/[MITgcm]/manual/s_ecco/text/ecco_costfunction.tex
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revision 1.2 by heimbach, Thu May 27 00:06:06 2004 UTC revision 1.3 by heimbach, Thu Jul 8 23:18:08 2004 UTC
# Line 59  field & file name & deccription & unit \ Line 59  field & file name & deccription & unit \
59  \begin{enumerate}  \begin{enumerate}
60  %  %
61  \item  \item
62  Compute 11yr model mean spatial distribution  Compute $nYears$ model mean spatial distribution
63  %  %
64  \begin{equation}  \begin{equation}
65  psmean(i,j)\, =\,  psmean(i,j)\, =\,
# Line 68  psbar(i,j) Line 68  psbar(i,j)
68  \end{equation}  \end{equation}
69  %  %
70  \item  \item
71  Compute global offset between 11-yr model and T/P mean:  Compute global offset between $nYears$ model and T/P mean:
72  %  %
73  \begin{equation}  \begin{equation}
74  \begin{split}  \begin{split}
# Line 289  field & file name & deccription & unit \ Line 289  field & file name & deccription & unit \
289  [ppt] \\  [ppt] \\
290  {\it wti, wsi} & {\tt data\_errfile} & vert. stdev. profile for $T$, $S$ &  {\it wti, wsi} & {\tt data\_errfile} & vert. stdev. profile for $T$, $S$ &
291  ~ \\  ~ \\
292  {\it wtheta2} & {\tt temperrfile} & ~ & [$^{\circ}$C] \\  {\it wtvar} & {\tt temperrfile} & spatially varying stdev. & [$^{\circ}$C] \\
293  {\it wsalt2} & {\tt salterrfile} & ~ & [ppt] \\  {\it wsvar} & {\tt salterrfile} & spatially varying stdev. & [ppt] \\
294  ~&~&~&~\\  ~&~&~&~\\
295  \hline \hline  \hline \hline
296  \end{tabular}  \end{tabular}
# Line 301  field & file name & deccription & unit \ Line 301  field & file name & deccription & unit \
301    
302  \begin{equation}  \begin{equation}
303  \begin{split}  \begin{split}
304  cost\_xbt\_t(i,j) & = \,  cost\_xbt\_t(i,j,k) & = \,
305  \left[ \, \frac{cosphi \cdot mask}{wtheta2} \sum_{\tau=1}^{nMonsRec}  \left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \sum_{\tau=1}^{nMonsRec}
306  \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)
307   \\   \\
308  \end{split}  \end{split}
309  \end{equation}  \end{equation}
# Line 312  cost\_xbt\_t(i,j) & = \, Line 312  cost\_xbt\_t(i,j) & = \,
312    
313  \begin{equation}  \begin{equation}
314  \begin{split}  \begin{split}
315  cost\_ctd\_t(i,j) & = \,  cost\_ctd\_t(i,j,k) & = \,
316  \left[ \, \frac{cosphi \cdot mask}{wtheta2} \sum_{\tau=1}^{nMonsRec}  \left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \sum_{\tau=1}^{nMonsRec}
