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\section{Linear Magnetic Inversion}\label{sec:forward magnetic}
For the magnetic inversion we use the anomaly of the magnetic flux
density~\index{magnetic flux density} of the Earth.
The controlling material parameter is the susceptibility~\index{susceptibility}
$k$ of the rock.
With magnetization $M$ and inducing magnetic field anomaly $H$, the magnetic
flux density anomaly $B$ is given as
\begin{equation}\label{ref:MAG:EQU:1}
B_i = \mu_0 \cdot ( H_i  + M_i )
\end{equation}
where $\mu_0 = 4 \pi \cdot 10^{-7} \frac{Vs}{Am}$.
In this forward model we make the simplifying assumption that the magnetization
is proportional to the known geomagnetic flux density $B^b$:
\begin{equation}\label{ref:MAG:EQU:4}
\mu_0  \cdot M_i = k \cdot B^b_i \;. 
\end{equation}
Values for the magnetic flux density can be obtained by the International
Geomagnetic Reference Field (IGRF)~\cite{IGRF}
(or the Australian Geomagnetic Reference Field (AGRF)~\cite{AGRF}).
In most cases it is reasonable to assume that that the background field is
constant across the domain.

The magnetic field anomaly $H$ can be represented by the gradient of a
magnetic scalar potential\index{scalar potential!magnetic} $\psi$.
We use the form 
\begin{equation}\label{ref:MAG:EQU:6}
\mu_0  \cdot H_i = - \psi_{,i}
\end{equation}
With this notation one gets from Equations~(\ref{ref:MAG:EQU:1}) and~(\ref{ref:MAG:EQU:4}):
\begin{equation}\label{ref:MAG:EQU:7}
B_i = - \psi_{,i}  + k \cdot B^b_i
\end{equation}
As the $B$ magnetic flux density anomaly is divergence free ($B_{i,i}=0$) we obtain the PDE
\begin{equation}\label{ref:MAG:EQU:8}
- \psi_{,ii} = - (k B^r_i)_{,i} 
\end{equation} 
with $B^r_i=B^b_i$ which needs to be solved for a given susceptibility $k$.
The magnetic scalar potential is set to zero at the top of the domain
$\Gamma_{0}$.
On all other faces the normal component of the magnetic flux density anomaly
$B_i$ is set to zero, i.e. $n_i \psi_{,i} = k \cdot n_i  B^b_i$ with outer
normal field $n_i$.

From the magnetic scalar potential we can calculate the magnetic flux density
anomaly via Equation~(\ref{ref:MAG:EQU:8}) to calculate the defect to the given
data.
If $B^{(s)}_i$ is a measurement of the magnetic flux density anomaly for
survey $s$ and $\omega^{(s)}_i$ is a weighting factor the data defect
$J^{mag}(k)$ in the notation of Chapter~\ref{chapter:ref:inversion cost function} is given as
\begin{equation}\label{ref:MAG:EQU:9}
J^{mag}(k) = \frac{1}{2}\sum_{s} \int_{\Omega} ( \omega^{(s)}_i \cdot (B_{i}- B^{(s)}_i) ) ^2 dx
\end{equation} 
Summation over $i$ is performed.
The cost function kernel\index{cost function!kernel} is given as
\begin{equation}\label{ref:MAG:EQU:10}
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
\end{equation} 
Notice that if magnetic flux density is measured in air one can ignore the
$k\cdot B^b_i$ as the susceptibility is zero.

In practice the magnetic flux density $b^{(s)}$ is measured along a certain
direction $d^{(s)}_i$ with a standard error deviation $\sigma^{(s)}$ at
certain locations in the domain.
In this case one sets $B^{(s)}_i=b^{(s)} \cdot d^{(s)}_i$ and the weighting
factors $\omega^{(s)}$ as
\begin{equation}\label{ref:MAG:EQU:11}
\omega^{(s)}_i 
= \left\{
\begin{array}{lcl}
f \cdot  \frac{d^{(s)}_i}{\sigma^{(s)}} & & \mbox{data are available} \\
& \mbox{ where } & \\
0 & & \mbox{ otherwise } \\
\end{array}
\right.
\end{equation} 
where it is assumed that $d^{(s)}_i \cdot d^{(s)}_i =1$. With the objective to control the 
gradient of the cost function the scaling factor $f$ is chosen in the way that 
\begin{equation}\label{ref:MAG:EQU:12}
\sum_{s} \int_{\Omega} ( \omega^{(s)}_i B^{(s)}_i ) 
 \cdot ( \omega^{(s)}_j \frac{1}{L_j} ) \cdot L^2 \cdot
( B^b_n \frac{1}{L_n} )
 \cdot k' \;
 dx =\alpha
\end{equation} 
where $\alpha$ defines a scaling factor which is typically set to one and $L$ is defined by equation~(\ref{ref:EQU:REG:6b}).
$k'$ is considering the 
derivative of the density with respect to the level set function. 


