. $n_i \psi_{,i} = k \cdot n_i B^b_i$ with outer
57 normal field $n_i$.
58
59 From the magnetic scalar potential we can calculate the magnetic flux density
60 anomaly via Equation~(\ref{ref:MAG:EQU:8}) to calculate the defect to the given
61 data.
62 If $B^{(s)}_i$ is a measurement of the magnetic flux density anomaly for
63 survey $s$ and $\omega^{(s)}_i$ is a weighting factor the data defect
64 $J^{mag}(k)$ in the notation of Chapter~\ref{chapter:ref:inversion cost function} is given as
65 \begin{equation}\label{ref:MAG:EQU:9}
66 J^{mag}(k) = \frac{1}{2}\sum_{s} \int_{\Omega} ( \omega^{(s)}_i \cdot (B_{i}- B^{(s)}_i) ) ^2 dx
67 \end{equation}
68 Summation over $i$ is performed.
69 The cost function kernel\index{cost function!kernel} is given as
70 \begin{equation}\label{ref:MAG:EQU:10}
71 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
72 \end{equation}
73 Notice that if magnetic flux density is measured in air one can ignore the
74 $k\cdot B^b_i$ as the susceptibility is zero.
75
76 In practice the magnetic flux density $b^{(s)}$ is measured along a certain
77 direction $d^{(s)}_i$ with a standard error deviation $\sigma^{(s)}$ at
78 certain locations in the domain.
79 In this case one sets $B^{(s)}_i=b^{(s)} \cdot d^{(s)}_i$ and the weighting
80 factors $\omega^{(s)}$ as
81 \begin{equation}\label{ref:MAG:EQU:11}
82 \omega^{(s)}_i
83 = \left\{
84 \begin{array}{lcl}
85 f \cdot \frac{d^{(s)}_i}{\sigma^{(s)}} & & \mbox{data are available} \\
86 & \mbox{ where } & \\
87 0 & & \mbox{ otherwise } \\
88 \end{array}
89 \right.
90 \end{equation}
91 where it is assumed that $d^{(s)}_i \cdot d^{(s)}_i =1$. With the objective to control the
92 gradient of the cost function the scaling factor $f$ is chosen in the way that
93 \begin{equation}\label{ref:MAG:EQU:12}
94 \sum_{s} \int_{\Omega} ( \omega^{(s)}_i B^{(s)}_i )
95 \cdot ( \omega^{(s)}_j \frac{1}{L_j} ) \cdot L^2 \cdot
96 ( B^b_n \frac{1}{L_n} )
97 \cdot k' \;
98 dx =\alpha
99 \end{equation}
100 where $\alpha$ defines a scaling factor which is typically set to one and $L$ is defined by equation~(\ref{ref:EQU:REG:6b}).
101 $k'$ is considering the
102 derivative of the density with respect to the level set function.
103
104 \subsection{Usage}
105
106 \LG{Add example}
107
108 \begin{classdesc}{MagneticModel}{domain, w, B, background_field,
109 \optional{, useSphericalCoordinates=False}
110 \optional{, fixPotentialAtBottom=False},
111 \optional{, tol=1e-8}}
112 opens a magnetic forward model over the \Domain \member{domain} with
113 weighting factors \member{w} ($=\omega^{(s)}$) and measured magnetic flux
114 density anomalies \member{B} ($=B^{(s)}$).
115 The weighting factors and the measured magnetic flux density anomalies must be vectors.
116 \member{background_field} defines the background magnetic flux density $B^b$
117 as a vector with north, east and vertical components.
118 If \member{useSphericalCoordinates} is \True spherical coordinates are used.
119 \member{tol} sets the tolerance for the solution of the PDE~(\ref{ref:MAG:EQU:8}).
120 If \member{fixPotentialAtBottom} is set to \True, the gravitational potential
121 at the bottom is set to zero in addition to the potential on the top.
122 \end{classdesc}
123
124 \begin{methoddesc}[MagneticModel]{rescaleWeights}{
125 \optional{scale=1.}
126 \optional{k_scale=1.}}
127 rescale the weighting factors such condition~(\ref{ref:MAG:EQU:12}) holds where
128 \member{scale} sets the scale $\alpha$
129 and \member{k_scale} sets $k'$. This method should be called before any inversion is started
130 in order to make sure that all components of the cost function are appropriately scaled.
