/[escript]/trunk/doc/inversion/ForwardGravity.tex
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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 76  derivative of the density with respect t Line 76  derivative of the density with respect t
76    
77  \begin{classdesc}{GravityModel}{domain,  \begin{classdesc}{GravityModel}{domain,
78  w, g,  w, g,
79  \optional{, reference=\None}  \optional{, coordinates=\None}
80  \optional{, fixPotentialAtBottom=False},  \optional{, fixPotentialAtBottom=False},
81  \optional{, tol=1e-8}  \optional{, tol=1e-8}
82  }  }
# Line 90  If \member{reference} defines the refere Line 90  If \member{reference} defines the refere
90  \member{tol} set the tolerance for the solution of the PDE~(\ref{ref:GRAV:EQU:100}).  \member{tol} set the tolerance for the solution of the PDE~(\ref{ref:GRAV:EQU:100}).
91  If \member{fixPotentialAtBottom} is set to  \True, the gravitational potential  If \member{fixPotentialAtBottom} is set to  \True, the gravitational potential
92  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.
93    \member{coordinates} set the reference coordinate system to be used. By the default the
94    Cartesian coordinate system is used.
95  \end{classdesc}  \end{classdesc}
96    
97  \begin{methoddesc}[GravityModel]{rescaleWeights}{  \begin{methoddesc}[GravityModel]{rescaleWeights}{
# Line 158  or Line 160  or
160  \end{equation}  \end{equation}
161    
162  \subsection{Geodetic Coordinates }  \subsection{Geodetic Coordinates }
163  For geodetic coordinates $(\phi, \lambda, h)$, see Appendix~\ref{sec:geodetic}, the solution process needs to be slightly modified.  For geodetic coordinates $(\phi, \lambda, h)$, see Chapter~\ref{Chp:ref:coordinates}, the solution process needs to be slightly modified.
164  Observations are recorded along the geodetic coordinates axes $\alpha$ rather than the Cartesian axes $i$. In fact we  Observations are recorded along the geodetic coordinates axes $\alpha$ rather than the Cartesian axes $i$. In fact we
165  have in equation~\ref{ref:GRAV:EQU:9}:  have in equation~\ref{ref:GRAV:EQU:9}:
166  \begin{equation}\label{ref:GRAV:EQU:300}  \begin{equation}\label{ref:GRAV:EQU:300}
167  \omega^{(s)}_i \cdot (g_{i}- g^{(s)}_i) = \omega^{(s)}_{\alpha} \cdot (g_{{\alpha}}- g^{(s)}_{\alpha})  \omega^{(s)}_i \cdot (g_{i}- g^{(s)}_i) = \omega^{(s)}_{\alpha} \cdot (g_{{\alpha}}- g^{(s)}_{\alpha})
168  \end{equation}  \end{equation}
169  where now $g^{(s)}_{\alpha}$ are the observational data with weighting factors $\omega^{(s)}_{\alpha}$.  Using the  where now $g^{(s)}_{\alpha}$ are the observational data with weighting factors $\omega^{(s)}_{\alpha}$.  Using the
170  fact that $g_{{\alpha}} = - \frac{1}{v_{\alpha \alpha}} \psi_{,\alpha}$  fact that $g_{{\alpha}} = - d_{\alpha \alpha} \psi_{,\alpha}$
171  equation~\ref{ref:GRAV:EQU:10} translates to  equation~\ref{ref:GRAV:EQU:10} translates to
172  \begin{equation}\label{ref:GRAV:EQU:301}  \begin{equation}\label{ref:GRAV:EQU:301}
