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

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

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-revision 1.4 by edhill,
Sat Oct 16 03:40:19 2004 UTC
+revision 1.8 by jmc,
Mon Aug 30 23:09:19 2010 UTC
 Line 4
  <!-- CMIREDIR:ecco_cost: -->
  \end{rawhtml}
+ Author: Patrick Heimbach
  The current ECCO state estimation covers an $nYears = 11$ year
  model trajectory.
  A variety of data sets enter a least squares cost function,
-Line 35 
 weighted by the full geoid error covaria
+Line 37 
 weighted by the full geoid error covaria
  \subsubsection{Input fields}
  ~
- \begin{table}[h!]
+ \begin{table}[!ht]
  \begin{center}
  \begin{tabular}{lllc}
  \hline \hline
-Line 74 
 psbar(i,j)
+Line 76 
 psbar(i,j)
  Compute global offset between $nYears$ model and T/P mean:
  %
  \begin{equation}
- \begin{split}
+ \begin{aligned}
  offset & = \, \overline{tpmean} \, - \, \overline{psmean} \\
  ~ & = \, \frac{1}{normaliz.} \sum_{i,j}
  \left\{ tpmean(i,j) \, - \, psmean(i,j) \right\}
  \cdot cosphi(i,j) \cdot tpmeanmask(i,j)
- \end{split}
+ \end{aligned}
  \end{equation}
  %
  \item
-Line 88 
 Misfits are computed w.r.t. global $offs
+Line 90 
 Misfits are computed w.r.t. global $offs
  First spatial distribution:
  %
  \begin{equation}
- \begin{split}
+ \begin{aligned}
  cost\_ssh\_mean(i,j) & = \,
  \frac{1}{wp^2} \left\{ \,
  \left[ \, psmean(i,j) - \overline{psmean} \, \right] \, - \,
  \left[ \, tpmean(i,j) - \overline{tpmean} \, \right] \, \right\}^2 \\
  ~ & = \, \frac{1}{wp^2} \left\{ \,
  psmean(i,j) \, - \, tpmean(i,j) \, + \, offset \, \right\}^2
- \end{split}
+ \end{aligned}
  \end{equation}
  %
-Line 120 
 Here we write out computation for T/P.
+Line 122 
 Here we write out computation for T/P.
  Compute difference in anomalies:
  \begin{equation}
- \begin{split}
+ \begin{aligned}
  cost\_ssh\_anom(i,j,t) & = \, \frac{1}{wtp^2} \left\{ \,
  \left[ \, psbar(i,j,t) - psmean(i,j) \, \right] \, - \,
  \left[ \, tpobs(i,j,t) \, \right] \,
  \right\}^2
- \end{split}
+ \end{aligned}
  \end{equation}
  %
  where $t$ denotes time (day) index, and
-Line 136 
 $tpmean(i,j)$ has already been removed f
+Line 138 
 $tpmean(i,j)$ has already been removed f
  Sum over all spatial points and all times
  \begin{equation}
- \begin{split}
+ \begin{aligned}
  \overline{cost\_ssh\_anom} & = \, \sum_{t} \sum_{i,j}
  cost\_ssh\_anom(i,j,t)
- \end{split}
+ \end{aligned}
  \end{equation}
  \end{enumerate}
-Line 259 
 $T$, $S$ from fields from Levitus (???)
+Line 261 
 $T$, $S$ from fields from Levitus (???)
  \subsubsection{Input fields}
  ~
- \begin{table}[h!]
+ \begin{table}[!ht]
  \begin{center}
  \begin{tabular}{lllc}
  \hline \hline
-Line 269 
 field & file name & deccription & unit \
+Line 271 
 field & file name & deccription & unit \
  \hline
  ~&~&~&~\\
  {\it tbar} & {\tt tbarfile} & monthly model mean pot. temperature &
- [$^{\circ}$C] \\
+ [$^{\circ}\mathrm{C}$] \\
  {\it sbar} & {\tt sbarfile} & monthly model mean salinity &
  [ppt] \\
  {\it tdat} & {\tt tdatfile} & monthly mean Levitus pot. temperature &
- [$^{\circ}$C] \\
+ [$^{\circ}\mathrm{C}$] \\
  {\it sdat} & {\tt sdatfile} & monthly mean Levitus salinity &
  [ppt] \\
  {\it ctdtobs}  & {\tt ctdtfile} & monthly WOCE CTD pot. temperature &
- [$^{\circ}$C] \\
+ [$^{\circ}\mathrm{C}$] \\
  {\it ctdsobs}  & {\tt ctdsfile} & monthly WOCE CTD salinity &
  [ppt] \\
  {\it xbtobs} & {\tt xbtfile} & monthly XBT in-situ(!) temperature &
- [$^{\circ}$C] \\
+ [$^{\circ}\mathrm{C}$] \\
  {\it sstdat}  & {\tt sstdatfile} & monthly Reynolds pot. SST &
