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\chapter{Magnetic Inversion}\label{chp:magnetic inversion} 
\chapter{Magnetic Inversion}\label{chp:magneticinversion} 
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We want to calculate the susceptibility $k$ over a rectangular domain as described in Chapter~\ref{chp:gravity inversion} from 
We want to calculate the susceptibility $k$ over a rectangular domain as described in Chapter~\ref{chp:gravityinversion} from 
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the measured magnetic field $B_i$. Under the assumption of linear, isotropic material the magnetic field is 
the measured magnetic field $B_i$. Under the assumption of linear, isotropic material the magnetic field is 
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given as 
given as 
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\begin{equation}\label{EQU:Hb} 
\begin{equation}\label{EQU:Hb} 
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< \nabla J_{data}(\Psi[C[m]]), q> = < ( Y_i[\psi]  (Y^*[\psi])_{,i} ) H^b_i, \frac{dC}{dm} \cdot q> 
< \nabla J_{data}(\Psi[C[m]]), q> = < ( Y_i[\psi]  (Y^*[\psi])_{,i} ) H^b_i, \frac{dC}{dm} \cdot q> 
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\end{equation} 
\end{equation} 
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Putting this all together we get 
Putting this all together we get 
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\begin{equation}\label{MAG:EQU:202g} 
\begin{align} 
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< \nabla f(m),q> = 
< \nabla f(m),q> = &\mu \cdot < 2 \omega \cdot (mm^{ref}), q>\nonumber\\ 
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\mu \cdot < 2 \omega \cdot (mm^{ref}), q> 
+ &\mu \cdot < 2 \omega_i \cdot L_i^2 \cdot (mm^{ref})_{,i}, q_{,i}>\nonumber\\ 
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+ \mu \cdot < 2 \omega_i \cdot L_i^2 \cdot (mm^{ref})_{,i}, q_{,i}> + 
+ &< ( Y_i[\psi]  (Y^*[\psi])_{,i} ) B^b_i, \frac{dC}{dm} \cdot q>\label{MAG:EQU:202g} 
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< ( Y_i[\psi]  (Y^*[\psi])_{,i} ) B^b_i, \frac{dC}{dm} \cdot q> 
\end{align} 

\end{equation} 

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