/[MITgcm]/manual/s_ecco/text/ecco_costfunction.tex
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revision 1.7 by edhill, Sat Apr 8 01:50:50 2006 UTC revision 1.8 by jmc, Mon Aug 30 23:09:19 2010 UTC
# Line 37  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 76  psbar(i,j) Line 76  psbar(i,j)
76  Compute global offset between $nYears$ 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 90  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 122  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 138  $tpmean(i,j)$ 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 261  $T$, $S$ 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 305  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,k) & = \,  cost\_xbt\_t(i,j,k) & = \,
310  \left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \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,k)  \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,k) & = \,  cost\_ctd\_t(i,j,k) & = \,
321  \left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \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,k)  \left\{ Tbar(\tau) \, - \, ctdTobs(\tau) \right\}^2 \, \right](i,j,k)
# Line 325  cost\_ctd\_s(i,j,k) & = \, Line 325  cost\_ctd\_s(i,j,k) & = \,
325  \left[ \, \frac{fac \cdot ratio}{wsi^2 + wsvar^2} \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,k)  \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,k) & = \,  cost\_argo\_t(i,j,k) & = \,
336  \left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \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,k)  \left\{ Tbar(\tau) \, - \, T2\theta[argoTobs(\tau)] \right\}^2 \, \right](i,j,k)
# Line 340  cost\_argo\_s(i,j,k) & = \, Line 340  cost\_argo\_s(i,j,k) & = \,
340  \left[ \, \frac{fac \cdot ratio}{wsi^2 + wsvar^2} \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,k)  \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[ \, 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)
# Line 355  cost\_sss(i,j) & = \, Line 355  cost\_sss(i,j) & = \,
355  \left[ \, 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}
# Line 417  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  \[  \[
420  \begin{split}  \begin{aligned}
421  wtheta(k) & = \, \frac{ratio}{wti(k)^2} \\  wtheta(k) & = \, \frac{ratio}{wti(k)^2} \\
422  wsalt(k) & = \, \frac{ratio}{wsi(k)^2} \\  wsalt(k) & = \, \frac{ratio}{wsi(k)^2} \\
423  \end{split}  \end{aligned}
424  \]  \]
425  %  %
426  \end{enumerate}  \end{enumerate}
# Line 438  $ {\tt salterrfile} \, \longrightarrow \ Line 438  $ {\tt salterrfile} \, \longrightarrow \
438  \item  \item
439  Weights are combination of spatially constant and varying parts:  Weights are combination of spatially constant and varying parts:
440  \[  \[
441  \begin{split}  \begin{aligned}
442  wtheta2(i,j,k) & = \, \frac{ratio}  wtheta2(i,j,k) & = \, \frac{ratio}
443  {wti(k)^2 \, + \,wtvar(i,j,k)^2 } \\  {wti(k)^2 \, + \,wtvar(i,j,k)^2 } \\
444  wsalt2(i,j,k) & = \,  wsalt2(i,j,k) & = \,
445  \frac{ratio}  \frac{ratio}
446  {wsi(k)^2 \, + \,wsvar(i,j,k)^2 } \\  {wsi(k)^2 \, + \,wsvar(i,j,k)^2 } \\
447  \end{split}  \end{aligned}
448  \]  \]
449  %  %
450  \end{enumerate}  \end{enumerate}
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