/[MITgcm]/manual/s_ecco/text/ecco_costfunction.tex
ViewVC logotype

Diff of /manual/s_ecco/text/ecco_costfunction.tex

Parent Directory Parent Directory | Revision Log Revision Log | View Revision Graph Revision Graph | View Patch Patch

revision 1.2 by heimbach, Thu May 27 00:06:06 2004 UTC revision 1.8 by jmc, Mon Aug 30 23:09:19 2010 UTC
# Line 1  Line 1 
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    Author: Patrick Heimbach
8    
9  The current ECCO state estimation covers an $nYears = 11$ year  The current ECCO state estimation covers an $nYears = 11$ year
10  model trajectory.  model trajectory.
# Line 32  weighted by the full geoid error covaria Line 37  weighted by the full geoid error covaria
37  \subsubsection{Input fields}  \subsubsection{Input fields}
38  ~  ~
39    
40  \begin{table}[h!]  \begin{table}[!ht]
41  \begin{center}  \begin{center}
42  \begin{tabular}{lllc}  \begin{tabular}{lllc}
43  \hline \hline  \hline \hline
# Line 59  field & file name & deccription & unit \ Line 64  field & file name & deccription & unit \
64  \begin{enumerate}  \begin{enumerate}
65  %  %
66  \item  \item
67  Compute 11yr model mean spatial distribution  Compute $nYears$ model mean spatial distribution
68  %  %
69  \begin{equation}  \begin{equation}
70  psmean(i,j)\, =\,  psmean(i,j)\, =\,
# Line 68  psbar(i,j) Line 73  psbar(i,j)
73  \end{equation}  \end{equation}
74  %  %
75  \item  \item
76  Compute global offset between 11-yr model and T/P mean:  Compute global offset between $nYears$ model and T/P mean:
77  %  %
78  \begin{equation}  \begin{equation}
79  \begin{split}  \begin{aligned}
80  offset & = \, \overline{tpmean} \, - \, \overline{psmean} \\  offset & = \, \overline{tpmean} \, - \, \overline{psmean} \\
81  ~ & = \, \frac{1}{normaliz.} \sum_{i,j}  ~ & = \, \frac{1}{normaliz.} \sum_{i,j}
82  \left\{ tpmean(i,j) \, - \, psmean(i,j) \right\}  \left\{ tpmean(i,j) \, - \, psmean(i,j) \right\}
83  \cdot cosphi(i,j) \cdot tpmeanmask(i,j)  \cdot cosphi(i,j) \cdot tpmeanmask(i,j)
84  \end{split}  \end{aligned}
85  \end{equation}  \end{equation}
86  %  %
87  \item  \item
# Line 85  Misfits are computed w.r.t. global $offs Line 90  Misfits are computed w.r.t. global $offs
90  First spatial distribution:  First spatial distribution:
91  %  %
92  \begin{equation}  \begin{equation}
93  \begin{split}  \begin{aligned}
94  cost\_ssh\_mean(i,j) & = \,  cost\_ssh\_mean(i,j) & = \,
95  \frac{1}{wp^2} \left\{ \,  \frac{1}{wp^2} \left\{ \,
96  \left[ \, psmean(i,j) - \overline{psmean} \, \right] \, - \,  \left[ \, psmean(i,j) - \overline{psmean} \, \right] \, - \,
97  \left[ \, tpmean(i,j) - \overline{tpmean} \, \right] \, \right\}^2 \\  \left[ \, tpmean(i,j) - \overline{tpmean} \, \right] \, \right\}^2 \\
98  ~ & = \, \frac{1}{wp^2} \left\{ \,  ~ & = \, \frac{1}{wp^2} \left\{ \,
99  psmean(i,j) \, - \, tpmean(i,j) \, + \, offset \, \right\}^2  psmean(i,j) \, - \, tpmean(i,j) \, + \, offset \, \right\}^2
