4DVar Normalization Tutorial

Error Covariance Normalization

The various files in the Normalization folder are needed to run the 4-Dimensional Variational (4D-Var) data assimilation error covariance model that computes normalization coefficients in the California Current System, 1/3° resolution, application (WC13).

The computation of 4D-Var error covariance normalization coefficients is very expensive and depends on the grid size. The ones computed here use the expensive "exact" method. For large grids, we need to use the "randomization" approach. The normalization coefficients need to be computed only once for a particular application provided that the grid, land/sea masking (if any), and decorrelation scales remain the same.

This page is under construction

Important CPP Options

NORMALIZATION 4D-Var error covariance normalization factors
WC13 Application CPP option

Input NetCDF Files

Grid File: ../Data/wc13_grd.nc
Nonlinear Initial File: wc13_ini.nc
Forcing File 01: ../Data/coamps_wc13_lwrad_down.nc
Forcing File 02: ../Data/coamps_wc13_Pair.nc
Forcing File 03: ../Data/coamps_wc13_Qair.nc
Forcing File 04: ../Data/coamps_wc13_rain.nc
Forcing File 05: ../Data/coamps_wc13_swrad.nc
Forcing File 06: ../Data/coamps_wc13_Tair.nc
Forcing File 07: ../Data/coamps_wc13_wind.nc
Boundary File: ../Data/wc13_ecco_bry.nc

Initial Conditions STD File: ../Data/wc13_std_i.nc
Model STD File: ../Data/wc13_std_m.nc
Boundary Conditions STD File: ../Data/wc13_std_b.nc
Surface Forcing STD File: ../Data/wc13_std_f.nc

Output NetCDF Files

Initial Conditions Norm File: wc13_nrm_i.nc
Model Norm File: wc13_nrm_m.nc
Boundary Conditions Norm File: wc13_nrm_b.nc
Surface Forcing Norm File: wc13_nrm_f.nc

Various Scripts and Include Files

build.bash bash shell script to compile application
build.sh csh Unix script to compile application
job_normalization.sh job configuration script
ocean_wc13.in ROMS standard input script for WC13
s4dvar.in 4D-Var standard input script template
wc13.h WC13 header with CPP options

Important Parameters

Since we are modeling the error covariance matrix, D, we need to compute the normalization coefficients to ensure that the diagonal elements of the associated correlation matrix C are equal to unity. There are two methods to compute normalization coefficients: exact and randomization (an approximation).

The exact method is very expensive on large grids. The normalization coefficients are computed by perturbing each model grid cell with a delta function scaled by the area (2D state variables) or volume (3D state variables), and then by convolving with the squared-root adjoint and tangent linear diffusion operators.

The approximate method is cheaper: the normalization coefficients are computed using the randomization approach of Fisher and Courtier (1995). The coefficients are initialized with random numbers having a uniform distribution (drawn from a normal distribution with zero mean and unit variance). Then, they are scaled by the inverse squared-root of the cell area (2D state variable) or volume (3D state variable) and convolved with the squared-root adjoint and tangent diffusion operators over a specified number of iterations, Nrandom.

Check following parameters in the 4D-Var input script s4dvar.in (see input script for details):

Nmethod == 0  ! normalization method
Nrandom == 5000  ! randomization iterations

LdefNRM == T T T T  ! Create a new normalization files
LwrtNRM == T T T T  ! Compute and write normalization

CnormI(isFsur) = T  ! 2D variable at RHO-points
CnormI(isUbar) = T  ! 2D variable at U-points
CnormI(isVbar) = T  ! 2D variable at V-points
CnormI(isUvel) = T  ! 3D variable at U-points
CnormI(isVvel) = T  ! 3D variable at V-points
CnormI(isTvar) = T T  ! NT tracers

CnormB(isFsur) = T  ! 2D variable at RHO-points
CnormB(isUbar) = T  ! 2D variable at U-points
CnormB(isVbar) = T  ! 2D variable at V-points
CnormB(isUvel) = T  ! 3D variable at U-points
CnormB(isVvel) = T  ! 3D variable at V-points
CnormB(isTvar) = T T  ! NT tracers

CnormF(isUstr) = T  ! surface U-momentum stress
CnormF(isVstr) = T  ! surface V-momentum stress
CnormF(isTsur) = T T  ! NT surface tracers flux

This directory computes the normalization coefficients using the exact method since this application has a small grid (54x53x30) an creates the following files:

wc13_nrm_i.nc initial conditions
wc13_nrm_m.nc model error
wc13_nrm_b.nc open boundary conditions
wc13_nrm_f.nc surface forcing (wind stress and net heat flux)

Notice that the switches LdefNRM and LwrtNRM are all false (F) since we already computed these coefficients.

