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heimbach |
1.1 |
\section{The ECCO state estimation cost function DRAFT!!! |
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\label{sectioneccocost}} |
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The current ECCO state estimation covers an $nYears = 11$ year |
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model trajectory. |
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A variety of data sets enter a least squares cost function, |
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in addition to penalty terms which constrain deviations |
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of control variables beyound their a priori errors. |
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\subsection{Sea surface height from TOPEX/Poseidon and ERS-1/2 altimetry} |
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Altimetric SSH contributions from T/P and ERS-1/2 are four-fold: |
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% |
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\begin{enumerate} |
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% |
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\item |
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an $nYears$ time mean SSH misfit between |
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model and T/P |
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% |
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\item |
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daily SSH anomaly misfits between T/P and model |
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% |
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\item |
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daily SSH anomaly misfits between ERS-1/2 and model |
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% |
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\item |
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daily absolute SSH misfit between T/P and model, |
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weighted by the full geoid error covariance. |
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% |
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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 psbar} & {\tt psbarfile} & daily model mean SSH fields & [m] \\ |
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{\it tpmean} & {\tt topexmeanfile} & $nYears$ T/P mean & [cm] \\ |
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{\it tpobs} & {\tt topexfile} & daily T/P SSH anomalies & [cm] \\ |
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{\it erspobs} & {\tt ersfile} & daily ERS-1/2 SSH anomalies & [cm] \\ |
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{\it wp} & {\tt geoid\_errfile} & diagonal of geoid error covariance & [m] \\ |
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{\it wtp, wers} & {\tt ssh\_errfile} & rms of SSH anomalies & [cm] \\ |
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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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heimbach |
1.2 |
\subsubsection{\textit{\textbf{nYears}} time mean SSH misfit} |
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heimbach |
1.1 |
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\begin{enumerate} |
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% |
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\item |
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heimbach |
1.3 |
Compute $nYears$ model mean spatial distribution |
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heimbach |
1.1 |
% |
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\begin{equation} |
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psmean(i,j)\, =\, |
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\frac{1}{nDaysRec} \sum_{i=1}^{nDaysRec} |
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psbar(i,j) |
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\end{equation} |
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% |
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\item |
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heimbach |
1.3 |
Compute global offset between $nYears$ model and T/P mean: |
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heimbach |
1.1 |
% |
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\begin{equation} |
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\begin{split} |
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offset & = \, \overline{tpmean} \, - \, \overline{psmean} \\ |
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~ & = \, \frac{1}{normaliz.} \sum_{i,j} |
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\left\{ tpmean(i,j) \, - \, psmean(i,j) \right\} |
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\cdot cosphi(i,j) \cdot tpmeanmask(i,j) |
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\end{split} |
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\end{equation} |
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% |
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\item |
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Misfits are computed w.r.t. global $offset$. |
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\\ |
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First spatial distribution: |
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% |
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\begin{equation} |
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\begin{split} |
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cost\_ssh\_mean(i,j) & = \, |
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\frac{1}{wp^2} \left\{ \, |
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\left[ \, psmean(i,j) - \overline{psmean} \, \right] \, - \, |