317  \left\{ Tbar(\tau) \, - \, ctdTobs(\tau) \right\}^2 \, \right](i,j)  \left\{ Tbar(\tau) \, - \, ctdTobs(\tau) \right\}^2 \, \right](i,j,k)
318   \\   \\
319  cost\_ctd\_s(i,j) & = \,  cost\_ctd\_s(i,j,k) & = \,
320  \left[ \, \frac{cosphi \cdot mask}{wsalt2} \sum_{\tau=1}^{nMonsRec}  \left[ \, \frac{fac \cdot ratio}{wsi^2 + wsvar^2} \sum_{\tau=1}^{nMonsRec}
321  \left\{ Sbar(\tau) \, - \, ctdSobs(\tau) \right\}^2 \, \right](i,j)  \left\{ Sbar(\tau) \, - \, ctdSobs(\tau) \right\}^2 \, \right](i,j,k)
322   \\   \\
323  \end{split}  \end{split}
324  \end{equation}  \end{equation}
# Line 327  cost\_ctd\_s(i,j) & = \, Line 327  cost\_ctd\_s(i,j) & = \,
327    
328  \begin{equation}  \begin{equation}
329  \begin{split}  \begin{split}
330  cost\_argo\_t(i,j) & = \,  cost\_argo\_t(i,j,k) & = \,
331  \left[ \, \frac{cosphi \cdot mask}{wtheta2} \sum_{\tau=1}^{nMonsRec}  \left[ \, \frac{fac \cdot ratio}{wti^2 + wvar^2} \sum_{\tau=1}^{nMonsRec}
332  \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)
333   \\   \\
334  cost\_argo\_s(i,j) & = \,  cost\_argo\_s(i,j,k) & = \,
335  \left[ \, \frac{cosphi \cdot mask}{wsalt2} \sum_{\tau=1}^{nMonsRec}  \left[ \, \frac{fac \cdot ratio}{wsi^2 + wsvar^2} \sum_{\tau=1}^{nMonsRec}
336  \left\{ Sbar(\tau) \, - \, argoSobs(\tau) \right\}^2 \, \right](i,j)  \left\{ Sbar(\tau) \, - \, argoSobs(\tau) \right\}^2 \, \right](i,j,k)
337   \\   \\
338  \end{split}  \end{split}
339  \end{equation}  \end{equation}
# Line 343  cost\_argo\_s(i,j) & = \, Line 343  cost\_argo\_s(i,j) & = \,
343  \begin{equation}  \begin{equation}
344  \begin{split}  \begin{split}
345  cost\_sst(i,j) & = \,  cost\_sst(i,j) & = \,
346  \left[ \, \frac{cosphi \cdot mask}{wsst} \sum_{\tau=1}^{nMonsRec}  \left[ \, wsst \sum_{\tau=1}^{nMonsRec}
347  \left\{ Tbar(\tau) \, - \, sstDat(\tau) \right\}^2 \, \right](i,j)  \left\{ Tbar(\tau) \, - \, sstDat(\tau) \right\}^2 \, \right](i,j)
348   \\   \\
349  cost\_sss(i,j) & = \,  cost\_sss(i,j) & = \,
350  \left[ \, \frac{cosphi \cdot mask}{wsss} \sum_{\tau=1}^{nMonsRec}  \left[ \, wsss \sum_{\tau=1}^{nMonsRec}
351  \left\{ Sbar(\tau) \, - \, sssDat(\tau) \right\}^2 \, \right](i,j)  \left\{ Sbar(\tau) \, - \, sssDat(\tau) \right\}^2 \, \right](i,j)
352   \\   \\
353  \end{split}  \end{split}
# Line 355  cost\_sss(i,j) & = \, Line 355  cost\_sss(i,j) & = \,
355    
356  \subsubsection{Levitus montly T, S climatological data}  \subsubsection{Levitus montly T, S climatological data}
357    
358    Model vs. data misfits are taken from $nYears$ monthly model means
359    vs. Levitus monthly data.
