(in directory: :filelink:`verification/tutorial_global_oce_optim/`)
This experiment illustrates the optimization capacity of the MITgcm: here, a high level description.
In this tutorial, a very simple case is used to illustrate the optimization 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 adjustment Q_\mathrm{netm} that attempts to bring the model climatology into consistency with observations (Levitus and Boyer (1994a,b) :cite:`levitus:94a,levitus:94b`).
This adjustment Q_\mathrm{netm} (a 2-D 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) :cite:`stammer:02` and Ferriera et a. (2005) :cite:`ferriera:05`.
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:
J_1=\frac{1}{N}\sum_{i=1}^N \left[ \frac{\overline{T}_i-\overline{T}_i^{lev}}{\sigma_i^T}\right]^2
where \overline{T}_i and \overline{T}_i^{lev} are, respectively, the model and observed potential temperature at each grid point i. The differences are weighted by an a priori uncertainty \sigma_i^T on observations (as provided by Levitus and Boyer (1994a) :cite:`levitus:94a`). The error \sigma_i^T is only a function of depth and varies from 0.5 K at the surface to 0.05 K at the bottom of the ocean, mainly reflecting the decreasing temperature variance with depth (see :numref:`tut_global_optim_errors`a). 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 adjustment to insure it is “reasonable”, i.e., of order of the uncertainties on the observed surface heat flux:
J_2 = \frac{1}{N} \sum_{i=1}^N \left[\frac{Q_\mathrm{netm}}{\sigma^Q_i} \right]^2
where \sigma^Q_i are the a priori errors on the observed heat flux as estimated by Stammer et al. (2002) :cite:`stammer:02` from 30% of local root-mean-square variability of the NCEP forcing field (see :numref:`tut_global_optim_errors`b).
The total cost function is defined 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 components. The adjoint model then yields the sensitivities \partial J/\partial Q_\mathrm{netm} of J relative to the 2-D fields Q_\mathrm{netm}. Using a line-searching algorithm (Gilbert and Lemaréchal 1989 :cite:`gil-lem:89`), Q_\mathrm{netm} is adjusted then in the sense to reduce J — the procedure is repeated until convergence.
:numref:`tut_global_optim_tutfig` shows the results of such an optimization. The model is started from rest and from January-mean temperature and salinity initial conditions taken from the Levitus dataset. The experiment is run a year and the averaged temperature over the whole run (i.e., annual mean) is used in the cost function :eq:`cost-temp` to evaluate the model [1]. Only the top 2 levels are used. The first guess Q_\mathrm{netm} is chosen to be zero. The weights \lambda_1 and \lambda_2 are set to 1 and 2, respectively. The total cost function converges after 15 iterations, decreasing from 6.1 to 2.7 (the temperature contribution decreases from 6.1 to 1.8 while the heat flux one increases from 0 to 0.42). The right panels of :numref:`tut_global_optim_tutfig` illustrate the evolution of the temperature error at the surface from iteration 0 to iteration 15. Unsurprisingly, the largest errors at iteration 0 (up to 6 oC, top left panels) are found in the Western boundary currents. After optimization, the departure of the model temperature from observations is reduced to 1 oC or less almost everywhere except in the Pacific equatorial cold tongue. Comparison of the initial temperature error (top, right) and heat flux adjustment (bottom, left) shows that the system basically increased the heat flux out of the ocean where temperatures were too warm and vice-versa. Obviously, heat flux uncertainties are not solely responsible for temperature errors, and the heat flux adjustment partly compensates the poor representation of narrow currents (Western boundary currents, equatorial currents) at 4\times4^\circ resolution. This is allowed by the large a priori error on the heat flux :numref:`tut_global_optim_errors`. The Pacific cold tongue is a counter example: there, heat fluxes uncertainties are fairly small (about 20 W m-2), and a large temperature errors remains after optimization.