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\left[ \, tpmean(i,j) - \overline{tpmean} \, \right] \, \right\}^2 \\ |
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~ & = \, \frac{1}{wp^2} \left\{ \, |
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psmean(i,j) \, - \, tpmean(i,j) \, + \, offset \, \right\}^2 |
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\end{split} |
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\end{equation} |
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% |
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Finally, sum over all spatial entries: |
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\begin{equation} |
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\overline{cost\_ssh\_mean} \, = \, |
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\sum_{i,j} cost\_ssh\_mean(i,j) |
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\end{equation} |
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\end{enumerate} |
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\subsubsection{Misfit of daily SSH anomalies} |
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Computation is same for T/P and ERS-1/2. |
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Here we write out computation for T/P. |
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\begin{enumerate} |
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% |
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\item |
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Compute difference in anomalies: |
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\begin{equation} |
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\begin{split} |
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cost\_ssh\_anom(i,j,t) & = \, \frac{1}{wtp^2} \left\{ \, |
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\left[ \, psbar(i,j,t) - psmean(i,j) \, \right] \, - \, |
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\left[ \, tpobs(i,j,t) \, \right] \, |
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\right\}^2 |
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\end{split} |
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\end{equation} |
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% |
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where $t$ denotes time (day) index, and |
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where it is assumed that $ nYears$ mean T/P spatial distribution |
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$tpmean(i,j)$ has already been removed from data $tpobs(i,j)$! |
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\item |
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Sum over all spatial points and all times |
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\begin{equation} |
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\begin{split} |
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\overline{cost\_ssh\_anom} & = \, \sum_{t} \sum_{i,j} |
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cost\_ssh\_anom(i,j,t) |
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\end{split} |
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\end{equation} |
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\end{enumerate} |
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\subsubsection{Flow chart} |
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\begin{verbatim} |
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cost_ssh |
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|- < compute nYears model mean > |
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|- < read nYears T/P mean > |
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| CALL COST_READTOPEXMEAN |
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|- < compute global T/P vs. model offset > |
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|- < compute cost_hmean > |
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| CALL COST_SSH_MEAN |
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|- < ... > |
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\end{verbatim} |
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heimbach |
1.2 |
\subsubsection{Weights and notes} |
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heimbach |
1.1 |
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\begin{itemize} |
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\item |
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All data are currently masked to zero where less than 13 depth levels, |
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mimicing no contribution for depth less than 1000m. |
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% |
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\item |
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$cosphi$ term in weights is set to 1. |
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% |
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\item |
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bad T/P and ERS-1/2 values are flagged $ \le \, -9990. $ |
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% |
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\item |
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T/P and ERS-1/2 data $ \le \, 1.\exp^{-8}$ cm are flagged as bad values |
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% |
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\item |
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$wp$ is read from {\tt geoid\_errfile} |
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and $1/wp^2$ is pre-computed in {\tt ecco\_cost\_weights} |
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% |
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\end{itemize} |
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\paragraph{$wp$ for SSH mean misfit} ~ |
