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\section{The ECCO state estimation cost function DRAFT!!! |
\section{The ECCO state estimation cost function DRAFT!!! |
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\label{sectioneccocost}} |
\label{sectioneccocost}} |
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\begin{rawhtml} |
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<!-- CMIREDIR:ecco_cost: --> |
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\end{rawhtml} |
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Author: Patrick Heimbach |
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The current ECCO state estimation covers an $nYears = 11$ year |
The current ECCO state estimation covers an $nYears = 11$ year |
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model trajectory. |
model trajectory. |
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\end{table} |
\end{table} |
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\subsubsection{$nYears$ time mean SSH misfit} |
\subsubsection{\textit{\textbf{nYears}} time mean SSH misfit} |
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\begin{enumerate} |
\begin{enumerate} |
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\item |
\item |
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Compute 11yr model mean spatial distribution |
Compute $nYears$ model mean spatial distribution |
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\begin{equation} |
\begin{equation} |
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psmean(i,j)\, =\, |
psmean(i,j)\, =\, |
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\end{equation} |
\end{equation} |
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\item |
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Compute global offset between 11-yr model and T/P mean: |
Compute global offset between $nYears$ model and T/P mean: |
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\begin{equation} |
\begin{equation} |
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\begin{split} |
\begin{split} |
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\end{verbatim} |
\end{verbatim} |
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\subsubsection{Weights} |
\subsubsection{Weights and notes} |
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\begin{itemize} |
\begin{itemize} |
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\frac{1}{\text{daily entries}} \sum_{i,j} cost\_ssh\_anom(i,j,t) |
\frac{1}{\text{daily entries}} \sum_{i,j} cost\_ssh\_anom(i,j,t) |
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\] |
\] |
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\end{itemize} |
\end{itemize} |
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\subsection{Hydrographic constraints} |
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Observation of temperature and salinity from various sources are |
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used to constrain the model. These are: |
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\begin{enumerate} |
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CTD obs. for $T$, $S$ from various WOCE sections |
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XBT obs. for $T$ |
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Sea surface temperature (SST) and salinity (SSS) from |
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Reynolds et al. (???) |
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$T$, $S$ from ARGO floats |
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$T$, $S$ from fields from Levitus (???) |
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\end{enumerate} |
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\subsubsection{Input fields} |
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~ |
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\begin{table}[h!] |
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\begin{center} |
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\begin{tabular}{lllc} |
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\hline \hline |
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~&~&~&~\\ |
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field & file name & deccription & unit \\ |
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~&~&~&~\\ |
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\hline |
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~&~&~&~\\ |
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{\it tbar} & {\tt tbarfile} & monthly model mean pot. temperature & |
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[$^{\circ}\mathrm{C}$] \\ |
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{\it sbar} & {\tt sbarfile} & monthly model mean salinity & |
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[ppt] \\ |
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{\it tdat} & {\tt tdatfile} & monthly mean Levitus pot. temperature & |
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[$^{\circ}\mathrm{C}$] \\ |
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{\it sdat} & {\tt sdatfile} & monthly mean Levitus salinity & |
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[ppt] \\ |
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{\it ctdtobs} & {\tt ctdtfile} & monthly WOCE CTD pot. temperature & |
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[$^{\circ}\mathrm{C}$] \\ |
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{\it ctdsobs} & {\tt ctdsfile} & monthly WOCE CTD salinity & |
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[ppt] \\ |
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{\it xbtobs} & {\tt xbtfile} & monthly XBT in-situ(!) temperature & |
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[$^{\circ}\mathrm{C}$] \\ |
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{\it sstdat} & {\tt sstdatfile} & monthly Reynolds pot. SST & |
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[$^{\circ}\mathrm{C}$] \\ |
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{\it sssdat} & {\tt sssdatfile} & monthly Reynolds SSS & |
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[ppt] \\ |
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{\it argotobs} & {\tt argotfile} & monthly ARGO in-situ(!) temperature & |
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[$^{\circ}\mathrm{C}$] \\ |
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{\it argosobs} & {\tt argosfile} & monthly ARGO salinity & |
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[ppt] \\ |
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{\it wti, wsi} & {\tt data\_errfile} & vert. stdev. profile for $T$, $S$ & |
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~ \\ |
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{\it wtvar} & {\tt temperrfile} & spatially varying stdev. & [$^{\circ}\mathrm{C}$] \\ |
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{\it wsvar} & {\tt salterrfile} & spatially varying stdev. & [ppt] \\ |
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~&~&~&~\\ |
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\hline \hline |
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\end{tabular} |
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\end{center} |
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\end{table} |
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\subsubsection{XBT data} |
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\begin{equation} |
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\begin{split} |
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cost\_xbt\_t(i,j,k) & = \, |
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\left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \sum_{\tau=1}^{nMonsRec} |
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\left\{ Tbar(\tau) \, - \, T2\theta[xbtobs(\tau)] \right\}^2 \, \right](i,j,k) |
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\\ |
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\end{split} |
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\end{equation} |
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\subsubsection{WOCE CTD data} |
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\begin{equation} |
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\begin{split} |
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cost\_ctd\_t(i,j,k) & = \, |
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\left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \sum_{\tau=1}^{nMonsRec} |
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\left\{ Tbar(\tau) \, - \, ctdTobs(\tau) \right\}^2 \, \right](i,j,k) |
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\\ |
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cost\_ctd\_s(i,j,k) & = \, |
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\left[ \, \frac{fac \cdot ratio}{wsi^2 + wsvar^2} \sum_{\tau=1}^{nMonsRec} |
