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14
15 \section{Gravity Inversion}\label{sec:forward gravity}
16 For the magnetic inversion we use the anomaly of the gravity acceleration~\index{gravity acceleration} of the Earth.
17 The controlling material parameter is the density~\index{density} $\rho$ of
18 the rock.
19 If the density field $\rho$ is known the gravitational potential $\psi$ is
20 given as the solution of the PDE
21 \begin{equation}\label{ref:GRAV:EQU:100}
22 -\psi_{,ii} = -4\pi G \cdot \rho
23 \end{equation}
24 where $G=6.6730 \cdot 10^{-11} \frac{m^3}{kg \cdot s^2}$ is the gravitational
25 constant.
26 The gravitational potential is set to zero at the top of the
27 domain $\Gamma_0$.
28 On all other faces the normal component of the gravity acceleration anomaly
29 $g_i$ is set to zero, i.e. $n_i \psi_{,i} = 0$ with outer normal field $n_i$.
30 The gravity force $g_i$ is given as the negative of the gradient of the gravity
31 potential $\psi$:
32 \begin{equation}\label{ref:GRAV:EQU:101}
33 g_i = - \psi_{,i}
34 \end{equation}
35 From the gravitational potential we can calculate the gravity acceleration
36 anomaly via Equation~(\ref{ref:GRAV:EQU:101}) to obtain the defect to the
37 given data.
38 If $g^{(s)}_i$ is a measurement of the gravity acceleration anomaly for
39 survey $s$ and $\omega^{(s)}_i$ is a weighting factor the data defect
40 $J^{grav}(k)$ in the notation of Chapter~\ref{chapter:ref:inversion cost function} is given as
41 \begin{equation}\label{ref:GRAV:EQU:9}
42 J^{grav}(k) = \frac{1}{2}\sum_{s} \int_{\Omega} ( \omega^{(s)}_i \cdot (g_{i}- g^{(s)}_i) ) ^2 dx
43 \end{equation}
44 Summation over $i$ is performed.
45 The cost function kernel\index{cost function!kernel} is given as
46 \begin{equation}\label{ref:GRAV:EQU:10}
47 K^{grav}(\psi_{,i},k) = \frac{1}{2}\sum_{s} ( \omega^{(s)}_i \cdot (\psi_{,i}+ g^{(s)}_i) ) ^2
48 \end{equation}
49 In practice the gravity acceleration $g^{(s)}$ is measured in vertical
50 direction $z$ with a standard error deviation $\sigma^{(s)}$ at certain
51 locations in the domain.
52 In this case one sets the weighting factors $\omega^{(s)}$ as
53 \begin{equation}\label{ref:GRAV:EQU:11}
54 \omega^{(s)}_i
55 = \left\{
56 \begin{array}{lcl}
57 f \cdot \frac{\delta_{iz}}{\sigma^{(s)}} & & \mbox{data are available} \\
58 & \mbox{ where } & \\
59 0 & & \mbox{ otherwise } \\
60 \end{array}
61 \right.
62 \end{equation}
63 With the objective to control the
64 gradient of the cost function
65 the scaling factor $f$ is chosen in the way that
66 \begin{equation}\label{ref:GRAV:EQU:12}
67 \sum_{s} \int_{\Omega} ( \omega^{(s)}_i g^{(s)}_i ) \cdot ( \omega^{(s)}_j \frac{1}{L_j} ) \cdot 4\pi G L^2 \cdot \rho' \; dx =\alpha
68 \end{equation}
69 where $\alpha$ defines a scaling factor which is typically set to one and $L$ is defined by equation~(\ref{ref:EQU:REG:6b}). $\rho'$ is considering the
70 derivative of the density with respect to the level set function.
71
72
73 \subsection{Usage}
74
75 \LG{Add example}
76
77 \begin{classdesc}{GravityModel}{domain,
78 w, g,
79 \optional{, useSphericalCoordinates=False}
80 \optional{, fixPotentialAtBottom=False},
81 \optional{, tol=1e-8}
82 }
83 opens a gravity forward model over the \Domain \member{domain} with
84 weighting factors \member{w} ($=\omega^{(s)}$) and measured gravity acceleration anomalies\member{g} ($=g^{(s)}$).
85 The weighting factors and the measured gravity acceleration anomalies\member must be vectors.
86 If \member{useSphericalCoordinates} is \True spherical coordinates are used.
87 \member{tol} set the tolerance for the solution of the PDE~(\ref{ref:GRAV:EQU:100}).
88 If \member{fixPotentialAtBottom} is set to \True, the gravitational potential
89 at the bottom is set to zero in addition to the potential on the top.
