Implementation of the Analysis Step for Hybrid 3D-Var with OMI without PDAFlocal

Overview

This documentation describes the implementation of 3D variational assimilation methods with OMI as it was standard since their introduction in PDAF version 2.0 and until and including PDAF version 2.2.1. With PDAF 2.3 we introduced the PDAFlocal interface, which simplifies the implemenation of the local analysis. This approach is described on the current page on the Implementation of the Analysis Step for hybrid 3D Var with OMI.

With Version 2.0 with introduced 3D variational assimilation methods to PDAF. There are genenerally three different variants of 3D variational assimilation methods in PDAF: parameterized 3D-Var, 3D Ensemble Var, and hybrid (parameterized + ensemble) 3D-Var.

This page describes the implementation of the analysis step for the Hybrid 3D-Var using PDAF-OMI.

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 PDAFomi_assimilate_hyb3dvar_* in the fully-parallel implementation (or PDAFomi_put_state_hyb3dvar_* for the 'flexible' implementation) that was discussed before. With regard to the parallelization, all these routines (except U_collect_state, U_distribute_state, and U_next_observation) are executed by the filter processes (filterpe=.true.) only.

For Hybrid 3D-Var the background covariance matrix B is represented by a combination of a parameterized covariance matrix with a covariance matrix part represented by the ensemble. In practive this means that in the square root of B one concatenates parameterized and ensemble columns. The ensemble perturbations need to be transformed by means of 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 completeness we discuss here all user-supplied routines that are specified in the interface to PDAFomi_assimilate_hyb3dvar_X. Thus, some of the user-supplied routines that are 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 PDAF_assimilate_* have been described on the page Modification of the model code for the ensemble integration and its sub-page on inserting the analysis step. The routine is used in the fully-parallel implementation variant of the data assimilation system. When the 'flexible' implementation variant is used, the routines PDAF_put_state_* is used as described further below. Here, we list the full interface of the routine. Subsequently, the user-supplied routines specified in the call is explained.

There are two variants that either compute the transformataion of the ensemble transformation using the local LESTKF method, or the global ESTKF.

PDAFomi_assimilate_hyb3dvar_lestkf

This routine is called for the case of transforming the ensemble perturbations using the local LESTKF.

The interface is:

SUBROUTINE PDAFomi_assimilate_hyb3dvar_lestkf(U_collect_state, U_distribute_state, &
                                 U_init_dim_obs_pdafomi, U_obs_op_pdafomi, &
                                 U_cvt_ens, U_cvt_adj_ens, U_cvt, U_cvt_adj, &
                                 U_obs_op_lin_pdafomi, U_obs_op_adj_pdafomi, &
                                 U_init_n_domains_p, U_init_dim_l, U_init_dim_obs_l_pdafomi, &
                                 U_g2l_state, U_l2g_state, U_prepoststep, U_next_observation, outflag)

with the following arguments:

• U_collect_state: 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 U_distribute_state used in PDAF_get_state as well as here.
• U_distribute_state: The name of a user supplied routine that initializes the model fields from the array holding the ensemble of model state vectors.
• U_init_dim_obs_pdafomi: The name of the user-supplied routine that initializes the observation information and provides the size of observation vector
• U_obs_op_pdafomi: The name of the user-supplied routine that acts as the observation operator on some state vector
• U_cvt_ens: 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.
• U_cvt_adj_ens: 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.
• U_cvt: 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.
• U_cvt_adj: 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.
• U_obs_op_lin_pdafomi: The name of the user-supplied routine that acts as the linearized observation operator on some state vector
• U_obs_op_lin_pdafomi: The name of the user-supplied routine that acts as the adjoint observation operator on some state vector
• U_init_n_domains: The name of the routine that provides the number of local analysis domains
• U_init_dim_l: The name of the routine that provides the state dimension for a local analysis domain
• U_init_dim_obs_l_pdafomi: The name of the routine that initializes the size of the observation vector for a local analysis domain
• U_g2l_state: The name of the routine that initializes a local state vector from the global state vector
• U_l2g_state: The name of the routine that initializes the corresponding part of the global state vector from the provided local state vector
• U_prepoststep: The name of the pre/poststep routine as in PDAF_get_state
• U_next_observation: The name of a user supplied routine that initializes the variables nsteps, timenow, and doexit. The same routine is also used in PDAF_get_state.
• status: The integer status flag. It is zero, if the routine is exited without errors.

Note:

• If your code shows a call to PDAFlocalomi_assimilate_hyb3dvar_lestkf, it uses the implementation variant with PDAFlocal. This is documented on the page on Implementing the Analysis Step for hybrid 3D Var with OMI with PDAFlocal (from V2.3 of PDAF).

PDAFomi_assimilate_hyb3dvar_estkf

This routine is called for the case of transforming the ensemble perturbations using the global ESTKF.