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$1/wp^2$ is pre-computed in {\tt ecco\_cost\_weights}; \\ |
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$wp$ is read from {\tt geoid\_errfile}; |
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\paragraph{$wtp$ and $wers$ for SSH anomaly misfit} ~ |
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$1/wtp^2$, $1/wers^2$ are pre-computed in {\tt ecco\_cost\_weights}; \\ |
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% |
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\begin{itemize} |
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% |
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\item |
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$wtp$, $wers$ are read from single {\tt ssh\_errfile} |
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% |
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\item |
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both are converted to meters and halved \\ |
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$ wtp \, \longrightarrow \, wtp \cdot 0.01 \cdot 0.5 $ |
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% |
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\item |
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ERS error is set to T/P error + 5cm \\ |
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$ wers \, = \, wtp \, + 0.5cm $ |
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% |
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\end{itemize} |
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\subsubsection{Cost diagnostics} |
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\begin{itemize} |
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% |
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\item |
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Map out $ cost\_ssh\_mean(i,j) $ |
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% |
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\item |
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Map out $ cost\_ssh\_anom(i,j,t) $ averaged over 1 month, i.e. |
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\[ |
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\frac{1}{\text{monthly entries}} \sum_{t}^{monthly} cost\_ssh\_anom(i,j,t) |
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\] |
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% |
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\item |
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sum over daily entries and plot daily average as function of time. i.e. |
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\[ |
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\frac{1}{\text{daily entries}} \sum_{i,j} cost\_ssh\_anom(i,j,t) |
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\] |
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\end{itemize} |
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heimbach |
1.2 |
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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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% |
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\begin{enumerate} |
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% |
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\item |
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CTD obs. for $T$, $S$ from various WOCE sections |
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% |
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\item |
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XBT obs. for $T$ |
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% |
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\item |
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Sea surface temperature (SST) and salinity (SSS) from |
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Reynolds et al. (???) |
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% |
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\item |
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$T$, $S$ from ARGO floats |
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% |
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\item |
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$T$, $S$ from fields from Levitus (???) |
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% |
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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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[$^{\circ}$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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[$^{\circ}$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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[$^{\circ}$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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[$^{\circ}$C] \\ |
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{\it sstdat} & {\tt sstdatfile} & monthly Reynolds pot. SST & |
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[$^{\circ}$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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[$^{\circ}$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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heimbach |
1.3 |
{\it wtvar} & {\tt temperrfile} & spatially varying stdev. & [$^{\circ}$C] \\ |
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{\it wsvar} & {\tt salterrfile} & spatially varying stdev. & [ppt] \\ |
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heimbach |
1.2 |
~&~&~&~\\ |
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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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heimbach |
1.3 |
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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heimbach |
