s_ecco/text/ecco_costfunction.tex

\section{The ECCO state estimation cost function DRAFT!!!
\label{sectioneccocost}}
\begin{rawhtml}
<!-- CMIREDIR:ecco_cost: -->
\end{rawhtml}

The current ECCO state estimation covers an $nYears = 11$ year
model trajectory.
A variety of data sets enter a least squares cost function,
in addition to penalty terms which constrain deviations
of control variables beyound their a priori errors.

\subsection{Sea surface height from TOPEX/Poseidon and ERS-1/2 altimetry}

Altimetric SSH contributions from T/P and ERS-1/2 are four-fold:
%
\begin{enumerate}
%
\item 
an $nYears$ time mean SSH misfit between
model and T/P
%
\item
daily SSH anomaly misfits between T/P and model
%
\item
daily SSH anomaly misfits between ERS-1/2 and model
%
\item
daily absolute SSH misfit between T/P and model,
weighted by the full geoid error covariance.
%
\end{enumerate}

\subsubsection{Input fields}
~

\begin{table}[h!]
\begin{center}
\begin{tabular}{lllc}
\hline \hline
~&~&~&~\\
field & file name & deccription & unit \\
~&~&~&~\\
\hline
~&~&~&~\\
{\it psbar} & {\tt psbarfile} & daily model mean SSH fields & [m] \\
{\it tpmean} & {\tt topexmeanfile} & $nYears$ T/P mean & [cm] \\
{\it tpobs}  & {\tt topexfile} & daily T/P SSH anomalies & [cm] \\
{\it erspobs}  & {\tt ersfile} & daily ERS-1/2 SSH anomalies & [cm] \\
{\it wp} & {\tt geoid\_errfile} & diagonal of geoid error covariance & [m] \\
{\it wtp, wers} & {\tt ssh\_errfile} & rms of SSH anomalies & [cm] \\
~&~&~&~\\
\hline \hline
\end{tabular}
\end{center}
\end{table}


\subsubsection{\textit{\textbf{nYears}} time mean SSH misfit}

\begin{enumerate}
%
\item
Compute $nYears$ model mean spatial distribution
%
\begin{equation}
psmean(i,j)\, =\, 
\frac{1}{nDaysRec} \sum_{i=1}^{nDaysRec}
psbar(i,j)
\end{equation}
%
\item
Compute global offset between $nYears$ model and T/P mean:
%
\begin{equation}
\begin{split}
offset & = \, \overline{tpmean} \, - \, \overline{psmean} \\
~ & = \, \frac{1}{normaliz.} \sum_{i,j}
\left\{ tpmean(i,j) \, - \, psmean(i,j) \right\} 
\cdot cosphi(i,j) \cdot tpmeanmask(i,j)
\end{split}
\end{equation}
%
\item
Misfits are computed w.r.t. global $offset$. 
\\
First spatial distribution:
%
\begin{equation}
\begin{split}
cost\_ssh\_mean(i,j) & = \,
\frac{1}{wp^2} \left\{ \,
\left[ \, psmean(i,j) - \overline{psmean} \, \right] \, - \,
\left[ \, tpmean(i,j) - \overline{tpmean} \, \right] \, \right\}^2 \\ 
~ & = \, \frac{1}{wp^2} \left\{ \,
psmean(i,j) \, - \, tpmean(i,j) \, + \, offset \, \right\}^2
\end{split}
\end{equation}

%
Finally, sum over all spatial entries:
\begin{equation}
\overline{cost\_ssh\_mean} \, = \,
\sum_{i,j} cost\_ssh\_mean(i,j)
\end{equation}


\end{enumerate}

\subsubsection{Misfit of daily SSH anomalies}

Computation is same for T/P and ERS-1/2.
Here we write out computation for T/P.

\begin{enumerate}
%
\item
Compute difference in anomalies:

\begin{equation}
\begin{split}
cost\_ssh\_anom(i,j,t) & = \, \frac{1}{wtp^2} \left\{ \, 
\left[ \, psbar(i,j,t) - psmean(i,j) \, \right] \, - \, 
\left[ \, tpobs(i,j,t) \, \right] \,
\right\}^2
\end{split}
\end{equation}
%
where $t$ denotes time (day) index, and
where it is assumed that $ nYears$ mean T/P spatial distribution
$tpmean(i,j)$ has already been removed from data $tpobs(i,j)$!

