= Lorenz-63 model with PDAF = 

[[PageOutline(2-3,Contents of this page)]]

The implementation of the Lorenz-63 model coupled to PDAF is in the directory `models/lorenz63/` of the PDAF package.
Provided is a full implementation of PDAF with the nonlinear Lorenz-63 model (E. N. Lorenz (1963) Deterministic non-periodic flows. J. Atmos. Sci. 20, 130-141) providing the possibility to assess various filter and smoother methods. The model has state dimenion 3. So it is not usable for
localized filters, but it is a good test case for the particle filter.

Next to the implementation of the Lorenz-96 model with PDAF, the test case provides tool programs and scripts  (in the subdirectory `tools/`) that
allow to run a test case and to display the outputs. Note, that this implementation runs without parallelization.


== Running the test case  ==

Runnning a data assimilation experiment with the Lorenz-63 model is a two step process: First one runs the model without PDAF to generate a file holding the trajectory of a forward run. Then, one generates a file with observations by perturbing the forward run.

Note, that in contrast to the Lorenz-96 model case, we initialize the ensemble by random sampling from the forward run. Using second-order exact sampling (see the [wiki:EnsembleGeneration page on ensemble generation]) is not useful for the Lorenz-63 model, as this would limit us to a maximum ensemble size of 4 states.

=== 1. Compile and run the forward model without assimilation ===

 First, we compile the Lorenz-96 model without assimilation. For this we need to deactivate the code parts that couple to PDAF. For this, change in the file make.arch/linux_gfortran.h the line

CPP_DEFS = -DUSE_PDAF

to

CPP_DEFS = #-DUSE_PDAF

Now build the forward model program with 
{{{
  cd models/lorenz63
  make lorenz_63 PDAF_ARCH=linux_gfortran
}}}
The executable is generated in the current directory.

You can replace `linux_gfortran` by any other make include file from `make.arch/`, e.g. specify `macos_gfortran_openmpi` for compiling on MacOS. 

'''Note''': The implementation uses the NetCDF library for file outputs. If the compilation above fails, please ensure the netCDF-library is installed. On computers running Linux, netCDF is usually available as a package of the operating system. On MacOS one can install the netcdf library e.g. using Homebrew or !MacPorts. NetCDF is a self-describing binary output format, but here it is not required that you know details about it. Anyway, if you like to look 'into' a NetCDF file, please try to use `ncdump FILENAME | less`. 

To run the forward model use
{{{
  ./lorenz_63 -total_steps 10000
}}}
This runs the Lorenz-63 model for 10000 time steps and the trajectory is written into a file `state_l63.nc`. 

=== 2. Generate observations ===

To build the executable to generate observations from the state trajectore use
{{{
  cd tools
  make generate_obs PDAF_ARCH=linux_gfortran
}}}

Now run
{{{
  ./generate_obs
}}}
to generate a file holding observations (`obs_l63.nc` in `models/lorenz63/`).


=== 3. Build and run the assimilation program ===

Now edit the file `make.arch/linux_gfortran.h` again and change the line
{{{
CPP_DEFS = #-DUSE_PDAF
}}}
back to 
{{{
CPP_DEFS = -DUSE_PDAF
}}}
to activate the coupling to PDAF in the model code. 

Now compile the Lorenz-63 model with activated
coupling to PDAF. First remove the binary files from the previous compilation using
{{{
  make clean PDAF_ARCH=linux_gfortran
}}}
Then build the assimilative program using
{{{
  make pdaf_lorenz_63 PDAF_ARCH=linux_gfortran
}}} 
The program `pdaf_lorenz_63` is generated in the current directory.


To run the assimilation program, do
{{{
  ./tools/run_ESTKF.sh 
}}}
The script run_ESTKF.sh runs an experiment with the ESTKF filter method for ensmeble size 20 in which only the variable 'X' of the model is observed at each 10th time step. The execution should only take seconds. The run will generate the output file `ESTKF_N20.nc`.

Further you can run
{{{
  ./tools/run_PF.sh 
}}}
to run a similar experiment with the particle filter. The run will generate the output file `PF_N20.nc`.



=== 4. Plot output from the assimilation experiments ===

To display the output of the assimilation experiments we provide plotting scripts for Matlab, Octave and Python. To use them do
{{{
cd plotting/
}}}

The following plot functions are available:
||= !Function/Script =||= Description =||
|| plot_eofs || Plot the mean state and the eigenvectors (EOFs) from the file holdig the covariance matrix that is generated with tools/generate_covar.F90. ||
|| plot_obs || Plot the observations for a specified time step. The observation file is generated using tools/generate_obs.F90. ||
|| plot_obs_series || Plot a time series of the observations for a specified variable of the model. The observation file is generated using tools/generate_obs.F90. ||
|| plot_rms || Plot the true and estimated RMS errors over time from an assimilation experiment. ||
|| plot_sigma || Plot the singular values from the file holdig the covariance matrix that is generated with tools/generate_covar.F90. ||
|| plot_state || Plot the state from the file holding a true state trajectory (state.nc) or the estimated forecast or analysis states from an assimilation experiment. ||
|| plot_state_series || Plot a time series of the state from the file holding a true state trajectory (state_l63.nc) or the estimated forecast or analysis states from an assimilation experiment for a selected state variable. ||