360    The description below is for potential temperature.
361    Procedure for salinity is fully analogous.
362    Spatial indices $(i,j,k)$ are omitted throughout.
363    %
364    \begin{enumerate}
365    %
366    \item
367    Compute $nYears$ monthly model means for each month $imon$:
368    \[
369    \overline{Tbar}(imon) \, = \, \frac{1}{nYears}
370    \sum_{iyear=1}^{nYears} Tbar(iyear,imon)
371    \]
372    %
373    \item
374    Compute misfit:
375    \[
376    cost\_theta(i,j,k) \, = \, \left[
377    \frac{fac \cdot ratio}{wti^2} \sum_{imon=1}^{12}
378    \left\{ \overline{Tbar}(imon) \, - \, Tdat(imon) \right\}^2  \right] (i,j,k)
379    \]
380    
381    \end{enumerate}
382    
383    
384  \subsubsection{Weights and notes}  \subsubsection{Weights and notes}
385    
# Line 370  cosphi(i,j) \, = \, 1 Line 395  cosphi(i,j) \, = \, 1
395  \]  \]
396  %  %
397  \item  \item
398  Spatially constant weights:  $ fac \, = \, cosphi \cdot mask $
399    %
400    \item
401    Spatially {\it constant} weights:
402  %  %
403  \begin{enumerate}  \begin{enumerate}
404  %  %
405  \item  \item
406  Read standard deviation fields \\  Read standard deviation vertical profiles for $T$, $S$ \\
407  $ {\tt data\_errfile} \, \longrightarrow \,  $ {\tt data\_errfile} \, \longrightarrow \,
408  wti(k), \,\, wsi(k) $ \\  wti(k), \,\, wsi(k) $ \\
409  $ {\tt data\_errfile} \, \longrightarrow \,  $ {\tt data\_errfile} \, \longrightarrow \,
# Line 385  ratio = 0.25 = \left( \frac{1}{2} \right Line 413  ratio = 0.25 = \left( \frac{1}{2} \right
413  Take inverse squares:  Take inverse squares:
414  \[  \[
415  \begin{split}  \begin{split}
416  wtheta(k) & = \, \frac{ratio}{wti(k) \cdot wti(k)} \\  wtheta(k) & = \, \frac{ratio}{wti(k)^2} \\
417  wsalt(k) & = \, \frac{ratio}{wsi(k) \cdot wsi(k)} \\  wsalt(k) & = \, \frac{ratio}{wsi(k)^2} \\
418  \end{split}  \end{split}
419  \]  \]
420  %  %
421  \end{enumerate}  \end{enumerate}
422  %  %
423  \item  \item
424  Spatially varying weights:  Spatially {\it varying} weights:
425  %  %
426  \begin{enumerate}  \begin{enumerate}
427  %  %
428  \item  \item
429  Read standard deviation fields \\  Read standard deviation fields \\
430  $ {\tt temperrfile} \, \longrightarrow \, wtheta2(i,j,k) $ \\  $ {\tt temperrfile} \, \longrightarrow \, wtvar(i,j,k) $ \\
431  $ {\tt salterrfile} \, \longrightarrow \, wsalt2(i,j,k) $ \\  $ {\tt salterrfile} \, \longrightarrow \, wsvar(i,j,k) $ \\
432  %  %
433  \item  \item
434  Weights are combination of spatially constant and varying parts:  Weights are combination of spatially constant and varying parts:
435  \[  \[
436  \begin{split}  \begin{split}
437  wtheta2(i,j,k) & = \, \frac{ratio}  wtheta2(i,j,k) & = \, \frac{ratio}
438  {wti(k) \cdot wti(k) \, + \,wtheta2(i,j,k) \cdot wtheta2(i,j,k) } \\  {wti(k)^2 \, + \,wtvar(i,j,k)^2 } \\
439  wsalt2(i,j,k) & = \,  wsalt2(i,j,k) & = \,
440  \frac{ratio}  \frac{ratio}
441  {wsi(k) \cdot wsi(k) \, + \,wsalt2(i,j,k) \cdot wsalt2(i,j,k) } \\  {wsi(k)^2 \, + \,wsvar(i,j,k)^2 } \\
442  \end{split}  \end{split}
443  \]  \]
444  %  %
# Line 420  wsalt2(i,j,k) & = \, Line 448  wsalt2(i,j,k) & = \,
448  Sea surface $T$, $S$ weights:  Sea surface $T$, $S$ weights:
449  \begin{itemize}  \begin{itemize}
450  \item  \item
451  SST: $ wsst \, = \, wtheta(1)$: horizontally constant  SST: $ wsst \, = \, fac \cdot wtheta(1)$: horizontally constant
452  \item  \item
453  SSS: $ wsss \, = \, wsalt2(i,j,1)$: horizontally varying  SSS: $ wsss \, = \, fac \cdot wsalt2(i,j,1)$: horizontally varying
454  \end{itemize}  \end{itemize}
455  (Why this difference? I don't know.)  (Why this difference? I don't know.)
456  %  %
# Line 434  SSS: $ wsss \, = \, wsalt2(i,j,1)$: hori Line 462  SSS: $ wsss \, = \, wsalt2(i,j,1)$: hori
462  \begin{itemize}  \begin{itemize}
463  %  %
464  \item  \item
465  Map out $wtheta2(i,j)$, $wsalt2(i,j)$.  Map out $wtheta2(i,j,k)$, $wsalt2(i,j,k)$.
466    
467  %  %
468  \end{itemize}  \end{itemize}
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