\subsection{Usage}

\begin{classdesc}{MagneticModel}{domain, w, B, background_field,
        \optional{, coordinates=\None}
        \optional{, fixPotentialAtBottom=False},
        \optional{, tol=1e-8},
}
opens a magnetic forward model over the \Domain \member{domain} with 
weighting factors \member{w} ($=\omega^{(s)}$) and measured magnetic flux
density anomalies \member{B} ($=B^{(s)}$).
The weighting factors and the  measured magnetic flux density anomalies must be vectors.
\member{background_field} defines the background magnetic flux density $B^b$
as a vector with north, east and vertical components. 
\member{tol} sets the tolerance for the solution of the PDE~(\ref{ref:MAG:EQU:8}).
If \member{fixPotentialAtBottom} is set to  \True, the magnetic potential 
at the bottom is set to zero in addition to the potential on the top. 
\member{coordinates} set the reference coordinate system to be used. By the default the 
Cartesian coordinate system is used.
\end{classdesc}

\begin{methoddesc}[MagneticModel]{rescaleWeights}{
        \optional{scale=1.}
 \optional{k_scale=1.}}
rescale the weighting factors such condition~(\ref{ref:MAG:EQU:12}) holds where 
\member{scale} sets the scale $\alpha$
and \member{k_scale} sets $k'$. This method should be called before any inversion is started
in order to make sure that all components of the cost function are appropriately scaled.
\end{methoddesc}


\subsection{Gradient Calculation}
This section briefly explains how the gradient
$\frac{\partial J^{mag}}{\partial k}$ of the cost function $J^{mag}$ with
respect to the susceptibility $k$ is calculated.  We follow the concept as outlined in section~\ref{chapter:ref:inversion cost function:gradient}.

The magnetic potential $\psi$ from PDE~(\ref{ref:MAG:EQU:8}) is solved in weak form:
\begin{equation}\label{ref:MAG:EQU:201}
\int_{\Omega} q_{,i} \psi_{,i} \; dx  = \int_{\Omega}  k \cdot q_{,i}  B^r_i \; dx 
\end{equation} 
for all $q$ with $q=0$ on $\Gamma_{0}$.
In the following we set $\Psi[k]=\psi$ for a given susceptibility $k$ as
solution of the variational problem~(\ref{ref:MAG:EQU:201}).
If $\Gamma_{k}$ denotes the region of the domain where the susceptibility is
known and for a given direction $p$ with $p=0$ on $\Gamma_{k}$ one has
\begin{equation}\label{ref:MAG:EQU:201aa}
\int_{\Omega}   \frac{\partial J^{mag}}{\partial k} \cdot p \; dx  = \int_{\Omega}  
\sum_{s} (\omega^{(s)}_j 
( B^{(s)}_j-B_{j}))  \cdot ( \omega^{(s)}_i ( \Psi[p]_{,i} - p  \cdot B^b_i  ) ) \; dx  
\end{equation} 
With
\begin{equation}\label{ref:MAG:EQU:202c}
Y_i[\psi]=   \sum_{s} (\omega^{(s)}_j 
(B^{(s)}_j - B_{j}) )  \cdot \omega^{(s)}_i  
\end{equation} 
this is written as 
\begin{equation}\label{ref:MAG:EQU:202cc}
\int_{\Omega}   \frac{\partial J^{mag}}{\partial k} \cdot p \;  dx  = \int_{\Omega}  
Y_i[\psi] \Psi[p]_{,i} - p \cdot Y_i[\psi]B^b_i   \; dx  
\end{equation} 
We then set adjoint function $Y^*[\psi]$ as the solution of the equation 
\begin{equation}\label{ref:MAG:EQU:202d}
\int_{\Omega} r_{,i} Y^*[\psi]_{,i} \; dx  =  \int_{\Omega} r_{,i} Y_i[\psi]  \; dx  \mbox{ for all } r \mbox{ with } r=0 \mbox{ on } \Gamma_{0}
\end{equation} 
with $Y^*[\psi]=0$ on $\Gamma_{0}$. With $r=\Psi[p]$ we get
\begin{equation}\label{ref:MAG:EQU:202dd}
\int_{\Omega} \Psi[p]_{,i} Y^*[\psi]_{,i} \; dx  =  \int_{\Omega} \Psi[p]_{,i} Y_i[\psi]  \; dx
\end{equation} 
and from Equation~(\ref{ref:MAG:EQU:201}) with $q=Y^*[\psi]$ we get
\begin{equation}\label{ref:MAG:EQU:20e}
\int_{\Omega} Y^*[\psi]_{,i}  \Psi[p]_{,i} \; dx  = \int_{\Omega}  p \cdot Y^*[\psi]_{,i}  B^r_i \; dx  
\end{equation}
which leads to 
\begin{equation}\label{ref:MAG:EQU:20ee}
\int_{\Omega} \Psi[p]_{,i} ,Y_i[\psi]  \; dx  = \int_{\Omega}  p \cdot Y^*[\psi]_{,i}  B^r_i \; dx  
\end{equation}
and finally
\begin{equation}\label{ref:MAG:EQU:201a}
\int_{\Omega}   \frac{\partial J^{mag}}{\partial k} \cdot p \;  dx  = \int_{\Omega}  
p \cdot (Y^*[\psi]_{,i}  B^r_i - Y_i[\psi] B^b_i) \; dx  
\end{equation} 
or 
\begin{equation}\label{ref:MAG:EQU:201b}
\frac{\partial J^{mag}}{\partial k} = Y^*[\psi]_{,i}  B^r_i - Y_i[\psi] B^b_i
\end{equation}