\item
Sum over all spatial points and all times

\begin{equation}
\begin{split}
\overline{cost\_ssh\_anom} & = \, \sum_{t} \sum_{i,j}
cost\_ssh\_anom(i,j,t)
\end{split}
\end{equation}

\end{enumerate}

\subsubsection{Flow chart}

\begin{verbatim}

cost_ssh
|
|- < compute nYears model mean >
|
|- < read nYears T/P mean >
|  CALL COST_READTOPEXMEAN
|
|- < compute global T/P vs. model offset >
|
|- < compute cost_hmean >
|  CALL COST_SSH_MEAN
|
|- < ... >

\end{verbatim}

\subsubsection{Weights and notes}

\begin{itemize}
%
\item
All data are currently masked to zero where less than 13 depth levels,
mimicing no contribution for depth less than 1000m.
%
\item
$cosphi$ term in weights is set to 1.
%
\item
bad T/P and ERS-1/2 values are flagged $ \le \, -9990. $
%
\item
T/P and ERS-1/2 data $ \le \, 1.\exp^{-8}$ cm are flagged as bad values
%
\item
$wp$ is read from {\tt geoid\_errfile}
and $1/wp^2$ is pre-computed in {\tt ecco\_cost\_weights}
%
\end{itemize}

\paragraph{$wp$ for SSH mean misfit} ~

$1/wp^2$ is pre-computed in {\tt ecco\_cost\_weights}; \\
$wp$ is read from {\tt geoid\_errfile};

\paragraph{$wtp$ and $wers$ for SSH anomaly misfit} ~

$1/wtp^2$, $1/wers^2$ are pre-computed in {\tt ecco\_cost\_weights}; \\
%
\begin{itemize}
%
\item
$wtp$, $wers$ are read from single {\tt ssh\_errfile}
%
\item
both are converted to meters and halved \\
$ wtp \, \longrightarrow \, wtp \cdot 0.01 \cdot 0.5 $
%
\item
ERS error is set to T/P error + 5cm \\
$ wers \, = \, wtp \, + 0.5cm $
%
\end{itemize}

\subsubsection{Cost diagnostics}

\begin{itemize}
%
\item
Map out $ cost\_ssh\_mean(i,j) $
%
\item
Map out $ cost\_ssh\_anom(i,j,t) $ averaged over 1 month, i.e.
\[
\frac{1}{\text{monthly entries}} \sum_{t}^{monthly} cost\_ssh\_anom(i,j,t)
\]
%
\item
sum over daily entries and plot daily average as function of time. i.e.
\[
\frac{1}{\text{daily entries}} \sum_{i,j} cost\_ssh\_anom(i,j,t)
\]
\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 & 
[$^{\circ}$C] \\
{\it sbar} & {\tt sbarfile} & monthly model mean salinity & 
[ppt] \\
{\it tdat} & {\tt tdatfile} & monthly mean Levitus pot. temperature & 
[$^{\circ}$C] \\
{\it sdat} & {\tt sdatfile} & monthly mean Levitus salinity & 
[ppt] \\
{\it ctdtobs}  & {\tt ctdtfile} & monthly WOCE CTD pot. temperature & 
[$^{\circ}$C] \\
{\it ctdsobs}  & {\tt ctdsfile} & monthly WOCE CTD salinity & 
[ppt] \\
{\it xbtobs} & {\tt xbtfile} & monthly XBT in-situ(!) temperature & 
[$^{\circ}$C] \\
{\it sstdat}  & {\tt sstdatfile} & monthly Reynolds pot. SST & 
[$^{\circ}$C] \\
{\it sssdat}  & {\tt sssdatfile} & monthly Reynolds SSS & 
[ppt] \\
{\it argotobs}  & {\tt argotfile} & monthly ARGO in-situ(!) temperature & 
[$^{\circ}$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. & [$^{\circ}$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 + wvar^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}

