s_examples/global_oce_optim/global_oce_estimation.tex

% $Header: /u/gcmpack/manual/part3/case_studies/global_oce_estimation/global_oce_estimation.tex,v 1.6 2005/08/03 15:48:06 edhill Exp $
% $Name:  $

\section[Global Ocean State Estimation Example]{Global Ocean State Estimation at $4^\circ$ Resolution}
\label{www:tutorials}
\label{sect:eg-global_state_estimate}
\begin{rawhtml}
<!-- CMIREDIR:eg-global_state_estimate: -->
\end{rawhtml}

\subsection{Overview}

Mean surface heat flux as a  control variable : $Q_\mathrm{netm}$

This experiment illustrates the optimization (or data-assimilation) capacity
of the MITgcm. Using an ocean configuration with realistic geography and bathymetry on a
$4\times4^circ$ spherical polar grid, we estimate a time-independent surface heat flux correction
$Q_\mathrm{netm}$ that brings the model climatology into consistency with observations (Levitus
climatology). The files for this experiment can be found in the verification directory under
tutorial\_global\_oce\_optim.

This correction $Q_\mathrm{netm}$ (a 2D field only function of longitude and latitude) is
the control variable of an optimization problem. It is inferred by an iterative
procedure using an `adjoint technique' and a least-squares method (see, for example, 
Stammer et al. (2002) and Ferreira et al. (2005)).

The ocean model is run forward in time and the quality of the solution is
determined by a cost function, $J_1$, a measure of the departure of the model
climatology from observations:
\begin{equation}
J_1=\frac{1}{N}\sum_{i=1}^N \left[ \frac{\overline{T}_i-\overline{T}_i^{lev}}{\sigma_i^T}\right]^2
\end{equation}
where $\overline{T}_i$ is the averaged model temperature and $\overline{T}_i^{lev}$
the annual mean observed temperature at each grid point $i$. The differences
are weighted by an a priori uncertainty $\sigma_i^T$ on observations (Levitus
and Boyer 1994). The error $\sigma_i^T$ is only a function of depth and varies
from 0.5 at the surface to 0.05~K at the bottom of the ocean, mainly reflecting
the decreasing temperature variance with depth. A value of $J_1$ of order 1 means
that the model is, on average, within observational uncertainties.

The cost function also places constraints on the correction to insure it is
"reasonnable", i.e. of order of the uncertainties on the observed surface heat
flux: 
\begin{equation}
J_2 = \frac{1}{N} \sum_{i=1}^N \left[\frac{Q_\mathrm{netm}}{\sigma^x_i} \right]^2
\end{equation}
where $\sigma^x_i$ are the a priori errors (2d field from ECCO ..... Fig ?).

The total cost function is obtained as $J=\lambda_1 J_1+ \lambda_2 J_2$ where
$\lambda_1$ and $\lambda_2$ are weights controlling the relative contribution
of the two mcomponents. The adjoint model then provides the sensitivities
$\partial J/\partial Q_\mathrm{netm}$ of $J$ relative to the 2D fields
$Q_\mathrm{netm}$. Using a line-searching algorithm (Gilbert and Lemar\'{e}chal 1989),
$Q_\mathrm{netm}$ is adjusted in the sense to reduce $J$ --- the procedure is 
repeated until convergence.

In the following example, the configuration is identical to the "Global ocean circulation"
tutorial where more details can be found. In each iteration, the model is started from
rest with temperature and salinity initial conditions taken from Levitus dataset and run
for a year. The first guess $Q_\mathrm{netm}$ is chosen to be zero.

The experiment employs two executables: one for the MITgcm and its adjoints and 
one for the line-search algorithmi (offline optimization). The implementation of
the control variable $Q_\mathrm{netm}$, the cost function $J$ and the I/O required
for the commmunication betwwen the model and the line-search are described in details 
in section 2. The compilation of the two executables is given in section 3.
A method to run the experiment is described in section 4.

Gilbert, J. C., and C. Lemar\'echal, 1989: Some numerical experiments with
variable-storage quasi-Newton algorithms. \textit{Math. Programm.,}
\textbf{45,} 407-435.

