| 1 |
\section{The ECCO state estimation cost function DRAFT!!! |
| 2 |
\label{sectioneccocost}} |
| 3 |
|
| 4 |
The current ECCO state estimation covers an $nYears = 11$ year |
| 5 |
model trajectory. |
| 6 |
A variety of data sets enter a least squares cost function, |
| 7 |
in addition to penalty terms which constrain deviations |
| 8 |
of control variables beyound their a priori errors. |
| 9 |
|
| 10 |
\subsection{Sea surface height from TOPEX/Poseidon and ERS-1/2 altimetry} |
| 11 |
|
| 12 |
Altimetric SSH contributions from T/P and ERS-1/2 are four-fold: |
| 13 |
% |
| 14 |
\begin{enumerate} |
| 15 |
% |
| 16 |
\item |
| 17 |
an $nYears$ time mean SSH misfit between |
| 18 |
model and T/P |
| 19 |
% |
| 20 |
\item |
| 21 |
daily SSH anomaly misfits between T/P and model |
| 22 |
% |
| 23 |
\item |
| 24 |
daily SSH anomaly misfits between ERS-1/2 and model |
| 25 |
% |
| 26 |
\item |
| 27 |
daily absolute SSH misfit between T/P and model, |
| 28 |
weighted by the full geoid error covariance. |
| 29 |
% |
| 30 |
\end{enumerate} |
| 31 |
|
| 32 |
\subsubsection{Input fields} |
| 33 |
~ |
| 34 |
|
| 35 |
\begin{table}[h!] |
| 36 |
\begin{center} |
| 37 |
\begin{tabular}{lllc} |
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\hline \hline |
| 39 |
~&~&~&~\\ |
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field & file name & deccription & unit \\ |
| 41 |
~&~&~&~\\ |
| 42 |
\hline |
| 43 |
~&~&~&~\\ |
| 44 |
{\it psbar} & {\tt psbarfile} & daily model mean SSH fields & [m] \\ |
| 45 |
{\it tpmean} & {\tt topexmeanfile} & $nYears$ T/P mean & [cm] \\ |
| 46 |
{\it tpobs} & {\tt topexfile} & daily T/P SSH anomalies & [cm] \\ |
| 47 |
{\it erspobs} & {\tt ersfile} & daily ERS-1/2 SSH anomalies & [cm] \\ |
| 48 |
{\it wp} & {\tt geoid\_errfile} & diagonal of geoid error covariance & [m] \\ |
| 49 |
{\it wtp, wers} & {\tt ssh\_errfile} & rms of SSH anomalies & [cm] \\ |
| 50 |
~&~&~&~\\ |
| 51 |
\hline \hline |
| 52 |
\end{tabular} |
| 53 |
\end{center} |
| 54 |
\end{table} |
| 55 |
|
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|
| 57 |
\subsubsection{\textit{\textbf{nYears}} time mean SSH misfit} |
| 58 |
|
| 59 |
\begin{enumerate} |
| 60 |
% |
| 61 |
\item |
| 62 |
Compute $nYears$ model mean spatial distribution |
| 63 |
% |
| 64 |
\begin{equation} |
| 65 |
psmean(i,j)\, =\, |
| 66 |
\frac{1}{nDaysRec} \sum_{i=1}^{nDaysRec} |
| 67 |
psbar(i,j) |
| 68 |
\end{equation} |
| 69 |
% |
| 70 |
\item |
| 71 |
Compute global offset between $nYears$ model and T/P mean: |
| 72 |
% |
| 73 |
\begin{equation} |
| 74 |
\begin{split} |
| 75 |
offset & = \, \overline{tpmean} \, - \, \overline{psmean} \\ |
| 76 |
~ & = \, \frac{1}{normaliz.} \sum_{i,j} |
| 77 |
\left\{ tpmean(i,j) \, - \, psmean(i,j) \right\} |
| 78 |
\cdot cosphi(i,j) \cdot tpmeanmask(i,j) |
| 79 |
\end{split} |
| 80 |
\end{equation} |
| 81 |
% |
| 82 |
\item |
| 83 |
Misfits are computed w.r.t. global $offset$. |
| 84 |
\\ |
| 85 |
First spatial distribution: |
| 86 |
% |
| 87 |
\begin{equation} |
| 88 |
\begin{split} |
| 89 |
cost\_ssh\_mean(i,j) & = \, |
| 90 |
\frac{1}{wp^2} \left\{ \, |
| 91 |
\left[ \, psmean(i,j) - \overline{psmean} \, \right] \, - \, |
| 92 |
\left[ \, tpmean(i,j) - \overline{tpmean} \, \right] \, \right\}^2 \\ |
| 93 |
~ & = \, \frac{1}{wp^2} \left\{ \, |
| 94 |
psmean(i,j) \, - \, tpmean(i,j) \, + \, offset \, \right\}^2 |
| 95 |
\end{split} |
| 96 |
\end{equation} |
| 97 |
