s_ecco/text/ecco_costfunction.tex

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

Author: Patrick Heimbach

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}[!ht]
\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{aligned}
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{aligned}
\end{equation}
%
\item
Misfits are computed w.r.t. global $offset$. 
\\
First spatial distribution:
%
\begin{equation}
\begin{aligned}
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{aligned}
\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{aligned}
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{aligned}
\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{aligned}
\overline{cost\_ssh\_anom} & = \, \sum_{t} \sum_{i,j}
cost\_ssh\_anom(i,j,t)
\end{aligned}
\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}[!ht]
\begin{center}
\begin{tabular}{lllc}
\hline \hline
~&~&~&~\\
field & file name & deccription & unit \\
~&~&~&~\\
\hline
~&~&~&~\\
{\it tbar} & {\tt tbarfile} & monthly model mean pot. temperature & 
[$^{\circ}\mathrm{C}$] \\
{\it sbar} & {\tt sbarfile} & monthly model mean salinity & 
[ppt] \\
{\it tdat} & {\tt tdatfile} & monthly mean Levitus pot. temperature & 
[$^{\circ}\mathrm{C}$] \\
{\it sdat} & {\tt sdatfile} & monthly mean Levitus salinity & 
[ppt] \\
{\it ctdtobs}  & {\tt ctdtfile} & monthly WOCE CTD pot. temperature & 
[$^{\circ}\mathrm{C}$] \\
{\it ctdsobs}  & {\tt ctdsfile} & monthly WOCE CTD salinity & 
[ppt] \\
{\it xbtobs} & {\tt xbtfile} & monthly XBT in-situ(!) temperature & 
[$^{\circ}\mathrm{C}$] \\
{\it sstdat}  & {\tt sstdatfile} & monthly Reynolds pot. SST & 
[$^{\circ}\mathrm{C}$] \\
{\it sssdat}  & {\tt sssdatfile} & monthly Reynolds SSS & 
[ppt] \\
{\it argotobs}  & {\tt argotfile} & monthly ARGO in-situ(!) temperature & 
[$^{\circ}\mathrm{C}$] \\
{\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. & [$^{\circ}\mathrm{C}$] \\
{\it wsvar} & {\tt salterrfile} & spatially varying stdev. & [ppt] \\
~&~&~&~\\
\hline \hline
\end{tabular}
\end{center}
\end{table}

\subsubsection{XBT data}

\begin{equation}
\begin{aligned}
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{aligned}
\end{equation}

\subsubsection{WOCE CTD data}

\begin{equation}
\begin{aligned}
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{aligned}
\end{equation}

\subsubsection{ARGO float data}

\begin{equation}
\begin{aligned}
cost\_argo\_t(i,j,k) & = \,
\left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^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{aligned}
\end{equation}

