--- manual/s_ecco/text/ecco_costfunction.tex 2004/05/11 21:55:14 1.1 +++ manual/s_ecco/text/ecco_costfunction.tex 2006/04/08 01:50:50 1.7 @@ -1,5 +1,10 @@ \section{The ECCO state estimation cost function DRAFT!!! \label{sectioneccocost}} +\begin{rawhtml} + +\end{rawhtml} + +Author: Patrick Heimbach The current ECCO state estimation covers an $nYears = 11$ year model trajectory. @@ -54,12 +59,12 @@ \end{table} -\subsubsection{$nYears$ time mean SSH misfit} +\subsubsection{\textit{\textbf{nYears}} time mean SSH misfit} \begin{enumerate} % \item -Compute 11yr model mean spatial distribution +Compute $nYears$ model mean spatial distribution % \begin{equation} psmean(i,j)\, =\, @@ -68,7 +73,7 @@ \end{equation} % \item -Compute global offset between 11-yr model and T/P mean: +Compute global offset between $nYears$ model and T/P mean: % \begin{equation} \begin{split} @@ -161,7 +166,7 @@ \end{verbatim} -\subsubsection{Weights} +\subsubsection{Weights and notes} \begin{itemize} % @@ -227,3 +232,243 @@ \frac{1}{\text{daily entries}} \sum_{i,j} cost\_ssh\_anom(i,j,t) \] \end{itemize} + +\subsection{Hydrographic constraints} + +Observation of temperature and salinity from various sources are +used to constrain the model. These are: +% +\begin{enumerate} +% +\item +CTD obs. for $T$, $S$ from various WOCE sections +% +\item +XBT obs. for $T$ +% +\item +Sea surface temperature (SST) and salinity (SSS) from +Reynolds et al. (???) +% +\item +$T$, $S$ from ARGO floats +% +\item +$T$, $S$ from fields from Levitus (???) +% +\end{enumerate} + +\subsubsection{Input fields} +~ + +\begin{table}[h!] +\begin{center} +\begin{tabular}{lllc} +\hline \hline +~&~&~&~\\ +field & file name & deccription & unit \\ +~&~&~&~\\ +\hline +~&~&~&~\\ +{\it tbar} & {\tt tbarfile} & monthly model mean pot. temperature & +[$^{\circ}\mathrm{C}$] \\ +{\it sbar} & {\tt sbarfile} & monthly model mean salinity & +[ppt] \\ +{\it tdat} & {\tt tdatfile} & monthly mean Levitus pot. temperature & +[$^{\circ}\mathrm{C}$] \\ +{\it sdat} & {\tt sdatfile} & monthly mean Levitus salinity & +[ppt] \\ +{\it ctdtobs} & {\tt ctdtfile} & monthly WOCE CTD pot. temperature & +[$^{\circ}\mathrm{C}$] \\ +{\it ctdsobs} & {\tt ctdsfile} & monthly WOCE CTD salinity & +[ppt] \\ +{\it xbtobs} & {\tt xbtfile} & monthly XBT in-situ(!) temperature & +[$^{\circ}\mathrm{C}$] \\ +{\it sstdat} & {\tt sstdatfile} & monthly Reynolds pot. SST & +[$^{\circ}\mathrm{C}$] \\ +{\it sssdat} & {\tt sssdatfile} & monthly Reynolds SSS & +[ppt] \\ +{\it argotobs} & {\tt argotfile} & monthly ARGO in-situ(!) temperature & +[$^{\circ}\mathrm{C}$] \\ +{\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. & [$^{\circ}\mathrm{C}$] \\ +{\it wsvar} & {\tt salterrfile} & spatially varying stdev. & [ppt] \\ +~&~&~&~\\ +\hline \hline +\end{tabular} +\end{center} +\end{table} + +\subsubsection{XBT data} + +\begin{equation} +\begin{split} +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{equation} + +\subsubsection{WOCE CTD