173  J^{grav}(k) = \frac{1}{2}\sum_{s} \int_{\widehat{\Omega}}  J^{grav}(k) = \frac{1}{2}\sum_{s} \int_{\widehat{\Omega}}
174  ( \omega^{(s)}_{\alpha} \cdot (\frac{1}{v_{\alpha \alpha}}  \psi_{,\alpha} + g^{(s)}_{\alpha} ) ) ^2 \; v \; d\widehat{x}  ( \omega^{(s)}_{\alpha} \cdot (d_{\alpha \alpha}  \psi_{,\alpha} + g^{(s)}_{\alpha} ) ) ^2 \; v \; d\widehat{x}
175  \end{equation}  \end{equation}
176  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
177  \begin{equation}\label{ref:GRAV:EQU:301}  \begin{equation}\label{ref:GRAV:EQU:301}
178  J^{grav}(k) = \frac{1}{2}\sum_{s} \int_{\widehat{\Omega}}  J^{grav}(k) = \frac{1}{2}\sum_{s} \int_{\widehat{\Omega}}
179  (  \frac{\omega^{(s)}_{\alpha} v^{\frac{1}{2}}}{v_{\alpha \alpha}} \cdot ( \psi_{,\alpha} + v_{\alpha \alpha} g^{(s)}_{\alpha} ) ) ^2 \; d\widehat{x}  (  \omega^{(s)}_{\alpha} v^{\frac{1}{2}} d_{\alpha \alpha} \cdot ( \psi_{,\alpha} + \frac{1}{d_{\alpha \alpha}} g^{(s)}_{\alpha} ) ) ^2 \; d\widehat{x}
180  =\frac{1}{2}\sum_{s} \int_{\widehat{\Omega}}  =\frac{1}{2}\sum_{s} \int_{\widehat{\Omega}}
181  (  {\widehat{\omega}}^{(s)}_{\alpha}\cdot ( \psi_{,\alpha} + \widehat{g}^{(s)}_{\alpha} ) ) ^2 \; d\widehat{x}  (  {\widehat{\omega}}^{(s)}_{\alpha}\cdot ( \psi_{,\alpha} + \widehat{g}^{(s)}_{\alpha} ) ) ^2 \; d\widehat{x}
182  \end{equation}  \end{equation}
183  with  with
184  \begin{equation}\label{ref:GRAV:EQU:301b}  \begin{equation}\label{ref:GRAV:EQU:301b}
185   \widehat{\omega}^{(s)}_{\alpha} = \frac{\omega^{(s)}_{\alpha} v^{\frac{1}{2}}}{v_{\alpha \alpha}} \mbox{ and }   \widehat{\omega}^{(s)}_{\alpha} =\omega^{(s)}_{\alpha} v^{\frac{1}{2}} d_{\alpha \alpha} \mbox{ and }
186  \widehat{ g}^{(s)}_{\alpha}=  \widehat{ g}^{(s)}_{\alpha}=
187  v_{\alpha \alpha} g^{(s)}_{\alpha}  \frac{1}{d_{\alpha \alpha}} g^{(s)}_{\alpha}
188  \end{equation}  \end{equation}
189  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.
190  The gravity potential is calculated from  The gravity potential is calculated from
191  \begin{equation}\label{ref:GRAV:EQU:302}  \begin{equation}\label{ref:GRAV:EQU:302}
192  \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}  
193  = - \int_{\widehat{\Omega}}  (4\pi G v) \cdot q \rho\;  d\widehat{x}  = - \int_{\widehat{\Omega}}  (4\pi G v) \cdot q \rho\;  d\widehat{x}
194  \end{equation}  \end{equation}
195  see equation~\ref{ref:GRAV:EQU:201}, and the adjoint function $Y^*[\psi]$ for $Y_{\alpha}[\psi]$ is given from  see equation~\ref{ref:GRAV:EQU:201}, and the adjoint function $Y^*[\psi]$ for $Y_{\alpha}[\psi]$ is given from
196  \begin{equation}\label{ref:GRAV:EQU:303}  \begin{equation}\label{ref:GRAV:EQU:303}
197  \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}  =
198    \int_{\widehat{\Omega}} q_{,\alpha} Y_{\alpha}[\psi]  \; d\widehat{x}    \int_{\widehat{\Omega}} q_{,\alpha} Y_{\alpha}[\psi]  \; d\widehat{x}
199  \end{equation}  \end{equation}
200  and finally  and finally

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