1	heimbach	1.1	\section{The ECCO state estimation cost function DRAFT!!!
2			\label{sectioneccocost}}
3	edhill	1.4	\begin{rawhtml}
4			<!-- CMIREDIR:ecco_cost: -->
5			\end{rawhtml}
6	heimbach	1.1
7			The current ECCO state estimation covers an $nYears = 11$ year
8			model trajectory.
9			A variety of data sets enter a least squares cost function,
10			in addition to penalty terms which constrain deviations
11			of control variables beyound their a priori errors.
12
13			\subsection{Sea surface height from TOPEX/Poseidon and ERS-1/2 altimetry}
14
15			Altimetric SSH contributions from T/P and ERS-1/2 are four-fold:
16			%
17			\begin{enumerate}
18			%
19			\item
20			an $nYears$ time mean SSH misfit between
21			model and T/P
22			%
23			\item
24			daily SSH anomaly misfits between T/P and model
25			%
26			\item
27			daily SSH anomaly misfits between ERS-1/2 and model
28			%
29			\item
30			daily absolute SSH misfit between T/P and model,
31			weighted by the full geoid error covariance.
32			%
33			\end{enumerate}
34
35			\subsubsection{Input fields}
36			~
37
38			\begin{table}[h!]
39			\begin{center}
40			\begin{tabular}{lllc}
41			\hline \hline
42			~&~&~&~\\
43			field & file name & deccription & unit \\
44			~&~&~&~\\
45			\hline
46			~&~&~&~\\
47			{\it psbar} & {\tt psbarfile} & daily model mean SSH fields & [m] \\
48			{\it tpmean} & {\tt topexmeanfile} & $nYears$ T/P mean & [cm] \\
49			{\it tpobs} & {\tt topexfile} & daily T/P SSH anomalies & [cm] \\
50			{\it erspobs} & {\tt ersfile} & daily ERS-1/2 SSH anomalies & [cm] \\
51			{\it wp} & {\tt geoid\_errfile} & diagonal of geoid error covariance & [m] \\
52			{\it wtp, wers} & {\tt ssh\_errfile} & rms of SSH anomalies & [cm] \\
53			~&~&~&~\\
54			\hline \hline
55			\end{tabular}
56			\end{center}
57			\end{table}
58
59
60	heimbach	1.2	\subsubsection{\textit{\textbf{nYears}} time mean SSH misfit}
61	heimbach	1.1
62			\begin{enumerate}
63			%
64			\item
65	heimbach	1.3	Compute $nYears$ model mean spatial distribution
66	heimbach	1.1	%
67			\begin{equation}
68			psmean(i,j)\, =\,
69			\frac{1}{nDaysRec} \sum_{i=1}^{nDaysRec}
70			psbar(i,j)
71			\end{equation}
72			%
73			\item
74	heimbach	1.3	Compute global offset between $nYears$ model and T/P mean:
75	heimbach	1.1	%
76			\begin{equation}
77			\begin{split}
78			offset & = \, \overline{tpmean} \, - \, \overline{psmean} \\
79			~ & = \, \frac{1}{normaliz.} \sum_{i,j}
80			\left\{ tpmean(i,j) \, - \, psmean(i,j) \right\}
81			\cdot cosphi(i,j) \cdot tpmeanmask(i,j)
82			\end{split}
83			\end{equation}
84			%
85			\item
86			Misfits are computed w.r.t. global $offset$.
87			\\
88			First spatial distribution:
89			%
90			\begin{equation}
91			\begin{split}
92			cost\_ssh\_mean(i,j) & = \,
93			\frac{1}{wp^2} \left\{ \,
94			\left[ \, psmean(i,j) - \overline{psmean} \, \right] \, - \,
95			\left[ \, tpmean(i,j) - \overline{tpmean} \, \right] \, \right\}^2 \\
96			~ & = \, \frac{1}{wp^2} \left\{ \,
97			psmean(i,j) \, - \, tpmean(i,j) \, + \, offset \, \right\}^2
98			\end{split}
99			\end{equation}
100
101			%
102			Finally, sum over all spatial entries:
103			\begin{equation}
104			\overline{cost\_ssh\_mean} \, = \,
105			\sum_{i,j} cost\_ssh\_mean(i,j)
106			\end{equation}
107
108
109
110			\end{enumerate}
111
112			\subsubsection{Misfit of daily SSH anomalies}
113
114			Computation is same for T/P and ERS-1/2.
115			Here we write out computation for T/P.
116
117			\begin{enumerate}
118			%