- [$^{\circ}$C] \\
+ [$^{\circ}\mathrm{C}$] \\
  {\it sssdat}  & {\tt sssdatfile} & monthly Reynolds SSS &
  [ppt] \\
  {\it argotobs}  & {\tt argotfile} & monthly ARGO in-situ(!) temperature &
- [$^{\circ}$C] \\
+ [$^{\circ}\mathrm{C}$] \\
  {\it argosobs}  & {\tt argosfile} & monthly ARGO salinity &
  [ppt] \\
  {\it wti, wsi} & {\tt data\_errfile} & vert. stdev. profile for $T$, $S$ &
  ~ \\
- {\it wtvar} & {\tt temperrfile} & spatially varying stdev. & [$^{\circ}$C] \\
+ {\it wtvar} & {\tt temperrfile} & spatially varying stdev. & [$^{\circ}\mathrm{C}$] \\
  {\it wsvar} & {\tt salterrfile} & spatially varying stdev. & [ppt] \\
  ~&~&~&~\\
  \hline \hline
-Line 303 
 field & file name & deccription & unit \
+Line 305 
 field & file name & deccription & unit \
  \subsubsection{XBT data}
  \begin{equation}
- \begin{split}
+ \begin{aligned}
  cost\_xbt\_t(i,j,k) & = \,
  \left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \sum_{\tau=1}^{nMonsRec}
  \left\{ Tbar(\tau) \, - \, T2\theta[xbtobs(\tau)] \right\}^2 \, \right](i,j,k)
   \\
- \end{split}
+ \end{aligned}
  \end{equation}
  \subsubsection{WOCE CTD data}
  \begin{equation}
- \begin{split}
+ \begin{aligned}
  cost\_ctd\_t(i,j,k) & = \,
  \left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \sum_{\tau=1}^{nMonsRec}
  \left\{ Tbar(\tau) \, - \, ctdTobs(\tau) \right\}^2 \, \right](i,j,k)
-Line 323 
 cost\_ctd\_s(i,j,k) & = \,
+Line 325 
 cost\_ctd\_s(i,j,k) & = \,
  \left[ \, \frac{fac \cdot ratio}{wsi^2 + wsvar^2} \sum_{\tau=1}^{nMonsRec}
  \left\{ Sbar(\tau) \, - \, ctdSobs(\tau) \right\}^2 \, \right](i,j,k)
   \\
- \end{split}
+ \end{aligned}
  \end{equation}
  \subsubsection{ARGO float data}
  \begin{equation}
- \begin{split}
+ \begin{aligned}
  cost\_argo\_t(i,j,k) & = \,
- \left[ \, \frac{fac \cdot ratio}{wti^2 + wvar^2} \sum_{\tau=1}^{nMonsRec}
+ \left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \sum_{\tau=1}^{nMonsRec}
  \left\{ Tbar(\tau) \, - \, T2\theta[argoTobs(\tau)] \right\}^2 \, \right](i,j,k)
   \\
  cost\_argo\_s(i,j,k) & = \,
  \left[ \, \frac{fac \cdot ratio}{wsi^2 + wsvar^2} \sum_{\tau=1}^{nMonsRec}
  \left\{ Sbar(\tau) \, - \, argoSobs(\tau) \right\}^2 \, \right](i,j,k)
   \\
- \end{split}
+ \end{aligned}
  \end{equation}
  \subsubsection{Reynolds sea surface T, S data}
  \begin{equation}
- \begin{split}
+ \begin{aligned}
  cost\_sst(i,j) & = \,
  \left[ \, wsst \sum_{\tau=1}^{nMonsRec}
  \left\{ Tbar(\tau) \, - \, sstDat(\tau) \right\}^2 \, \right](i,j)
-Line 353 
 cost\_sss(i,j) & = \,
+Line 355 
 cost\_sss(i,j) & = \,
  \left[ \, wsss \sum_{\tau=1}^{nMonsRec}
  \left\{ Sbar(\tau) \, - \, sssDat(\tau) \right\}^2 \, \right](i,j)
   \\
- \end{split}
+ \end{aligned}
  \end{equation}
  \subsubsection{Levitus montly T, S climatological data}
-Line 415 
 ratio = 0.25 = \left( \frac{1}{2} \right
+Line 417 
 ratio = 0.25 = \left( \frac{1}{2} \right
  \item
  Take inverse squares:
  \[
- \begin{split}
+ \begin{aligned}
  wtheta(k) & = \, \frac{ratio}{wti(k)^2} \\
  wsalt(k) & = \, \frac{ratio}{wsi(k)^2} \\
- \end{split}
+ \end{aligned}
  \]
  %
  \end{enumerate}
-Line 436 
 $ {\tt salterrfile} \, \longrightarrow \
+Line 438 
 $ {\tt salterrfile} \, \longrightarrow \
  \item
  Weights are combination of spatially constant and varying parts:
  \[
- \begin{split}
+ \begin{aligned}
  wtheta2(i,j,k) & = \, \frac{ratio}
  {wti(k)^2 \, + \,wtvar(i,j,k)^2 } \\
  wsalt2(i,j,k) & = \,
  \frac{ratio}
  {wsi(k)^2 \, + \,wsvar(i,j,k)^2 } \\
- \end{split}
+ \end{aligned}
  \]
  %
  \end{enumerate}

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