Implementation of the Analysis Step of 3D-Var using the universal PDAF3 interface

Overview

This page describes the recommended implementation of the analysis step for the different 3D-Var schemes using the universal interface of PDAF3.

There are genenerally three different variants of 3D-Var provided by PDAF: parameterized 3D-Var, 3D Ensemble Var, and hybrid (parameterized + ensemble) 3D-Var. All can be called using the universal interface routines described here.

The different 3D-Var methods in PDAF were explained on the page providing the verview of the Analysis Step for 3D-Var Methods. Depending the type of 3D-Var, the background covariance matrix B is represented either in a parameterized form, by an ensemble, or by a combination of both. The 3D-Var methods that use an ensemble need to transform the ensemble perturbations using an ensemble Kalman filter. PDAF uses for this the error-subspace transform filter ESTKF. There are two variants: The first uses the localized filter LESTKF, while the second uses the global filter ESTKF.

For the analysis step of 3D-Var we need different operations related to the observations. These operations are requested by PDAF by call-back routines supplied by the user and provided in the OMI structure. The names of the routines that are provided by the user are specified in the call to the routine PDAF3_assimilate_3dvar_all for the online mode of PDAF or PDAF3_assim_offline_3dvar_all for the offline mode.

The universal interface has more arguments than the specific interfaces for the parameterized 3D-Var or the 3D Ensemble Var methods. The universal interface is useful if one implements both the 3D-Var with parameterized covariances and the 3D ensemble Var. The hybrid 3D-Var using the LESTKF is always called using this unversal interface.

For completeness we discuss here all user-supplied routines that are specified as arguments. Thus, some of the user-supplied routines, which were explained on the page describing the modification of the model code for the ensemble integration, are repeated here.

Analysis Routines

The general aspects of the filter (or solver) specific routines for the 3D-Var analysis step have been described on the page Modification of the model code for the ensemble integration. Here, we list the full interface of the routine. Subsequently, the user-supplied routines specified in the call is explained.

PDAF3_assimilate_3dvar_all

This routine is used both in the fully-parallel and the flexible implementation variants of the data assimilation system. (See the page Modification of the model code for the ensemble integration for these variants)

The interface is:

SUBROUTINE PDAF3_assimilate_3dvar_all(collect_state_pdaf, distribute_state_pdaf, &
                                 init_dim_obs_pdafomi, obs_op_pdafomi, &
                                 cvt_ens_pdaf, cvt_adj_ens_pdaf, cvt_pdaf, cvt_adj_pdaf, &
                                 obs_op_lin_pdafomi, obs_op_adj_pdafomi, &
                                 init_n_domains_p_pdaf, init_dim_l_pdaf, init_dim_obs_l_pdafomi, &
                                 prepoststep_pdaf, next_observation_pdaf, outflag)

where all arguments, except the last one, are the names of call-back routines:

• collect_state_pdaf: The name of the user-supplied routine that initializes a state vector from the array holding the ensemble of model states from the model fields. This is basically the inverse operation to distribute_state used in PDAF_init_forecast as well as here.
• distribute_state_pdaf: The name of a user supplied routine that initializes the model fields from the array holding the ensemble of model state vectors.
• init_dim_obs_pdafomi: The name of the user-supplied routine that initializes the observation information and provides the size of observation vector
• obs_op_pdafomi: The name of the user-supplied routine that acts as the observation operator on some state vector
• cvt_ens_pdaf: The name of the user-supplied routine that applies the ensemble control-vector transformation (square-root of the B-matrix) on some control vector to obtain a state vector.
• cvt_adj_ens_pdaf: The name of the user-supplied routine that applies the adjoint ensemble control-vector transformation (with square-root of the B-matrix) on some state vector to obtain the control vector.
• cvt_pdaf: The name of the user-supplied routine that applies the control-vector transformation (square-root of the B-matrix) on some control vector to obtain a state vector.
• cvt_adj_pdaf: The name of the user-supplied routine that applies the adjoint control-vector transformation (with square-root of the B-matrix) on some state vector to obtain the control vector.
• obs_op_lin_pdafomi: The name of the user-supplied routine that acts as the linearized observation operator on some state vector
• obs_op_lin_pdafomi: The name of the user-supplied routine that acts as the adjoint observation operator on some state vector
• init_n_domains_pdaf: The name of the routine that provides the number of local analysis domains
• init_dim_l_pdaf: The name of the routine that provides the state dimension for a local analysis domain
• init_dim_obs_l_pdafomi: The name of the routine that initializes the size of the observation vector for a local analysis domain
• prepoststep_pdaf: The name of the pre/poststep routine as in PDAF_init_forecast
• next_observation_pdaf: The name of a user supplied routine that initializes the variables nsteps, timenow, and doexit. The same routine is also used in PDAF_init_forecast.
• status: The integer status flag. It is zero, if the routine is exited without errors.