|
| 98 |
% |
| 99 |
Finally, sum over all spatial entries: |
| 100 |
\begin{equation} |
| 101 |
\overline{cost\_ssh\_mean} \, = \, |
| 102 |
\sum_{i,j} cost\_ssh\_mean(i,j) |
| 103 |
\end{equation} |
| 104 |
|
| 105 |
|
| 106 |
|
| 107 |
\end{enumerate} |
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|
| 109 |
\subsubsection{Misfit of daily SSH anomalies} |
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|
| 111 |
Computation is same for T/P and ERS-1/2. |
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Here we write out computation for T/P. |
| 113 |
|
| 114 |
\begin{enumerate} |
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% |
| 116 |
\item |
| 117 |
Compute difference in anomalies: |
| 118 |
|
| 119 |
\begin{equation} |
| 120 |
\begin{split} |
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cost\_ssh\_anom(i,j,t) & = \, \frac{1}{wtp^2} \left\{ \, |
| 122 |
\left[ \, psbar(i,j,t) - psmean(i,j) \, \right] \, - \, |
| 123 |
\left[ \, tpobs(i,j,t) \, \right] \, |
| 124 |
\right\}^2 |
| 125 |
\end{split} |
| 126 |
\end{equation} |
| 127 |
% |
| 128 |
where $t$ denotes time (day) index, and |
| 129 |
where it is assumed that $ nYears$ mean T/P spatial distribution |
| 130 |
$tpmean(i,j)$ has already been removed from data $tpobs(i,j)$! |
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|
| 132 |
\item |
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Sum over all spatial points and all times |
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|
| 135 |
\begin{equation} |
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\begin{split} |
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\overline{cost\_ssh\_anom} & = \, \sum_{t} \sum_{i,j} |
| 138 |
cost\_ssh\_anom(i,j,t) |
| 139 |
\end{split} |
| 140 |
\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 and notes} |
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|
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\begin{itemize} |
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% |
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\item |
| 169 |
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 |
| 173 |
$cosphi$ 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}; \\ |
| 190 |
$wp$ is read from {\tt geoid\_errfile}; |
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|
| 192 |
\paragraph{$wtp$ and $wers$ for SSH anomaly misfit} ~ |
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|
| 194 |
$1/wtp^2$, $1/wers^2$ are pre-computed in {\tt ecco\_cost\_weights}; \\ |
| 195 |
% |
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\begin{itemize} |
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% |
| 198 |
\item |
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$wtp$, $wers$ are read from single {\tt ssh\_errfile} |
| 200 |
% |
| 201 |
\item |
| 202 |
both are converted to meters and halved \\ |
| 203 |
$ wtp \, \longrightarrow \, wtp \cdot 0.01 \cdot 0.5 $ |
| 204 |
% |
| 205 |
\item |
| 206 |
ERS error is set to T/P error + 5cm \\ |
| 207 |
$ wers \, = \, wtp \, + 0.5cm $ |
| 208 |
% |
| 209 |
\end{itemize} |
| 210 |
|
| 211 |
\subsubsection{Cost diagnostics} |
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|
| 213 |
\begin{itemize} |
| 214 |
% |
| 215 |
\item |
| 216 |
Map out $ cost\_ssh\_mean(i,j) $ |
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% |
| 218 |
\item |
| 219 |
Map out $ cost\_ssh\_anom(i,j,t) $ averaged over 1 month, i.e. |