\subsubsection{Reynolds sea surface T, S data}

\begin{equation}
\begin{aligned}
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{aligned}
\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:
\[
\begin{aligned}
wtheta(k) & = \, \frac{ratio}{wti(k)^2} \\
wsalt(k) & = \, \frac{ratio}{wsi(k)^2} \\
\end{aligned}
\]
%
\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:
\[
\begin{aligned}
wtheta2(i,j,k) & = \, \frac{ratio}
{wti(k)^2 \, + \,wtvar(i,j,k)^2 } \\
wsalt2(i,j,k) & = \, 
\frac{ratio}
{wsi(k)^2 \, + \,wsvar(i,j,k)^2 } \\
\end{aligned}
\]
%
\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	heimbach	1.6	Author: Patrick Heimbach
8
9	heimbach	1.1	The current ECCO state estimation covers an $nYears = 11$ year
10			model trajectory.
11			A variety of data sets enter a least squares cost function,
12			in addition to penalty terms which constrain deviations
13			of control variables beyound their a priori errors.
14
15			\subsection{Sea surface height from TOPEX/Poseidon and ERS-1/2 altimetry}
16
17			Altimetric SSH contributions from T/P and ERS-1/2 are four-fold:
18			%
19			\begin{enumerate}
20			%
21			\item
22			an $nYears$ time mean SSH misfit between
23			model and T/P
24			%
25			\item
26			daily SSH anomaly misfits between T/P and model
27			%
28			\item
29			daily SSH anomaly misfits between ERS-1/2 and model
30			%
31			\item
32			daily absolute SSH misfit between T/P and model,
33			weighted by the full geoid error covariance.
34			%
35			\end{enumerate}
36
37			\subsubsection{Input fields}
38			~
39
40	jmc	1.8	\begin{table}[!ht]
41	heimbach	1.1	\begin{center}
42			\begin{tabular}{lllc}
43			\hline \hline
44			~&~&~&~\\
45			field & file name & deccription & unit \\
46			~&~&~&~\\
47			\hline
48			~&~&~&~\\
49			{\it psbar} & {\tt psbarfile} & daily model mean SSH fields & [m] \\
50			{\it tpmean} & {\tt topexmeanfile} & $nYears$ T/P mean & [cm] \\
51			{\it tpobs} & {\tt topexfile} & daily T/P SSH anomalies & [cm] \\
52			{\it erspobs} & {\tt ersfile} & daily ERS-1/2 SSH anomalies & [cm] \\
53			{\it wp} & {\tt geoid\_errfile} & diagonal of geoid error covariance & [m] \\
54			{\it wtp, wers} & {\tt ssh\_errfile} & rms of SSH anomalies & [cm] \\
55			~&~&~&~\\
56			\hline \hline
57			\end{tabular}
58			\end{center}
59			\end{table}
60
61
62	heimbach	1.2	\subsubsection{\textit{\textbf{nYears}} time mean SSH misfit}
63	heimbach	1.1
64			\begin{enumerate}
65			%
66			\item
67	heimbach	1.3	Compute $nYears$ model mean spatial distribution
68	heimbach	1.1	%
69			\begin{equation}
70			psmean(i,j)\, =\,
71			\frac{1}{nDaysRec} \sum_{i=1}^{nDaysRec}
72			psbar(i,j)
73			\end{equation}
74			%
75			\item
76	heimbach	1.3	Compute global offset between $nYears$ model and T/P mean:
77	heimbach	1.1	%
78			\begin{equation}
79	jmc	1.8	\begin{aligned}
80	heimbach	1.1	offset & = \, \overline{tpmean} \, - \, \overline{psmean} \\
81			~ & = \, \frac{1}{normaliz.} \sum_{i,j}
82			\left\{ tpmean(i,j) \, - \, psmean(i,j) \right\}
83			\cdot cosphi(i,j) \cdot tpmeanmask(i,j)
84	jmc	1.8	\end{aligned}
85	heimbach	1.1	\end{equation}
86			%
87			\item
88			Misfits are computed w.r.t. global $offset$.
89			\\
90			First spatial distribution:
91			%
92			\begin{equation}
93	jmc	1.8	\begin{aligned}
94	heimbach	1.1	cost\_ssh\_mean(i,j) & = \,
95			\frac{1}{wp^2} \left\{ \,
96			\left[ \, psmean(i,j) - \overline{psmean} \, \right] \, - \,
97			\left[ \, tpmean(i,j) - \overline{tpmean} \, \right] \, \right\}^2 \\
98			~ & = \, \frac{1}{wp^2} \left\{ \,
99			psmean(i,j) \, - \, tpmean(i,j) \, + \, offset \, \right\}^2
100	jmc	1.8	\end{aligned}
101	heimbach	1.1	\end{equation}
102
103			%
104			Finally, sum over all spatial entries:
105			\begin{equation}