PDAF3_assim_offline_3dvar_all

This routine is used to perform the analysis step for the offline mode of PDAF. The interface of the routine is identical with that of PDAF3_assimilate_3dvar_all, except that the user-supplied routines distribute_state_pdaf, collect_state and next_observation_pdaf are missing.

The interface is:

  SUBROUTINE PDAF3_assim_offline_3dvar_all( &
                                 init_dim_obs_pdafomi, obs_op_pdafomi, &
                                 cvt_ens_pdaf, cvt_adj_ens_pdaf, cvt_pdaf, cvt_adj_pdaf, &
                                 obs_op_lin_pdafomi, obs_op_adj_pdafomi, &
                                 init_n_domains_p_pdaf, init_dim_l_pdaf, init_dim_obs_l_pdafomi, &
                                 prepoststep_pdaf, outflag)

where all arguments, except the last one, are the names of call-back routines. See the description of the arguments for PDAF3_assimilate_3dvar_all.

PDAF3_put_state_3dvar_all

This routine exists for backward-compatibility. In implementations that were done before the release of PDAF V3.0, a 'put_state' routine was used for the flexible parallelization variant and for the offline mode. When the flexible implementation variant is chosen for the assimilation system, this routine allows to port such implementations to the PDAF3 interface with minimal changes. The interface of the routine is identical with that of PDAF3_assimilate_3dvar_all, except that the user-supplied routines distribute_state_pdaf and next_observation_padf are missing.

The interface is:

  SUBROUTINE PDAF3_put_state_3dvar_all(collect_state_pdaf, &
                                 init_dim_obs_pdafomi, obs_op_pdafomi, &
                                 cvt_ens_pdaf, cvt_adj_ens_pdaf, cvt_pdaf, cvt_adj_pdaf, &
                                 obs_op_lin_pdafomi, obs_op_adj_pdafomi, &
                                 init_n_domains_p_pdaf, init_dim_l_pdaf, init_dim_obs_l_pdafomi, &
                                 prepoststep_pdaf, outflag)

where all arguments, except the last one, are the names of call-back routines. See the description of the arguments for PDAF3_assimilate_3dvar_all.

User-supplied routines

Here, all user-supplied routines are described that are required in the call to the assimilation routines for hybrid 3D-Var. For some of the generic routines, we link to the page on modifying the model code for the ensemble integration.

The names of the user-suppled routines routines ending on _pdaf relate to operations on the model state, while those ensing on _pdafomi handle observations using the structured appraoch guided by PDAF-OMI. The user-routines relating to PDAF-OMI are collected in the file callback_obs_pdafomi.F90. In the section titles below we provide the name of the template file in parentheses.

In the subroutine interfaces some variables appear with the suffix _p. This suffix indicates that the variable is particular to a model sub-domain, if a domain decomposed model is used. Thus, the value(s) in the variable will be different for different model sub-domains. Further, the suffix _l indices variables that are specific for a local analysis domain in the LESTKF.

collect_state_pdaf (collect_state_pdaf.F90)

This routine is independent of the filter algorithm used.

See the page on modifying the model code for the ensemble integration for the description of this routine.

distribute_state_pdaf (distribute_state_pdaf.F90)

This routine is independent of the filter algorithm used.

