 # Contents of /branches/doubleplusgood/doc/inversion/Mapping.tex

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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