106			\overline{cost\_ssh\_mean} \, = \,
107			\sum_{i,j} cost\_ssh\_mean(i,j)
108			\end{equation}
109
110
111
112			\end{enumerate}
113
114			\subsubsection{Misfit of daily SSH anomalies}
115
116			Computation is same for T/P and ERS-1/2.
117			Here we write out computation for T/P.
118
119			\begin{enumerate}
120			%
121			\item
122			Compute difference in anomalies:
123
124			\begin{equation}
125	jmc	1.8	\begin{aligned}
126	heimbach	1.1	cost\_ssh\_anom(i,j,t) & = \, \frac{1}{wtp^2} \left\{ \,
127			\left[ \, psbar(i,j,t) - psmean(i,j) \, \right] \, - \,
128			\left[ \, tpobs(i,j,t) \, \right] \,
129			\right\}^2
130	jmc	1.8	\end{aligned}
131	heimbach	1.1	\end{equation}
132			%
133			where $t$ denotes time (day) index, and
134			where it is assumed that $ nYears$ mean T/P spatial distribution
135			$tpmean(i,j)$ has already been removed from data $tpobs(i,j)$!
136
137			\item
138			Sum over all spatial points and all times
139
140			\begin{equation}
141	jmc	1.8	\begin{aligned}
142	heimbach	1.1	\overline{cost\_ssh\_anom} & = \, \sum_{t} \sum_{i,j}
143			cost\_ssh\_anom(i,j,t)
144	jmc	1.8	\end{aligned}
145	heimbach	1.1	\end{equation}
146
147			\end{enumerate}
148
149			\subsubsection{Flow chart}
150
151			\begin{verbatim}
152
153			cost_ssh
154			\|
155			\|- < compute nYears model mean >
156			\|
157			\|- < read nYears T/P mean >
158			\| CALL COST_READTOPEXMEAN
159			\|
160			\|- < compute global T/P vs. model offset >
161			\|
162			\|- < compute cost_hmean >
163			\| CALL COST_SSH_MEAN
164			\|
165			\|- < ... >
166
167			\end{verbatim}
168
169	heimbach	1.2	\subsubsection{Weights and notes}
170	heimbach	1.1
171			\begin{itemize}
172			%
173			\item
174			All data are currently masked to zero where less than 13 depth levels,
175			mimicing no contribution for depth less than 1000m.
176			%
177			\item
178			$cosphi$ term in weights is set to 1.
179			%
180			\item
181			bad T/P and ERS-1/2 values are flagged $ \le \, -9990. $
182			%
183			\item
184			T/P and ERS-1/2 data $ \le \, 1.\exp^{-8}$ cm are flagged as bad values
185			%
186			\item
187			$wp$ is read from {\tt geoid\_errfile}
188			and $1/wp^2$ is pre-computed in {\tt ecco\_cost\_weights}
189			%
190			\end{itemize}
191
192			\paragraph{$wp$ for SSH mean misfit} ~
193
194			$1/wp^2$ is pre-computed in {\tt ecco\_cost\_weights}; \\
195			$wp$ is read from {\tt geoid\_errfile};
196
197			\paragraph{$wtp$ and $wers$ for SSH anomaly misfit} ~
198
199			$1/wtp^2$, $1/wers^2$ are pre-computed in {\tt ecco\_cost\_weights}; \\
200			%
201			\begin{itemize}
202			%
203			\item
204			$wtp$, $wers$ are read from single {\tt ssh\_errfile}
205			%
206			\item
207			both are converted to meters and halved \\
208			$ wtp \, \longrightarrow \, wtp \cdot 0.01 \cdot 0.5 $
209			%
210			\item
211			ERS error is set to T/P error + 5cm \\
212			$ wers \, = \, wtp \, + 0.5cm $
213			%
214			\end{itemize}
215
216			\subsubsection{Cost diagnostics}
217
218			\begin{itemize}
219			%
220			\item
221			Map out $ cost\_ssh\_mean(i,j) $
222			%
223			\item
224			Map out $ cost\_ssh\_anom(i,j,t) $ averaged over 1 month, i.e.
225			\[
226			\frac{1}{\text{monthly entries}} \sum_{t}^{monthly} cost\_ssh\_anom(i,j,t)
227			\]
228			%
229			\item
230			sum over daily entries and plot daily average as function of time. i.e.
231			\[
232			\frac{1}{\text{daily entries}} \sum_{i,j} cost\_ssh\_anom(i,j,t)
233			\]
234			\end{itemize}
235	heimbach	1.2
236			\subsection{Hydrographic constraints}
237
238			Observation of temperature and salinity from various sources are
239			used to constrain the model. These are:
240			%
241			\begin{enumerate}
242			%
243			\item
244			CTD obs. for $T$, $S$ from various WOCE sections
245			%
246			\item