See the page on modifying the model code for the ensemble integration for the description of this routine.

init_dim_obs_pdafomi (callback_obs_pdafomi.F90)

This is a call-back routine initializing the observation information. The routine just calls a routine from the observation module for each observation type.

See the documentation on callback_obs_pdafomi.F90 for more information.

obs_op_pdafomi (callback_obs_pdafomi.F90)

This is a call-back routine applying the observation operator to the state vector. The routine calls a routine from the observation module for each observation type.

See the documentation on callback_obs_pdafomi.F90 for more information.

cvt_ens_pdaf (cvt_ens_pdaf.F90)

The interface for this routine is:

SUBROUTINE cvt_ens_pdaf(iter, dim_p, dim_ens, dim_cv_ens_p, ens_p, cv_p, Vcv_p)

  INTEGER, INTENT(in) :: iter               ! Iteration of optimization
  INTEGER, INTENT(in) :: dim_p              ! PE-local observation dimension
  INTEGER, INTENT(in) :: dim_ens            ! Ensemble size
  INTEGER, INTENT(in) :: dim_cv_ens_p       ! Dimension of control vector
  REAL, INTENT(in) :: ens_p(dim_p, dim_ens) ! PE-local ensemble
  REAL, INTENT(in) :: cv_p(dim_cv_ens_p)    ! PE-local control vector
  REAL, INTENT(inout) :: Vcv_p(dim_p)       ! PE-local state increment

The routine is called during the analysis step during the iterative minimization of the cost function. It has to apply the control vector transformation to the control vector and return the transformed result state vector. Usually this transformation is the multiplication with the square-root of the background error covariance matrix B. For the 3D Ensemble Var, this square root is usually expressed through the ensemble. More complex transformation, including the combination with a parameterized covariance matrix, are possible and the routine permits the flexiblity to implement any transformation.

If the control vector is decomposed in case of parallelization it first needs to the gathered on each processor and afterwards the transformation is computed on the potentially domain-decomposed state vector.

cvt_adj_ens_pdaf (cvt_adj_ens_pdaf.F90)

The interface for this routine is:

SUBROUTINE cvt_adj_ens_pdaf(iter, dim_p, dim_ens, dim_cv_ens_p, ens_p, Vcv_p, cv_p)

  INTEGER, INTENT(in) :: iter                ! Iteration of optimization
  INTEGER, INTENT(in) :: dim_p               ! PE-local observation dimension
  INTEGER, INTENT(in) :: dim_ens             ! Ensemble size
  INTEGER, INTENT(in) :: dim_cv_ens_p        ! PE-local dimension of control vector
  REAL, INTENT(in) :: ens_p(dim_p, dim_ens)  ! PE-local ensemble
  REAL, INTENT(in)    :: Vcv_p(dim_p)        ! PE-local input vector
  REAL, INTENT(inout) :: cv_p(dim_cv_ens_p)  ! PE-local result vector

The routine is called during the analysis step during the iterative minimization of the cost function. It has to apply the adjoint control vector transformation to a state vector and return the control vector. Usually this transformation is the multiplication with transpose of the square-root of the background error covariance matrix B. For the 3D Ensemble Var, this square root is usually expressed through the ensemble. More complex transformation, including the combination with a parameterized covariance matrix, are possible and the routine permits the flexiblity to implement any transformation.

If the state vector is decomposed in case of parallelization one needs to take care that the application of the trasformation is complete. This usually requries a comminucation with MPI_Allreduce to obtain a global sun.

cvt_pdaf (cvt_pdaf.F90)

The interface for this routine is:

SUBROUTINE cvt_pdaf(iter, dim_p, dim_cvec, cv_p, Vv_p)

  INTEGER, INTENT(in) :: iter           ! Iteration of optimization
  INTEGER, INTENT(in) :: dim_p          ! PE-local observation dimension
  INTEGER, INTENT(in) :: dim_cvec       ! Dimension of control vector
  REAL, INTENT(in)    :: cv_p(dim_cvec) ! PE-local control vector
  REAL, INTENT(inout) :: Vv_p(dim_p)    ! PE-local result vector (state vector increment)

The routine is called during the analysis step during the iterative minimization of the cost function. It has to apply the control vector transformation to the control vector and return the transformed result state vector. Usually this transformation is the multiplication with the square-root of the background error covariance matrix B in its parameterized form.