\subsection{Geodetic Coordinates}
For geodetic coordinates $(\phi, \lambda, h)$, see Chapter~\ref{Chp:ref:coordinates}, the solution process needs to be slightly modified.
Observations are recorded along the geodetic coordinates axes $\alpha$ rather than the Cartesian axes $i$. In fact we
have in equation~\ref{ref:MAG:EQU:9}:
\begin{equation}\label{ref:MAG:EQU:300}
\omega^{(s)}_i \cdot (B_{i}- B^{(s)}_i) = \omega^{(s)}_{\alpha} \cdot (B_{{\alpha}}- B^{(s)}_{\alpha}) 
\end{equation} 
where now $B^{(s)}_{\alpha}$ are the observational data with weighting factors $\omega^{(s)}_{\alpha}$.  Using the 
fact that $B_{{\alpha}} = k \cdot B^b_{{\alpha}} -d_{\alpha \alpha} \psi_{,\alpha}$ 
equation~\ref{ref:MAG:EQU:10} translates to 
\begin{equation}\label{ref:MAG:EQU:301}
J^{mag}(k) = \frac{1}{2}\sum_{s} \int_{\widehat{\Omega}} 
( \omega^{(s)}_{\alpha} \cdot (d_{\alpha \alpha}  \psi_{,\alpha} - k \cdot B^b_{{\alpha}}  + B^{(s)}_{\alpha} ) ) ^2 \; v \; d\widehat{x}
\end{equation} 
where $\widehat{\Omega}$ and $d\widehat{x}$ refer to integration over the geodetic coordinates axes. This can be rearranged to 
\begin{equation}\label{ref:MAG:EQU:301bb}
J^{mag}(k) = \frac{1}{2}\sum_{s} \int_{\widehat{\Omega}} 
(  \omega^{(s)}_{\alpha} v^{\frac{1}{2}} d_{\alpha \alpha} \cdot ( 
 \psi_{,\alpha} -  k \cdot v_{\alpha \alpha} B^b_{{\alpha}} + v_{\alpha \alpha} B^{(s)}_{\alpha} ) ) ^2 \; d\widehat{x}
=\frac{1}{2}\sum_{s} \int_{\widehat{\Omega}} 
(  {\widehat{\omega}}^{(s)}_{\alpha}\cdot ( \psi_{,\alpha} -  k \cdot \widehat{B}^b_{{\alpha}}+  \widehat{B}^{(s)}_{\alpha} ) ) ^2 \; d\widehat{x}
\end{equation} 
with 
\begin{equation}\label{ref:MAG:EQU:301b}
 \widehat{\omega}^{(s)}_{\alpha} = \omega^{(s)}_{\alpha} v^{\frac{1}{2}} d_{\alpha \alpha} \mbox{ , }
\widehat{B}^{(s)}_{\alpha}=
\frac{1}{d_{\alpha \alpha}} B^{(s)}_{\alpha}  \mbox{ and } \widehat{B}^b_{{\alpha}} = \frac{1}{d_{\alpha \alpha}}  B^b_{{\alpha}} 
\end{equation} 
which means one can apply the Cartesian formulation to the geodetic coordinates using modified data. 


The magnetic potential is calculated from 
\begin{equation}\label{ref:MAG:EQU:302}
\int_{\widehat{\Omega}} v \; d_{\alpha \alpha}^2 q_{,\alpha} \psi_{,\alpha} \;  d\widehat{x}  
=  \int_{\widehat{\Omega}} v \; d_{\alpha \alpha} k \cdot q_{,\alpha}  B^r_{\alpha} \; d\widehat{x}   
=  \int_{\widehat{\Omega}}  k \cdot q_{,\alpha}  \widehat{B}^r_{\alpha} \; d\widehat{x}   
\end{equation} 
with 
\begin{equation}\label{ref:MAG:EQU:302b}
\widehat{B}^r_{\alpha}  =v \; d_{\alpha \alpha} \widehat{B}^r_{\alpha}
\end{equation} 
see equation~\ref{ref:MAG:EQU:201}, and the adjoint function $Y^*[\psi]$ for $Y_{\alpha}[\psi]$ is given from
\begin{equation}\label{ref:MAG:EQU:303}
\int_{\widehat{\Omega}} v \; d_{\alpha \alpha}^2 q_{,\alpha} Y^*[\psi]_{,\alpha } \;d\widehat{x}  =
\int_{\widehat{\Omega}} r_{,{\alpha}} ,Y_{\alpha}[\psi]  \;d\widehat{x}
\end{equation} 
and finally 
\begin{equation}\label{ref:MAG:EQU:310}
\frac{\partial J^{mag}}{\partial k} = Y^*[\psi]_{,{\alpha}}  B^r_{\alpha} - Y_i[\psi] B^b_{\alpha}
\end{equation}