The interface is:

SUBROUTINE PDAFomi_assimilate_hyb3dvar_estkf(U_collect_state, U_distribute_state, &
                                 U_init_dim_obs_pdafomi, U_obs_op_pdafomi, &
                                 U_cvt_ens, U_cvt_adj_ens, U_cvt, U_cvt_adj, &
                                 U_obs_op_lin_pdafomi, U_obs_op_adj_pdafomi, &
                                 U_prepoststep, U_next_observation, outflag)

with the following arguments:

• U_collect_state: 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 U_distribute_state used in PDAF_get_state as well as here.
• U_distribute_state: The name of a user supplied routine that initializes the model fields from the array holding the ensemble of model state vectors.
• U_init_dim_obs_pdafomi: The name of the user-supplied routine that initializes the observation information and provides the size of observation vector
• U_cvt_ens: 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.
• U_cvt_adj_ens: 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.
• U_cvt: 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.
• U_cvt_adj: 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.
• U_obs_op_pdafomi: The name of the user-supplied routine that acts as the observation operator on some state vector
• U_obs_op_lin_pdafomi: The name of the user-supplied routine that acts as the linearized observation operator on some state vector
• U_obs_op_lin_pdafomi: The name of the user-supplied routine that acts as the adjoint observation operator on some state vector
• U_prepoststep: The name of the pre/poststep routine as in PDAF_get_state
• U_next_observation: The name of a user supplied routine that initializes the variables nsteps, timenow, and doexit. The same routine is also used in PDAF_get_state.
• status: The integer status flag. It is zero, if the routine is exited without errors.

Note that the interface of PDAFomi_assimilate_en3dvar_estkf is identical to that of PDAFomi_assimilate_3dvar apart from using the routines U_cvt_ens and U_cvt_adj_ens in case of the ensemble variational method.

PDAFomi_put_state_hyb3dvar_lestkf

When the 'flexible' implementation variant is chosen for the assimilation system, the routine PDAFomi_put_state_* has to be used instead of PDAFomi_assimilate_*. The general aspects of the filter specific routines PDAF_put_state_* have been described on the page Modification of the model code for the ensemble integration. The interface of the routine is identical with that of PDAF_assimilate_* with the exception the specification of the user-supplied routines U_distribute_state and U_next_observation are missing.

The interface when using one of the global filters is the following:

  SUBROUTINE PDAFomi_put_state_hyb3dvar_lestkf(U_collect_state, &
                                 U_init_dim_obs_pdafomi, U_obs_op_pdafomi, &
                                 U_cvt_ens, U_cvt_adj_ens, U_cvt, U_cvt_adj, &
                                 U_obs_op_lin_pdafomi, U_obs_op_adj_pdafomi, &
                                 U_init_n_domains_p, U_init_dim_l, U_init_dim_obs_l_pdafomi, &
                                 U_g2l_state, U_l2g_state, U_prepoststep, outflag)

Note:

• If your code shows a call to PDAFlocalomi_put_state_hyb3dvar_lestkf, it uses the implementation variant with PDAFlocal. This is documented on the page on Implementing the Analysis Step for hybrid 3D Var with OMI with PDAFlocal (from V2.3 of PDAF).

PDAFomi_put_state_hyb3dvar_estkf

The interface of this routine is analogous to that of PDAFomi_assimilate_en3dvar_estkf'. Thus it is identical to this routine with the exception the specification of the user-supplied routines U_distribute_state and U_next_observation` are missing.

The interface when using one of the global filters is the following:

  SUBROUTINE PDAFomi_put_state_hyb3dvar_estkf(U_collect_state, &
                                 U_init_dim_obs_pdafomi, U_obs_op_pdafomi, &
                                 U_cvt_ens, U_cvt_adj_ens, U_cvt, U_cvt_adj, &
                                 U_obs_op_lin_pdafomi, U_obs_op_adj_pdafomi, &
                                 U_prepoststep, outflag)

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.

To indicate user-supplied routines we use the prefix U_. In the template directory templates/ as well as in the tutorial implementations in tutorial/ these routines exist without the prefix, but with the extension _pdaf.F90. The user-routines relating to 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.

U_collect_state (collect_state_pdaf.F90)

This routine is independent of the filter algorithm used.

See the page on inserting the analysis step for the description of this routine.

U_distribute_state (distribute_state_pdaf.F90)

This routine is independent of the filter algorithm used.

See the page on inserting the analysis step for the description of this routine.

U_init_dim_obs_pdafomi (callback_obs_pdafomi.F90)

This is a call-back routine for PDAF-OMI 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.

U_obs_op_pdafomi (callback_obs_pdafomi.F90)

This is a call-back routine for PDAF-OMI 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.

U_cvt_ens (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 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.

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.

U_cvt_adj (cvt_adj_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. or the 3D Ensemble Var, this square root is usually expressed through the ensemble.

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.

U_cvt (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 vector. Usually this transformation is the multiplication with the square-root of the background error covariance matrix B.