100  \end{split}  \end{aligned}
101  \end{equation}  \end{equation}
102    
103  %  %
# Line 117  Here we write out computation for T/P. Line 122  Here we write out computation for T/P.
122  Compute difference in anomalies:  Compute difference in anomalies:
123    
124  \begin{equation}  \begin{equation}
125  \begin{split}  \begin{aligned}
126  cost\_ssh\_anom(i,j,t) & = \, \frac{1}{wtp^2} \left\{ \,  cost\_ssh\_anom(i,j,t) & = \, \frac{1}{wtp^2} \left\{ \,
127  \left[ \, psbar(i,j,t) - psmean(i,j) \, \right] \, - \,  \left[ \, psbar(i,j,t) - psmean(i,j) \, \right] \, - \,
128  \left[ \, tpobs(i,j,t) \, \right] \,  \left[ \, tpobs(i,j,t) \, \right] \,
129  \right\}^2  \right\}^2
130  \end{split}  \end{aligned}
131  \end{equation}  \end{equation}
132  %  %
133  where $t$ denotes time (day) index, and  where $t$ denotes time (day) index, and
# Line 133  $tpmean(i,j)$ has already been removed f Line 138  $tpmean(i,j)$ has already been removed f
138  Sum over all spatial points and all times  Sum over all spatial points and all times
139    
140  \begin{equation}  \begin{equation}
141  \begin{split}  \begin{aligned}
142  \overline{cost\_ssh\_anom} & = \, \sum_{t} \sum_{i,j}  \overline{cost\_ssh\_anom} & = \, \sum_{t} \sum_{i,j}
143  cost\_ssh\_anom(i,j,t)  cost\_ssh\_anom(i,j,t)
144  \end{split}  \end{aligned}
145  \end{equation}  \end{equation}
146    
147  \end{enumerate}  \end{enumerate}
# Line 256  $T$, $S$ from fields from Levitus (???) Line 261  $T$, $S$ from fields from Levitus (???)
261  \subsubsection{Input fields}  \subsubsection{Input fields}
262  ~  ~
263    
264  \begin{table}[h!]  \begin{table}[!ht]
265  \begin{center}  \begin{center}
266  \begin{tabular}{lllc}  \begin{tabular}{lllc}
267  \hline \hline  \hline \hline
# Line 266  field & file name & deccription & unit \ Line 271  field & file name & deccription & unit \
271  \hline  \hline
272  ~&~&~&~\\  ~&~&~&~\\
273  {\it tbar} & {\tt tbarfile} & monthly model mean pot. temperature &  {\it tbar} & {\tt tbarfile} & monthly model mean pot. temperature &
274  [$^{\circ}$C] \\  [$^{\circ}\mathrm{C}$] \\
275  {\it sbar} & {\tt sbarfile} & monthly model mean salinity &  {\it sbar} & {\tt sbarfile} & monthly model mean salinity &
276  [ppt] \\  [ppt] \\
277  {\it tdat} & {\tt tdatfile} & monthly mean Levitus pot. temperature &  {\it tdat} & {\tt tdatfile} & monthly mean Levitus pot. temperature &
278  [$^{\circ}$C] \\  [$^{\circ}\mathrm{C}$] \\
279  {\it sdat} & {\tt sdatfile} & monthly mean Levitus salinity &  {\it sdat} & {\tt sdatfile} & monthly mean Levitus salinity &
280  [ppt] \\  [ppt] \\
281  {\it ctdtobs}  & {\tt ctdtfile} & monthly WOCE CTD pot. temperature &  {\it ctdtobs}  & {\tt ctdtfile} & monthly WOCE CTD pot. temperature &
282  [$^{\circ}$C] \\  [$^{\circ}\mathrm{C}$] \\
283  {\it ctdsobs}  & {\tt ctdsfile} & monthly WOCE CTD salinity &  {\it ctdsobs}  & {\tt ctdsfile} & monthly WOCE CTD salinity &
284  [ppt] \\  [ppt] \\
285  {\it xbtobs} & {\tt xbtfile} & monthly XBT in-situ(!) temperature &  {\it xbtobs} & {\tt xbtfile} & monthly XBT in-situ(!) temperature &