Initial annual mean surface heat flux (top right in W m-2) and adjustment obtained at iteration 15 (bottom right). Averaged difference between model and observed potential temperatures at the surface (in ^\circC) before optimization (iteration 0, top right) and after optimization (iteration 15, bottom right). Contour intervals for heat flux and temperature are 25 W m-2 and 1 oC, respectively. A positive flux is out of the ocean.
One of the goals of this tutorial is to illustrate how to implement a new
control variable. Most of this is fairly generic and is done in :filelink:`pkg/ctrl`
and :filelink:`pkg/cost`. The modifications can be
tracked by the CPP option :varlink:`ALLOW_HFLUXM_CONTROL` or the comment
cHFLUXM_CONTROL
. The more specific modifications required for the
experiment are found in
:filelink:`verification/tutorial_global_oce_optim/code_ad`. Here follows a brief
description of the implementation.
The adjustment Q_\mathrm{netm} is activated by setting #define
:varlink:`ALLOW_HFLUXM_CONTROL` in :filelink:`code_ad/CTRL_OPTIONS.h <verification/tutorial_global_oce_optim/code_ad//CTRL_OPTIONS.h>`.
It is first implemented as a “normal” forcing variable. It is defined in :filelink:`model/inc/FFIELDS.h`, initialized to zero in :filelink:`model/src/ini_forcing.F`, and then used in :filelink:`model/src/external_forcing_surf.F`. Q_\mathrm{netm} is made a control variable in :filelink:`pkg/ctrl` by modifying the following subroutines:
- :filelink:`pkg/ctrl/ctrl_init.F` where Q_\mathrm{netm} is defined as the control variable number 24,
- :filelink:`pkg/ctrl/ctrl_pack.F` which writes, at the end of each iteration, the sensitivity of the cost function \partial J/\partial Q_\mathrm{netm} in to a file to be used by the line-search algorithm,
- :filelink:`pkg/ctrl/ctrl_unpack.F` which reads, at the start of each iteration, the updated adjustment as provided by the line-search algorithm,
- :filelink:`pkg/ctrl/ctrl_map_forcing.F` in which the updated adjustment is added to the first guess Q_\mathrm{netm}.
Note also some minor changes in :filelink:`pkg/ctrl/ctrl.h`, :filelink:`pkg/ctrl/ctrl_readparms.F`, and :filelink:`pkg/ctrl/ctrl_dummy.h` (:varlink:`xx_hfluxm_file`, :varlink:`fname_hfluxm`, :varlink:`xx_hfluxm_dummy`).
The cost functions are implemented using :filelink:`pkg/cost`.
- The temperature cost function J_1 which measures the drift of
the mean model temperature from the Levitus climatology is
implemented in :filelink:`/verification/tutorial_global_oce_optim/code_ad/cost_temp.F`.
It is activated by
#define
:varlink:`ALLOW_COST_TEMP` in :filelink:`code_ad/COST_OPTIONS.h <verification/tutorial_global_oce_optim/code_ad//COST_OPTIONS.h>`. It requires the mean temperature of the model which is obtained by accumulating the temperature in :filelink:`pkg/cost/cost_tile.F` (called at each time step). The value of the cost function is stored in :varlink:`objf_temp` and its weight \lambda_1 in :varlink:`mult_temp`. - The heat flux cost function, penalizing the departure of the surface
heat flux from observations is implemented in :filelink:`/verification/tutorial_global_oce_optim/code_ad/cost_hflux.F`, and
activated by
#define
:varlink:`ALLOW_COST_HFLUXM` in :filelink:`code_ad/COST_OPTIONS.h <verification/tutorial_global_oce_optim/code_ad//COST_OPTIONS.h>`. The value of the cost function is stored in :varlink:`objf_hfluxm` and its weight \lambda_2 in :varlink:`mult_hflux`. - The subroutine :filelink:`pkg/cost/cost_final.F` calls the cost function subroutines and makes the (weighted) sum of the various contributions.