119			\item
120			Compute difference in anomalies:
121
122			\begin{equation}
123			\begin{split}
124			cost\_ssh\_anom(i,j,t) & = \, \frac{1}{wtp^2} \left\{ \,
125			\left[ \, psbar(i,j,t) - psmean(i,j) \, \right] \, - \,
126			\left[ \, tpobs(i,j,t) \, \right] \,
127			\right\}^2
128			\end{split}
129			\end{equation}
130			%
131			where $t$ denotes time (day) index, and
132			where it is assumed that $ nYears$ mean T/P spatial distribution
133			$tpmean(i,j)$ has already been removed from data $tpobs(i,j)$!
134
135			\item
136			Sum over all spatial points and all times
137
138			\begin{equation}
139			\begin{split}
140			\overline{cost\_ssh\_anom} & = \, \sum_{t} \sum_{i,j}
141			cost\_ssh\_anom(i,j,t)
142			\end{split}
143			\end{equation}
144
145			\end{enumerate}
146
147			\subsubsection{Flow chart}
148
149			\begin{verbatim}
150
151			cost_ssh
152			\|
153			\|- < compute nYears model mean >
154			\|
155			\|- < read nYears T/P mean >
156			\| CALL COST_READTOPEXMEAN
157			\|
158			\|- < compute global T/P vs. model offset >
159			\|
160			\|- < compute cost_hmean >
161			\| CALL COST_SSH_MEAN
162			\|
163			\|- < ... >
164
165			\end{verbatim}
166
167	heimbach	1.2	\subsubsection{Weights and notes}
168	heimbach	1.1
169			\begin{itemize}
170			%
171			\item
172			All data are currently masked to zero where less than 13 depth levels,
173			mimicing no contribution for depth less than 1000m.
174			%
175			\item
176			$cosphi$ term in weights is set to 1.
177			%
178			\item
179			bad T/P and ERS-1/2 values are flagged $ \le \, -9990. $
180			%
181			\item
182			T/P and ERS-1/2 data $ \le \, 1.\exp^{-8}$ cm are flagged as bad values
183			%
184			\item
185			$wp$ is read from {\tt geoid\_errfile}
186			and $1/wp^2$ is pre-computed in {\tt ecco\_cost\_weights}
187			%
188			\end{itemize}
189
190			\paragraph{$wp$ for SSH mean misfit} ~
191
192			$1/wp^2$ is pre-computed in {\tt ecco\_cost\_weights}; \\
193			$wp$ is read from {\tt geoid\_errfile};
194
195			\paragraph{$wtp$ and $wers$ for SSH anomaly misfit} ~
196
197			$1/wtp^2$, $1/wers^2$ are pre-computed in {\tt ecco\_cost\_weights}; \\
198			%
199			\begin{itemize}
200			%
201			\item
202			$wtp$, $wers$ are read from single {\tt ssh\_errfile}
203			%
204			\item
205			both are converted to meters and halved \\
206			$ wtp \, \longrightarrow \, wtp \cdot 0.01 \cdot 0.5 $
207			%
208			\item
209			ERS error is set to T/P error + 5cm \\
210			$ wers \, = \, wtp \, + 0.5cm $
211			%
212			\end{itemize}
213
214			\subsubsection{Cost diagnostics}
215
216			\begin{itemize}
217			%
218			\item
219			Map out $ cost\_ssh\_mean(i,j) $
220			%
221			\item
222			Map out $ cost\_ssh\_anom(i,j,t) $ averaged over 1 month, i.e.
223			\[
224			\frac{1}{\text{monthly entries}} \sum_{t}^{monthly} cost\_ssh\_anom(i,j,t)
225			\]
226			%
227			\item
228			sum over daily entries and plot daily average as function of time. i.e.
229			\[
230			\frac{1}{\text{daily entries}} \sum_{i,j} cost\_ssh\_anom(i,j,t)
231			\]
232			\end{itemize}
233	heimbach	1.2
234			\subsection{Hydrographic constraints}
235
236			Observation of temperature and salinity from various sources are
237			used to constrain the model. These are:
238			%
239			\begin{enumerate}
240			%
241			\item
242			CTD obs. for $T$, $S$ from various WOCE sections
243			%
244			\item
245			XBT obs. for $T$
246			%
247			\item
248			Sea surface temperature (SST) and salinity (SSS) from
249			Reynolds et al. (???)
