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

revision 4375 by gross, Mon Apr 22 03:16:24 2013 UTC revision 4376 by gross, Mon Apr 22 08:44:45 2013 UTC
# Line 107  derivative of the density with respect t Line 107  derivative of the density with respect t
108
109  \begin{classdesc}{MagneticModel}{domain, w, B, background_field,  \begin{classdesc}{MagneticModel}{domain, w, B, background_field,
110          \optional{, useSphericalCoordinates=False}          \optional{, coordinates=\None}
111          \optional{, fixPotentialAtBottom=False},          \optional{, fixPotentialAtBottom=False},
112          \optional{, tol=1e-8}}          \optional{, tol=1e-8},
113    }
114  opens a magnetic forward model over the \Domain \member{domain} with  opens a magnetic forward model over the \Domain \member{domain} with
115  weighting factors \member{w} ($=\omega^{(s)}$) and measured magnetic flux  weighting factors \member{w} ($=\omega^{(s)}$) and measured magnetic flux
116  density anomalies \member{B} ($=B^{(s)}$).  density anomalies \member{B} ($=B^{(s)}$).
117  The weighting factors and the  measured magnetic flux density anomalies must be vectors.  The weighting factors and the  measured magnetic flux density anomalies must be vectors.
118  \member{background_field} defines the background magnetic flux density $B^b$  \member{background_field} defines the background magnetic flux density $B^b$
119  as a vector with north, east and vertical components.  as a vector with north, east and vertical components.
If \member{useSphericalCoordinates} is \True spherical coordinates are used.
120  \member{tol} sets the tolerance for the solution of the PDE~(\ref{ref:MAG:EQU:8}).  \member{tol} sets the tolerance for the solution of the PDE~(\ref{ref:MAG:EQU:8}).
121  If \member{fixPotentialAtBottom} is set to  \True, the gravitational potential  If \member{fixPotentialAtBottom} is set to  \True, the gravitational potential
122  at the bottom is set to zero in addition to the potential on the top.  at the bottom is set to zero in addition to the potential on the top.
123    \member{coordinates} set the reference coordinate system to be used. By the default the
124    Cartesian coordinate system is used.
125  \end{classdesc}  \end{classdesc}
126
127  \begin{methoddesc}[MagneticModel]{rescaleWeights}{  \begin{methoddesc}[MagneticModel]{rescaleWeights}{
# Line 195  have in equation~\ref{ref:MAG:EQU:9}: Line 197  have in equation~\ref{ref:MAG:EQU:9}:
197  \omega^{(s)}_i \cdot (B_{i}- B^{(s)}_i) = \omega^{(s)}_{\alpha} \cdot (B_{{\alpha}}- B^{(s)}_{\alpha})  \omega^{(s)}_i \cdot (B_{i}- B^{(s)}_i) = \omega^{(s)}_{\alpha} \cdot (B_{{\alpha}}- B^{(s)}_{\alpha})
198
199  where now $B^{(s)}_{\alpha}$ are the observational data with weighting factors $\omega^{(s)}_{\alpha}$.  Using the  where now $B^{(s)}_{\alpha}$ are the observational data with weighting factors $\omega^{(s)}_{\alpha}$.  Using the
200  fact that $B_{{\alpha}} = k \cdot B^b_{{\alpha}} - \frac{1}{v_{\alpha \alpha}} \psi_{,\alpha}$  fact that $B_{{\alpha}} = k \cdot B^b_{{\alpha}} -d_{\alpha \alpha} \psi_{,\alpha}$
201  equation~\ref{ref:MAG:EQU:10} translates to  equation~\ref{ref:MAG:EQU:10} translates to
202  \label{ref:MAG:EQU:301}  \label{ref:MAG:EQU:301}
203  J^{mag}(k) = \frac{1}{2}\sum_{s} \int_{\widehat{\Omega}}  J^{mag}(k) = \frac{1}{2}\sum_{s} \int_{\widehat{\Omega}}
204  ( \omega^{(s)}_{\alpha} \cdot (\frac{1}{v_{\alpha \alpha}}  \psi_{,\alpha} - k \cdot B^b_{{\alpha}}  + B^{(s)}_{\alpha} ) ) ^2 \; v \; d\widehat{x}  ( \omega^{(s)}_{\alpha} \cdot (d_{\alpha \alpha}  \psi_{,\alpha} - k \cdot B^b_{{\alpha}}  + B^{(s)}_{\alpha} ) ) ^2 \; v \; d\widehat{x}
205
206  where $\widehat{\Omega}$ and $d\widehat{x}$ refer to integration over the geodetic coordinates axes. This can be rearranged to  where $\widehat{\Omega}$ and $d\widehat{x}$ refer to integration over the geodetic coordinates axes. This can be rearranged to