90 \end{classdesc}
91
92 \begin{methoddesc}[GravityModel]{rescaleWeights}{
93 \optional{scale=1.}
94 \optional{rho_scale=1.}}
95 rescale the weighting factors such condition~(\ref{ref:GRAV:EQU:12}) holds where
96 \member{scale} sets the scale $\alpha$
97 and \member{rho_scale} sets $\rho'$. This method should be called before any inversion is started
98 in order to make sure that all components of the cost function are appropriately scaled.
99 \end{methoddesc}
100
101
102 \subsection{Gradient Calculation}
103 This section briefly explains how the gradient
104 $\frac{\partial J^{grav}}{\partial \rho}$ of the cost function $J^{grav}$ with
105 respect to the density $\rho$ is calculated. We follow the concept as outlined in section~\ref{chapter:ref:inversion cost function:gradient}.
106 The magnetic potential $\psi$ from PDE~(\ref{ref:GRAV:EQU:100}) is solved in
107 weak form:
108 \begin{equation}\label{ref:GRAV:EQU:201}
109 \int_{\Omega} q_{,i} \psi_{,i} \; dx = - \int_{\Omega} 4\pi G \cdot q \rho\; dx
110 \end{equation}
111 for all $q$ with $q=0$ on $\Gamma_0$.
112 In the following we set $\Psi[\cdot]=\psi$ for a given density $\cdot$ as
113 solution of the variational problem~(\ref{ref:GRAV:EQU:201}).
114 If $\Gamma_{\rho}$ denotes the region of the domain where the density is known
115 and for a given direction $p$ with $p=0$ on $\Gamma_{\rho}$ one has
116 \begin{equation}\label{ref:GRAV:EQU:201aa}
117 \int_{\Omega} \frac{\partial J^{grav}}{\partial \rho} \cdot p \; dx = \int_{\Omega}
118 \sum_{s} (\omega^{(s)}_j \cdot
119 (g^{(s)}_j-g_{j}) ) \cdot ( \omega^{(s)}_i \Psi[p]_{,i}) \; dx
120 \end{equation}
121 with
122 \begin{equation}\label{ref:GRAV:EQU:202c}
123 Y_i[\psi]= \sum_{s} (\omega^{(s)}_j \cdot
124 (g^{(s)}_j-g_{j}) ) \cdot \omega^{(s)}_i
125 \end{equation}
126 This is written as
127 \begin{equation}\label{ref:GRAV:EQU:202cc}
128 \int_{\Omega} \frac{\partial J^{grav}}{\partial \rho} \cdot p \; dx = \int_{\Omega}
129 Y_i[\psi] \Psi[p]_{,i} \; dx
130 \end{equation}
131 We then set $Y^*[\psi]$ as the solution of the equation
132 \begin{equation}\label{ref:GRAV:EQU:202d}
133 \int_{\Omega} r_{,i} Y^*[\psi]_{,i} \; dx = \int_{\Omega} r_{,i} ,Y_i[\psi] \; dx \mbox{ for all } p \mbox{ with } r=0 \mbox{ on } \Gamma_{top}
134 \end{equation}
135 with $Y^*[\psi]=0$ on $\Gamma_0$. With $r=\Psi[p]$ we get
136 \begin{equation}\label{ref:GRAV:EQU:202dd}
137 \int_{\Omega} \Psi[p]_{,i} Y^*[\psi]_{,i} \; dx = \int_{\Omega} \Psi[p]_{,i} ,Y_i[\psi] \; dx
138 \end{equation}
139 and from Equation~(\ref{ref:GRAV:EQU:201}) with $q=Y^*[\psi]$ we get
140 \begin{equation}\label{ref:GRAV:EQU:20e}
141 \int_{\Omega} Y^*[\psi]_{,i} \Psi[p]_{,i} \; dx = - \int_{\Omega} 4\pi G \cdot Y^*[\psi] \cdot p\; dx
142 \end{equation}
143 which leads to
144 \begin{equation}\label{ref:GRAV:EQU:20ee}
145 \int_{\Omega} \Psi[p]_{,i} ,Y_i[\psi] \; dx = - \int_{\Omega} 4\pi G \cdot Y^*[\psi] \cdot p \; dx
146 \end{equation}
147 and finally
148 \begin{equation}\label{ref:GRAV:EQU:201a}
149 \int_{\Omega} \frac{\partial J^{grav}}{\partial \rho} \cdot p \; dx = - \int_{\Omega}
150 4\pi G \cdot Y^*[\psi] \cdot p \; dx
151 \end{equation}
152 or
153 \begin{equation}\label{ref:GRAV:EQU:201b}
154 \frac{\partial J^{grav}}{\partial \rho} =- 4\pi G \cdot Y^*[\psi]
155 \end{equation}
156

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