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revision 1.2 by heimbach, Thu May 27 00:06:06 2004 UTC revision 1.5 by heimbach, Thu Jul 14 21:02:29 2005 UTC
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1  \section{The ECCO state estimation cost function DRAFT!!!  \section{The ECCO state estimation cost function DRAFT!!!
2  \label{sectioneccocost}}  \label{sectioneccocost}}
3    \begin{rawhtml}
4    <!-- CMIREDIR:ecco_cost: -->
5    \end{rawhtml}
6    
7  The current ECCO state estimation covers an $nYears = 11$ year  The current ECCO state estimation covers an $nYears = 11$ year
8  model trajectory.  model trajectory.
# Line 59  field & file name & deccription & unit \ Line 62  field & file name & deccription & unit \
62  \begin{enumerate}  \begin{enumerate}
63  %  %
64  \item  \item
65  Compute 11yr model mean spatial distribution  Compute $nYears$ model mean spatial distribution
66  %  %
67  \begin{equation}  \begin{equation}
68  psmean(i,j)\, =\,  psmean(i,j)\, =\,
# Line 68  psbar(i,j) Line 71  psbar(i,j)
71  \end{equation}  \end{equation}
72  %  %
73  \item  \item
74  Compute global offset between 11-yr model and T/P mean:  Compute global offset between $nYears$ model and T/P mean:
75  %  %
76  \begin{equation}  \begin{equation}
77  \begin{split}  \begin{split}
# Line 289  field & file name & deccription & unit \ Line 292  field & file name & deccription & unit \
292  [ppt] \\  [ppt] \\
293  {\it wti, wsi} & {\tt data\_errfile} & vert. stdev. profile for $T$, $S$ &  {\it wti, wsi} & {\tt data\_errfile} & vert. stdev. profile for $T$, $S$ &
294  ~ \\  ~ \\
295  {\it wtheta2} & {\tt temperrfile} & ~ & [$^{\circ}$C] \\  {\it wtvar} & {\tt temperrfile} & spatially varying stdev. & [$^{\circ}$C] \\
296  {\it wsalt2} & {\tt salterrfile} & ~ & [ppt] \\  {\it wsvar} & {\tt salterrfile} & spatially varying stdev. & [ppt] \\
297  ~&~&~&~\\  ~&~&~&~\\
298  \hline \hline  \hline \hline
299  \end{tabular}  \end{tabular}
# Line 301  field & file name & deccription & unit \ Line 304  field & file name & deccription & unit \
304    
305  \begin{equation}  \begin{equation}
306  \begin{split}  \begin{split}
307  cost\_xbt\_t(i,j) & = \,  cost\_xbt\_t(i,j,k) & = \,
308  \left[ \, \frac{cosphi \cdot mask}{wtheta2} \sum_{\tau=1}^{nMonsRec}  \left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \sum_{\tau=1}^{nMonsRec}
309  \left\{ Tbar(\tau) \, - \, T2\theta[xbtobs(\tau)] \right\}^2 \, \right](i,j)  \left\{ Tbar(\tau) \, - \, T2\theta[xbtobs(\tau)] \right\}^2 \, \right](i,j,k)
310   \\   \\
311  \end{split}  \end{split}
312  \end{equation}  \end{equation}
# Line 312  cost\_xbt\_t(i,j) & = \, Line 315  cost\_xbt\_t(i,j) & = \,
315    
316  \begin{equation}  \begin{equation}
317  \begin{split}  \begin{split}
318  cost\_ctd\_t(i,j) & = \,  cost\_ctd\_t(i,j,k) & = \,