247			XBT obs. for $T$
248			%
249			\item
250			Sea surface temperature (SST) and salinity (SSS) from
251			Reynolds et al. (???)
252			%
253			\item
254			$T$, $S$ from ARGO floats
255			%
256			\item
257			$T$, $S$ from fields from Levitus (???)
258			%
259			\end{enumerate}
260
261			\subsubsection{Input fields}
262			~
263
264	jmc	1.8	\begin{table}[!ht]
265	heimbach	1.2	\begin{center}
266			\begin{tabular}{lllc}
267			\hline \hline
268			~&~&~&~\\
269			field & file name & deccription & unit \\
270			~&~&~&~\\
271			\hline
272			~&~&~&~\\
273			{\it tbar} & {\tt tbarfile} & monthly model mean pot. temperature &
274	edhill	1.7	[$^{\circ}\mathrm{C}$] \\
275	heimbach	1.2	{\it sbar} & {\tt sbarfile} & monthly model mean salinity &
276			[ppt] \\
277			{\it tdat} & {\tt tdatfile} & monthly mean Levitus pot. temperature &
278	edhill	1.7	[$^{\circ}\mathrm{C}$] \\
279	heimbach	1.2	{\it sdat} & {\tt sdatfile} & monthly mean Levitus salinity &
280			[ppt] \\
281			{\it ctdtobs} & {\tt ctdtfile} & monthly WOCE CTD pot. temperature &
282	edhill	1.7	[$^{\circ}\mathrm{C}$] \\
283	heimbach	1.2	{\it ctdsobs} & {\tt ctdsfile} & monthly WOCE CTD salinity &
284			[ppt] \\
285			{\it xbtobs} & {\tt xbtfile} & monthly XBT in-situ(!) temperature &
286	edhill	1.7	[$^{\circ}\mathrm{C}$] \\
287	heimbach	1.2	{\it sstdat} & {\tt sstdatfile} & monthly Reynolds pot. SST &
288	edhill	1.7	[$^{\circ}\mathrm{C}$] \\
289	heimbach	1.2	{\it sssdat} & {\tt sssdatfile} & monthly Reynolds SSS &
290			[ppt] \\
291			{\it argotobs} & {\tt argotfile} & monthly ARGO in-situ(!) temperature &
292	edhill	1.7	[$^{\circ}\mathrm{C}$] \\
293	heimbach	1.2	{\it argosobs} & {\tt argosfile} & monthly ARGO salinity &
294			[ppt] \\
295			{\it wti, wsi} & {\tt data\_errfile} & vert. stdev. profile for $T$, $S$ &
296			~ \\
297	edhill	1.7	{\it wtvar} & {\tt temperrfile} & spatially varying stdev. & [$^{\circ}\mathrm{C}$] \\
298	heimbach	1.3	{\it wsvar} & {\tt salterrfile} & spatially varying stdev. & [ppt] \\
299	heimbach	1.2	~&~&~&~\\
300			\hline \hline
301			\end{tabular}
302			\end{center}
303			\end{table}
304
305			\subsubsection{XBT data}
306
307			\begin{equation}
308	jmc	1.8	\begin{aligned}
309	heimbach	1.3	cost\_xbt\_t(i,j,k) & = \,
310			\left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \sum_{\tau=1}^{nMonsRec}
311			\left\{ Tbar(\tau) \, - \, T2\theta[xbtobs(\tau)] \right\}^2 \, \right](i,j,k)
312	heimbach	1.2	\\
313	jmc	1.8	\end{aligned}
314	heimbach	1.2	\end{equation}
315
316			\subsubsection{WOCE CTD data}
317
318			\begin{equation}
319	jmc	1.8	\begin{aligned}
320	heimbach	1.3	cost\_ctd\_t(i,j,k) & = \,
321			\left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \sum_{\tau=1}^{nMonsRec}
322			\left\{ Tbar(\tau) \, - \, ctdTobs(\tau) \right\}^2 \, \right](i,j,k)
323	heimbach	1.2	\\
324	heimbach	1.3	cost\_ctd\_s(i,j,k) & = \,
325			\left[ \, \frac{fac \cdot ratio}{wsi^2 + wsvar^2} \sum_{\tau=1}^{nMonsRec}
326			\left\{ Sbar(\tau) \, - \, ctdSobs(\tau) \right\}^2 \, \right](i,j,k)
327	heimbach	1.2	\\
328	jmc	1.8	\end{aligned}
329	heimbach	1.2	\end{equation}
330
331			\subsubsection{ARGO float data}
332
333			\begin{equation}
334	jmc	1.8	\begin{aligned}
335	heimbach	1.3	cost\_argo\_t(i,j,k) & = \,
336	heimbach	1.5	\left[ \, \frac{fac \cdot ratio}{wti^2 + wtvar^2} \sum_{\tau=1}^{nMonsRec}
337	heimbach	1.3	\left\{ Tbar(\tau) \, - \, T2\theta[argoTobs(\tau)] \right\}^2 \, \right](i,j,k)
338	heimbach	1.2	\\
339	heimbach	1.3	cost\_argo\_s(i,j,k) & = \,
340			\left[ \, \frac{fac \cdot ratio}{wsi^2 + wsvar^2} \sum_{\tau=1}^{nMonsRec}
341			\left\{ Sbar(\tau) \, - \, argoSobs(\tau) \right\}^2 \, \right](i,j,k)
342	heimbach	1.2	\\
343	jmc	1.8	\end{aligned}
344	heimbach	1.2	\end{equation}
