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\section{The ECCO state estimation cost function DRAFT!!! |
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\label{sectioneccocost}} |
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|
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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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|
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\subsection{Sea surface height from TOPEX/Poseidon and ERS-1/2 altimetry} |
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|
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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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|
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\subsubsection{Input fields} |
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~ |
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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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|
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|
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\subsubsection{$nYears$ time mean SSH misfit} |
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|
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\begin{enumerate} |
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% |
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\item |
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Compute 11yr model mean spatial distribution |
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% |
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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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Compute global offset between 11-yr model and T/P mean: |
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% |
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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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% |
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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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|
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|
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|
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\end{enumerate} |
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|
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\subsubsection{Misfit of daily SSH anomalies} |
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|
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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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|
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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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|
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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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|
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\item |
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Sum over all spatial points and all times |
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|
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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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|
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\end{enumerate} |
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|
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\subsubsection{Flow chart} |
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|
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\begin{verbatim} |
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|
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cost_ssh |
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| |
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|- < compute nYears model mean > |
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| |
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|- < read nYears T/P mean > |
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| CALL COST_READTOPEXMEAN |
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| |
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|- < compute global T/P vs. model offset > |
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| |
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|- < compute cost_hmean > |
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| CALL COST_SSH_MEAN |
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| |
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|- < ... > |
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|
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\end{verbatim} |
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|
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\subsubsection{Weights} |
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|
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\begin{itemize} |
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% |
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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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|
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\paragraph{$wp$ for SSH mean misfit} ~ |
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|
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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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|
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\paragraph{$wtp$ and $wers$ for SSH anomaly misfit} ~ |
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|
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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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|
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\subsubsection{Cost diagnostics} |
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|
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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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