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\[ |
| 221 |
\frac{1}{\text{monthly entries}} \sum_{t}^{monthly} cost\_ssh\_anom(i,j,t) |
| 222 |
\] |
| 223 |
% |
| 224 |
\item |
| 225 |
sum over daily entries and plot daily average as function of time. i.e. |
| 226 |
\[ |
| 227 |
\frac{1}{\text{daily entries}} \sum_{i,j} cost\_ssh\_anom(i,j,t) |
| 228 |
\] |
| 229 |
\end{itemize} |
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|
| 231 |
\subsection{Hydrographic constraints} |
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|
| 233 |
Observation of temperature and salinity from various sources are |
| 234 |
used to constrain the model. These are: |
| 235 |
% |
| 236 |
\begin{enumerate} |
| 237 |
% |
| 238 |
\item |
| 239 |
CTD obs. for $T$, $S$ from various WOCE sections |
| 240 |
% |
| 241 |
\item |
| 242 |
XBT obs. for $T$ |
| 243 |
% |
| 244 |
\item |
| 245 |
Sea surface temperature (SST) and salinity (SSS) from |
| 246 |
Reynolds et al. (???) |
| 247 |
% |
| 248 |
\item |
| 249 |
$T$, $S$ from ARGO floats |
| 250 |
% |
| 251 |
\item |
| 252 |
$T$, $S$ from fields from Levitus (???) |
| 253 |
% |
| 254 |
\end{enumerate} |
| 255 |
|
| 256 |
\subsubsection{Input fields} |
| 257 |
~ |
| 258 |
|
| 259 |
\begin{table}[h!] |
| 260 |
\begin{center} |
| 261 |
\begin{tabular}{lllc} |
| 262 |
\hline \hline |
| 263 |
~&~&~&~\\ |
| 264 |
field & file name & deccription & unit \\ |
| 265 |
~&~&~&~\\ |
| 266 |
\hline |
| 267 |
~&~&~&~\\ |
| 268 |
{\it tbar} & {\tt tbarfile} & monthly model mean pot. temperature & |
| 269 |
[$^{\circ}$C] \\ |
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{\it sbar} & {\tt sbarfile} & monthly model mean salinity & |
| 271 |
[ppt] \\ |
| 272 |
{\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 & |
| 275 |
[ppt] \\ |
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{\it ctdtobs} & {\tt ctdtfile} & monthly WOCE CTD pot. temperature & |
| 277 |
[$^{\circ}$C] \\ |
| 278 |
{\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] \\ |
| 282 |
{\it sstdat} & {\tt sstdatfile} & monthly Reynolds pot. SST & |
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[$^{\circ}$C] \\ |
| 284 |
{\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 & |
| 287 |
[$^{\circ}$C] \\ |
| 288 |
{\it argosobs} & {\tt argosfile} & monthly ARGO salinity & |
| 289 |
[ppt] \\ |
| 290 |
{\it wti, wsi} & {\tt data\_errfile} & vert. stdev. profile for $T$, $S$ & |
| 291 |
~ \\ |
| 292 |
{\it wtvar} & {\tt temperrfile} & spatially varying stdev. & [$^{\circ}$C] \\ |
| 293 |
{\it wsvar} & {\tt salterrfile} & spatially varying stdev. & [ppt] \\ |
| 294 |
~&~&~&~\\ |
| 295 |
\hline \hline |
| 296 |
\end{tabular} |
| 297 |
\end{center} |
| 298 |
\end{table} |
| 299 |
|
| 300 |
\subsubsection{XBT data} |
| 301 |
|
| 302 |
\begin{equation} |
| 303 |
\begin{split} |
| 304 |
cost\_xbt\_t(i,j,k) & = \, |
| 305 |
\left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \sum_{\tau=1}^{nMonsRec} |
| 306 |
\left\{ Tbar(\tau) \, - \, T2\theta[xbtobs(\tau)] \right\}^2 \, \right](i,j,k) |
| 307 |
\\ |
| 308 |
\end{split} |
| 309 |
\end{equation} |
| 310 |
|
| 311 |
\subsubsection{WOCE CTD data} |
| 312 |
|
| 313 |
\begin{equation} |
| 314 |
\begin{split} |
| 315 |
cost\_ctd\_t(i,j,k) & = \, |
| 316 |
\left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \sum_{\tau=1}^{nMonsRec} |
| 317 |
\left\{ Tbar(\tau) \, - \, ctdTobs(\tau) \right\}^2 \, \right](i,j,k) |
