RBL4D-Var Tutorial
Tutorial Menu |
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1. Introduction |
2. Error Covariance Normalization |
3. I4D-Var |
4. RBL4D-Var |
5. RBL4D-Var Analysis Observation Impact |
6. RBL4D-Var Analysis Observation Sensitivity |
7. Array Modes |
8. RBL4D-Var Forecast Observation Impact |
9. RBL4D-Var Forecast Observation Sensitivity |
Contents
Introduction
In this tutorial you will apply the strong/weak constraint, dual form of 4-Dimensional Variational (4D-Var) data assimilation based on the Physical-space Statistical Analysis System (PSAS) algorithm (4D-PSAS) to ROMS configured for the U.S. west coast and the California Current System (WC13). In 4D-PSAS the search for the best ocean circulation estimate proceeds in the space spanned only by the observations, as opposed to the full space spanned by the model (i.e. the primal form, I4D-Var). Formally, the primal and dual formulations yield identical estimates of the ocean circulation so one might wonder if there is any advantage of one form over the other? The practical advantages and disadvantages to both approaches are discussed in Moore et al. (2011b, c).
Model Set-up
The WC13 model domain is shown in Fig. 1 and has open boundaries along the northern, western, and southern edges of the model domain.
In the tutorial, you will perform a 4D-Var data assimilation cycle that spans the period 3-6 January, 2004. The 4D-Var control vector δz is comprised of increments to the initial conditions, δx(t_{0}), surface forcing, δf(t), and open boundary conditions, δb(t). The prior initial conditions, x_{b}(t_{0}), are taken from the sequence of 4D-Var experiments described by Moore et al. (2011b) in which data were assimilated every 7 days during the period July 2002- December 2004. The prior surface forcing, f_{b}(t), takes the form of surface wind stress, heat flux, and a freshwater flux computed using the ROMS bulk flux formulation, and using near surface air data from COAMPS (Doyle et al., 2009). Clamped open boundary conditions are imposed on (u,v) and tracers, and the prior boundary conditions, b_{b}(t), are taken from the global ECCO product (Wunsch and Heimbach, 2007). The free-surface height and vertically integrated velocity components are subject to the usual Chapman and Flather radiation conditions at the open boundaries. The prior surface forcing and open boundary conditions are provided daily and linearly interpolated in time. Similarly, the increments δf(t) and δb(t) are also computed daily and linearly interpolated in time.
The observations assimilated into the model are satellite SST, satellite SSH in the form of a gridded product from Aviso, and hydrographic observations of temperature and salinity collected from Argo floats and during the GLOBEC/LTOP and CalCOFI cruises off the coast of Oregon and southern California, respectively. The observation locations are illustrated in Fig. 2.
Running 4D-PSAS
To run this tutorial, go first to the directory WC13/PSAS. Instructions for compiling and running the model are provided below or can be found in the Readme file. The recommended configuration for this exercise is one outer-loop and 26 inner-loops, and roms_wc13.in is configured for this default case. The number of inner-loops is controlled by the parameter Ninner in roms_wc13.in.
Important CPP Options
The following C-preprocessing options are activated in the build script:
ANA_SPONGE Analytical enhanced viscosity/diffusion sponge
BGQC Backgound quality control of observations
MINRES Minimal Residual Method for 4D-Var minimization
RPCG Restricted B-preconditioned Lanczos minimization
POSTERIOR_EOFS Estimate posterior analysis error
POSTERIOR_ERROR_I Estimate initial posterior analysis error
TIME_CONV Weak-constraint 4D-Var time convolution
WC13 Application CPP option
Input NetCDF Files
WC13 requires the following input NetCDF files:
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
Initial Conditions Norm File: ../Data/wc13_nrm_i.nc
Model Norm File: ../Data/wc13_nrm_m.nc
Boundary Conditions Norm File: ../Data/wc13_nrm_b.nc
Surface Forcing Norm File: ../Data/wc13_nrm_f.nc
Observations File: wc13_obs.nc
Various Scripts and Include Files
The following files will be found in WC13/PSAS directory after downloading from ROMS test cases SVN repository:
build_roms.bash bash shell script to compile application
build_roms.sh csh Unix script to compile application
job_psas.sh job configuration script
roms_wc13.in ROMS standard input script for WC13
s4dvar.in 4D-Var standard input script template
wc13.h WC13 header with CPP options
Instructions
To run this application you need to take the following steps:
- 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=diag(B_{x}, B_{b}, B_{f}, Q), is very large and not well known. B is modeled as the solution of a diffusion equation as in Weaver and Courtier (2001). Each covariance matrix is factorized as B = K Σ C Σ^{T} K^{T}, where C is a univariate correlation matrix, Σ is a diagonal matrix of error standard deviations, and K is a multivariate balance operator.../Data/wc13_std_i.nc initial conditions
../Data/wc13_std_m.nc model error (if weak constraint)
../Data/wc13_std_b.nc open boundary conditions
../Data/wc13_std_f.nc surface forcing (wind stress and net heat flux) - 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).Nmethod == 0 ! normalization method: 0=Exact (expensive) or 1=Approximated (randomization)These normalization coefficients have already been computed for you (../Normalization) using the exact method since this application has a small grid (54x53x30):
Nrandom == 5000 ! randomization iterations
LdefNRM == T T T T ! Create a new normalization files
LwrtNRM == T T T T ! Compute and write normalization
CnormM(isFsur) = T ! model error covariance, 2D variable at RHO-points
CnormM(isUbar) = T ! model error covariance, 2D variable at U-points
CnormM(isVbar) = T ! model error covariance, 2D variable at V-points
CnormM(isUvel) = T ! model error covariance, 3D variable at U-points
CnormM(isVvel) = T ! model error covariance, 3D variable at V-points
CnormM(isTvar) = T T ! model error covariance, NT tracers
CnormI(isFsur) = T ! IC error covariance, 2D variable at RHO-points
CnormI(isUbar) = T ! IC error covariance, 2D variable at U-points
CnormI(isVbar) = T ! IC error covariance, 2D variable at V-points
CnormI(isUvel) = T ! IC error covariance, 3D variable at U-points
CnormI(isVvel) = T ! IC error covariance, 3D variable at V-points
CnormI(isTvar) = T T ! IC error covariance, NT tracers
CnormB(isFsur) = T ! BC error covariance, 2D variable at RHO-points
CnormB(isUbar) = T ! BC error covariance, 2D variable at U-points
CnormB(isVbar) = T ! BC error covariance, 2D variable at V-points
CnormB(isUvel) = T ! BC error covariance, 3D variable at U-points
CnormB(isVvel) = T ! BC error covariance, 3D variable at V-points
CnormB(isTvar) = T T ! BC error covariance, NT tracers
CnormF(isUstr) = T ! surface forcing error covariance, U-momentum stress
CnormF(isVstr) = T ! surface forcing error covariance, V-momentum stress
CnormF(isTsur) = T T ! surface forcing error covariance, NT tracers fluxes../Data/wc13_nrm_i.nc initial conditionsNotice that the switches LdefNRM and LwrtNRM are all false (F) since we already computed these coefficients.
