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More on magnetic inversion
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14    
15    
16     \chapter{Magnetic Inversion}\label{chp:magnetic inversion}
17    
18     We want to calculate the susceptibility $k$ over a rectangular domain as described in Chapter~\ref{chp:gravity inversion} from
19     the measured magnetic field $B_i$. Under the assumption of linear, isotropic material the magnetic field is
20     given as
21     \begin{equation}\label{EQU:Hb}
22     B_i = H_i + k \cdot B^b_i
23     \end{equation}
24     where $H_i$ is the unknown is the magnetizing field and
25     $B^b_i$ is the background magnetic field of the Earth.
26     \footnote{Notation: the superscript ``b'' following a symbol indicates
27     a background field quantity.}. We assume that the background magnetizing field is constant across the domain
28     where for a domain at latitude $theta$ the latitude and radial components of magnetizing field are given as
29     \begin{equation}\label{EQU:Hb 2}
30     \begin{array}{rcl}
31     B^b_{\theta} & = & \displaystyle{ \frac{\mu_0 \cdot m}{4 \pi \cdot R_{earth}^3} sin(\theta) } \\
32     B^b_r & = & \displaystyle{ \frac{\mu_0 \cdot m}{1 \pi \cdot R_{earth}^3} cos(\theta) }
33     \end{array}
34     \end{equation}
35     with the vacuum permeability $\mu_0 = 4 \pi \cdot 10^{-7} \frac{Vs}{Am}$,
36     the magnetic dipole moment of Earth $m= 8.22 \cdot 10^22 Am^2$ and earth radius $R_{earth}= 6378137m$. We assume a flat domain
37     and associate where longitude is associated with $x_0$-direction,
38     the latitude is associated with $x_1$-direction and the radial direction is associated with $x_2$-direction:
39     \begin{equation}\label{EQU:Hb 3}
40     \begin{array}{rcl}
41     B^b_0 & = & 0 \\
42     B^b_1 & = & -B^b_{\theta} \\
43     B^b_2 & = & -B^b_r \\
44     \end{array}
45     \end{equation}
46     We introduce the potential $\psi$ for the deviation of the magnetizing field from the background magnetizing field:
47     \begin{equation}\label{EQU:Hb 4}
48     H_i = - \psi_{,i} + B^b
49     \end{equation}
50     With the is notation we get from equation~(\ref{EQU:Hb}):
51     \begin{equation}\label{EQU:Hb 5}
52     B_i = (1+k) B^b_i - \psi_{,i}
53     \end{equation}
54     As the magnetic field $B$ is divergence free we get the PDE
55     \begin{equation}\label{EQU:Hb 5}
56     - \psi_{,ii} = - ((1+k) B^b_i)_{,i}
57     \end{equation}
58     for the magnetic potential $\psi$.
59    
60     The problem is to calculate the susceptibility $k$ from the magnetic field $B$ known at some parts of the region of interest
61     $\Omega$. In fact we want to minimize the value
62     \begin{equation}\label{GRAV:EQU:102}
63     J_{data}(\psi) = \frac{1}{2}\sum_{s} \int_{\Omega} \chi^{(s)}_i \cdot ( B_{i}- B^{(s)}_i)^2 dx
64     \footnote{Summation over $i$ is performed by Einstein notation.}
65     \end{equation}
66     where $B^{(s)}_i$ is a measurement of the gravity force $B_i = (1+k) H^b_i - \psi_{,i} $
67     and $\chi^{(s)}_i$ is a weighting factor.
68     $s$ indexes the surveys included in the inversion.
69    
70    
71    
72     \section{Solution methods}
73     To apply the Nonlinear Conjugate Gradient method (NLCG), see Appendix~\ref{sec:NLCG} or the L-BFGS method, see Appendix~\ref{sec:LBFGS} we need
74     to define an inner product $<.,.>$ and need to calculate the gradient of of $f$.
75    
76     As inner product we use
77     \begin{equation}\label{MAG:EQU:200}
78     <p,q> = \int_{\Omega} p \cdot q \; dx
79     \end{equation}
80     With this notation the magnetic potential is given as $\psi=\Psi[1+k]$ where
81     \begin{equation}\label{MAG:EQU:201}
82     < q_{,i},\Psi[p]_{,i} > = < q_{,i} , p \cdot H^b_i> \mbox{ for all } q \mbox{ with } q=0 \mbox{ on } \Gamma_{z}
83     \end{equation}
84     where $\Gamma_{z}$ denotes the top and bottom surface of the domain for case (D)
85     and the top surface of the domain for case (N).