286  [$^{\circ}$C] \\  [$^{\circ}\mathrm{C}$] \\
287  {\it sstdat}  & {\tt sstdatfile} & monthly Reynolds pot. SST &  {\it sstdat}  & {\tt sstdatfile} & monthly Reynolds pot. SST &
288  [$^{\circ}$C] \\  [$^{\circ}\mathrm{C}$] \\
289  {\it sssdat}  & {\tt sssdatfile} & monthly Reynolds SSS &  {\it sssdat}  & {\tt sssdatfile} & monthly Reynolds SSS &
290  [ppt] \\  [ppt] \\
291  {\it argotobs}  & {\tt argotfile} & monthly ARGO in-situ(!) temperature &  {\it argotobs}  & {\tt argotfile} & monthly ARGO in-situ(!) temperature &
292  [$^{\circ}$C] \\  [$^{\circ}\mathrm{C}$] \\
293  {\it argosobs}  & {\tt argosfile} & monthly ARGO salinity &  {\it argosobs}  & {\tt argosfile} & monthly ARGO salinity &
294  [ppt] \\  [ppt] \\
295  {\it wti, wsi} & {\tt data\_errfile} & vert. stdev. profile for $T$, $S$ &  {\it wti, wsi} & {\tt data\_errfile} & vert. stdev. profile for $T$, $S$ &
296  ~ \\  ~ \\
297  {\it wtheta2} & {\tt temperrfile} & ~ & [$^{\circ}$C] \\  {\it wtvar} & {\tt temperrfile} & spatially varying stdev. & [$^{\circ}\mathrm{C}$] \\
298  {\it wsalt2} & {\tt salterrfile} & ~ & [ppt] \\  {\it wsvar} & {\tt salterrfile} & spatially varying stdev. & [ppt] \\
299  ~&~&~&~\\  ~&~&~&~\\
300  \hline \hline  \hline \hline
301  \end{tabular}  \end{tabular}
# Line 300  field & file name & deccription & unit \ Line 305  field & file name & deccription & unit \
305  \subsubsection{XBT data}  \subsubsection{XBT data}
306    
307  \begin{equation}  \begin{equation}
308  \begin{split}  \begin{aligned}
309  cost\_xbt\_t(i,j) & = \,  cost\_xbt\_t(i,j,k) & = \,
310  \left[ \, \frac{cosphi \cdot mask}{wtheta2} \sum_{\tau=1}^{nMonsRec}  \left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \sum_{\tau=1}^{nMonsRec}
311  \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)
312   \\   \\
313  \end{split}  \end{aligned}
314  \end{equation}  \end{equation}
315    
316  \subsubsection{WOCE CTD data}  \subsubsection{WOCE CTD data}
317    
318  \begin{equation}  \begin{equation}
319  \begin{split}  \begin{aligned}
320  cost\_ctd\_t(i,j) & = \,  cost\_ctd\_t(i,j,k) & = \,
321  \left[ \, \frac{cosphi \cdot mask}{wtheta2} \sum_{\tau=1}^{nMonsRec}  \left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \sum_{\tau=1}^{nMonsRec}
322  \left\{ Tbar(\tau) \, - \, ctdTobs(\tau) \right\}^2 \, \right](i,j)  \left\{ Tbar(\tau) \, - \, ctdTobs(\tau) \right\}^2 \, \right](i,j,k)
323   \\   \\
324  cost\_ctd\_s(i,j) & = \,  cost\_ctd\_s(i,j,k) & = \,
325  \left[ \, \frac{cosphi \cdot mask}{wsalt2} \sum_{\tau=1}^{nMonsRec}  \left[ \, \frac{fac \cdot ratio}{wsi^2 + wsvar^2} \sum_{\tau=1}^{nMonsRec}
326  \left\{ Sbar(\tau) \, - \, ctdSobs(\tau) \right\}^2 \, \right](i,j)  \left\{ Sbar(\tau) \, - \, ctdSobs(\tau) \right\}^2 \, \right](i,j,k)
327   \\   \\
328  \end{split}  \end{aligned}
329  \end{equation}  \end{equation}
330    
331  \subsubsection{ARGO float data}  \subsubsection{ARGO float data}
332    
333  \begin{equation}  \begin{equation}