- The various weights used in the cost functions are read in :filelink:`/verification/tutorial_global_oce_optim/code_ad/cost_weights.F`. The weight of the cost functions are read in :filelink:`pkg/cost/cost_readparms.F` from the input file :filelink:`verification/tutorial_global_oce_optim/input_ad/data.cost`.
The experiment files in :filelink:`verification/tutorial_global_oce_optim/code_ad/` and :filelink:`verification/tutorial_global_oce_optim/input_ad/` contain the code customizations and parameter settings. Most of them are identical to those used in the Global Ocean ( experiment :filelink:`verification/tutorial_global_oce_latlon/`). Below, we describe some of the customizations required for this experiment.
Compilation-time customizations in :filelink:`code_ad <verification/tutorial_global_oce_optim/code_ad/>`
In :filelink:`code_ad/CTRL_OPTIONS.h <verification/tutorial_global_oce_optim/code_ad//CTRL_OPTIONS.h>`:
Running-time customizations in :filelink:`input_ad <verification/tutorial_global_oce_optim/input_ad/>`
- :filelink:`input_ad/data <verification/tutorial_global_oce_optim/input_ad/data>`: note the smaller :varlink:`cg2dTargetResidual` than in the forward-only experiment,
- :filelink:`input_ad/data.optim <verification/tutorial_global_oce_optim/input_ad/data.optim>` specifies the iteration number,
- :filelink:`input_ad/data.ctrl <verification/tutorial_global_oce_optim/input_ad/data.ctrl>` is used, in particular, to specify the name of the sensitivity and adjustment files associated to a control variable,
- :filelink:`input_ad/data.cost <verification/tutorial_global_oce_optim/input_ad/data.cost>`: parameters of the cost functions, in particular :varlink:`lastinterval` specifies the length of time-averaging for the model temperature to be used in the cost function :eq:`cost-temp`,
- :filelink:`input_ad/data.pkg <verification/tutorial_global_oce_optim/input_ad/data.pkg>`: note that the Gradient Check package is turned on by
default (:varlink:`useGrdchk`
=.TRUE.
), Err_hflux.bin
andErr_levitus_15layer.bin
are the files containing the heat flux and potential temperature uncertainties, respectively.
The optimization experiment requires two executables: 1) the MITgcm and
its adjoint (mitgcmuv_ad
) and 2) the line-search algorithm
(optim.x
).
Before compiling, first note that in the directory :filelink:`code_ad <verification/tutorial_global_oce_optim/code_ad/>`, two files must be updated:
- :filelink:`code_ad/code_ad_diff.list <verification/tutorial_global_oce_optim/code_ad/code_ad_diff.list>` which lists new subroutines to be compiled by the TAF software (:filelink:`code_ad/cost_temp.F <verification/tutorial_global_oce_optim/code_ad/cost_temp.F>` and :filelink:`code_ad/cost_hflux.F <verification/tutorial_global_oce_optim/code_ad/cost_hflux.F>`),
- the file :filelink:`code_ad/ad_optfile.local <verification/tutorial_global_oce_optim/code_ad/ad_optfile.local>` provides a list of the control variables and the name of cost function to the TAF software.
Then, in the directory :filelink:`build <verification/tutorial_global_oce_optim/build/>`, type:
% ../../../tools/genmake2 -mods=../code_ad -adof=../code_ad/ad_optfile.local % make depend % make adall
to generate the MITgcm executable mitgcmuv_ad
.
This is done from the directories :filelink:`lsopt/` and :filelink:`optim/` (found in the top MITgcm directory). In
:filelink:`lsopt/`, unzip the blash1
library adapted to your platform (see :filelink:`lsopt/README`), and change
the Makefile
accordingly. Compile with:
% make all
(more details in :filelink:`lsopt/lsopt_doc.txt`)
In :filelink:`optim/`, the path of the directory where mitgcm_ad
was compiled
must be specified in the Makefile
in the variable :varlink:`INCLUDEDIRS`. The file
name of the control variable (here, :varlink:`xx_hfluxm_file`) must be added to
the namelist read by :filelink:`optim/optim_numbmod.F`. Then use
% make depend
and
% make
to generate the line-search executable optim.x
.