The normalization coefficients need to be computed only once for a particular application provided that the grid, land/sea masking (if any), and decorrelation scales (HdecayI, VdecayI, HdecayB, VdecayV, and HdecayF) remain the same. Notice that large spatial changes in the normalization coefficient structure are observed near the open boundaries and land/sea masking regions.

Instructions

To run this application you need to take the following steps:

1. We need to run the model application for a period that is long enough to compute meaningful circulation statistics, like mean and standard deviations for all prognostic state variables (zeta, u, v, T, and S). The standard deviations are written to NetCDF files and are read by the 4D-Var algorithm to convert modeled error correlations to error covariances. The error covariance matrix, D, is very large and not well known. It is modeled as the solution of a diffusion equation as in Weaver and Courtier (2001).
    
   In this application, we need standard deviations for initial conditions, surface forcing (ADJUST_WSTRESS and ADJUST_STFLUX), and open boundary conditions (ADJUST_BOUNDARY). The standard deviations for the initial and open boundary conditions are in terms of the unbalanced error covariance (K D_u K^T) since the balanced operator is activated (BALANCE_OPERATOR and ZETA_ELLIPTIC).
    
   The balance operator imposes a multivariate constraint on the error covariance such that the unobserved variable information is extracted from observed data by establishing balance relationships (i.e., T-S empirical formulas, hydrostactic balance, and geostrophic balance) with other state variables (Weaver et al., 2005).
    
   These standard deviations have already been created for you:
   ../Data/wc13_std_i.nc initial conditions
   ../Data/wc13_std_b.nc open boundary conditions
   ../Data/wc13_std_f.nc surface forcing (wind stress and net heat flux)
2. Customize your preferred build script and provide the appropriate values for:
   • Root directory, MY_ROOT_DIR
   • ROMS source code, MY_ROMS_SRC
   • Fortran compiler, FORT
   • MPI flags, USE_MPI and USE_MPIF90
   • Path of MPI, NetCDF, and ARPACK libraries according to the compiler. Notice that you need to provide the correct places of these libraries for your computer. If you want to ignore this section, comment out the assignment for the variable USE_MY_LIBS.
3. Notice that the most important CPP option for this application is specified in the build script instead of wc13.h:
   setenv MY_CPP_FLAGS "-DNORMALIZATION"
   This is to allow flexibility with different CPP options.
    
   For this to work, however, any #undef directives MUST be avoided in the header file wc13.h since it has precedence during C-preprocessing.
4. You MUST use the build script to compile.
5. Customize the ROMS input script ocean_wc13.in and specify the appropriate values for the distributed-memory partition. It is set by default to:
   NtileI == 1  ! I-direction partition
   NtileJ == 8  ! J-direction partition
   Notice that the adjoint-based algorithms can only be run in parallel using MPI. This is because of the way that the adjoint model is constructed.
6. Customize the configuration script job_normalization.sh and provide the appropriate place for the substitute Perl script:
   set SUBSTITUTE=${ROMS_ROOT}/ROMS/Bin/substitute
   This script is distributed with ROMS and it is found in the ROMS/Bin sub-directory. Alternatively, you can define ROMS_ROOT environmental variable in your .cshrc login script. For example, I have:
   setenv ROMS_ROOT /home/arango/ocean/toms/repository/trunk
7. Execute the configuration job_normalization.sh before running the model. It copies the required files and creates the c4dvar.in input script from template s4dvar.in.
8. Run ROMS with data assimilation:
   mpirun -np 8 oceanM ocean_wc13.in > & log &

References

The technical description of the algorithms and application used in this tutorial are described in Moore et al. (2010a, b, c).