\subsection{Implemention of the control variable and the cost function}

All subroutines that require modifications are found in verifications/Optim/code\_ad

\subsubsection{The control variable}

The correction $Q_\mathrm{netm}$ is activated by setting ALLOW\_HFLUXM\_CONTROL to "define" in ECCO\_OPTIONS.h.

It is first implemented as a forcing variable. It is defined in FFIELDS.h,
initialized to zero in ini\_forcing.F, and then used in external\_forcing\_surf.F.

$Q_\mathrm{netm}$ is made a control variable in the ctrl package by modifying the following subroutines:

\begin{itemize}
\item ctrl\_init.F where $Q_\mathrm{netm}$ is defined as the control variable number 24,

\item ctrl\_pack.F which writes, at the end of iteration, the sensitivity of the cost function
$\partial J/\partial Q_\mathrm{netm}$ into a file to be used by the lins-search algorithm,

\item ctrl\_unpack.F which reads, at the start of each iteration, the updated perturbations as
provided by the line-search algorithm,

\item ctrl\_map\_forcing.F where the updated perturbation is added to the first guess $Q_\mathrm{netm}$.
\end{itemize}

Note also some minor changes in ctrl.h, ctrl\_readparams.F, and ctrl\_dummy.h (xx\_hfluxm\_file,
fname\_hfluxm, xx\_hfluxm\_dummy).

\subsubsection{Cost functions}

The cost functions are implemented using the cost package.

\begin{itemize}

\item The temperature cost function $J_1$ which measures the drift of the mean model
temperature from the Levitus climatology is implemented cost\_temp.F. It is
activated by ALLOW\_COST\_TEMP in ECCO\_OPTIONS.h. It requires the mean temperature of
the model which is obtained by accumulating the temperature in cost\_tile.F (called at
each time step).
The value of the cost function is stored in objf\_temp and its weight $\lambda_1$
in mult\_temp.

\item The heat flux cost function penalizing the departure of the surface heat flux from
observations is implemented in cost\_hflux.F, and activated by activated
ALLOW\_COST\_HFLUXM in ECCO\_OPTIONS.h. The value of the cost function is stored in
objf\_hfluxm and its weight $\lambda_2$ in mult\_hfluxm,

\item The subroutine cost\_final.F calls the cost\_functions subroutines
and make the (weighted) sum of the different contributions.

\item The weights used in the cost functions are read is cost\_weights.F.
The weigth of the cost functions are read in cost\_readparams.F from the input file
data.cost.    

\end{itemize}


\subsection{Code Configuration}
\label{www:tutorials}
\label{SEC:eg_fourl_code_config}

The model configuration for this experiment resides under the
directory {\it verification/???/}.  The experiment files in code\_ad/
and input\_ad/ contain the code customisations and parameter settings
for this experiment. Most of them are identical to those used in
the Global ocean experiment. Below we describe the customisations to
these files associated with this experiment.

\subsubsection{File {\it input/data}}


\subsection{Compiling} 

The optimization experiment requires two executables: 1) the 
MITgcm and its adjoints (it{mitgcmuv\_ad}) and 2) the line-search
algorithm ({\it optim.x}) 

\subsubsection{Compilation of MITgcm and its adjoint: {\it mitcgmuv\_ad}}

Before compiling, first note that, in the directory code\_ad/, two files
must be updated:
\begin{itemize}
\item code\_ad\_diff.list which lists new subroutines to be compiled
by the TAF software (cost\_temp.f and cost\_hflux.f here),

\item  the adjoint\_hfluxm files which provides a list of the control variables
and the name of cost function to the TAF sotware.

\end{itemize}

Then, in the directory build\_ad/, type:
\begin{verbatim}
% ../../../tools/genmake2 -mods=../code\_ad -adof=../code\_ad/adjoint\_hfluxm
% make depend
% make adall
\end{verbatim}
to generate the MITgcm executable mitgcmuv\_ad

\subsubsection{Compilation of the line-search algorithm: {\it optim.x}}

This is done from the directories lsopt/ and optim/ (under MITgcm/)

In lsopt/, unzip the blash1 library you need, and change accordingly the
library in the Makefile. Compile with:
\begin{verbatim}
% make all
\end{verbatim}
(more details in lsopt\_doc.txt)

In optim/,  the path of the directory where {\it mitgcm\_ad} was compliled must be specified
in the Makefile in the variable INCLUDEDIRS. The file name of the controle variable
(xx\_hfluxm\_file here) must be added to the namelist read by optim\_num.F

Then use
\begin{verbatim}
% make depend
\end{verbatim} 
and
\begin{verbatim}
% make
\end{verbatim}
to generate the line-search executable {\it optim.x}.