131 \end{methoddesc}
132
133
134 \subsection{Gradient Calculation}
135 This section briefly explains how the gradient
136 $\frac{\partial J^{mag}}{\partial k}$ of the cost function $J^{mag}$ with
137 respect to the susceptibility $k$ is calculated. We follow the concept as outlined in section~\ref{chapter:ref:inversion cost function:gradient}.
138
139 The magnetic potential $\psi$ from PDE~(\ref{ref:MAG:EQU:8}) is solved in weak form:
140 \begin{equation}\label{ref:MAG:EQU:201}
141 \int_{\Omega} q_{,i} \psi_{,i} \; dx = \int_{\Omega} k \cdot q_{,i} B^b_i \; dx
142 \end{equation}
143 for all $q$ with $q=0$ on $\Gamma_{0}$.
144 In the following we set $\Psi[k]=\psi$ for a given susceptibility $k$ as
145 solution of the variational problem~(\ref{ref:MAG:EQU:201}).
146 If $\Gamma_{k}$ denotes the region of the domain where the susceptibility is
147 known and for a given direction $p$ with $p=0$ on $\Gamma_{k}$ one has
148 \begin{equation}\label{ref:MAG:EQU:201aa}
149 \int_{\Omega} \frac{\partial J^{mag}}{\partial k} \cdot p \; dx = \int_{\Omega}
150 \sum_{s} (\omega^{(s)}_j
151 ( B^{(s)}_j-B_{j})) \cdot ( \omega^{(s)}_i ( \Psi[p]_{,i} - p \cdot B^b_i ) ) \; dx
152 \end{equation}
153 with
154 \begin{equation}\label{ref:MAG:EQU:202c}
155 Y_i[\psi]= \sum_{s} (\omega^{(s)}_j
156 (B^{(s)}_j - B_{j}) ) \cdot \omega^{(s)}_i
157 \end{equation}
158 This is written as
159 \begin{equation}\label{ref:MAG:EQU:202cc}
160 \int_{\Omega} \frac{\partial J^{mag}}{\partial k} \cdot p \; dx = \int_{\Omega}
161 Y_i[\psi] \Psi[p]_{,i} - p \cdot Y_i[\psi]B^b_i \; dx
162 \end{equation}
163 We then set $Y^*[\psi]$ as the solution of the equation
164 \begin{equation}\label{ref:MAG:EQU:202d}
165 \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}
166 \end{equation}
167 with $Y^*[\psi]=0$ on $\Gamma_{0}$. With $r=\Psi[p]$ we get
168 \begin{equation}\label{ref:MAG:EQU:202dd}
169 \int_{\Omega} \Psi[p]_{,i} Y^*[\psi]_{,i} \; dx = \int_{\Omega} \Psi[p]_{,i} ,Y_i[\psi] \; dx
170 \end{equation}
171 and from Equation~(\ref{ref:MAG:EQU:201}) with $q=Y^*[\psi]$ we get
172 \begin{equation}\label{ref:MAG:EQU:20e}
173 \int_{\Omega} Y^*[\psi]_{,i} \Psi[p]_{,i} \; dx = \int_{\Omega} p \cdot Y^*[\psi]_{,i} B^b_i \; dx
174 \end{equation}
175 which leads to
176 \begin{equation}\label{ref:MAG:EQU:20ee}
177 \int_{\Omega} \Psi[p]_{,i} ,Y_i[\psi] \; dx = \int_{\Omega} p \cdot Y^*[\psi]_{,i} B^b_i \; dx
178 \end{equation}
179 and finally
180 \begin{equation}\label{ref:MAG:EQU:201a}
181 \int_{\Omega} \frac{\partial J^{mag}}{\partial k} \cdot p \; dx = \int_{\Omega}
182 p \cdot (Y_i[\psi] - Y^*[\psi]_{,i}) B^b_i \; dx
183 \end{equation}
184 or
185 \begin{equation}\label{ref:MAG:EQU:201b}
186 \frac{\partial J^{mag}}{\partial k} = (Y^*[\psi]_{,i}-Y_i[\psi]) B^b_i
187 \end{equation}
188

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