Implementation of the Analysis Step for parameterized 3D-Var

Overview

This page describes the implementation of the analysis step for the parameterized 3D-Var using the particular interface routines.

The interface routines for the parameterized 3D-Var are specialized versions of the universal interface with a reduced number of arguments. If one implements both 3D Ensemble Var and parameterlized 3D-Var we recommend to use the universal interface routines for 3D-Var.

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.

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 PDAF-OMI structure. in the call to the assimilation routines as was examplained on the page providing the verview of the Analysis Step for 3D-Var Methods.

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.

AssimilationRoutines

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

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(collect_state_pdaf, distribute_state_pdaf, &
                                 init_dim_obs_pdafomi, obs_op_pdafomi, &
                                 cvt_pdaf, cvt_adj_pdaf, &
                                 obs_op_lin_pdafomi, obs_op_adj_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_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
• 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

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, 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( &
                                 init_dim_obs_pdafomi, obs_op_pdafomi, &
                                 cvt_pdaf, cvt_adj_pdaf, &
                                 obs_op_lin_pdafomi, obs_op_adj_pdafomi, &
                                 prepoststep_pdaf, next_observation_pdaf, outflag)

PDAF3_put_state_3dvar

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, the routine. 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, except that the user-supplied routines distribute_state_pdaf and next_observation_pdaf are missing.

The interface is:

  SUBROUTINE PDAF3_put_state_3dvar(collect_state_pdaf, &
                                 init_dim_obs_pdafomi, obs_op_pdafomi, &
                                 cvt_pdaf, cvt_adj_pdaf, &
                                 obs_op_lin_pdafomi, obs_op_adj_pdafomi, &
                                 prepoststep_pdaf, next_observation_pdaf, outflag)

User-supplied routines

Here, all user-supplied routines are described that are required in the call to the assimilation routines for for parameterized 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.

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.

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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.

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 essentially executed in the order they are listed in the interface to PDAF3_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. 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. obs_op_lin_pdafomi
3. obs_op_adj_pdafomi
4. cvt_adj_pdaf

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

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

In case of the routine PDAF3_assimilate_3dvar, the following routines are executed after the analysis step:

1. distribute_state_pdaf
2. next_observation_pdaf