If the control vector is decomposed in case of parallelization it first needs to the gathered on each processor and afterwards the transformation is computed on the potentially domain-decomposed state vector.

cvt_adj_pdaf (cvt_adj_pdaf.F90)

The interface for this routine is:

SUBROUTINE cvt_adj_pdaf(iter, dim_p, dim_cvec, Vv_p, cv_p)

  INTEGER, INTENT(in) :: iter           ! Iteration of optimization
  INTEGER, INTENT(in) :: dim_p          ! PE-local observation dimension
  INTEGER, INTENT(in) :: dim_cvec       ! Dimension of control vector
  REAL, INTENT(in)    :: Vv_p(dim_p)    ! PE-local result vector (state vector increment)
  REAL, INTENT(inout) :: cv_p(dim_cvec) ! PE-local control vector

The routine is called during the analysis step during the iterative minimization of the cost function. It has to apply the adjoint control vector transformation to a state vector and return the control vector. Usually this transformation is the multiplication with transposed of the square-root of the background error covariance matrix B in its parameterized form.

If the state vector is decomposed in case of parallelization one needs to take care that the application of the trasformation is complete. This usually requries a comminucation with MPI_Allreduce to obtain a global sum.

obs_op_lin_pdafomi (callback_obs_pdafomi.F90)

This is a call-back routine for PDAF-OMI applying the linearized observation operator to the state vector. The routine calls a routine from the observation module for each observation type. If the full observation operator is lineaer the same operator can be used here.

See the documentation on callback_obs_pdafomi.F90 for more information.

obs_op_adj_pdafomi (callback_obs_pdafomi.F90)

This is a call-back routine for PDAF-OMI applying the adjoint observation operator to some vector inthe observation space. The routine calls a routine from the observation module for each observation type.

See the documentation on callback_obs_pdafomi.F90 for more information.

init_n_domains_pdaf (init_n_domains_pdaf.F90)

This routine is used in the LESTKF.

The routine called during the analysis step before the loop over the local analysis domains is entered. It has to provide the number of local analysis domains. In case of a domain-decomposed model, the number of local analysis domain for the model sub-domain of the calling process has to be initialized.

The interface for this routine is:

SUBROUTINE init_n_domains_pdaf(step, n_domains_p)

  INTEGER, INTENT(in)  :: step        ! Current time step
  INTEGER, INTENT(out) :: n_domains_p ! Number of analysis domains for local model sub-domain

Hints:

• As a simple case, if the localization is only performed horizontally, the local analysis domains can be single vertical columns of the model grid. In this case, n_domains_p is simply the number of vertical columns in the process-local model sub-domain.

init_dim_l_pdaf (init_dim_l_pdaf.F90)

This routine is used in the LESTKF.

The interface for this routine is:

SUBROUTINE init_dim_l_pdaf(step, domain_p, dim_l)

  INTEGER, INTENT(in)  :: step        ! Current time step
  INTEGER, INTENT(in)  :: domain_p    ! Current local analysis domain
  INTEGER, INTENT(out) :: dim_l       ! Local state dimension

The routine is called during the loop over the local analysis domains in the analysis step.

It provides in dim_l the dimension of the state vector for the local analysis domain with index domain_p to PDAF.

In the recommended implementation shown in the tutorial and template codes, there are two further initializations:

1. The routine has initialize the index array id_lstate_in_pstate containing the indices of the elements of the local state vector in the global (or domain-decomposed) state vector. Then it has to provide this array to PDAF by calling PDAFlocal_set_indices (see below).
2. The routine initializes an array coords_l containing the coordinates of the local analysis domain. This is shared with U_init_dim_obs_l_pdafomi via the module mod_assimilation.