319  \left[ \, \frac{cosphi \cdot mask}{wtheta2} \sum_{\tau=1}^{nMonsRec}  \left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \sum_{\tau=1}^{nMonsRec}
320  \left\{ Tbar(\tau) \, - \, ctdTobs(\tau) \right\}^2 \, \right](i,j)  \left\{ Tbar(\tau) \, - \, ctdTobs(\tau) \right\}^2 \, \right](i,j,k)
321   \\   \\
322  cost\_ctd\_s(i,j) & = \,  cost\_ctd\_s(i,j,k) & = \,
323  \left[ \, \frac{cosphi \cdot mask}{wsalt2} \sum_{\tau=1}^{nMonsRec}  \left[ \, \frac{fac \cdot ratio}{wsi^2 + wsvar^2} \sum_{\tau=1}^{nMonsRec}
324  \left\{ Sbar(\tau) \, - \, ctdSobs(\tau) \right\}^2 \, \right](i,j)  \left\{ Sbar(\tau) \, - \, ctdSobs(\tau) \right\}^2 \, \right](i,j,k)
325   \\   \\
326  \end{split}  \end{split}
327  \end{equation}  \end{equation}
# Line 327  cost\_ctd\_s(i,j) & = \, Line 330  cost\_ctd\_s(i,j) & = \,
330    
331  \begin{equation}  \begin{equation}
332  \begin{split}  \begin{split}
333  cost\_argo\_t(i,j) & = \,  cost\_argo\_t(i,j,k) & = \,
334  \left[ \, \frac{cosphi \cdot mask}{wtheta2} \sum_{\tau=1}^{nMonsRec}  \left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \sum_{\tau=1}^{nMonsRec}
335  \left\{ Tbar(\tau) \, - \, T2\theta[argoTobs(\tau)] \right\}^2 \, \right](i,j)  \left\{ Tbar(\tau) \, - \, T2\theta[argoTobs(\tau)] \right\}^2 \, \right](i,j,k)
336   \\   \\
337  cost\_argo\_s(i,j) & = \,  cost\_argo\_s(i,j,k) & = \,
338  \left[ \, \frac{cosphi \cdot mask}{wsalt2} \sum_{\tau=1}^{nMonsRec}  \left[ \, \frac{fac \cdot ratio}{wsi^2 + wsvar^2} \sum_{\tau=1}^{nMonsRec}
339  \left\{ Sbar(\tau) \, - \, argoSobs(\tau) \right\}^2 \, \right](i,j)  \left\{ Sbar(\tau) \, - \, argoSobs(\tau) \right\}^2 \, \right](i,j,k)
340   \\   \\
341  \end{split}  \end{split}
342  \end{equation}  \end{equation}
# Line 343  cost\_argo\_s(i,j) & = \, Line 346  cost\_argo\_s(i,j) & = \,
346  \begin{equation}  \begin{equation}
347  \begin{split}  \begin{split}
348  cost\_sst(i,j) & = \,  cost\_sst(i,j) & = \,
349  \left[ \, \frac{cosphi \cdot mask}{wsst} \sum_{\tau=1}^{nMonsRec}  \left[ \, wsst \sum_{\tau=1}^{nMonsRec}
350  \left\{ Tbar(\tau) \, - \, sstDat(\tau) \right\}^2 \, \right](i,j)  \left\{ Tbar(\tau) \, - \, sstDat(\tau) \right\}^2 \, \right](i,j)
351   \\   \\
352  cost\_sss(i,j) & = \,  cost\_sss(i,j) & = \,
353  \left[ \, \frac{cosphi \cdot mask}{wsss} \sum_{\tau=1}^{nMonsRec}  \left[ \, wsss \sum_{\tau=1}^{nMonsRec}
354  \left\{ Sbar(\tau) \, - \, sssDat(\tau) \right\}^2 \, \right](i,j)  \left\{ Sbar(\tau) \, - \, sssDat(\tau) \right\}^2 \, \right](i,j)
355   \\   \\
356  \end{split}  \end{split}
# Line 355  cost\_sss(i,j) & = \, Line 358  cost\_sss(i,j) & = \,
358    
359  \subsubsection{Levitus montly T, S climatological data}  \subsubsection{Levitus montly T, S climatological data}
360    
361    Model vs. data misfits are taken from $nYears$ monthly model means
362    vs. Levitus monthly data.