250			%
251			\item
252			$T$, $S$ from ARGO floats
253			%
254			\item
255			$T$, $S$ from fields from Levitus (???)
256			%
257			\end{enumerate}
258
259			\subsubsection{Input fields}
260			~
261
262			\begin{table}[h!]
263			\begin{center}
264			\begin{tabular}{lllc}
265			\hline \hline
266			~&~&~&~\\
267			field & file name & deccription & unit \\
268			~&~&~&~\\
269			\hline
270			~&~&~&~\\
271			{\it tbar} & {\tt tbarfile} & monthly model mean pot. temperature &
272			[$^{\circ}$C] \\
273			{\it sbar} & {\tt sbarfile} & monthly model mean salinity &
274			[ppt] \\
275			{\it tdat} & {\tt tdatfile} & monthly mean Levitus pot. temperature &
276			[$^{\circ}$C] \\
277			{\it sdat} & {\tt sdatfile} & monthly mean Levitus salinity &
278			[ppt] \\
279			{\it ctdtobs} & {\tt ctdtfile} & monthly WOCE CTD pot. temperature &
280			[$^{\circ}$C] \\
281			{\it ctdsobs} & {\tt ctdsfile} & monthly WOCE CTD salinity &
282			[ppt] \\
283			{\it xbtobs} & {\tt xbtfile} & monthly XBT in-situ(!) temperature &
284			[$^{\circ}$C] \\
285			{\it sstdat} & {\tt sstdatfile} & monthly Reynolds pot. SST &
286			[$^{\circ}$C] \\
287			{\it sssdat} & {\tt sssdatfile} & monthly Reynolds SSS &
288			[ppt] \\
289			{\it argotobs} & {\tt argotfile} & monthly ARGO in-situ(!) temperature &
290			[$^{\circ}$C] \\
291			{\it argosobs} & {\tt argosfile} & monthly ARGO salinity &
292			[ppt] \\
293			{\it wti, wsi} & {\tt data\_errfile} & vert. stdev. profile for $T$, $S$ &
294			~ \\
295	heimbach	1.3	{\it wtvar} & {\tt temperrfile} & spatially varying stdev. & [$^{\circ}$C] \\
296			{\it wsvar} & {\tt salterrfile} & spatially varying stdev. & [ppt] \\
297	heimbach	1.2	~&~&~&~\\
298			\hline \hline
299			\end{tabular}
300			\end{center}
301			\end{table}
302
303			\subsubsection{XBT data}
304
305			\begin{equation}
306			\begin{split}
307	heimbach	1.3	cost\_xbt\_t(i,j,k) & = \,
308			\left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \sum_{\tau=1}^{nMonsRec}
309			\left\{ Tbar(\tau) \, - \, T2\theta[xbtobs(\tau)] \right\}^2 \, \right](i,j,k)
310	heimbach	1.2	\\
311			\end{split}
312			\end{equation}
313
314			\subsubsection{WOCE CTD data}
315
316			\begin{equation}
317			\begin{split}
318	heimbach	1.3	cost\_ctd\_t(i,j,k) & = \,
319			\left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \sum_{\tau=1}^{nMonsRec}
320			\left\{ Tbar(\tau) \, - \, ctdTobs(\tau) \right\}^2 \, \right](i,j,k)
321	heimbach	1.2	\\
322	heimbach	1.3	cost\_ctd\_s(i,j,k) & = \,
323			\left[ \, \frac{fac \cdot ratio}{wsi^2 + wsvar^2} \sum_{\tau=1}^{nMonsRec}
324			\left\{ Sbar(\tau) \, - \, ctdSobs(\tau) \right\}^2 \, \right](i,j,k)
325	heimbach	1.2	\\
326			\end{split}
327			\end{equation}
328
329			\subsubsection{ARGO float data}
330
331			\begin{equation}
332			\begin{split}
333	heimbach	1.3	cost\_argo\_t(i,j,k) & = \,
334			\left[ \, \frac{fac \cdot ratio}{wti^2 + wvar^2} \sum_{\tau=1}^{nMonsRec}
335			\left\{ Tbar(\tau) \, - \, T2\theta[argoTobs(\tau)] \right\}^2 \, \right](i,j,k)
336	heimbach	1.2	\\
337	heimbach	1.3	cost\_argo\_s(i,j,k) & = \,
338			\left[ \, \frac{fac \cdot ratio}{wsi^2 + wsvar^2} \sum_{\tau=1}^{nMonsRec}
339			\left\{ Sbar(\tau) \, - \, argoSobs(\tau) \right\}^2 \, \right](i,j,k)
340	heimbach	1.2	\\
341			\end{split}
342			\end{equation}
343
344			\subsubsection{Reynolds sea surface T, S data}
345
346			\begin{equation}
347			\begin{split}
348			cost\_sst(i,j) & = \,
349	heimbach	1.3	\left[ \, wsst \sum_{\tau=1}^{nMonsRec}