../Data/wc13_nrm_m.nc model error (if weak constraint)
../Data/wc13_nrm_b.nc open boundary conditions
../Data/wc13_nrm_f.nc surface forcing (wind stress and
net heat flux) - 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 are set in my_build_paths.sh. Notice that you need to provide the correct places of these libraries for your computer. If you want to ignore this section, set USE_MY_LIBS value to no.
- Notice that the most important CPP options for this application are specified in the build script instead of wc13.h:setenv MY_CPP_FLAGS "${MY_CPP_FLAGS} -DW4DPSAS"This is to allow flexibility with different CPP options.
setenv MY_CPP_FLAGS "${MY_CPP_FLAGS} -DANA_SPONGE"
setenv MY_CPP_FLAGS "${MY_CPP_FLAGS} -DBGQC"
setenv MY_CPP_FLAGS "${MY_CPP_FLAGS} -DMINRES"
setenv MY_CPP_FLAGS "${MY_CPP_FLAGS} -DRPCG"
setenv MY_CPP_FLAGS "${MY_CPP_FLAGS} -DPOSTERIOR_EOFS"
setenv MY_CPP_FLAGS "${MY_CPP_FLAGS} -DPOSTERIOR_ERROR_I"
setenv MY_CPP_FLAGS "${MY_CPP_FLAGS} -DTIME_CONV" - You MUST use the build script to compile.
- Customize the ROMS input script roms_wc13.in and specify the appropriate values for the distributed-memory partition. It is set by default to: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.
- Customize the configuration script job_psas.sh and provide the appropriate place for the substitute Perl script:set SUBSTITUTE=${ROMS_ROOT}/ROMS/Bin/substituteThis 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
- Execute the configuration job_psas.sh before running the model. It copies the required files and creates psas.in input script from template s4dvar.in. This has to be done every time that you run this application. We need a clean and fresh copy of the initial conditions and observation files since they are modified by ROMS during execution.
- Run ROMS with data assimilation:mpirun -np 8 romsM roms_wc13.in > & log &
- We recommend creating a new subdirectory for each Case in this exercise (EX3_CONGRAD, EX3_MINRES, EX3_RPCG, and EX4_WEAK), and saving the solution in it for analysis and plotting to avoid overwriting solutions when playing with different parameters. For examplemkdir EX3_CONGRADwhere log is the ROMS standard output specified in the previous step.
mv Build_roms psas.in *.nc log EX3_CONGRAD
cp -p romsM roms_wc13.in EX3_CONGRAD
Plotting your Results
Several Matlab scripts are provided in the directory WC13/plotting which will allow you to plot some of the 4D-PSAS output.
Recall that 4D-PSAS minimizes the cost function given by:
Plot first the 4D-PSAS cost function and its components , and the theoretical minimum value using the Matlab script plot_psas_cost.m.
Next, plot the surface initial conditions increments and the surface forcing increments at initial time using Matlab script plot_psas_increments.m or ROMS plotting package script ccnt_psas_increments.in for horizontal plots at 100 m or csec_psas_increments.in for cross-sections along 37°N.
Results
The 4D-PSAS cost function value for each inner loop iteration is shown below for various minimization algorithms:
The convergence of various minimization algorithms (PSAS default, MINRES, and RPCG) is compared with the primal formulation I4DVAR.
In the above plots, the prior nonlinear trajectory is saved every 2 hours instead of daily (NHIS = NTLM = NSFF = NOBC = 4). It shows the convergence of various minimization algorithms (PSAS default, MINRES, and RPCG) is compared with the primal formulation I4DVAR.
The 4D-PSAS initial conditions increments for free-surface (m), surface wind stress components (Pa), and surface net heat flux (W/m^{2}) are shown below:
The 4D-PSAS initial conditions increments at the surface for temperature (°C), salinity, and momentum components (m/s) are shown below:
The 4D-PSAS initial conditions increments at 100m for temperature (°C), salinity, and momentum components (m/s) are shown below:
A cross-section along 37°N for the 4D-PSAS initial conditions increments is shown below.