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GitHub Pull Request: ​https://github.com/myroms/roms/pull/25

In 4D-Var, the background (prior) error covariance, B, is a large matrix that cannot be computed or stored directly. Still, its effects can be modeled using spatial correlations, C, and spatial convolutions via diffusion operators. To convert correlations into error covariance, we multiply by a diagonal matrix of the background error standard deviations, S. Hence,

B = L S C S^T L^T

where L is a balance operator, if BALANCE_OPERATOR is activated in ROMS. It allows the information on unobserved state variables to be extracted from directly observed quantities by imposing linear balance relationships between temperature and other state variables using T-S empirical formulas, the linear equation of state, hydrostatic balance, and geostrophic balance. Please check Moore et al. (2011) for more information.

This update includes an approach to computing the standard deviation, S, directly from the background state field as an alternative to climatological values read from the input NetCDF files. It follows the work of Mogensen et al. (2012) by assuming the background errors are proportional to the vertical derivatives of the background field. The field error has a similar profile shape, but the difference with the actual error values is due to a vertical displacement.

In the past, the standard deviation was read from input NetCDF files and computed from a long simulation of the ROMS application. The climatological S can be categorized as monthly, seasonal, or annual values.

The modeling of the standard deviation from the background (prior) is activated with the STD_MODEL option and done in the new module background_std.F. For further information about this capability in ROMS, check Moore et al. (2020).

New parameters are added to 4D-Var input script s4dvar.in to constraint the standard deviation profile at the mixed-layer depth and deep ocean:

! Modeled standard deviation (STD) of Background Error Covariance parameters.
!
! The Mogensen et al. (2012) formulation assumes that the background errors
! are proportional to vertical derivatives of the state vector field. Its
! error has a similar field profile shape, but the difference with its
! ture error value is due to a vertical displacement.
!
! If COMPUTE_MLD is activated, the mixed-layer depth is computed using the
! criterion from Kara et al. (2000). Otherwise, it will be set to a uniform
! value provided below.

Sigma_max(isFsur) == 0.025d0           ! free surface  maximum STD value

Sigma_max(isUvel) == 0.06d0            ! U-velocity maximum STD value
 Sigma_ml(isUvel) == 0.05d0            ! U-velocity minimum STD at mixed layer
 Sigma_do(isUvel) == 0.02d0            ! U-velocity minimum STD in deep ocean
 Sigma_dz(isUvel) == 500.0d0           ! U-velocity vertical displacement

Sigma_max(isVvel) == 0.06d0            ! V-velocity maximum STD
 Sigma_ml(isVvel) == 0.05d0            ! V-velocity minimum STD at mixed layer
 Sigma_do(isVvel) == 0.02d0            ! V-velocity minimum STD in deep ocean
 Sigma_dz(isVvel) == 500.0d0           ! V-velocity vertical displacement

Sigma_max(isTvar) == 0.33d0   0.056d0  ! 1:NT tracers maximum STD
 Sigma_ml(isTvar) == 0.05d0   0.05d0   ! 1:NT tracers minimum STD at the mixed layer
 Sigma_do(isTvar) == 0.02d0   0.0028d0 ! 1:NT tracers minimum STD in the deep ocean
 Sigma_dz(isTvar) == 40.0d0   40.0d0   ! 1:NT tracer vertical displacement

      mld_uniform == -75.0d0           ! Uniform mixed layer depth value

Notice we have the option COMPUTE_MLD to compute the mixed-layer depth using the approach of Kara et al. (2000) or set a constant value of mld_uniform.

Two new routines are added, def_std.F and wrt_std.F, to write the standard deviation modeled from the background into an output NetCDF file. It will be needed in the split 4D-Var algorithms and for postprocessing. The new filename is also specified in s4dvar.in:

! If computing the standard deviation from the background (prior) state
! vector as an alternative to climatological values read from the
! input NetCDF file, enter the output standard deviation file name,
! [1:Ngrids].

       STDnameC == roms_std_c.nc

We foresee enhancing this capability in the future. We only model the standard deviation for adjusting the initial state vector (zeta, u, v, T, and S). We don't have options for background error on the model (weak constraint), lateral model boundary conditions, surface tracer fluxes, and surface momentum stress. Thus, those standard deviation values are still read from input files.

Notice that the standard deviation structure for I/O management increased its inner dimension from 4 to 5 in mod_iounits.F:

       IF (.not.allocated(STD)) THEN
         allocate ( STD(5,Ngrids) )
       END IF

WARNING:

• The s4dvar.in has the additional parameters mentioned above. Please update your 4D-Var input script to use his new capability.

References:

• Kara A., P. Rochford, and E. Hulburt, 2000: An optimal definition for ocean mixed layer depth, J. Geophys. Res., 105, NoC7, pp 16, 803-16, 821.

• Mogensen, K., M.A. Balmaseda, and A.T. Weaver, 2012: The NEMOVAR ocean data assimilation system implemented in the ECMWF ocean analysis for system 4, ECMWF Tech. Memorandum 668, 59.

• Moore, A.M., H.G. Arango, G. Broquet, B.S. Powell, A.T. Weaver, and J. Zavala-Garay, 2011: The Regional Ocean Modeling System (ROMS) 4-dimensional variational data assimilations systems, Part I - System overview and formulation, Prog. Oceanogr., 91, 34-49, ​https://doi:10.1016/j.pocean.2011.05.004.

• Moore, A., J. Zavala-Garay, H.G. Arango, C.A. Edwards, J. Anderson, and T. Hoar, 2020: Regional and basin scale applications of ensemble adjustment Kalman filter and 4D-Var ocean data assimilation systems, Progress in Oceanography, 189, 102450, ​https://doi.org/10.1016/j.pocean.2020.102450.

Copyright (c) 2002-2024 The ROMS/TOMS Group
    Licensed under a MIT/X style license
    See License_ROMS.md

• The metadata indices for writing Eastward/Northward wind components into the station's output NetCDF file were misspelled. We must use idUaiE and idVaiN instead of idUairE and idVairN. Many thanks to Aijun Zhang for bringing this issue to my attention.

• Updated documentation in cppdefs.h preamble.