345
346			\subsubsection{Reynolds sea surface T, S data}
347
348			\begin{equation}
349	jmc	1.8	\begin{aligned}
350	heimbach	1.2	cost\_sst(i,j) & = \,
351	heimbach	1.3	\left[ \, wsst \sum_{\tau=1}^{nMonsRec}
352	heimbach	1.2	\left\{ Tbar(\tau) \, - \, sstDat(\tau) \right\}^2 \, \right](i,j)
353			\\
354			cost\_sss(i,j) & = \,
355	heimbach	1.3	\left[ \, wsss \sum_{\tau=1}^{nMonsRec}
356	heimbach	1.2	\left\{ Sbar(\tau) \, - \, sssDat(\tau) \right\}^2 \, \right](i,j)
357			\\
358	jmc	1.8	\end{aligned}
359	heimbach	1.2	\end{equation}
360
361			\subsubsection{Levitus montly T, S climatological data}
362
363	heimbach	1.3	Model vs. data misfits are taken from $nYears$ monthly model means
364			vs. Levitus monthly data.
365			The description below is for potential temperature.
366			Procedure for salinity is fully analogous.
367			Spatial indices $(i,j,k)$ are omitted throughout.
368			%
369			\begin{enumerate}
370			%
371			\item
372			Compute $nYears$ monthly model means for each month $imon$:
373			\[
374			\overline{Tbar}(imon) \, = \, \frac{1}{nYears}
375			\sum_{iyear=1}^{nYears} Tbar(iyear,imon)
376			\]
377			%
378			\item
379			Compute misfit:
380			\[
381			cost\_theta(i,j,k) \, = \, \left[
382			\frac{fac \cdot ratio}{wti^2} \sum_{imon=1}^{12}
383			\left\{ \overline{Tbar}(imon) \, - \, Tdat(imon) \right\}^2 \right] (i,j,k)
384			\]
385
386			\end{enumerate}
387
388	heimbach	1.2
389			\subsubsection{Weights and notes}
390
391			\begin{itemize}
392			%
393			\item
394			$T2\theta$ is an operator mapping in-situ to potential temperatures
395			%
396			\item
397			Latitudinal weight not used:
398			\[
399			cosphi(i,j) \, = \, 1
400			\]
401			%
402			\item
403	heimbach	1.3	$ fac \, = \, cosphi \cdot mask $
404			%
405			\item
406			Spatially {\it constant} weights:
407	heimbach	1.2	%
408			\begin{enumerate}
409			%
410			\item
411	heimbach	1.3	Read standard deviation vertical profiles for $T$, $S$ \\
412	heimbach	1.2	$ {\tt data\_errfile} \, \longrightarrow \,
413			wti(k), \,\, wsi(k) $ \\
414			$ {\tt data\_errfile} \, \longrightarrow \,
415			ratio = 0.25 = \left( \frac{1}{2} \right)^2 $
416			%
417			\item
418			Take inverse squares:
419			\[
420	jmc	1.8	\begin{aligned}
421	heimbach	1.3	wtheta(k) & = \, \frac{ratio}{wti(k)^2} \\
422			wsalt(k) & = \, \frac{ratio}{wsi(k)^2} \\
423	jmc	1.8	\end{aligned}
424	heimbach	1.2	\]
425			%
426			\end{enumerate}
427			%
428			\item
429	heimbach	1.3	Spatially {\it varying} weights:
430	heimbach	1.2	%
431			\begin{enumerate}
432			%
433			\item
434			Read standard deviation fields \\
435	heimbach	1.3	$ {\tt temperrfile} \, \longrightarrow \, wtvar(i,j,k) $ \\
436			$ {\tt salterrfile} \, \longrightarrow \, wsvar(i,j,k) $ \\
437	heimbach	1.2	%
438			\item
439			Weights are combination of spatially constant and varying parts:
440			\[
441	jmc	1.8	\begin{aligned}
442	heimbach	1.2	wtheta2(i,j,k) & = \, \frac{ratio}
443	heimbach	1.3	{wti(k)^2 \, + \,wtvar(i,j,k)^2 } \\
444	heimbach	1.2	wsalt2(i,j,k) & = \,
445			\frac{ratio}
446	heimbach	1.3	{wsi(k)^2 \, + \,wsvar(i,j,k)^2 } \\
447	jmc	1.8	\end{aligned}
448	heimbach	1.2	\]
449			%
450			\end{enumerate}
451			%
452			\item
453			Sea surface $T$, $S$ weights:
454			\begin{itemize}
455			\item
456	heimbach	1.3	SST: $ wsst \, = \, fac \cdot wtheta(1)$: horizontally constant
457	heimbach	1.2	\item
458	heimbach	1.3	SSS: $ wsss \, = \, fac \cdot wsalt2(i,j,1)$: horizontally varying
459	heimbach	1.2	\end{itemize}
460			(Why this difference? I don't know.)
461			%
462			\end{itemize}
463
464
465			\subsubsection{Diagnostics}
466
467			\begin{itemize}
468			%
469			\item
470	heimbach	1.3	Map out $wtheta2(i,j,k)$, $wsalt2(i,j,k)$.
471	heimbach	1.2
472			%
473			\end{itemize}
474