207  \label{ref:MAG:EQU:301}  \label{ref:MAG:EQU:301}
208  J^{mag}(k) = \frac{1}{2}\sum_{s} \int_{\widehat{\Omega}}  J^{mag}(k) = \frac{1}{2}\sum_{s} \int_{\widehat{\Omega}}
209  (  \frac{\omega^{(s)}_{\alpha} v^{\frac{1}{2}}}{v_{\alpha \alpha}} \cdot (  (  \omega^{(s)}_{\alpha} v^{\frac{1}{2}} d_{\alpha \alpha} \cdot (
210   \psi_{,\alpha} -  k \cdot v_{\alpha \alpha} B^b_{{\alpha}} + v_{\alpha \alpha} B^{(s)}_{\alpha} ) ) ^2 \; d\widehat{x}   \psi_{,\alpha} -  k \cdot v_{\alpha \alpha} B^b_{{\alpha}} + v_{\alpha \alpha} B^{(s)}_{\alpha} ) ) ^2 \; d\widehat{x}
211  =\frac{1}{2}\sum_{s} \int_{\widehat{\Omega}}  =\frac{1}{2}\sum_{s} \int_{\widehat{\Omega}}
212  (  {\widehat{\omega}}^{(s)}_{\alpha}\cdot ( \psi_{,\alpha} -  k \cdot \widehat{B}^b_{{\alpha}}+  \widehat{B}^{(s)}_{\alpha} ) ) ^2 \; d\widehat{x}  (  {\widehat{\omega}}^{(s)}_{\alpha}\cdot ( \psi_{,\alpha} -  k \cdot \widehat{B}^b_{{\alpha}}+  \widehat{B}^{(s)}_{\alpha} ) ) ^2 \; d\widehat{x}
213
214  with  with
215  \label{ref:MAG:EQU:301b}  \label{ref:MAG:EQU:301b}
216   \widehat{\omega}^{(s)}_{\alpha} = \frac{\omega^{(s)}_{\alpha} v^{\frac{1}{2}}}{v_{\alpha \alpha}} \mbox{ , }   \widehat{\omega}^{(s)}_{\alpha} = \omega^{(s)}_{\alpha} v^{\frac{1}{2}} d_{\alpha \alpha} \mbox{ , }
217  \widehat{B}^{(s)}_{\alpha}=  \widehat{B}^{(s)}_{\alpha}=
218  v_{\alpha \alpha} B^{(s)}_{\alpha}  \mbox{ and } \widehat{B}^b_{{\alpha}} = v_{\alpha \alpha} B^b_{{\alpha}}  \frac{1}{d_{\alpha \alpha}} B^{(s)}_{\alpha}  \mbox{ and } \widehat{B}^b_{{\alpha}} = \frac{1}{d_{\alpha \alpha}}  B^b_{{\alpha}}
219
220  which means one can apply the Cartesian formulation to the geodetic coordinates using modified data.  which means one can apply the Cartesian formulation to the geodetic coordinates using modified data.
221
222
223  The magnetic potential is calculated from  The magnetic potential is calculated from
224  \label{ref:MAG:EQU:302}  \label{ref:MAG:EQU:302}
225  \int_{\widehat{\Omega}} \frac{v}{v_{\alpha \alpha}^2} q_{,\alpha} \psi_{,\alpha} \;  d\widehat{x}    \int_{\widehat{\Omega}} v \; d_{\alpha \alpha}^2 q_{,\alpha} \psi_{,\alpha} \;  d\widehat{x}
226  =  \int_{\widehat{\Omega}}  \frac{v}{v_{\alpha \alpha}} k \cdot q_{,\alpha}  B^r_{\alpha} \; d\widehat{x}    =  \int_{\widehat{\Omega}} v \; d_{\alpha \alpha} k \cdot q_{,\alpha}  B^r_{\alpha} \; d\widehat{x}
227  =  \int_{\widehat{\Omega}}  k \cdot q_{,\alpha}  \widehat{B}^r_{\alpha} \; d\widehat{x}    =  \int_{\widehat{\Omega}}  k \cdot q_{,\alpha}  \widehat{B}^r_{\alpha} \; d\widehat{x}
228
229  with  with
230  \label{ref:MAG:EQU:302b}  \label{ref:MAG:EQU:302b}
231  \widehat{B}^r_{\alpha}  = \frac{v}{v_{\alpha \alpha}} \widehat{B}^r_{\alpha}  \widehat{B}^r_{\alpha}  =v \; d_{\alpha \alpha} \widehat{B}^r_{\alpha}
232
233  see equation~\ref{ref:MAG:EQU:201}, and the adjoint function $Y^*[\psi]$ for $Y_{\alpha}[\psi]$ is given from  see equation~\ref{ref:MAG:EQU:201}, and the adjoint function $Y^*[\psi]$ for $Y_{\alpha}[\psi]$ is given from
234  \label{ref:MAG:EQU:303}  \label{ref:MAG:EQU:303}
235  \int_{\widehat{\Omega}} \frac{v}{v_{\alpha \alpha}^2} q_{,\alpha} Y^*[\psi]_{,\alpha } \;d\widehat{x}  =  \int_{\widehat{\Omega}} v \; d_{\alpha \alpha}^2 q_{,\alpha} Y^*[\psi]_{,\alpha } \;d\widehat{x}  =
236  \int_{\widehat{\Omega}} r_{,{\alpha}} ,Y_{\alpha}[\psi]  \;d\widehat{x}  \int_{\widehat{\Omega}} r_{,{\alpha}} ,Y_{\alpha}[\psi]  \;d\widehat{x}
237
238  and finally  and finally
239  \label{ref:GRAV:EQU:201a}  \label{ref:MAG:EQU:201a}
240  \frac{\partial J^{mag}}{\partial k} = Y^*[\psi]_{,{\alpha}}  B^r_{\alpha} - Y_i[\psi] B^b_{\alpha}  \frac{\partial J^{mag}}{\partial k} = Y^*[\psi]_{,{\alpha}}  B^r_{\alpha} - Y_i[\psi] B^b_{\alpha}
241
242

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