350	heimbach	1.2	\left\{ Tbar(\tau) \, - \, sstDat(\tau) \right\}^2 \, \right](i,j)
351			\\
352			cost\_sss(i,j) & = \,
353	heimbach	1.3	\left[ \, wsss \sum_{\tau=1}^{nMonsRec}
354	heimbach	1.2	\left\{ Sbar(\tau) \, - \, sssDat(\tau) \right\}^2 \, \right](i,j)
355			\\
356			\end{split}
357			\end{equation}
358
359			\subsubsection{Levitus montly T, S climatological data}
360
361	heimbach	1.3	Model vs. data misfits are taken from $nYears$ monthly model means
362			vs. Levitus monthly data.
363			The description below is for potential temperature.
364			Procedure for salinity is fully analogous.
365			Spatial indices $(i,j,k)$ are omitted throughout.
366			%
367			\begin{enumerate}
368			%
369			\item
370			Compute $nYears$ monthly model means for each month $imon$:
371			\[
372			\overline{Tbar}(imon) \, = \, \frac{1}{nYears}
373			\sum_{iyear=1}^{nYears} Tbar(iyear,imon)
374			\]
375			%
376			\item
377			Compute misfit:
378			\[
379			cost\_theta(i,j,k) \, = \, \left[
380			\frac{fac \cdot ratio}{wti^2} \sum_{imon=1}^{12}
381			\left\{ \overline{Tbar}(imon) \, - \, Tdat(imon) \right\}^2 \right] (i,j,k)
382			\]
383
384			\end{enumerate}
385
386	heimbach	1.2
387			\subsubsection{Weights and notes}
388
389			\begin{itemize}
390			%
391			\item
392			$T2\theta$ is an operator mapping in-situ to potential temperatures
393			%
394			\item
395			Latitudinal weight not used:
396			\[
397			cosphi(i,j) \, = \, 1
398			\]
399			%
400			\item
401	heimbach	1.3	$ fac \, = \, cosphi \cdot mask $
402			%
403			\item
404			Spatially {\it constant} weights:
405	heimbach	1.2	%
406			\begin{enumerate}
407			%
408			\item
409	heimbach	1.3	Read standard deviation vertical profiles for $T$, $S$ \\
410	heimbach	1.2	$ {\tt data\_errfile} \, \longrightarrow \,
411			wti(k), \,\, wsi(k) $ \\
412			$ {\tt data\_errfile} \, \longrightarrow \,
413			ratio = 0.25 = \left( \frac{1}{2} \right)^2 $
414			%
415			\item
416			Take inverse squares:
417			\[
418			\begin{split}
419	heimbach	1.3	wtheta(k) & = \, \frac{ratio}{wti(k)^2} \\
420			wsalt(k) & = \, \frac{ratio}{wsi(k)^2} \\
421	heimbach	1.2	\end{split}
422			\]
423			%
424			\end{enumerate}
425			%
426			\item
427	heimbach	1.3	Spatially {\it varying} weights:
428	heimbach	1.2	%
429			\begin{enumerate}
430			%
431			\item
432			Read standard deviation fields \\
433	heimbach	1.3	$ {\tt temperrfile} \, \longrightarrow \, wtvar(i,j,k) $ \\
434			$ {\tt salterrfile} \, \longrightarrow \, wsvar(i,j,k) $ \\
435	heimbach	1.2	%
436			\item
437			Weights are combination of spatially constant and varying parts:
438			\[
439			\begin{split}
440			wtheta2(i,j,k) & = \, \frac{ratio}
441	heimbach	1.3	{wti(k)^2 \, + \,wtvar(i,j,k)^2 } \\
442	heimbach	1.2	wsalt2(i,j,k) & = \,
443			\frac{ratio}
444	heimbach	1.3	{wsi(k)^2 \, + \,wsvar(i,j,k)^2 } \\
445	heimbach	1.2	\end{split}
446			\]
447			%
448			\end{enumerate}
449			%
450			\item
451			Sea surface $T$, $S$ weights:
452			\begin{itemize}
453			\item
454	heimbach	1.3	SST: $ wsst \, = \, fac \cdot wtheta(1)$: horizontally constant
455	heimbach	1.2	\item
456	heimbach	1.3	SSS: $ wsss \, = \, fac \cdot wsalt2(i,j,1)$: horizontally varying
457	heimbach	1.2	\end{itemize}
458			(Why this difference? I don't know.)
459			%
460			\end{itemize}
461
462
463			\subsubsection{Diagnostics}
464
465			\begin{itemize}
466			%
467			\item
468	heimbach	1.3	Map out $wtheta2(i,j,k)$, $wsalt2(i,j,k)$.
469	heimbach	1.2
470			%
471			\end{itemize}
472