Hints:

• The coordinates in coords_l have to describe one location in space that is used for localization to compute the distance from observations.
• The coordinates in coords_l have the same units as those used for the observations
• For geographic distance computations, the unit of the coordinates needs to be radian, thus (0, 2*pi) or (-pi,pi) for longitude and (-pi/2, pi/2) for latitude.

• Any form of local domain is possible as long as it can be describe as a single location.
  • If the local domain is a single grid point, dim_l will be the number of model variables at this grid point.
  • The local analysis domain can also be a single vertical column of the model grid if observations are only horizontally distributed (a common situation with satellite data in the ocean).
    • In this case, dim_l will be the number of vertical grid points at this location times the number of model fields that exist in the vertical, plus possible variables at e.g. the surface.
    • In this case only the horizontal coordinates are used in coords_l.

The index array id_lstate_in_pstate is an integer array in form of a one-dimensional vector. One initializes this vector by determining the indices of the elements of the local state vector in the global, or domain decomposed, state vector. After initializing id_lstate_in_pstate, one has to provided it to PDAF by calling `PDAFlocal_set_indices'. The interface interface is:

SUBROUTINE PDAFlocal_set_indices(dim_l, id_lstate_in_pstate)

  INTEGER, INTENT(in) :: dim_l                          ! Dimension of local state vector
  INTEGER, INTENT(in) :: id_lstate_in_pstate(dim_l)     ! Index array for mapping

Hint for id_lstate_in_pstate:

• The initialization of the index vector id_lstate_to_pstate is analogous to a loop that directly performs the initialization of a local state vector. However, here only the indices are stored.
• See the PDAFlocal overview page for more information on the functionality of PDAFlocal.

init_dim_obs_l_pdafomi (callback_obs_pdafomi.F90)

This routine is used in the LESTKF.

It is a call-back routine for PDAF-OMI that initializes the local observation vector. The routine calls a routine from the observation module for each observation type.

See the documentation on callback_obs_pdafomi.F90 for more information.

prepoststep_pdaf (prepoststep_ens_pdaf.F90)

The routine has already been described for modifying the model for the ensemble integration and for inserting the analysis step.

See the page on modifying the model code for the ensemble integration for the description of this routine.

next_observation_pdaf (next_observation_pdaf.F90)

This routine is independent of the filter algorithm used.

See the page on modifying the model code for the ensemble integration for the description of this routine.

Execution order of user-supplied routines

The user-supplied routines are executed in the order listed below. The order can be important as some routines can perform preparatory work for later routines. For example, init_dim_obs_pdafomi prepares an index array that provides the information for executing the observation operator in obs_op_pdafomi. How this information is initialized is described in the documentation of OMI.

Before the analysis step is called the following routine is executed:

1. collect_state_pdaf

The analysis step is executed when the ensemble integration of the forecast is completed. During the analysis step the following routines are executed in the given order:

1. prepoststep_pdaf (Call to act on the forecast ensemble, called with negative value of the time step)
2. init_dim_obs_pdafomi
3. obs_op_pdafomi (multiple calls, one for each ensemble member)

Inside the analysis step the interative optimization is computed. This involves the repeated call of the routines:

1. cvt_pdaf
2. cvt_ens_pdaf
3. obs_op_lin_pdafomi
4. obs_op_adj_pdafomi
5. cvt_adj_ens_pdaf
6. cvt_adj_pdaf

After the iterative optimization the following routines are executes to complte the analysis step:

1. cvt_pdaf (Call to the parameterized part of the control vector transform to compute the final state vector increment)
2. cvt_ens_pdaf (Call to the eensemble-part of the control vector transform to compute the final state vector increment)
3. prepoststep_pdaf (Call to act on the analysis ensemble, called with (positive) value of the time step)

The iterative optimization above computes an updated ensemble mean state. Subsequently, the ensemble perturbations are updated using the LESTKF or ESTKF. The execution of the routines for these filters is described on the page on implementing the local filter analysis step .

In case of the routine PDAF3_assimilate_3dvar_all, the following routines are executed after the analysis step:

1. distribute_state_pdaf
2. next_observation_pdaf