334  \begin{split}  \begin{aligned}
335  cost\_argo\_t(i,j) & = \,  cost\_argo\_t(i,j,k) & = \,
336  \left[ \, \frac{cosphi \cdot mask}{wtheta2} \sum_{\tau=1}^{nMonsRec}  \left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \sum_{\tau=1}^{nMonsRec}
337  \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)
338   \\   \\
339  cost\_argo\_s(i,j) & = \,  cost\_argo\_s(i,j,k) & = \,
340  \left[ \, \frac{cosphi \cdot mask}{wsalt2} \sum_{\tau=1}^{nMonsRec}  \left[ \, \frac{fac \cdot ratio}{wsi^2 + wsvar^2} \sum_{\tau=1}^{nMonsRec}
341  \left\{ Sbar(\tau) \, - \, argoSobs(\tau) \right\}^2 \, \right](i,j)  \left\{ Sbar(\tau) \, - \, argoSobs(\tau) \right\}^2 \, \right](i,j,k)
342   \\   \\
343  \end{split}  \end{aligned}
344  \end{equation}  \end{equation}
345    
346  \subsubsection{Reynolds sea surface T, S data}  \subsubsection{Reynolds sea surface T, S data}
347    
348  \begin{equation}  \begin{equation}
349  \begin{split}  \begin{aligned}
350  cost\_sst(i,j) & = \,  cost\_sst(i,j) & = \,
351  \left[ \, \frac{cosphi \cdot mask}{wsst} \sum_{\tau=1}^{nMonsRec}  \left[ \, wsst \sum_{\tau=1}^{nMonsRec}
352  \left\{ Tbar(\tau) \, - \, sstDat(\tau) \right\}^2 \, \right](i,j)  \left\{ Tbar(\tau) \, - \, sstDat(\tau) \right\}^2 \, \right](i,j)
353   \\   \\
354  cost\_sss(i,j) & = \,  cost\_sss(i,j) & = \,
355  \left[ \, \frac{cosphi \cdot mask}{wsss} \sum_{\tau=1}^{nMonsRec}  \left[ \, wsss \sum_{\tau=1}^{nMonsRec}
356  \left\{ Sbar(\tau) \, - \, sssDat(\tau) \right\}^2 \, \right](i,j)  \left\{ Sbar(\tau) \, - \, sssDat(\tau) \right\}^2 \, \right](i,j)
357   \\   \\
358  \end{split}  \end{aligned}
359  \end{equation}  \end{equation}
360    
361  \subsubsection{Levitus montly T, S climatological data}  \subsubsection{Levitus montly T, S climatological data}
362    
363    Model vs. data misfits are taken from $nYears$ monthly model means
364    vs. Levitus monthly data.
365    The description below is for potential temperature.
366    Procedure for salinity is fully analogous.
367    Spatial indices $(i,j,k)$ are omitted throughout.
368    %
369    \begin{enumerate}
370    %
371    \item
372    Compute $nYears$ monthly model means for each month $imon$:
373    \[
374    \overline{Tbar}(imon) \, = \, \frac{1}{nYears}
375    \sum_{iyear=1}^{nYears} Tbar(iyear,imon)
376    \]
377    %
378    \item
379    Compute misfit:
380    \[
381    cost\_theta(i,j,k) \, = \, \left[
382    \frac{fac \cdot ratio}{wti^2} \sum_{imon=1}^{12}
383    \left\{ \overline{Tbar}(imon) \, - \, Tdat(imon) \right\}^2  \right] (i,j,k)
384    \]
385    
386    \end{enumerate}
387    
388    
389  \subsubsection{Weights and notes}  \subsubsection{Weights and notes}
390    
# Line 370  cosphi(i,j) \, = \, 1 Line 400  cosphi(i,j) \, = \, 1
400  \]  \]
401  %  %
402  \item  \item
403  Spatially constant weights:  $ fac \, = \, cosphi \cdot mask $
404    %
405    \item
406    Spatially {\it constant} weights:
407  %  %
408  \begin{enumerate}  \begin{enumerate}
409  %  %
410  \item  \item
