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More on magnetic inversion

 1 gross 4047 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 % Copyright (c) 2003-2012 by University of Queensland 4 5 % 6 % Primary Business: Queensland, Australia 7 % Licensed under the Open Software License version 3.0 8 9 % 10 % Development until 2012 by Earth Systems Science Computational Center (ESSCC) 11 % Development since 2012 by School of Earth Sciences 12 % 13 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 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 = \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