86    
87    
88    
89     For any $q$ with $q=0$ on $\Omega^{ref}$ we get
90     \begin{equation}\label{MAG:EQU:202}
91     < \nabla f(m),q> = < \nabla J_{data}(\Psi[1+C[m]]), q> + \mu \cdot < \nabla J_{reg}(m),q>
92     \end{equation}
93     with
94     \begin{equation}\label{MAG:EQU:202a}
95     < \nabla J_{reg}(m),q> =
96     \int_{\Omega} 2 \omega q \cdot (m-m^{ref}) + 2 \omega_i \cdot L_i^2 \cdot q_{,i} (m-m^{ref})_{,i} dx
97     \end{equation}
98     and
99     \begin{equation}\label{MAG:EQU:202b}
100     < \nabla J_{data}(\Psi[C[m]]), q> = \sum_{s} \int_{\Omega} \chi^{(s)}_i \cdot ( B_i - B^{(s)}_i )
101     \cdot ( \frac{dC}{dm} \cdot q \cdot B^b_i - \Psi[\frac{dC}{dm} \cdot q]_{,i}) dx
102     \end{equation}
103     where $k=C(m)$ and $B_i=(1+k) H^b_i - \Psi[1+k]_{,i}$
104     With
105     \begin{equation}\label{MAG:EQU:202c}
106     Y_i[\psi]= \sum_{s} \chi^{(s)}_i \cdot ( B_i - B^{(s)}_i )
107     \end{equation}
108     we get
109     \begin{equation}\label{MAG:EQU:202bb}
110     < \nabla J_{data}(\Psi[C[m]]), q> =
111     < Y_i[\psi] B^b_i, \frac{dC}{dm} q> - < Y_i[\psi] , \Psi[\frac{dC}{dm} \cdot q]_{,i} >
112     \end{equation}
113     and $Y^*[\psi]$ as the solution of the equation
114     \begin{equation}\label{MAG:EQU:202d}
115     < p_{,i},Y^*[\psi]_{,i} > = < p_{,i} ,Y_i[\psi] > \mbox{ for all } p \mbox{ with } p=0 \mbox{ on } \Gamma_{z}
116     \end{equation}
117     with $Y^*[\psi]=0$ on $\Gamma_{z}$.
118     By stetting $p=\Psi[\frac{dC}{dm} \cdot q]$ we get
119     \begin{equation}\label{MAG:EQU:202e}
120     <(\Psi[\frac{dC}{dm} \cdot q]) _{,i},Y^*[\psi]_{,i} > = < (\Psi[\frac{dC}{dm} \cdot q])_{,i} ,Y_i[\psi] >
121     \end{equation}
122     and from~\ref{MAG:EQU:201} with $q=Y^*[\psi]$ we get
123     \begin{equation}\label{MAG:EQU:20e}
124     < (Y^*[\psi])_{,i},(\Psi[\frac{dC}{dm} \cdot q])_{,i} > = <(Y^*[\psi])_{,i} , \frac{dC}{dm} \cdot q \cdot H^b_i>
125     \end{equation}
126     which leads to
127     \begin{equation}\label{MAG:EQU:202f}
128     < \nabla J_{data}(\Psi[C[m]]), q> = < ( Y_i[\psi] - (Y^*[\psi])_{,i} ) H^b_i, \frac{dC}{dm} \cdot q>
129     \end{equation}
130     Putting this all together we get
131     \begin{equation}\label{MAG:EQU:202g}
132     < \nabla f(m),q> =
133     \mu \cdot < 2 \omega \cdot (m-m^{ref}), q>
134     + \mu \cdot < 2 \omega_i \cdot L_i^2 \cdot (m-m^{ref})_{,i}, q_{,i}> +
135     < ( Y_i[\psi] - (Y^*[\psi])_{,i} ) B^b_i, \frac{dC}{dm} \cdot q>
136     \end{equation}
137    
138    

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