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1 \chapter{Mapping}\label{Chp:ref:mapping}
2
3 Mapping classes map a level set function $m$ as described in Chapter~\ref{Chp:ref:regularization}
4 onto a physical parameter such as density and susceptibility.
5
6 \section{Density Map}\label{Chp:ref:mapping density}
7 For density we use the form
8 \begin{equation}\label{EQU:MAP:1}
9 \rho = \rho_{0} + \Delta \rho \cdot \left( \frac{z_0-x_2}{l_z} \right)^{\frac{\beta}{2}} \cdot m
10 \end{equation}
11 where $\rho_{0}$ is the reference density, $\Delta \rho$ is the density scaling, $z_0$ an offset, $l_z$ vertical expansion
12 of the domain and $\beta$ is a suitable exponent.
13
14 \begin{classdesc}{DensityMapping}{domain
15 \optional{, z0=None}
16 \optional{, rho0=0}
17 \optional{, drho=$2750 \cdot kg \cdot m^{-3}$}
18 \optional{, beta=2.}}
19 a linear density mapping including depth weighting. \member{domain} is the
20 domain of the inversion, \member{z0} reference depth in the depth weighting
21 factor, \member{drho} is the density scaling factor (by default the density of
22 granite is used) and \member{beta} is the exponent in the depth weighting factor.
23 If no reference depth \member{z0} is given no depth weighting is applied.
24 \member{rho0} is the reference density which may be a function of its location
25 in the domain.
26 \end{classdesc}
27
28 \begin{methoddesc}[DensityMapping]{getValue}{m}
29 returns the density for level set function $m$
30 \end{methoddesc}
31
32 \begin{methoddesc}[DensityMapping]{getDerivative}{m}
33 return the derivative of density with respect to the level set function.
34 \end{methoddesc}
35
36 \begin{methoddesc}[DensityMapping]{getInverse}{p}
37 returns the value level set function $m$ for given density value $p$.
38 \end{methoddesc}
39
40
41 \section{Susceptibility Map}\label{Chp:ref:mapping susceptibility}
42 For the magnetic susceptibility $k$ the following mapping is used:
43 \begin{equation}\label{EQU:MAP:2}
44 k= k_{0} + \Delta k \cdot \left( \frac{z_0-x_2}{l_z} \right)^{\frac{\beta}{2}} \cdot m
45 \end{equation}
46 where $k_{0}$ is the reference density and $\Delta k$ is the density scaling.
47
48 \begin{classdesc}{SusceptibilityMapping}{domain
49 \optional{, z0=None}
50 \optional{, k0=0}
51 \optional{, dk=1}
52 \optional{, beta=2.}}
53 a linear susceptibility mapping including depth weighting.
54 \member{domain} is the domain of the inversion, \member{z0} reference depth in
55 the depth weighting factor, \member{dk} is the susceptibility scaling factor
56 (by default one is used) and \member{beta} is the exponent in the depth
57 weighting factor. If no reference depth \member{z0} is given no depth
58 weighting is applied.
59 \member{k0} is the reference susceptibility which may be a function of its
60 location in the domain.
61 \end{classdesc}
62
63 \begin{methoddesc}[SusceptibilityMapping]{getValue}{m}
64 returns the susceptibility for level set function $m$
65 \end{methoddesc}
66
67 \begin{methoddesc}[SusceptibilityMapping]{getDerivative}{m}
68 return the derivative of susceptibility with respect to the level set function.
69 \end{methoddesc}
70
71 \begin{methoddesc}[SusceptibilityMapping]{getInverse}{p}
72 returns the value level set function $m$ for given susceptibility value $p$.
73 \end{methoddesc}
74
75
76 \section{General Mapping Class}
77 Users can define their own mapping $p=\Psi(m)$.
78 The following interface needs to be served
79
80 \begin{classdesc}{Mapping}{}
81 mapping of a level set function onto a physical parameter to be used by a
82 forward model.
83 \end{classdesc}
84
85 \begin{methoddesc}[Mapping]{getValue}{m}
86 returns the result $\Psi(m)$ of the mapping for level set function $m$
87 \end{methoddesc}
88
89 \begin{methoddesc}[Mapping]{getDerivative}{m}
90 return the derivative $\frac{\partial \Psi}{\partial m}$ of the mapping with respect to the level set function for
91 the level set function $m$.
92 \end{methoddesc}
93
94 \begin{methoddesc}[Mapping]{getInverse}{p}
95 returns the value level set function $m$ for given value $p$ of the physical parameter, ie $p=\Psi(m)$.
96 \end{methoddesc}
97
98

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