# Diff of /trunk/doc/inversion/CostFunctions.tex

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1  \chapter{Introduction}\label{Chp:ref:introduction}  \chapter{Introduction}\label{Chp:ref:introduction}
2  Our task in the inversion\index{inversion} is to find the geological structure within a given three-dimensional region $\Omega$ from given geophysical  Our task in the inversion\index{inversion} is to find the geological structure within a given three-dimensional region $\Omega$ from given geophysical
3  observations\index{observation}.  observations\index{observation}.
4  The structure is described by level set function $m$\index{level set function}. The level set  The structure is described by a \emph{level set function} $m$\index{level set function}.
5  function can be a scalar function or may have several components, see Chapter~\ref{Chp:ref:regularization} for more details.  This function can be a scalar function or may have several components,
6  Its values are dimensionless and should be between zero and one. The latter condition is not enforced.      see Chapter~\ref{Chp:ref:regularization} for more details.
7  Through a mapping, see Chapter~\ref{Chp:ref:mapping}\index{mapping}, the values of the level set function are mapped  Its values are dimensionless and should be between zero and one.
8  onto physical parameter $p^f$\index{physical parameter}. The physical parameter feeds into forward models\index{forward model}  However, the latter condition is not enforced.
9  which returns a prediction for the observations, see Chapter~\ref{Chp:ref:forward models}. An inversion may consider  Through a mapping (see Chapter~\ref{Chp:ref:mapping}\index{mapping}) the values
10  several forward models which we call joint inversion\index{joint inversion}.  of the level set function are mapped onto physical parameter $p^f$\index{physical parameter}.
11    The physical parameter feeds into one or more forward models\index{forward model}
12    which return a prediction for the observations, see Chapter~\ref{Chp:ref:forward models}.
13  The level set function describing the actual geological structure is given as those level set function which minimizes  An inversion may consider several forward models at once which we call
14  a particular cost function $J$\index{cost function} which is a composition of the  \emph{joint inversion}\index{joint inversion}.
15  difference of the predicted observations to the actual observations for the relevant forward models
16  and the regularization term\index{regularization} which controls the smoothness of the level set function. In general
17  the  cost function $J$ takes the form  The level set function describing the actual geological structure is given as
18    the function which minimizes a particular \emph{cost function}
19    $J$\index{cost function}.
20    This cost function is a composition of the difference of the predicted
21    observations to the actual observations for the relevant forward models, and
22    the regularization term\index{regularization} which controls the smoothness of
23    the level set function.
24    In general the cost function $J$ takes the form
25  \label{REF:EQU:INTRO 1}  \label{REF:EQU:INTRO 1}
26  J(m) = J^{reg}(m) + \sum_{f} \mu^{data}_{f} \cdot J^{f}(p^f)  J(m) = J^{reg}(m) + \sum_{f} \mu^{data}_{f} \cdot J^{f}(p^f)
27
28  where $J^{f}(p)$ is a measure of the defect of the observations predicted for the parameter $p^f$  where $J^{f}(p)$ is a measure of the defect of the observations predicted for
29  against the observations for forward model $f$ and $J^{reg}(m)$ is the regularization term.  Notice  the parameter $p^f$ against the observations for forward model $f$, and
30  that the physical parameter $p^f$ depend on the level set function $m$ in a known form.  $J^{reg}(m)$ is the regularization term.
31  The weighting factors $\mu^{data}_{f}$ are dimensionless, non-negative trade-off factors\index{trade-off factor}. Potentially  Notice that the physical parameter $p^f$ depends on the level set function
32  values for the trade-off factors are altered during the inversion process in order to improve the balance  $m$ in a known form.
33  between the regularization term and the data defect terms\footnote{The current version does not support an automated selection  The weighting factors $\mu^{data}_{f}$ are dimensionless, non-negative
35    Potentially, values for the trade-off factors are altered during the inversion
36    process in order to improve the balance between the regularization term and
37    the data defect terms\footnote{The current version does not support an automated selection
39
40  The regularization cost function $J^{reg}$ is given through a cost function kernel\index{cost function!kernel} $K^{reg}$ in the form  The regularization cost function $J^{reg}$ is given through a cost function
41    kernel\index{cost function!kernel} $K^{reg}$ in the form
42  \label{REF:EQU:INTRO 2a}  \label{REF:EQU:INTRO 2a}
43  J^{reg}(m) = \int_{\Omega} K^{reg}(m, \nabla m) \; dx  J^{reg}(m) = \int_{\Omega} K^{reg}(m, \nabla m) \; dx
44
45  where $\Omega$ is the region of interest. $K^{reg}$ is a given function of the level set function $m$ and its gradient $\nabla m$.  where $\Omega$ is the region of interest. $K^{reg}$ is a given function of the
46  In case of a multi-component level set function the kernel may consider cross correlation terms between the components.  level set function $m$ and its gradient $\nabla m$.
47    In case of a multi-component level set function the kernel may consider cross
48    correlation terms between the components.
49  Similar for the data defect for forward model $f$ we use a cost function kernel $K^{f}$  Similar for the data defect for forward model $f$ we use a cost function kernel $K^{f}$
50  \label{REF:EQU:INTRO 2b}  \label{REF:EQU:INTRO 2b}
51  J^{f}(p^f) = \int_{\Omega} K^{f}(u^f, \nabla u^f,p^f) \; dx  J^{f}(p^f) = \int_{\Omega} K^{f}(u^f, \nabla u^f,p^f) \; dx
52
53  where $u_f$ is a solution of a partial differential equation (PDE) involving the physical parameter $p^f$. The  where $u_f$ is a solution of a partial differential equation (PDE) involving
54  cost function kernel may directly depend on $u^f$, its gradient $\nabla u^f$ and the physical parameter $p^f$.  the physical parameter $p^f$.
55    The cost function kernel may directly depend on $u^f$, its gradient
56    $\nabla u^f$ and the physical parameter $p^f$.
57
58
59  \subsection{Cartesian Domain}  \section{Cartesian Domain}
60  For the Cartesian domain\index{Cartesian Domain} $\Omega$ we assume a flat Earth in the form  For the Cartesian domain\index{Cartesian Domain} $\Omega$ we assume a flat Earth in the form
61   \label{REF:EQU:INTRO 8}   \label{REF:EQU:INTRO 8}
62  \Omega = [x^{min}_0, x^{max}_0] \times  \Omega = [x^{min}_0, x^{max}_0] \times
# Line 62  Q_2  & = & -Q_r \\ Line 78  Q_2  & = & -Q_r \\
78  \end{array}  \end{array}
79
80
81  \subsection{Sphere Shell Segment}  \section{Sphere Shell Segment}
82

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