1.2 |
\\ |
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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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heimbach |
1.3 |
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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heimbach |
1.2 |
\\ |
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heimbach |
1.3 |
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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heimbach |
1.2 |
\\ |
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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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heimbach |
1.3 |
cost\_argo\_t(i,j,k) & = \, |
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\left[ \, \frac{fac \cdot ratio}{wti^2 + wvar^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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heimbach |
1.2 |
\\ |
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heimbach |
1.3 |
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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heimbach |
1.2 |
\\ |
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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) & = \, |
| 346 |
heimbach |
1.3 |
\left[ \, wsst \sum_{\tau=1}^{nMonsRec} |
| 347 |
heimbach |
1.2 |
\left\{ Tbar(\tau) \, - \, sstDat(\tau) \right\}^2 \, \right](i,j) |
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\\ |
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cost\_sss(i,j) & = \, |
| 350 |
heimbach |
1.3 |
\left[ \, wsss \sum_{\tau=1}^{nMonsRec} |
| 351 |
heimbach |
1.2 |
\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} |
| 355 |
|
|
|
| 356 |
|
|
\subsubsection{Levitus montly T, S climatological data} |
| 357 |
|
|
|
| 358 |
heimbach |
1.3 |
Model vs. data misfits are taken from $nYears$ monthly model means |
| 359 |
|
|
vs. Levitus monthly data. |
| 360 |
|
|
The description below is for potential temperature. |
| 361 |
|
|
Procedure for salinity is fully analogous. |
| 362 |
|
|
Spatial indices $(i,j,k)$ are omitted throughout. |
| 363 |
|
|
% |
| 364 |
|
|
\begin{enumerate} |
| 365 |
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|
% |
| 366 |
|
|
\item |
| 367 |
|
|
Compute $nYears$ monthly model means for each month $imon$: |
| 368 |
|
|
\[ |
| 369 |
|
|
\overline{Tbar}(imon) \, = \, \frac{1}{nYears} |
| 370 |
|
|
\sum_{iyear=1}^{nYears} Tbar(iyear,imon) |
| 371 |
|
|
\] |
| 372 |
|
|
% |
| 373 |
|
|
\item |
| 374 |
|
|
Compute misfit: |
| 375 |
|
|
\[ |
| 376 |
|
|
cost\_theta(i,j,k) \, = \, \left[ |
| 377 |
|
|
\frac{fac \cdot ratio}{wti^2} \sum_{imon=1}^{12} |
| 378 |
|
|
\left\{ \overline{Tbar}(imon) \, - \, Tdat(imon) \right\}^2 \right] (i,j,k) |
| 379 |
|
|
\] |
| 380 |
|
|
|
| 381 |
|
|
\end{enumerate} |
| 382 |
|
|
|
| 383 |
heimbach |
1.2 |
|
| 384 |
|
|
\subsubsection{Weights and notes} |
| 385 |
|
|
|
| 386 |
|
|
\begin{itemize} |
| 387 |
|
|
% |
| 388 |
|
|
\item |
| 389 |
|
|
$T2\theta$ is an operator mapping in-situ to potential temperatures |
| 390 |
|
|
% |
| 391 |
|
|
\item |
| 392 |
|
|
Latitudinal weight not used: |
| 393 |
|
|
\[ |
| 394 |
|
|
cosphi(i,j) \, = \, 1 |
| 395 |
|
|
\] |
| 396 |
|
|
% |
| 397 |
|
|
\item |
| 398 |
heimbach |
1.3 |
$ fac \, = \, cosphi \cdot mask $ |
| 399 |
|
|
% |
| 400 |
|
|
\item |
| 401 |
|
|
Spatially {\it constant} weights: |
| 402 |
heimbach |
1.2 |
% |
| 403 |
|
|
\begin{enumerate} |
| 404 |
|
|
% |
| 405 |
|
|
\item |
| 406 |
heimbach |
1.3 |
Read standard deviation vertical profiles for $T$, $S$ \\ |
| 407 |
heimbach |
1.2 |
$ {\tt data\_errfile} \, \longrightarrow \, |
| 408 |
|
|
wti(k), \,\, wsi(k) $ \\ |
| 409 |
|
|
$ {\tt data\_errfile} \, \longrightarrow \, |
| 410 |
|
|
ratio = 0.25 = \left( \frac{1}{2} \right)^2 $ |
| 411 |
|
|
% |
| 412 |
|
|
\item |
| 413 |
|
|
Take inverse squares: |
| 414 |
|
|
\[ |
| 415 |
|
|
\begin{split} |
| 416 |
heimbach |
1.3 |
wtheta(k) & = \, \frac{ratio}{wti(k)^2} \\ |
| 417 |
|
|
wsalt(k) & = \, \frac{ratio}{wsi(k)^2} \\ |
| 418 |
heimbach |
1.2 |
\end{split} |
| 419 |
|
|
\] |
| 420 |
|
|
% |
| 421 |
|
|
\end{enumerate} |
| 422 |
|
|
% |
| 423 |
|
|
\item |
| 424 |
heimbach |
1.3 |
Spatially {\it varying} weights: |
| 425 |
heimbach |
1.2 |
% |
| 426 |
|
|
\begin{enumerate} |
| 427 |
|
|
% |
| 428 |
|
|
\item |
| 429 |
|
|
Read standard deviation fields \\ |
| 430 |
heimbach |
1.3 |
$ {\tt temperrfile} \, \longrightarrow \, wtvar(i,j,k) $ \\ |
| 431 |
|
|
$ {\tt salterrfile} \, \longrightarrow \, wsvar(i,j,k) $ \\ |
| 432 |
heimbach |
1.2 |
% |
| 433 |
|
|
\item |
| 434 |
|
|
Weights are combination of spatially constant and varying parts: |
| 435 |
|
|
\[ |
| 436 |
|
|
\begin{split} |
| 437 |
|
|
wtheta2(i,j,k) & = \, \frac{ratio} |
| 438 |
heimbach |
1.3 |
{wti(k)^2 \, + \,wtvar(i,j,k)^2 } \\ |
| 439 |
heimbach |
1.2 |
wsalt2(i,j,k) & = \, |
| 440 |
|
|
\frac{ratio} |
| 441 |
heimbach |
1.3 |
{wsi(k)^2 \, + \,wsvar(i,j,k)^2 } \\ |
| 442 |
heimbach |
1.2 |
\end{split} |
| 443 |
|
|
\] |
| 444 |
|
|
% |
| 445 |
|
|
\end{enumerate} |
| 446 |
|
|
% |
| 447 |
|
|
\item |
| 448 |
|
|
Sea surface $T$, $S$ weights: |
| 449 |
|
|
\begin{itemize} |
| 450 |
|
|
\item |
| 451 |
heimbach |
1.3 |
SST: $ wsst \, = \, fac \cdot wtheta(1)$: horizontally constant |
| 452 |
heimbach |
1.2 |
\item |
| 453 |
heimbach |
1.3 |
SSS: $ wsss \, = \, fac \cdot wsalt2(i,j,1)$: horizontally varying |
| 454 |
heimbach |
1.2 |
\end{itemize} |
| 455 |
|
|
(Why this difference? I don't know.) |
| 456 |
|
|
% |
| 457 |
|
|
\end{itemize} |
| 458 |
|
|
|
| 459 |
|
|
|
| 460 |
|
|
\subsubsection{Diagnostics} |
| 461 |
|
|
|
| 462 |
|
|
\begin{itemize} |
| 463 |
|
|
% |
| 464 |
|
|
\item |
| 465 |
heimbach |
1.3 |
Map out $wtheta2(i,j,k)$, $wsalt2(i,j,k)$. |
| 466 |
heimbach |
1.2 |
|
| 467 |
|
|
% |
| 468 |
|
|
\end{itemize} |
| 469 |
|
|
|
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