\subsection{Running the estimation}

Copy the {\it mitgcmuv\_ad} executable to input\_ad and {\it optim.x}
to the subdirectory input\_ad/OPTIM. Move into input\_ad/.

The first iteration has be done "by hand". Check that the iteration number is set
to 0 in data.optim and run the Mitgcm
\begin{verbatim}
% ./mitgcmuv_ad
\end{verbatim}

The output files adxx\_hfluxm.0000000000.* and xx\_hfluxm.0000000000.* contain
the sensitivity of the cost function to $Q_\mathrm{netm}$ and the perturbations
to $Q_\mathrm{netm}$ (zero at the first iteration), respectively. Two other files
called ecco\_cost.. and ecco\_ctrl are also generated. They essentially contains
the same information as the adxx\_* and xx\_* files, but in a compressed format.
These two file are the only ones involved in the communication between the adjoint
model {\it mitgcmuv\_ad} and the line-search algorithm {\it optim.x}. The ecco\_cost*n
is an ouput of the adjoint model at iteration $n$ and an input of the line-search. The
latter returns an updated perturbation in ecco\_ctrl*n+1 to be used as an input of
the adjoint model at iteration n+1. 

At the first iteration, move these two files ecco\_cost and ecco\_ctrl 
to OPTIM/, open data.optim and check the iteration number is set to 0.
The target cost function {\it fmin} needs to be specified: a rule of thumb 
suggest it should be set to about 0.95-0.90 times the value of the cost
function at the first iteration. This value is only used at the first
iteration and does not need to be updated afterwards. 
However, it implicitly specifies the "pace" at which the cost function is
going down (if you are lucky and it goes down). More in the ECCO section maybe ?

Once this is done, run the line-search algorithm:
\begin{verbatim}
% ./optim.x
\end{verbatim}
which computes the updated perturbations for iteration 1 ecco\_ctrl\_1.

The following iterations can be executed automatically using the shell
script {\it cycsh}
found in input\_ad/. This script will take care of changing the iteration numbers in the
data.optim, launch the adjoint model, clean and store the ouputs, move the
ecco\_cost* and ecco\_ctrl* files, and run the line-search algotrithm.
Edit {\it cycsh} to specify the prefix of the directories used to store the ouputs and
the maximum number of iteration.

