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16 gross 4131 \section{Linear Magnetic Inversion}\label{sec:forward magnetic}
17 caltinay 4095 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 gross 4093 \begin{equation}\label{ref:MAG:EQU:1}
24 gross 4091 B_i = \mu_0 \cdot ( H^s_i + M_i )
25 gross 4047 \end{equation}
26 gross 4091 where $\mu_0 = 4 \pi \cdot 10^{-7} \frac{Vs}{Am}$.
27 caltinay 4095 In this forward model we make the simplifying assumption that the magnetization
28     is proportional to the known geomagnetic flux density $B^b$:
29 gross 4093 \begin{equation}\label{ref:MAG:EQU:4}
30 gross 4091 \mu_0 \cdot M_i = k \cdot B^b_i \;.
31 gross 4047 \end{equation}
32 caltinay 4095 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 gross 4091
38 caltinay 4095 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 gross 4093 \begin{equation}\label{ref:MAG:EQU:6}
42 gross 4091 \mu_0 \cdot H^s_i = - \psi_{,i}
43 gross 4047 \end{equation}
44 caltinay 4095 With this notation one gets from Equations~(\ref{ref:MAG:EQU:1}) and~(\ref{ref:MAG:EQU:4}):
45 gross 4093 \begin{equation}\label{ref:MAG:EQU:7}
46 gross 4091 B_i = - \psi_{,i} + k \cdot B^b_i
47     \end{equation}
48 caltinay 4095 As the $B^s$ magnetic flux density anomaly we obtain the PDE
49 gross 4093 \begin{equation}\label{ref:MAG:EQU:8}
50 gross 4091 - \psi_{,ii} = - (k B^b_i)_{,i}
51 gross 4047 \end{equation}
52 caltinay 4095 which needs to be solved for a given susceptibility $k$.
53 gross 4125 The magnetic scalar potential is set to zero at the top of the domain
54 gross 4121 $\Gamma_{0}$.
55 caltinay 4095 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 gross 4047
59 caltinay 4095 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 gross 4142 $J^{mag}(k)$ in the notation of Chapter~\ref{chapter:ref:inversion cost function} is given as
65 gross 4093 \begin{equation}\label{ref:MAG:EQU:9}
66 gross 4099 J^{mag}(k) = \frac{1}{2}\sum_{s} \int_{\Omega} ( \omega^{(s)}_i \cdot (B_{i}- B^{(s)}_i) ) ^2 dx
67 gross 4047 \end{equation}
68 gross 4099 Summation over $i$ is performed.
69 gross 4091 The cost function kernel\index{cost function!kernel} is given as
70 gross 4093 \begin{equation}\label{ref:MAG:EQU:10}
71 gross 4099 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 gross 4091 \end{equation}
73 caltinay 4095 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 gross 4047
76 caltinay 4095 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 gross 4093 \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 gross 4102 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 gross 4093 \begin{equation}\label{ref:MAG:EQU:12}
94 gross 4102 \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 gross 4093 \end{equation}
100 gross 4102 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 gross 4047
104 gross 4093 \subsection{Usage}
105 gross 4047
106 gross 4093 \LG{Add example}
107 gross 4047
108 caltinay 4095 \begin{classdesc}{MagneticModel}{domain, w, B, background_field,
109     \optional{, useSphericalCoordinates=False}
110 jfenwick 4345 \optional{, fixPotentialAtBottom=False},
111 caltinay 4095 \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 gross 4093 The weighting factors and the measured magnetic flux density anomalies must be vectors.
116 caltinay 4095 \member{background_field} defines the background magnetic flux density $B^b$
117 jfenwick 4345 as a vector with north, east and vertical components.
118 caltinay 4095 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 jfenwick 4345 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 gross 4093 \end{classdesc}
123 gross 4047
124 gross 4102 \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 gross 4047
133 gross 4102
134 gross 4093 \subsection{Gradient Calculation}
135 caltinay 4095 This section briefly explains how the gradient
136     $\frac{\partial J^{mag}}{\partial k}$ of the cost function $J^{mag}$ with
137 gross 4142 respect to the susceptibility $k$ is calculated. We follow the concept as outlined in section~\ref{chapter:ref:inversion cost function:gradient}.
138    
139 caltinay 4095 The magnetic potential $\psi$ from PDE~(\ref{ref:MAG:EQU:8}) is solved in weak form:
140 gross 4093 \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 gross 4047 \end{equation}
143 gross 4121 for all $q$ with $q=0$ on $\Gamma_{0}$.
144 caltinay 4095 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 gross 4121 \begin{equation}\label{ref:MAG:EQU:201aa}
149 gross 4099 \int_{\Omega} \frac{\partial J^{mag}}{\partial k} \cdot p \; dx = \int_{\Omega}
150 gross 4093 \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 gross 4047 \end{equation}
153 caltinay 4095 with
154 gross 4093 \begin{equation}\label{ref:MAG:EQU:202c}
155 gross 4099 Y_i[\psi]= \sum_{s} (\omega^{(s)}_j
156 gross 4093 (B^{(s)}_j - B_{j}) ) \cdot \omega^{(s)}_i
157 gross 4047 \end{equation}
158 gross 4093 This is written as
159 gross 4121 \begin{equation}\label{ref:MAG:EQU:202cc}
160 gross 4093 \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 gross 4047 \end{equation}
163 gross 4093 We then set $Y^*[\psi]$ as the solution of the equation
164     \begin{equation}\label{ref:MAG:EQU:202d}
165 gross 4121 \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 gross 4047 \end{equation}
167 gross 4121 with $Y^*[\psi]=0$ on $\Gamma_{0}$. With $r=\Psi[p]$ we get
168     \begin{equation}\label{ref:MAG:EQU:202dd}
169 gross 4093 \int_{\Omega} \Psi[p]_{,i} Y^*[\psi]_{,i} \; dx = \int_{\Omega} \Psi[p]_{,i} ,Y_i[\psi] \; dx
170 gross 4047 \end{equation}
171 caltinay 4095 and from Equation~(\ref{ref:MAG:EQU:201}) with $q=Y^*[\psi]$ we get
172 gross 4093 \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 gross 4047 \end{equation}
175     which leads to
176 gross 4093 \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 gross 4047 \end{equation}
184 gross 4093 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 caltinay 4095 \end{equation}
188    

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