The scripts need the specification of the input file and the time step or variable to plot. The syntax of the functions is identical for Matlab and Python (except for `plot_rms`):

||= Python =||= !Matlab/Octave =||= Comment =||
|| plot_eofs(filename, index) || plot_eofs(filename, index) || index=0 for mean state, index>1 for EOFs ||
|| plot_obs(filename, timestep) || plot_obs(filename, timestep) || ||
|| plot_obs_series(filename, variable) || plot_obs_series(filename, variable) || ||
|| plot_rms(filename [, plot_forecast=True, plot_analysis=True]) ||plot_rms(filename [, plot_forecast=1, plot_analysis=1]) ||  plot_forecast/plot_analysis are optional and switch on/off a plot ||
|| plot_sigma(filename) || plot_sigma(filename) || ||
|| plot_state(filename, timestep [, choice='t']) || plot_state(filename, timestep [, choice='t']) || `choice` is optional; options for `choice`: 't' true (use with truth file `state.nc`); for use with assimilation experiment output files: 'f' forecast, 'a' analysis, 'i' initial ||
|| plot_state_series(filename, variable, choice='t') || plot_state(filename, variable [, choice='t']) || options for `choice`: 't' true (use with truth file `state_l63.nc`); for use with assimilation experiment output files: 'f' forecast, 'a' analysis ||

Here `filename` is the name of the file including its path. 
In Matlab use 'help' to display the information about required input. 

'''Python plotting'''[[BR]]
For Python the scripts are provided by `plot_l63.py`.

The module can either be imported, e.g. to use its functions interactively:[[BR]]
`>>> import plot_l63`[[BR]]
`>>> plot_l63.plot_obs('../obs_l63.nc', 4)`[[BR]]

Alternatively the script can be run from the command line, providing the function
name and its argument as command line parameters:[[BR]]
`./plot_l63.py plot_obs ../obs_l63.nc 4`[[BR]]
(If this fails you can also try to run the script as
`python plot_l63.py plot_obs ../obs_l63.nc 4`). [[BR]]
To plot only the analysis RMS error one can use[[BR]]
`./plot_l63.py plot_rms ../ESTKF_N20.nc False True`

'''!Matlab/Octave plotting examples:'''[[BR]]
`plot_obs('../obs.nc',100)` plots the observation at time step 100[[BR]]
`plot_obs_series('../obs.nc',1)` plots the time series of observations for variable X (likely set 2=Y, 3=Z) [[BR]]
`plot_state('../ESTKF_N20.nc',100,'f')` plots the forecast state estimate from the ESTKF run example at the 100th analysis step[[BR]]
`plot_rms('../t1_N30_f0.97.nc')` plots the true and estimated RMS errors over time for the chosen experiment[[BR]]
`plot_state('../state_l63.nc',1101)` plots the true state at model time step 1101 (= analysis step 100)[[BR]] 
`plot_state_series('../ESTKF_N20.nc',1,'f')` plots the time series for the ESTKF run example of the forecast state for variable X[[BR]] 


== Run options ==

The implementation for the Lorenz-63 model has a wide range of options. For the full set, we recommend to check the list in the file init_pdaf.F90 in the directory lorenz63. Essentially all of the options can be specified on the command line in the format '-VARIABLE VALUE', where VARIABLE is the name of the variable in the program and VALUE is its value. The experiment in `run_ESTKF.sh` is executed like
{{{
./pdaf_lorenz_63 -total_steps 5000 -step_null 1000 -dim_ens 20 \
      -forget 0.8 -filtertype 6 -file_asml ESTKF_N20.nc
}}}
and the experiment in `run_PF.sh` is executed like
{{{
./pdaf_lorenz_63 -total_steps 5000 -step_null 1000 -dim_ens 20 \
      -pf_res_type 2 -pf_noise_type 2 -pf_noise_amp 0.2 -filtertype 12 -file_asml PF_N20.nc
}}}
The meaning of the different options is the following:

||= Variable =||= Description =||= Default value =||
|| dim_ens || ensemble size || 20
|| file_asml || the name of the output file || ESTKF_N20.nc ||
|| filtertype || Specifies the filter to use; 6 sets the ESTKF filter, 12 the particle filter || 6 || 
|| forget  || value of the forgetting factor for multiplicative variance inflation || 1.0 ||
|| step_null || initial time step of experiment. This specifies the offset of the observations which are generated from a model forward run. A value 1000 avoids the spin-up phase of the model || 0 ||
|| total_steps || number of time step in the experiment || 5000 ||
|| pf_res_type || type of resampling scheme in the particle filter (2 set stochastic unversal resampling) || 1 ||
|| pf_noise_type || type of noise to perturb resampled particles (2 sets noise with amplitude relative to ensemble standard deviation) || 0 ||
|| pf_noise_amp || amplitude of noise to perturb resampled particles || 0.0 ||

Some further options:

||= Variable =||= Description =||= Default value =||
|| delt_obs  || time interval between observations, i.e. the forecast length || 10 ||
|| dim_lag   || set the time lag for the smoother (the smoother is active for dim_lag>0) || 0 ||
|| dx_obs    || distance between observations on the grid; let's you specify an incompletely observated state (only used when `use_obs_mask=.true.`) || 3 ||
|| numobs || number of observed grid points; the points 1 to numobs are observed (can be combined with dx_obs; only used when `use_obs_mask=.true.`) || 1 ||
|| use_obs_mask || whether to use incomplete observations (need to be .true. for any other settings on the observation density to be used) || .true. ||

For the full set of options, please see init_pdaf.F90 in the lorenz63 directory. There, also the possible settings e.g. for `filtertype` are described.