363    The description below is for potential temperature.
364    Procedure for salinity is fully analogous.
365    Spatial indices $(i,j,k)$ are omitted throughout.
366    %
367    \begin{enumerate}
368    %
369    \item
370    Compute $nYears$ monthly model means for each month $imon$:
371    \[
372    \overline{Tbar}(imon) \, = \, \frac{1}{nYears}
373    \sum_{iyear=1}^{nYears} Tbar(iyear,imon)
374    \]
375    %
376    \item
377    Compute misfit:
378    \[
379    cost\_theta(i,j,k) \, = \, \left[
380    \frac{fac \cdot ratio}{wti^2} \sum_{imon=1}^{12}
381    \left\{ \overline{Tbar}(imon) \, - \, Tdat(imon) \right\}^2  \right] (i,j,k)
382    \]
383    
384    \end{enumerate}
385    
386    
387  \subsubsection{Weights and notes}  \subsubsection{Weights and notes}
388    
# Line 370  cosphi(i,j) \, = \, 1 Line 398  cosphi(i,j) \, = \, 1
398  \]  \]
399  %  %
400  \item  \item
401  Spatially constant weights:  $ fac \, = \, cosphi \cdot mask $
402    %
403    \item
404    Spatially {\it constant} weights:
405  %  %
406  \begin{enumerate}  \begin{enumerate}
407  %  %
408  \item  \item
409  Read standard deviation fields \\  Read standard deviation vertical profiles for $T$, $S$ \\
410  $ {\tt data\_errfile} \, \longrightarrow \,  $ {\tt data\_errfile} \, \longrightarrow \,
411  wti(k), \,\, wsi(k) $ \\  wti(k), \,\, wsi(k) $ \\
412  $ {\tt data\_errfile} \, \longrightarrow \,  $ {\tt data\_errfile} \, \longrightarrow \,
# Line 385  ratio = 0.25 = \left( \frac{1}{2} \right Line 416  ratio = 0.25 = \left( \frac{1}{2} \right
416  Take inverse squares:  Take inverse squares:
417  \[  \[
418  \begin{split}  \begin{split}
419  wtheta(k) & = \, \frac{ratio}{wti(k) \cdot wti(k)} \\  wtheta(k) & = \, \frac{ratio}{wti(k)^2} \\
420  wsalt(k) & = \, \frac{ratio}{wsi(k) \cdot wsi(k)} \\  wsalt(k) & = \, \frac{ratio}{wsi(k)^2} \\
421  \end{split}  \end{split}
422  \]  \]
423  %  %
424  \end{enumerate}  \end{enumerate}
425  %  %
426  \item  \item
427  Spatially varying weights:  Spatially {\it varying} weights:
428  %  %
429  \begin{enumerate}  \begin{enumerate}
430  %  %
431  \item  \item
432  Read standard deviation fields \\  Read standard deviation fields \\
433  $ {\tt temperrfile} \, \longrightarrow \, wtheta2(i,j,k) $ \\  $ {\tt temperrfile} \, \longrightarrow \, wtvar(i,j,k) $ \\
434  $ {\tt salterrfile} \, \longrightarrow \, wsalt2(i,j,k) $ \\  $ {\tt salterrfile} \, \longrightarrow \, wsvar(i,j,k) $ \\
435  %  %
436  \item  \item
437  Weights are combination of spatially constant and varying parts:  Weights are combination of spatially constant and varying parts:
438  \[  \[
439  \begin{split}  \begin{split}
440  wtheta2(i,j,k) & = \, \frac{ratio}  wtheta2(i,j,k) & = \, \frac{ratio}
441  {wti(k) \cdot wti(k) \, + \,wtheta2(i,j,k) \cdot wtheta2(i,j,k) } \\  {wti(k)^2 \, + \,wtvar(i,j,k)^2 } \\
442  wsalt2(i,j,k) & = \,  wsalt2(i,j,k) & = \,
443  \frac{ratio}  \frac{ratio}
444  {wsi(k) \cdot wsi(k) \, + \,wsalt2(i,j,k) \cdot wsalt2(i,j,k) } \\  {wsi(k)^2 \, + \,wsvar(i,j,k)^2 } \\
445  \end{split}  \end{split}
446  \]  \]
447  %  %
# Line 420  wsalt2(i,j,k) & = \, Line 451  wsalt2(i,j,k) & = \,
451  Sea surface $T$, $S$ weights:  Sea surface $T$, $S$ weights:
452  \begin{itemize}  \begin{itemize}
453  \item  \item
454  SST: $ wsst \, = \, wtheta(1)$: horizontally constant  SST: $ wsst \, = \, fac \cdot wtheta(1)$: horizontally constant
455  \item  \item
456  SSS: $ wsss \, = \, wsalt2(i,j,1)$: horizontally varying  SSS: $ wsss \, = \, fac \cdot wsalt2(i,j,1)$: horizontally varying
457  \end{itemize}  \end{itemize}
458  (Why this difference? I don't know.)  (Why this difference? I don't know.)
459  %  %
# Line 434  SSS: $ wsss \, = \, wsalt2(i,j,1)$: hori Line 465  SSS: $ wsss \, = \, wsalt2(i,j,1)$: hori
465  \begin{itemize}  \begin{itemize}
466  %  %
467  \item  \item
468  Map out $wtheta2(i,j)$, $wsalt2(i,j)$.  Map out $wtheta2(i,j,k)$, $wsalt2(i,j,k)$.
469    
470  %  %
471  \end{itemize}  \end{itemize}
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