data} + +\begin{equation} +\begin{split} +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) + \\ +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{equation} + +\subsubsection{ARGO float data} + +\begin{equation} +\begin{split} +cost\_argo\_t(i,j,k) & = \, +\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{equation} + +\subsubsection{Reynolds sea surface T, S data} + +\begin{equation} +\begin{split} +cost\_sst(i,j) & = \, +\left[ \, wsst \sum_{\tau=1}^{nMonsRec} +\left\{ Tbar(\tau) \, - \, sstDat(\tau) \right\}^2 \, \right](i,j) + \\ +cost\_sss(i,j) & = \, +\left[ \, wsss \sum_{\tau=1}^{nMonsRec} +\left\{ Sbar(\tau) \, - \, sssDat(\tau) \right\}^2 \, \right](i,j) + \\ +\end{split} +\end{equation} + +\subsubsection{Levitus montly T, S climatological data} + +Model vs. data misfits are taken from $nYears$ monthly model means +vs. Levitus monthly data. +The description below is for potential temperature. +Procedure for salinity is fully analogous. +Spatial indices $(i,j,k)$ are omitted throughout. +% +\begin{enumerate} +% +\item +Compute $nYears$ monthly model means for each month $imon$: +\[ +\overline{Tbar}(imon) \, = \, \frac{1}{nYears} +\sum_{iyear=1}^{nYears} Tbar(iyear,imon) +\] +% +\item +Compute misfit: +\[ +cost\_theta(i,j,k) \, = \, \left[ +\frac{fac \cdot ratio}{wti^2} \sum_{imon=1}^{12} +\left\{ \overline{Tbar}(imon) \, - \, Tdat(imon) \right\}^2 \right] (i,j,k) +\] + +\end{enumerate} + + +\subsubsection{Weights and notes} + +\begin{itemize} +% +\item +$T2\theta$ is an operator mapping in-situ to potential temperatures +% +\item +Latitudinal weight not used: +\[ +cosphi(i,j) \, = \, 1 +\] +% +\item +$ fac \, = \, cosphi \cdot mask $ +% +\item +Spatially {\it constant} weights: +% +\begin{enumerate} +% +\item +Read standard deviation vertical profiles for $T$, $S$ \\ +$ {\tt data\_errfile} \, \longrightarrow \, +wti(k), \,\, wsi(k) $ \\ +$ {\tt data\_errfile} \, \longrightarrow \, +ratio = 0.25 = \left( \frac{1}{2} \right)^2 $ +% +\item +Take inverse squares: +\[ +\begin{split} +wtheta(k) & = \, \frac{ratio}{wti(k)^2} \\ +wsalt(k) & = \, \frac{ratio}{wsi(k)^2} \\ +\end{split} +\] +% +\end{enumerate} +% +\item +Spatially {\it varying} weights: +% +\begin{enumerate} +% +\item +Read standard deviation fields \\ +$ {\tt temperrfile} \, \longrightarrow \, wtvar(i,j,k) $ \\ +$ {\tt salterrfile} \, \longrightarrow \, wsvar(i,j,k) $ \\ +% +\item +Weights are combination of spatially constant and varying parts: +\[ +\begin{split} +wtheta2(i,j,k) & = \, \frac{ratio} +{wti(k)^2 \, + \,wtvar(i,j,k)^2 } \\ +wsalt2(i,j,k) & = \, +\frac{ratio} +{wsi(k)^2 \, + \,wsvar(i,j,k)^2 } \\ +\end{split} +\] +% +\end{enumerate} +% +\item +Sea surface $T$, $S$ weights: +\begin{itemize} +\item +SST: $ wsst \, = \, fac \cdot wtheta(1)$: horizontally constant +\item +SSS: $ wsss \, = \, fac \cdot wsalt2(i,j,1)$: horizontally varying +\end{itemize} +(Why this difference? I don't know.) +% +\end{itemize} + + +\subsubsection{Diagnostics} + +\begin{itemize} +% +\item +Map out $wtheta2(i,j,k)$, $wsalt2(i,j,k)$. + +% +\end{itemize} +