1	molod	1.7	% $Header: /u/gcmpack/manual/part3/case_studies/global_oce_estimation/global_oce_estimation.tex,v 1.6 2005/08/03 15:48:06 edhill Exp $
2	dfer	1.1	% $Name: $
3
4	dfer	1.5	\section[Global Ocean State Estimation Example]{Global Ocean State Estimation at $4^\circ$ Resolution}
5	dfer	1.1	\label{www:tutorials}
6			\label{sect:eg-global_state_estimate}
7			\begin{rawhtml}
8			<!-- CMIREDIR:eg-global_state_estimate: -->
9			\end{rawhtml}
10
11	dfer	1.3	\subsection{Overview}
12
13	dfer	1.5	Mean surface heat flux as a control variable : $Q_\mathrm{netm}$
14	dfer	1.1
15			This experiment illustrates the optimization (or data-assimilation) capacity
16			of the MITgcm. Using an ocean configuration with realistic geography and bathymetry on a
17	dfer	1.5	$4\times4^circ$ spherical polar grid, we estimate a time-independent surface heat flux correction
18			$Q_\mathrm{netm}$ that brings the model climatology into consistency with observations (Levitus
19	molod	1.7	climatology). The files for this experiment can be found in the verification directory under
20			tutorial\_global\_oce\_optim.
21	dfer	1.1
22	dfer	1.5	This correction $Q_\mathrm{netm}$ (a 2D field only function of longitude and latitude) is
23	dfer	1.1	the control variable of an optimization problem. It is inferred by an iterative
24			procedure using an `adjoint technique' and a least-squares method (see, for example,
25			Stammer et al. (2002) and Ferreira et al. (2005)).
26
27			The ocean model is run forward in time and the quality of the solution is
28			determined by a cost function, $J_1$, a measure of the departure of the model
29			climatology from observations:
30			\begin{equation}
31			J_1=\frac{1}{N}\sum_{i=1}^N \left[ \frac{\overline{T}_i-\overline{T}_i^{lev}}{\sigma_i^T}\right]^2
32			\end{equation}
33			where $\overline{T}_i$ is the averaged model temperature and $\overline{T}_i^{lev}$
34			the annual mean observed temperature at each grid point $i$. The differences
35			are weighted by an a priori uncertainty $\sigma_i^T$ on observations (Levitus
36			and Boyer 1994). The error $\sigma_i^T$ is only a function of depth and varies
37			from 0.5 at the surface to 0.05~K at the bottom of the ocean, mainly reflecting
38			the decreasing temperature variance with depth. A value of $J_1$ of order 1 means
39			that the model is, on average, within observational uncertainties.
40
41			The cost function also places constraints on the correction to insure it is
42			"reasonnable", i.e. of order of the uncertainties on the observed surface heat
43			flux:
44			\begin{equation}
45			J_2 = \frac{1}{N} \sum_{i=1}^N \left[\frac{Q_\mathrm{netm}}{\sigma^x_i} \right]^2
46			\end{equation}
47			where $\sigma^x_i$ are the a priori errors (2d field from ECCO ..... Fig ?).
48
49			The total cost function is obtained as $J=\lambda_1 J_1+ \lambda_2 J_2$ where
50			$\lambda_1$ and $\lambda_2$ are weights controlling the relative contribution
51			of the two mcomponents. The adjoint model then provides the sensitivities
52			$\partial J/\partial Q_\mathrm{netm}$ of $J$ relative to the 2D fields
53			$Q_\mathrm{netm}$. Using a line-searching algorithm (Gilbert and Lemar\'{e}chal 1989),
54			$Q_\mathrm{netm}$ is adjusted in the sense to reduce $J$ --- the procedure is
55			repeated until convergence.
56
57			In the following example, the configuration is identical to the "Global ocean circulation"
58			tutorial where more details can be found. In each iteration, the model is started from
59			rest with temperature and salinity initial conditions taken from Levitus dataset and run
60	dfer	1.5	for a year. The first guess $Q_\mathrm{netm}$ is chosen to be zero.
61	dfer	1.1
62			The experiment employs two executables: one for the MITgcm and its adjoints and
63			one for the line-search algorithmi (offline optimization). The implementation of
64			the control variable $Q_\mathrm{netm}$, the cost function $J$ and the I/O required
65			for the commmunication betwwen the model and the line-search are described in details
66			in section 2. The compilation of the two executables is given in section 3.
67	dfer	1.3	A method to run the experiment is described in section 4.
68	dfer	1.1