Make a new subdirectory input_ad/OPTIM
.
Copy the mitgcmuv_ad
executable to :filelink:`input_ad <verification/tutorial_global_oce_optim/input_ad/>`
and optim.x
to this subdirectory.
cd
into :filelink:`input_ad/<verification/tutorial_global_oce_optim/input_ad/>`. The first iteration
is somewhat particular and is best done “by hand” while the following
iterations can be run automatically (see below). Check that the
iteration number is set to 0 in :filelink:`input_ad/data.optim <verification/tutorial_global_oce_optim/input_ad/data.optim>` and run MITgcm:
% ./mitgcmuv_ad
The output files adxx_hfluxm.0000000000.*
and xx_hfluxm.0000000000.*
contain the sensitivity of the cost function to Q_\mathrm{netm}
and the adjustment to Q_\mathrm{netm} (zero at the first
iteration), respectively. Two other files called
costhflux_tut_MITgcm.opt0000
and ctrlhflux_tut_MITgcm.opt0000
are
also generated. They essentially contain the same information as the
adxx_.hfluxm*
and xx_hfluxm*
files, but in a compressed format.
These two files are the only ones involved in the communication between
the adjoint model mitgcmuv_ad
and the line-search algorithm
optim.x
. Only at the first iteration, are they both generated by
mitgcmuv_ad
. Subsequently, costhflux_tut_MITgcm.opt
n is
an output of the adjoint model at iteration n and an input of
the line-search. The latter returns an updated adjustment in
ctrlhflux_tut_MITgcm.opt
n+1 to be used as an input of the
adjoint model at iteration n+1.
At the first iteration, move costhflux_tut_MITgcm.opt0000
and
ctrlhflux_tut_MITgcm.opt0000
to input_ad/OPTIM
,
move into this directory and link :filelink:`input_ad/data.optim <verification/tutorial_global_oce_optim/input_ad/data.optim>`
and :filelink:`input_ad/data.ctrl <verification/tutorial_global_oce_optim/input_ad/data.ctrl>` locally:
% cd OPTIM/ % ln -s ../data.optim . % ln -s ../data.ctrl .
The target cost function :varlink:`fmin` needs to be specified in :filelink:`input_ad/data.optim <verification/tutorial_global_oce_optim/input_ad/data.optim>`: as a rule of thumb, it should be 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 afterward. However, it implicitly specifies the “pace” at which the cost function is going down (if you are lucky and it does indeed diminish!).
Once this is done, run the line-search algorithm:
% ./optim.x
which computes the updated adjustment for iteration 1,
ctrlhflux_tut_MITgcm.opt0001
.
The following iterations can be executed automatically using the shell
script :filelink:`input_ad/cycsh <verification/tutorial_global_oce_optim/input_ad/cycsh>`. This script will take care of
changing the iteration numbers in :filelink:`input_ad/data.optim <verification/tutorial_global_oce_optim/input_ad/data.optim>`, launch the adjoint
model, clean and store the outputs, move the costhflux*
and ctrlhflux*
files, and run the line-search algorithm. Edit :filelink:`input_ad/cycsh <verification/tutorial_global_oce_optim/input_ad/cycsh>` to specify the
prefix of the directories used to store the outputs and the maximum
number of iteration.
[1] | Because of the daily automatic testing, the experiment as found in the repository is set-up with a very small number of time-steps. To reproduce the results shown here, one needs to set :varlink:`nTimeSteps` = 360 and :varlink:`lastinterval` =31104000 (both corresponding to a year, see :numref:`tut_global_oce_runsect` for further details). |