| 318 |
\\ |
| 319 |
cost\_ctd\_s(i,j,k) & = \, |
| 320 |
\left[ \, \frac{fac \cdot ratio}{wsi^2 + wsvar^2} \sum_{\tau=1}^{nMonsRec} |
| 321 |
\left\{ Sbar(\tau) \, - \, ctdSobs(\tau) \right\}^2 \, \right](i,j,k) |
| 322 |
\\ |
| 323 |
\end{split} |
| 324 |
\end{equation} |
| 325 |
|
| 326 |
\subsubsection{ARGO float data} |
| 327 |
|
| 328 |
\begin{equation} |
| 329 |
\begin{split} |
| 330 |
cost\_argo\_t(i,j,k) & = \, |
| 331 |
\left[ \, \frac{fac \cdot ratio}{wti^2 + wvar^2} \sum_{\tau=1}^{nMonsRec} |
| 332 |
\left\{ Tbar(\tau) \, - \, T2\theta[argoTobs(\tau)] \right\}^2 \, \right](i,j,k) |
| 333 |
\\ |
| 334 |
cost\_argo\_s(i,j,k) & = \, |
| 335 |
\left[ \, \frac{fac \cdot ratio}{wsi^2 + wsvar^2} \sum_{\tau=1}^{nMonsRec} |
| 336 |
\left\{ Sbar(\tau) \, - \, argoSobs(\tau) \right\}^2 \, \right](i,j,k) |
| 337 |
\\ |
| 338 |
\end{split} |
| 339 |
\end{equation} |
| 340 |
|
| 341 |
\subsubsection{Reynolds sea surface T, S data} |
| 342 |
|
| 343 |
\begin{equation} |
| 344 |
\begin{split} |
| 345 |
cost\_sst(i,j) & = \, |
| 346 |
\left[ \, wsst \sum_{\tau=1}^{nMonsRec} |
| 347 |
\left\{ Tbar(\tau) \, - \, sstDat(\tau) \right\}^2 \, \right](i,j) |
| 348 |
\\ |
| 349 |
cost\_sss(i,j) & = \, |
| 350 |
\left[ \, wsss \sum_{\tau=1}^{nMonsRec} |
| 351 |
\left\{ Sbar(\tau) \, - \, sssDat(\tau) \right\}^2 \, \right](i,j) |
| 352 |
\\ |
| 353 |
\end{split} |
| 354 |
\end{equation} |
| 355 |
|
| 356 |
\subsubsection{Levitus montly T, S climatological data} |
| 357 |
|
| 358 |
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 |
% |
| 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 |
|
| 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 |
$ fac \, = \, cosphi \cdot mask $ |
| 399 |
% |
| 400 |
\item |
| 401 |
Spatially {\it constant} weights: |
| 402 |
% |
| 403 |
\begin{enumerate} |
| 404 |
% |
| 405 |
\item |
| 406 |
Read standard deviation vertical profiles for $T$, $S$ \\ |
| 407 |
$ {\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 |
\[ |
| 415 |
\begin{split} |
| 416 |
wtheta(k) & = \, \frac{ratio}{wti(k)^2} \\ |
| 417 |
wsalt(k) & = \, \frac{ratio}{wsi(k)^2} \\ |
| 418 |
\end{split} |
| 419 |
\] |
| 420 |
% |
| 421 |
\end{enumerate} |
| 422 |
% |
| 423 |
\item |
| 424 |
Spatially {\it varying} weights: |
| 425 |
% |
| 426 |
\begin{enumerate} |
| 427 |
% |
| 428 |
\item |
| 429 |
Read standard deviation fields \\ |
| 430 |
$ {\tt temperrfile} \, \longrightarrow \, wtvar(i,j,k) $ \\ |
| 431 |
$ {\tt salterrfile} \, \longrightarrow \, wsvar(i,j,k) $ \\ |
| 432 |
% |
| 433 |
\item |
| 434 |
Weights are combination of spatially constant and varying parts: |
| 435 |
\[ |
| 436 |
\begin{split} |
| 437 |
wtheta2(i,j,k) & = \, \frac{ratio} |
| 438 |
{wti(k)^2 \, + \,wtvar(i,j,k)^2 } \\ |
| 439 |
wsalt2(i,j,k) & = \, |
| 440 |
\frac{ratio} |
| 441 |
{wsi(k)^2 \, + \,wsvar(i,j,k)^2 } \\ |
| 442 |
\end{split} |
| 443 |
\] |
| 444 |
% |
| 445 |
\end{enumerate} |
| 446 |
% |
| 447 |
\item |
| 448 |
Sea surface $T$, $S$ weights: |
| 449 |
\begin{itemize} |
| 450 |
\item |
| 451 |
SST: $ wsst \, = \, fac \cdot wtheta(1)$: horizontally constant |
| 452 |
\item |
| 453 |
SSS: $ wsss \, = \, fac \cdot wsalt2(i,j,1)$: horizontally varying |
| 454 |
\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 |
Map out $wtheta2(i,j,k)$, $wsalt2(i,j,k)$. |
| 466 |
|
| 467 |
% |
| 468 |
\end{itemize} |
| 469 |
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