69			Gilbert, J. C., and C. Lemar\'echal, 1989: Some numerical experiments with
70			variable-storage quasi-Newton algorithms. \textit{Math. Programm.,}
71			\textbf{45,} 407-435.
72
73	dfer	1.3	\subsection{Implemention of the control variable and the cost function}
74	dfer	1.1
75			All subroutines that require modifications are found in verifications/Optim/code\_ad
76
77	dfer	1.3	\subsubsection{The control variable}
78
79	dfer	1.5	The correction $Q_\mathrm{netm}$ is activated by setting ALLOW\_HFLUXM\_CONTROL to "define" in ECCO\_OPTIONS.h.
80	dfer	1.1
81			It is first implemented as a forcing variable. It is defined in FFIELDS.h,
82			initialized to zero in ini\_forcing.F, and then used in external\_forcing\_surf.F.
83
84	dfer	1.5	$Q_\mathrm{netm}$ is made a control variable in the ctrl package by modifying the following subroutines:
85	dfer	1.1
86	dfer	1.3	\begin{itemize}
87	dfer	1.5	\item ctrl\_init.F where $Q_\mathrm{netm}$ is defined as the control variable number 24,
88	dfer	1.1
89	dfer	1.3	\item ctrl\_pack.F which writes, at the end of iteration, the sensitivity of the cost function
90	dfer	1.1	$\partial J/\partial Q_\mathrm{netm}$ into a file to be used by the lins-search algorithm,
91
92	dfer	1.3	\item ctrl\_unpack.F which reads, at the start of each iteration, the updated perturbations as
93	dfer	1.1	provided by the line-search algorithm,
94
95	dfer	1.5	\item ctrl\_map\_forcing.F where the updated perturbation is added to the first guess $Q_\mathrm{netm}$.
96	dfer	1.3	\end{itemize}
97	dfer	1.1
98			Note also some minor changes in ctrl.h, ctrl\_readparams.F, and ctrl\_dummy.h (xx\_hfluxm\_file,
99			fname\_hfluxm, xx\_hfluxm\_dummy).
100
101	dfer	1.3	\subsubsection{Cost functions}
102
103			The cost functions are implemented using the cost package.
104	dfer	1.1
105	dfer	1.3	\begin{itemize}
106	dfer	1.1
107	dfer	1.3	\item The temperature cost function $J_1$ which measures the drift of the mean model
108	dfer	1.1	temperature from the Levitus climatology is implemented cost\_temp.F. It is
109			activated by ALLOW\_COST\_TEMP in ECCO\_OPTIONS.h. It requires the mean temperature of
110			the model which is obtained by accumulating the temperature in cost\_tile.F (called at
111			each time step).
112			The value of the cost function is stored in objf\_temp and its weight $\lambda_1$
113			in mult\_temp.
114
115	dfer	1.3	\item The heat flux cost function penalizing the departure of the surface heat flux from
116	dfer	1.1	observations is implemented in cost\_hflux.F, and activated by activated
117			ALLOW\_COST\_HFLUXM in ECCO\_OPTIONS.h. The value of the cost function is stored in
118			objf\_hfluxm and its weight $\lambda_2$ in mult\_hfluxm,
119
120	dfer	1.3	\item The subroutine cost\_final.F calls the cost\_functions subroutines
121	dfer	1.1	and make the (weighted) sum of the different contributions.
122
123	dfer	1.3	\item The weights used in the cost functions are read is cost\_weights.F.
124	dfer	1.1	The weigth of the cost functions are read in cost\_readparams.F from the input file
125			data.cost.
126
127	dfer	1.3	\end{itemize}
128
129	dfer	1.5
130			\subsection{Code Configuration}
131			\label{www:tutorials}
132			\label{SEC:eg_fourl_code_config}
133
134			The model configuration for this experiment resides under the
135			directory {\it verification/???/}. The experiment files in code\_ad/
136			and input\_ad/ contain the code customisations and parameter settings
137			for this experiment. Most of them are identical to those used in
138			the Global ocean experiment. Below we describe the customisations to
139			these files associated with this experiment.
140
141			\subsubsection{File {\it input/data}}
142
143
144	dfer	1.3	\subsection{Compiling}
145	dfer	1.1
146			The optimization experiment requires two executables: 1) the
147	dfer	1.5	MITgcm and its adjoints (it{mitgcmuv\_ad}) and 2) the line-search
148	edhill	1.6	algorithm ({\it optim.x})
149	dfer	1.1
150	edhill	1.6	\subsubsection{Compilation of MITgcm and its adjoint: {\it mitcgmuv\_ad}}
151	dfer	1.1
152			Before compiling, first note that, in the directory code\_ad/, two files
153			must be updated:
154	dfer	1.5	\begin{itemize}
155			\item code\_ad\_diff.list which lists new subroutines to be compiled