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.

U_cvt_adj (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.

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.

U_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.

U_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.

U_init_n_domains (init_n_domains_pdaf.F90)

The interface for this routine is:

SUBROUTINE init_n_domains(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

The routine is 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.

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 local model sub-domain.

U_init_dim_l (init_dim_l_pdaf.F90)

The interface for this routine is:

SUBROUTINE init_dim_l(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 has to provide in dim_l the dimension of the state vector for the local analysis domain with index domain_p.

Hints:

• For sharing through the module 'mod_assimilation', we further initialize an array 'coords_l' containing the coordinates that describe the local domain. These coordinates have to describe one location in space that is used in the OMI observation modules to compute the distance from observations. This requires that the coordinates in 'coords_l' have the same units as those used for the observations.
• Any form of local domain is possible as long as it can be describe as a single location. If observations are only horizontally distributed (a common situation with satellite data in the ocean), the local analysis domain can be a single vertical column of the model grid. In this case, the size of the state in the local analysis domain will be just the number of vertical grid points at this location and the horizontal coordinates are used in 'coords_l'
• Further, we recommend to initialize an array containing the indices of the elements of the local state vector in the global (or domain-decomposed) state vector (id_lstate_in_pstate in the template files). This array is also shared through 'mod_assimilation'.

U_init_dim_obs_l_pdafomi (callback_obs_pdafomi.F90)

This 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.

U_g2l_state (g2l_state_pdaf.F90)

The interface for this routine is:

SUBROUTINE g2l_state(step, domain_p, dim_p, state_p, dim_l, state_l)

  INTEGER, INTENT(in) :: step           ! Current time step
  INTEGER, INTENT(in) :: domain_p       ! Current local analysis domain
  INTEGER, INTENT(in) :: dim_p          ! State dimension for model sub-domain
  INTEGER, INTENT(in) :: dim_l          ! Local state dimension
  REAL, INTENT(in)    :: state_p(dim_p) ! State vector for model sub-domain 
  REAL, INTENT(out)   :: state_l(dim_l) ! State vector on local analysis domain

The routine is called during the loop over the local analysis domains in the analysis step. It has to provide the local state vector state_l that corresponds to the local analysis domain with index domain_p. Provided to the routine is the state vector state_p. With a domain decomposed model, this is the state for the local model sub-domain.

Hints:

• In the simple case that a local analysis domain is a single vertical column of the model grid, the operation in this routine would be to take out of state_p the data for the vertical column indexed by domain_p.
• Usually, one can initialize the indices of the local state vector elements in the global state vector in U_init_dim_l and just use these here.

U_l2g_state (l2g_state_pdaf.F90)

The interface for this routine is:

SUBROUTINE l2g_state(step, domain_p, dim_l, state_l, dim_p, state_p)

  INTEGER, INTENT(in) :: step           ! Current time step
  INTEGER, INTENT(in) :: domain_p       ! Current local analysis domain
  INTEGER, INTENT(in) :: dim_p          ! State dimension for model sub-domain
  INTEGER, INTENT(in) :: dim_l          ! Local state dimension
  REAL, INTENT(in)    :: state_p(dim_p) ! State vector for model sub-domain 
  REAL, INTENT(out)   :: state_l(dim_l) ! State vector on local analysis domain

The routine is called during the loop over the local analysis domains in the analysis step. It has to initialize the part of the global state vector state_p that corresponds to the local analysis domain with index domain_p. Provided to the routine is the state vector state_l for the local analysis domain.

Hints:

• In the simple case that a local analysis domain is a single vertical column of the model grid, the operation in this routine would be to write into state_p the data for the vertical column indexed by domain_p.
• Usually, one can initialize the indices of the local state vector elements in the global state vector in U_init_dim_l and just use these here.

U_prepoststep (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 inserting the analysis step for the description of this routine.

U_next_observation (next_observation_pdaf.F90)

This routine is independent of the filter algorithm used.

See the page on inserting the analysis step for the description of this routine.

Execution order of user-supplied routines

The user-supplied routines are essentially executed in the order they are listed in the interface to PDAFomi_assimilate_3dvar. The order can be important as some routines can perform preparatory work for later routines. For example, U_init_dim_obs_pdafomi prepares an index array that provides the information for executing the observation operator in U_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. U_collect_state

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. U_prepoststep (Call to act on the forecast ensemble, called with negative value of the time step)
2. U_init_dim_obs_pdafomi
3. U_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. U_cvt
2. U_cvt_ens
3. U_obs_op_lin_pdafomi
4. U_obs_op_adj_pdafomi
5. U_cvt_adj
6. U_cvt_adj_ens

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

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

The iterative optimization abovve 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 for the LESTKF on the page on implementing the local filter analysis step and for the ESTKF on the page on implementing the global filter analysis step.

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

1. U_distribute_state
2. U_next_observation