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\left\{ Sbar(\tau) \, - \, ctdSobs(\tau) \right\}^2 \, \right](i,j,k) |
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\\ |
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\end{split} |
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\end{equation} |
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\subsubsection{ARGO float data} |
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\begin{equation} |
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\begin{split} |
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cost\_argo\_t(i,j,k) & = \, |
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\left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \sum_{\tau=1}^{nMonsRec} |
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\left\{ Tbar(\tau) \, - \, T2\theta[argoTobs(\tau)] \right\}^2 \, \right](i,j,k) |
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\\ |
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cost\_argo\_s(i,j,k) & = \, |
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\left[ \, \frac{fac \cdot ratio}{wsi^2 + wsvar^2} \sum_{\tau=1}^{nMonsRec} |
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\left\{ Sbar(\tau) \, - \, argoSobs(\tau) \right\}^2 \, \right](i,j,k) |
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\\ |
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\end{split} |
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\end{equation} |
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\subsubsection{Reynolds sea surface T, S data} |
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\begin{equation} |
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\begin{split} |
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cost\_sst(i,j) & = \, |
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\left[ \, wsst \sum_{\tau=1}^{nMonsRec} |
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\left\{ Tbar(\tau) \, - \, sstDat(\tau) \right\}^2 \, \right](i,j) |
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\\ |
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cost\_sss(i,j) & = \, |
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\left[ \, wsss \sum_{\tau=1}^{nMonsRec} |
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\left\{ Sbar(\tau) \, - \, sssDat(\tau) \right\}^2 \, \right](i,j) |
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\\ |
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\end{split} |
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\end{equation} |
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\subsubsection{Levitus montly T, S climatological data} |
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Model vs. data misfits are taken from $nYears$ monthly model means |
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vs. Levitus monthly data. |
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The description below is for potential temperature. |
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Procedure for salinity is fully analogous. |
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Spatial indices $(i,j,k)$ are omitted throughout. |
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\begin{enumerate} |
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Compute $nYears$ monthly model means for each month $imon$: |
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\[ |
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\overline{Tbar}(imon) \, = \, \frac{1}{nYears} |
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\sum_{iyear=1}^{nYears} Tbar(iyear,imon) |
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\] |
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\item |
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Compute misfit: |
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\[ |
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cost\_theta(i,j,k) \, = \, \left[ |
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\frac{fac \cdot ratio}{wti^2} \sum_{imon=1}^{12} |
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\left\{ \overline{Tbar}(imon) \, - \, Tdat(imon) \right\}^2 \right] (i,j,k) |
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\] |
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\end{enumerate} |
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\subsubsection{Weights and notes} |
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\begin{itemize} |
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$T2\theta$ is an operator mapping in-situ to potential temperatures |
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Latitudinal weight not used: |
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\[ |
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cosphi(i,j) \, = \, 1 |
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\] |
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\item |
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$ fac \, = \, cosphi \cdot mask $ |
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Spatially {\it constant} weights: |
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\begin{enumerate} |
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\item |
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Read standard deviation vertical profiles for $T$, $S$ \\ |
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$ {\tt data\_errfile} \, \longrightarrow \, |
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wti(k), \,\, wsi(k) $ \\ |
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$ {\tt data\_errfile} \, \longrightarrow \, |
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ratio = 0.25 = \left( \frac{1}{2} \right)^2 $ |
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\item |
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Take inverse squares: |
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\[ |
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\begin{split} |
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wtheta(k) & = \, \frac{ratio}{wti(k)^2} \\ |
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wsalt(k) & = \, \frac{ratio}{wsi(k)^2} \\ |
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\end{split} |
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\] |
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\end{enumerate} |
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\item |
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Spatially {\it varying} weights: |
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\begin{enumerate} |
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\item |
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Read standard deviation fields \\ |
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$ {\tt temperrfile} \, \longrightarrow \, wtvar(i,j,k) $ \\ |
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$ {\tt salterrfile} \, \longrightarrow \, wsvar(i,j,k) $ \\ |
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Weights are combination of spatially constant and varying parts: |
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\[ |
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\begin{split} |
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wtheta2(i,j,k) & = \, \frac{ratio} |
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{wti(k)^2 \, + \,wtvar(i,j,k)^2 } \\ |
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wsalt2(i,j,k) & = \, |
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\frac{ratio} |
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{wsi(k)^2 \, + \,wsvar(i,j,k)^2 } \\ |
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\end{split} |
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\] |
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\end{enumerate} |
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\item |
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Sea surface $T$, $S$ weights: |
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\begin{itemize} |
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SST: $ wsst \, = \, fac \cdot wtheta(1)$: horizontally constant |
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SSS: $ wsss \, = \, fac \cdot wsalt2(i,j,1)$: horizontally varying |
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\end{itemize} |
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(Why this difference? I don't know.) |
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\end{itemize} |
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\subsubsection{Diagnostics} |
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\begin{itemize} |
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Map out $wtheta2(i,j,k)$, $wsalt2(i,j,k)$. |
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\end{itemize} |
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