| 54 |
\end{table} |
\end{table} |
| 55 |
|
|
| 56 |
|
|
| 57 |
\subsubsection{$nYears$ time mean SSH misfit} |
\subsubsection{\textit{\textbf{nYears}} time mean SSH misfit} |
| 58 |
|
|
| 59 |
\begin{enumerate} |
\begin{enumerate} |
| 60 |
% |
% |
| 61 |
\item |
\item |
| 62 |
Compute 11yr model mean spatial distribution |
Compute $nYears$ model mean spatial distribution |
| 63 |
% |
% |
| 64 |
\begin{equation} |
\begin{equation} |
| 65 |
psmean(i,j)\, =\, |
psmean(i,j)\, =\, |
| 68 |
\end{equation} |
\end{equation} |
| 69 |
% |
% |
| 70 |
\item |
\item |
| 71 |
Compute global offset between 11-yr model and T/P mean: |
Compute global offset between $nYears$ model and T/P mean: |
| 72 |
% |
% |
| 73 |
\begin{equation} |
\begin{equation} |
| 74 |
\begin{split} |
\begin{split} |
| 161 |
|
|
| 162 |
\end{verbatim} |
\end{verbatim} |
| 163 |
|
|
| 164 |
\subsubsection{Weights} |
\subsubsection{Weights and notes} |
| 165 |
|
|
| 166 |
\begin{itemize} |
\begin{itemize} |
| 167 |
% |
% |
| 227 |
\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) |
| 228 |
\] |
\] |
| 229 |
\end{itemize} |
\end{itemize} |
| 230 |
|
|
| 231 |
|
\subsection{Hydrographic constraints} |
| 232 |
|
|
| 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] \\ |
| 270 |
|
{\it sbar} & {\tt sbarfile} & monthly model mean salinity & |
| 271 |
|
[ppt] \\ |
| 272 |
|
{\it tdat} & {\tt tdatfile} & monthly mean Levitus pot. temperature & |
| 273 |
|
[$^{\circ}$C] \\ |
| 274 |
|
{\it sdat} & {\tt sdatfile} & monthly mean Levitus salinity & |
| 275 |
|
[ppt] \\ |
| 276 |
|
{\it ctdtobs} & {\tt ctdtfile} & monthly WOCE CTD pot. temperature & |
| 277 |
|
[$^{\circ}$C] \\ |
| 278 |
|
{\it ctdsobs} & {\tt ctdsfile} & monthly WOCE CTD salinity & |
| 279 |
|
[ppt] \\ |
| 280 |
|
{\it xbtobs} & {\tt xbtfile} & monthly XBT in-situ(!) temperature & |
| 281 |
|
[$^{\circ}$C] \\ |
| 282 |
|
{\it sstdat} & {\tt sstdatfile} & monthly Reynolds pot. SST & |
| 283 |
|
[$^{\circ}$C] \\ |
| 284 |
|
{\it sssdat} & {\tt sssdatfile} & monthly Reynolds SSS & |
| 285 |
|
[ppt] \\ |
| 286 |
|
{\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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