411  Read standard deviation fields \\  Read standard deviation vertical profiles for $T$, $S$ \\
412  $ {\tt data\_errfile} \, \longrightarrow \,  $ {\tt data\_errfile} \, \longrightarrow \,
413  wti(k), \,\, wsi(k) $ \\  wti(k), \,\, wsi(k) $ \\
414  $ {\tt data\_errfile} \, \longrightarrow \,  $ {\tt data\_errfile} \, \longrightarrow \,
# Line 384  ratio = 0.25 = \left( \frac{1}{2} \right Line 417  ratio = 0.25 = \left( \frac{1}{2} \right
417  \item  \item
418  Take inverse squares:  Take inverse squares:
419  \[  \[
420  \begin{split}  \begin{aligned}
421  wtheta(k) & = \, \frac{ratio}{wti(k) \cdot wti(k)} \\  wtheta(k) & = \, \frac{ratio}{wti(k)^2} \\
422  wsalt(k) & = \, \frac{ratio}{wsi(k) \cdot wsi(k)} \\  wsalt(k) & = \, \frac{ratio}{wsi(k)^2} \\
423  \end{split}  \end{aligned}
424  \]  \]
425  %  %
426  \end{enumerate}  \end{enumerate}
427  %  %
428  \item  \item
429  Spatially varying weights:  Spatially {\it varying} weights:
430  %  %
431  \begin{enumerate}  \begin{enumerate}
432  %  %
433  \item  \item
434  Read standard deviation fields \\  Read standard deviation fields \\
435  $ {\tt temperrfile} \, \longrightarrow \, wtheta2(i,j,k) $ \\  $ {\tt temperrfile} \, \longrightarrow \, wtvar(i,j,k) $ \\
436  $ {\tt salterrfile} \, \longrightarrow \, wsalt2(i,j,k) $ \\  $ {\tt salterrfile} \, \longrightarrow \, wsvar(i,j,k) $ \\
437  %  %
438  \item  \item
439  Weights are combination of spatially constant and varying parts:  Weights are combination of spatially constant and varying parts:
440  \[  \[
441  \begin{split}  \begin{aligned}
442  wtheta2(i,j,k) & = \, \frac{ratio}  wtheta2(i,j,k) & = \, \frac{ratio}
443  {wti(k) \cdot wti(k) \, + \,wtheta2(i,j,k) \cdot wtheta2(i,j,k) } \\  {wti(k)^2 \, + \,wtvar(i,j,k)^2 } \\
444  wsalt2(i,j,k) & = \,  wsalt2(i,j,k) & = \,
445  \frac{ratio}  \frac{ratio}
446  {wsi(k) \cdot wsi(k) \, + \,wsalt2(i,j,k) \cdot wsalt2(i,j,k) } \\  {wsi(k)^2 \, + \,wsvar(i,j,k)^2 } \\
447  \end{split}  \end{aligned}
448  \]  \]
449  %  %
450  \end{enumerate}  \end{enumerate}
# Line 420  wsalt2(i,j,k) & = \, Line 453  wsalt2(i,j,k) & = \,
453  Sea surface $T$, $S$ weights:  Sea surface $T$, $S$ weights:
454  \begin{itemize}  \begin{itemize}
455  \item  \item
456  SST: $ wsst \, = \, wtheta(1)$: horizontally constant  SST: $ wsst \, = \, fac \cdot wtheta(1)$: horizontally constant
457  \item  \item
458  SSS: $ wsss \, = \, wsalt2(i,j,1)$: horizontally varying  SSS: $ wsss \, = \, fac \cdot wsalt2(i,j,1)$: horizontally varying
459  \end{itemize}  \end{itemize}
460  (Why this difference? I don't know.)  (Why this difference? I don't know.)
461  %  %
# Line 434  SSS: $ wsss \, = \, wsalt2(i,j,1)$: hori Line 467  SSS: $ wsss \, = \, wsalt2(i,j,1)$: hori
467  \begin{itemize}  \begin{itemize}
468  %  %
469  \item  \item
470  Map out $wtheta2(i,j)$, $wsalt2(i,j)$.  Map out $wtheta2(i,j,k)$, $wsalt2(i,j,k)$.
471    
472  %  %
473  \end{itemize}  \end{itemize}
Legend:
Removed from v.1.2  
changed lines
  Added in v.1.8
  ViewVC Help
Powered by ViewVC 1.1.22