 Diff of /trunk/doc/inversion/magnetism.tex

revision 4047 by gross, Tue Oct 30 09:10:07 2012 UTC revision 4051 by caltinay, Wed Oct 31 03:40:47 2012 UTC
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16  \chapter{Magnetic Inversion}\label{chp:magnetic inversion}  \chapter{Magnetic Inversion}\label{chp:magneticinversion}
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18  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
19  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
20  given as  given as
21  \begin{equation}\label{EQU:Hb}  \begin{equation}\label{EQU:Hb}
# Line 128  which leads to Line 128  which leads to
128  < \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>
129  \end{equation}  \end{equation}
130  Putting this all together we get  Putting this all together we get
131  \begin{equation}\label{MAG:EQU:202g}  \begin{align}
132  < \nabla f(m),q> =  < \nabla f(m),q> = &\mu \cdot < 2 \omega \cdot (m-m^{ref}), q>\nonumber\\
133  \mu \cdot < 2 \omega \cdot (m-m^{ref}), q>  + &\mu \cdot < 2 \omega_i \cdot L_i^2 \cdot (m-m^{ref})_{,i}, q_{,i}>\nonumber\\
134  + \mu \cdot < 2 \omega_i \cdot L_i^2 \cdot (m-m^{ref})_{,i}, q_{,i}> +  + &< ( Y_i[\psi] - (Y^*[\psi])_{,i} ) B^b_i,  \frac{dC}{dm} \cdot q>\label{MAG:EQU:202g}
135  < ( Y_i[\psi] - (Y^*[\psi])_{,i} ) B^b_i,  \frac{dC}{dm} \cdot q>  \end{align}
\end{equation}
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