156	dfer	1.1	by the TAF software (cost\_temp.f and cost\_hflux.f here),
157
158	dfer	1.5	\item the adjoint\_hfluxm files which provides a list of the control variables
159			and the name of cost function to the TAF sotware.
160
161			\end{itemize}
162	dfer	1.1
163			Then, in the directory build\_ad/, type:
164	dfer	1.5	\begin{verbatim}
165			% ../../../tools/genmake2 -mods=../code\_ad -adof=../code\_ad/adjoint\_hfluxm
166			% make depend
167			% make adall
168			\end{verbatim}
169			to generate the MITgcm executable mitgcmuv\_ad
170	dfer	1.1
171	edhill	1.6	\subsubsection{Compilation of the line-search algorithm: {\it optim.x}}
172	dfer	1.1
173	dfer	1.3	This is done from the directories lsopt/ and optim/ (under MITgcm/)
174	dfer	1.1
175	dfer	1.5	In lsopt/, unzip the blash1 library you need, and change accordingly the
176			library in the Makefile. Compile with:
177			\begin{verbatim}
178			% make all
179			\end{verbatim}
180			(more details in lsopt\_doc.txt)
181	dfer	1.1
182	edhill	1.6	In optim/, the path of the directory where {\it mitgcm\_ad} was compliled must be specified
183	dfer	1.1	in the Makefile in the variable INCLUDEDIRS. The file name of the controle variable
184			(xx\_hfluxm\_file here) must be added to the namelist read by optim\_num.F
185
186	dfer	1.5	Then use
187			\begin{verbatim}
188			% make depend
189			\end{verbatim}
190			and
191			\begin{verbatim}
192			% make
193			\end{verbatim}
194	edhill	1.6	to generate the line-search executable {\it optim.x}.
195	dfer	1.1
196	dfer	1.3	\subsection{Running the estimation}
197	dfer	1.1
198	edhill	1.6	Copy the {\it mitgcmuv\_ad} executable to input\_ad and {\it optim.x}
199	dfer	1.5	to the subdirectory input\_ad/OPTIM. Move into input\_ad/.
200	dfer	1.1
201	dfer	1.3	The first iteration has be done "by hand". Check that the iteration number is set
202			to 0 in data.optim and run the Mitgcm
203			\begin{verbatim}
204			% ./mitgcmuv_ad
205			\end{verbatim}
206	dfer	1.1
207	dfer	1.5	The output files adxx\_hfluxm.0000000000.* and xx\_hfluxm.0000000000.* contain
208			the sensitivity of the cost function to $Q_\mathrm{netm}$ and the perturbations
209			to $Q_\mathrm{netm}$ (zero at the first iteration), respectively. Two other files
210			called ecco\_cost.. and ecco\_ctrl are also generated. They essentially contains
211			the same information as the adxx\_* and xx\_* files, but in a compressed format.
212			These two file are the only ones involved in the communication between the adjoint
213	edhill	1.6	model {\it mitgcmuv\_ad} and the line-search algorithm {\it optim.x}. The ecco\_cost*n
214	dfer	1.5	is an ouput of the adjoint model at iteration $n$ and an input of the line-search. The
215			latter returns an updated perturbation in ecco\_ctrl*n+1 to be used as an input of
216			the adjoint model at iteration n+1.
217	dfer	1.4
218			At the first iteration, move these two files ecco\_cost and ecco\_ctrl
219			to OPTIM/, open data.optim and check the iteration number is set to 0.
220	edhill	1.6	The target cost function {\it fmin} needs to be specified: a rule of thumb
221	dfer	1.4	suggest it should be set to about 0.95-0.90 times the value of the cost
222			function at the first iteration. This value is only used at the first
223			iteration and does not need to be updated afterwards.
224			However, it implicitly specifies the "pace" at which the cost function is
225			going down (if you are lucky and it goes down). More in the ECCO section maybe ?
226
227			Once this is done, run the line-search algorithm:
228			\begin{verbatim}
229			% ./optim.x
230			\end{verbatim}
231	dfer	1.5	which computes the updated perturbations for iteration 1 ecco\_ctrl\_1.
232	dfer	1.4
233	edhill	1.6	The following iterations can be executed automatically using the shell
234			script {\it cycsh}
235	dfer	1.5	found in input\_ad/. This script will take care of changing the iteration numbers in the
236			data.optim, launch the adjoint model, clean and store the ouputs, move the
237			ecco\_cost* and ecco\_ctrl* files, and run the line-search algotrithm.
238	edhill	1.6	Edit {\it cycsh} to specify the prefix of the directories used to store the ouputs and
239	dfer	1.5	the maximum number of iteration.
240	dfer	1.1